Euler-Bernoulli Beams: Bending, Buckling, and VibrationLinear Elastic Beam TheoryBeam Theory:Slice Equilibrium RelationsEuler-Bernoulli Beam Theory:Displacement, strain, and stress distributionsSlice Equilibrium:Section Axial Force N(x) and Bending Moment M(x) in terms of Displacement fieldsCentroidal CoordinatesTip-Loaded Cantilever Beam: EquilibriumExercise: Cantilever Beam Under Self-WeightTip-Loaded Cantilever: Lateral DeflectionsTip-Loaded Cantilever: Axial Strain DistributionEuler Column Buckling:Non-uniqueness of deformed configurationEuler Column Buckling, Cont.Euler Column Buckling: General ObservationsEuler-Bernoulli Beam VibrationEuler-Bernoulli Beam Vibration, Cont.(1)Euler-Bernoulli Beam Vibration, Cont(2)Euler-Bernoulli Beams:Bending, Buckling, and VibrationDavid M. Parks2.002 Mechanics and Materials IIDepartment of Mechanical EngineeringMITFebruary 9, 2004Linear Elastic Beam Theory• Basics of beams–Geometry of deformation–Equilibrium of “slices”–Constitutive equations•Applications:–Cantilever beam deflection–Buckling of beams under axial compression–Vibration of beamsBeam Theory:Slice Equilibrium RelationsAxial force balance:•q(x): distributed load/length•N(x): axial force•V(x): shear force•M(x): bending momentTransverse force balance:Moment balance about ‘x+dx’:Euler-Bernoulli Beam Theory:Displacement, strain, and stress distributionsBeam theory assumptions on spatialvariation of displacement components:Axial strain distribution in beam:1-D stress/strain relation:Stress distribution in terms of Displacement field:yAxial strain varies linearlyThrough-thickness at section ‘x’ε0ε0- κ h/2εxx(y)ε0+ κ h/2Slice Equilibrium:Section Axial Force N(x) and Bending Moment M(x) in terms of Displacement fieldsN(x): x-component of force equilibriumon slice at location ‘x’:σxxM(x): z-component of moment equilibriumon slice at location ‘x’:Centroidal Coordinateschoice:Tip-Loaded Cantilever Beam: EquilibriumPFree body diagrams:•statically determinant:support reactions R, M0from equilibrium alone•reactions “present” because of x=0 geometricalboundary conditions v(0)=0;v’(0)=φ(0)=0•general equilibrium equations (CDL 3.11-12)satisfiedHow to determine lateral displacementv(x); especially at tip (x=L)?Exercise: Cantilever Beam Under Self-WeightFree body diagrams:•Weight per unit lenth: q0•q0= ρAg=ρbhgFind:•Reactions: R and M0•Shear force: V(x)•Bending moment: M(x)Tip-Loaded Cantilever: Lateral Deflectionscurvature / moment relations:geometric boundary conditionstip deflection and rotation: stiffness and modulus:Tip-Loaded Cantilever: Axial Strain Distributionstrain field (no axial force):εxxTOPεxxTOPεxxTOPεxxBOTTOMεxxBOTTOMtop/bottom axial strain distribution:strain-gauged estimate of E:Euler Column Buckling:Non-uniqueness of deformed configurationmoment/curvature:ode for buckled shape:free body diagram(note: evaluated in deformedconfiguration):Note: linear 2ndorder ode;Constant coefficients (butparametric: k2= P/EIEuler Column Buckling, Cont.ode for buckled shape:general solution to ode:boundary conditions:parametric consequences: non-trivial buckled shape only whenbuckling-based estimate of E:Euler Column Buckling: General Observations•buckling load, Pcrit, is proportional to EI/L2•proportionality constant depends stronglyon boundary conditions atboth ends: •the more kinematicallyrestrained the ends are, the larger the constant and the higher the critical buckling load (see Lab 1 handout)•safe design of long slender columnsrequires adequate margins with respectto buckling•buckling load may occur a a compressive stress value (σ=P/A) that is less than yieldstress, σyEuler-Bernoulli Beam Vibrationassume time-dependent lateral motion:lateral velocity of slice at ‘x’:lateral acceleration of slice at ‘x’:mass of dx-thickness slice:moment balance:net lateral force (q(x,t)=0):linear momentum balance (Newton):Euler-Bernoulli Beam Vibration, Cont.(1)linear momentum balance:moment/curvature:ode for mode shape, v(x), and vibrationfrequency, ω:general solution to ode:Euler-Bernoulli Beam Vibration, Cont(2)general solution to ode:pinned/pinned boundary conditions:pinned/pinned restricted solution:τ1: period offirst mode:Solution (n=1, first mode):A1: ‘arbitrary’ (but small)vibration