Bending Deflection – Differential Equation Method

Bending Deflection –Differential Equation MethodAE1108-II: Aerospace Mechanics of MaterialsDr. Calvin RansDr. Sofia Teixeira De FreitasAerospace Structures& MaterialsFaculty of Aerospace Engineering

Recap So far, for symmetric beams, we have: Looked at internal shear force and bending momentdistributions Determined normal stress distribution due to bending moments Determined shear stress distribution due to shear force Need to determine deflections and slopes of beamsunder load Important in many design applications Essential in the analysis of statically indeterminate beams2

Deformation of a BeamAssumptionsShear deformationV V Moment deformationVMMM Negligible (for long beams)Bending Deformation Shear Deformation Moment Deformation

Deformation of a BeamAssumptions For long beams (length much greater than beamdepth), shear deformation is negligible This is the case for most engineering structures Will consider moment deformation only in this course Recap of sign conventionzN.A.y v wx My M V V

Deformation of a BeamVisualizing Bending DeformationElastic curve: plot of the deflection of the neutral axis of a beamHow does this beam deform?We can gain insight into thedeformation by looking at thebending moment diagramM- zM- MMAnd by considering boundaryconditions at supportsQualitatively can determine elastic curve!

Moment-Curvature Relationshipzdzm1m2(-ve M)( ve v)v(z) vertical deflection at z dv (z) slope at z dz zdz

Moment-Curvature Relationshipzds R d For small d : orcurvatureFor small :dz1 d R dsds dz dzcos cos 1when is smallRecall 2d dvd1 d dz v dz 2dzR dz(negative sign a result of sign convention)M E IRd 2vM EI 2 EIv dz

Deflection by Method of Integrationd 2vM EI 2dzLets consider a prismatic beam(ie: EI constant) dv1 M dz dzEI 1v EI M dzIndefinite integrals result inconstants of integration thatcan be determined fromboundary conditions of theproblemConstant of integrationie: z dz 1 2z C2

Determining Constants of IntegrationSupport Conditionsv 0v 0v 0 0

Determining Constants of IntegrationContinuity ConditionszaPCAPabLbDeformedshapeBDiscontinuity at z az aM AC M 0PbzLz aM CB v AC a vCB a AC a CB a Pa L z L

Determining Constants of IntegrationSymmetry ConditionsPACB Symmetry implies reflection ofdeformation across symmetryplane v is equal is opposite Continuity implies equaldeformation at symmetry plane v is equal is equal C 0

Procedure for AnalysisDeflection by Integration Draw a FBD including reaction forces Determine V and M relations for the beam Integrate Moment-displacement differential equation Select appropriate support, symmetry, and continuityconditions to solve for constants of integration Calculate desired deflection (v) and slopes (θ)

Example 1aqProblem StatementDetermine the deflection and slopeat point B in a prismatic beam dueto the distributed load qEIALBSolution1) FBD & Equilibriumzq F 0 Rz MA F 0 RRzRycw M Ay qL Ry qLLqL2 0 M A qL M A 22Aerospace Mechanics of Materials (AE1108-II) – Example Problem13

qEIExample 1aSolutionA2) Determine M and V @ zzqqLqVqL2 2MAqL Vccw zMB F 0 qL qz V V q L z MBL-qL2/2 qL2zL2 z 2 0 M qLz qz M q Lz 222 2 Aerospace Mechanics of Materials (AE1108-II) – Example Problem14

qEIExample 1aSolutionA3) Boundary ConditionsBLAt z 0:v 0, v′ 0Two boundary conditionsd 2vThus can solve by integrating: M EI 2dzd 2v1 2 MdzEIAerospace Mechanics of Materials (AE1108-II) – Example Problem15

qEIExample 1aSolutionA4) Solve Differential Equation qz 2qL2M qLz 22d v 1 qzqL qLz 2dzEI 22 222Boundary Conditiondv 1 qz 3 qLz 2 qL2 z 0 C1 dz EI 622 0 1 qz 4 qLz 3 qL2 z 2v C2 EI 2464 qz 2v z 2 4 Lz 6 L2 24 EIBLBC: At z 0, θ 0 C1 0BC: At z 0, v 0 C2 0dvqz 22zLzL 3 3 dz 6 EIAerospace Mechanics of Materials (AE1108-II) – Example Problem16

qEIExample 1aSolutionABL5) Calculate slopes and deflectionsDetermine deflection and slope at B:qz 2v z 2 4 Lz 6 L2 24 EIvB v(z L )qL4 8EI qz 2z 3Lz 3L2 6 EI B (z L )qL3 6 EIAerospace Mechanics of Materials (AE1108-II) – Example Problem17

Relating Deformation to LoadingShear Force-Moment Diagram Relationships Recall from Statics(refer to Hibbler Ch. 6.2 for refresher, but be careful of coordinate system) w M M V VdV wdzdM VdzMoment-CurvatureRelationship (Eq. 10.1)d 2vM EI 2dzd 4 v w(z) v 4dzEId 3vV (z) v 3dzEId 2vM (z) v 2dzEI

zqEIExample 1bWe can also solve Example 1 in analternative way:AWatch negative sign! EIv q -w(z) EIv qz C1 V(z)qz 2 EIv C1 z C2 M(z)2qz 3z2 EIv C1 C2 z C3 -θ(z)EI62qz 4z3z2 EIv C1 C2 C3 z C42462BLd 4 v w( z ) v 4dzEId 3vV ( z) v 3dzEId 2vM ( z) v 2dzEIWe have 4 unknownconstants of integration,thus need 4 BCsAerospace Mechanics of Materials (AE1108-II) – Example Problem19

zqEIExample 1bWe can also solve this problem analternative way:ABLBoundary Condition: EIv qz C1 V(z)At z L, V 0 C1 qLC1qz 2 EIv qLz C2 M(z)2C2qz 3 qL 2 qL2 EIv z z C3 -θ(z)EI622At z L, M 0 qL2qL22 C2 qL 2 2 At z 0, θ 0 C3 0Aerospace Mechanics of Materials (AE1108-II) – Example Problem20

zqEIExample 1bWe can also solve this problem analternative way:ABLBoundary Condition: EIv 42qzqL 3 qL 2 z z 0 C42464qz 2v z 2 4 Lz 6 L2 24 EIAt x 0, v 0 C4 0v qx 2z 3Lz 3L2 6 EISame result as before!Aerospace Mechanics of Materials (AE1108-II) – Example Problem21

Example 2z A EIv q -w(z) EIv EIv 2qz C1 z C2 M(z)23243LBWe will apply Approach 2 EIv qz C1 V(z) EIv qEIDetermine deflection andslope at B: qLqzz C1 C2 z C3 -θ(z)EI622qzzz C1 C2 C3 z C42462Exact same differentialequations as before!!What makes theproblem different?Boundary Conditions!Aerospace Mechanics of Materials (AE1108-II) – Example Problem22

zExample 2qLqEIDetermine deflection andslope at B:ABLBoundary Condition: EIv qz C1 V(z)At z L, V qLqqEIEIALAB C1 2qLqLLB2qLqLqLVVMM-qL2/2-3qL2/2Aerospace Mechanics of Materials (AE1108-II) – Example Problem23

zExample 2qLqEIDetermine deflection andslope at B:ABLBoundary Condition: EIv qz C1 V(z)At z L, V qL C1 2qLC1qz 2 EIv 2qLz C2 M(z)2C2qz 33qL22 EIv qLz z C3 -θ(x)EI62At z L, M 02 qL2 3qL C2 2qL2 2 2 At z 0, θ 0 C3 0Aerospace Mechanics of Materials (AE1108-II) – Example Problem24

zExample 2EIDetermine deflection andslope at B:Aqz 222v z Lz L818 24 EI11qL4 24 EIBLBoundary Condition:At z 0, v 0qz 4 qL 3 3qL2 2 EIv z z 0 C42434vB v( z L )qLq C4 0v qz 2z 6 Lz 9 L2 6 EI B v '( z L )2qL3 3EIAerospace Mechanics of Materials (AE1108-II) – Example Problem25

Example 2xzqqLxzqEIEIALABqz 2v z 2 8Lz 18L2 24 EILBqz 2v z 2 4 Lz 6 L2 24 EI0qL48 EI4qLqL4 EI22EI011qL424 EILAerospace Mechanics of Materials (AE1108-II) – Example Problem26

PzExample 3EIDetermine deflection at C interms of EI:To save time, reactions are providedBAP/2CLL/23P/2Since reaction forces act at B (discontinuity), we must split the differentialequation into parts for AB and BCWe can easily see by inspection that: EIv V P2 EIv V P(0 z L)(L z 3L/2)Integrate to find MAerospace Mechanics of Materials (AE1108-II) – Example Problem27

PzExample 3EIBACDetermine deflection at C:P/2Moments:M EIv Pz C12M EIv Pz C2LL/23P/2(0 z L)(L z 3L/2)Moment BC’s:At z 0, M 0 C1 0At z 3L/2, M 0 C2 3PL2Integrate to find θAerospace Mechanics of Materials (AE1108-II) – Example Problem28

PzExample 3EIBACDetermine deflection at C:P/2Slopes:P 2z C34P3PL EIv z 2 z C422 EIv LL/23P/2(0 z L)(L z 3L/2)Slope Continuity Condition:At z L, θAB θBCPL2 C3 PL2 C44Integrate to find vAerospace Mechanics of Materials (AE1108-II) – Example Problem29

PzExample 3EIBACDetermine deflection at C:P/2Deflections:P 3 EIv z C3 z C512P3PL 2 EIv z 3 z C4 z C664 C5 02At z L, v 0L/23P/2(0 z L)(L z 3L/2)From last slidePL2 C3 PL2 C44Deflection BC’s:At z 0, v 0LPL C3 125 PL2C4 6Aerospace Mechanics of Materials (AE1108-II) – Example ProblemPL3C6 430

PzExample 3EIBACDetermine deflection at C:Deflections:LP/23P/2Pzv L2 z 2 12 EIv L/2(0 z L)P3L3 10 L2 z 9 Lz 2 2 z 3 12 EI(L z 3L/2)PL33PL0.5 12EIEI123LPL3v(C ) v( ) 28EI00.51 0.51.5 L 1 1.5 2Aerospace Mechanics of Materials (AE1108-II) – Example Problem31

Example 4PzWhat about beams with anon loaded free end?EIALCLBPWill it work itself out?ACBCurved partStraight partAerospace Mechanics of Materials (AE1108-II) – Example Problem32

PzExample 4EI-PLAWhat about beams with anon loaded free end?To save time, reactions are providedLCBLPPMoments:VM EIv P z L (0 z L)M EIv 0M(L z 2L)-PLAerospace Mechanics of Materials (AE1108-II) – Example Problem33

PzExample 4EI-PLAWhat about beams with anon loaded free end?BLPSlopes: EIv LCP 2z PLz C12 EIv C2(0 z L)(L z 2L)Slope BC’s: C1 0At z 0, θ 0Slope CC’s:At z L, θAC θCBPL2 C2 2Aerospace Mechanics of Materials (AE1108-II) – Example Problem34

PzExample 4EI-PLAWhat about beams with anon loaded free end?LCBLPDisplacements:P 3 PL 2 EIv z z C362(0 z L)PL2 EIv z C42(L z 2L)Displacement BC’s: C3 0At z 0, v 0Displacement CC’s:At z L, vAC vCBPL3 C4 6Aerospace Mechanics of Materials (AE1108-II) – Example Problem35

PzExample 4EI-PLAWhat about beams with anon loaded free end?CLBLPDisplacements:Pz 2v z 3L 6 EI(0 z L)PL2 L v z 2 EI 3 (L z 2L)0Formula for a straight line!No curvature, it does work out! 0.5 101LAerospace Mechanics of Materials (AE1108-II) – Example Problem2L36

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Need to determine deflections and slopes of beams under load Important in many design applications Essential in the analysis of statically indeterminate beams 2. . Determine the deflection and slope at point B in