formula sheet

# formula sheet - 1D stress components = max = P A = My I =...

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1D stress components σ = P A = My I = VQ It = Tr J τ max = 3 V 2 A rectangle max = 4 V 3 A solid circle max = 2 V A thin tube shear stress distribution in a rectangle τ = 3 V 2 A ± 1 y c 2 ² where y is measured from the neutral axis and c is the half height of the beam ( c = h/ 2) Axial Torsion Beams Δ= PL EA φ = TL GJ 1 ρ = κ = d 2 v dx 2 = M EI dM dx = V dV dx = w Q x = Z ydA, Q y = Z xdA ¯ y = Q x A , ¯ x = Q y A I x = Z y 2 dA , I y = Z x 2 dA Solid circle: I x = I y = π r 4 4 ,J = π r 4 2 Plane stress σ xx σ yy σ xy = E 1 ν 2 νE 1 ν 2 0 1 ν 2 E 1 ν 2 0 00 E 2(1 + ν ) ε xx ε γ xy Plane strain σ xx σ σ xy = λ +2 μλ 0 λλ μ 0 μ ε xx ε γ xy λ = (1 + ν )(1 2 ν ) μ = G = E 2(1 + ν ) 1

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Stress transformations in 2D σ 0 xx = σ xx + σ yy 2 + σ xx σ 2 cos 2 θ + σ xy sin 2 θ σ 0 xy = σ xx σ 2 sin 2 θ
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formula sheet - 1D stress components = max = P A = My I =...

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