Example: Pure Bending of a BeamAy4M2cMX2/1=2c/3Elasticity SolutionStrengthof Materials SolutionUsesEuler-BernoullibeamtheorytofindM.y.bending stress and deflection of beam11centerlineC,=T=0;M0yMxyuEIC, =Tx =0;MMVV=V(x,O)2EI2E1Two solutions areidentical, with the exception of thex-displacements11
Example: Pure Bending of a Beam ( ) 2 22 , 0; , . 2 x y xy M y I Mxy u EI M v yxl EI σ σ τ ν = − = = = − = +− Elasticity Solution Strength of Materials Solution Uses Euler-Bernoulli beam theory to find bending stress and deflection of beam centerline ( ) 2 2 , 0; ( ,0) . 2 x y xy M y I M v vx x l EI σ σ τ = − = = = = − • Two solutions are identical, with the exception of the x-displacements 3 I c = 2 /3 11
Example: Beam under Uniform Transverse Loading+W++★+:Boundary conditions个WlWI AT(x,±c)=0[Fv(±l,0)= 02c,(x,c)=0M(±l,0) = 0xFs(±l,0) =干wl[o,(x,-c)=-wvy21Solve by the semi-inverse methodAnalyze the sources of individual stress components and proposean appropriate form for Airy Stress FunctionaM→or, Fs→txy， w→o,, w=constant→o,=f(yax?f(y)+xf.(y)+ f2(y)212
Example: Beam under Uniform Transverse Loading ( ) 2 2 2 1 2 , , , constant () () () 2 M F w w x S xy y y f y x x f y xf y f y ψ στσ σ ψ ∂ → → → = →= = ∂ ⇒= + + (, ) 0 ( ,0) 0 ( , ) 0 ( ,0) 0 (, ) ( ,0) xy N y y S x c F l xc M l xc w F l wl τ σ σ ± = ± = = ±= − =− ± = • Boundary conditions • Solve by the semi-inverse method • Analyze the sources of individual stress components and propose an appropriate form for Airy Stress Function 12