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EM 319 - Formula Sheet for Final

EM 319 - Formula Sheet for Final - Axial Tension and...

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Axial Tension and Torsion ( ) TL dx x EA x P E L A P x axial lateral Δ + = = = = = = α δ ε σ ε ε ε ε ν δ ε σ 0 ) ( ) ( , If P, A are constant: ) ( TL EA PL Δ + = α δ dx x GJ x T E G G r L r J Tr x = + = = = = = 0 ) ( ) ( ) 1 ( 2 φ ν γ τ θ φ γ τ If T, J are constant: GJ TL = φ , 32 2 4 4 d r J π π = = 32 ) ( 2 ) ( 4 4 4 4 i o i o d d r r J = = π π (hollow cylinder) Thin Walled: m m m m m L tA J GtA TL dx d tA T 2 2 4 , 4 , 2 = = = φ τ P δ A m L m T φ
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Beam bending: Shear forces and bending moments – normal and shear stresses – deflections Shown are the positive moments and shears: ) ( ) ( ) ( ) ( x V dx x dM x q dx x dV = = ) ( ) ( ) ( ) ( ) ( ) ( ' ' ' ' ' ' ' ' ' x M x EIv x V x EIv x q x EIv = = = = i i i A y A y where y at axis Neutral : , d A Q circle rect A V Ib VQ Ad I I circle d I rect bh I I My * 2 4 3 , ) ( 3 4 , .) ( 2 3 , , ) ( 64 , .) ( 12 , = = = = = + = = = = α α α τ π σ Plane Stress (Stress transformations - Mohr’s circle) c ys xs p s xy y x s y x xy p c p p xy y x y x c R shear plane in Maximum R stresses incipal R σ σ σ π θ θ τ σ σ θ τ σ σ τ θ σ σ τ σ σ σ σ σ = = = = = = ± = + = + = 4 , 2 2 tan , 2 2 tan , Pr 2 2 max 2 , 1 2 2 θ τ θ σ σ τ θ τ θ σ σ σ σ σ σ 2 cos 2 sin 2 , 2 sin 2 cos 2 2 , 1 1 1 1 xy y x y x xy y x y x y x + = ± ± + = } 2 , 2 { ) ( : 2 1 max σ σ τ Max Plane of Out shear plane of out Max =
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