Theory of elasticity and plasticity (Equations sheet, Part 3) Att 9035
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Theory of elasticity and plasticity (Equations sheet, Part 3)
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Theory of elasticity and plasticity (Equations sheet, Part 3) Att 9035
- 1. Theory of elasticity and plasticity Equations sheet, Part 3 Circular and annular plates Equilibrium equation ∂2 ∂r2 + 1 r2 ∂2 ∂θ2 + 1 r ∂ ∂r ∂2 w ∂r2 + 1 r2 ∂2 w ∂θ2 + 1 r ∂w ∂r = q(r, θ) D Moment resultants Mr = −D ∂2 w ∂r2 + ν 1 r2 ∂2 w ∂θ2 + 1 r ∂w ∂r , Mθ = −D 1 r ∂w ∂r + 1 r2 ∂2 w ∂θ2 + ν ∂2 w ∂r2 , Mrθ = −(1 − ν)D 1 r ∂2 w ∂r∂θ − 1 r2 ∂w ∂θ Axi-symmetrical bending Equilibrium equation d4 w dr4 + 2 r d3 w dr3 − 1 r2 d2 w dr2 + 1 r3 dw dr = q(r) D Homogeneous solution wh = C1 + 1 2 r2 C2 Bending moments Mr = −D d2 w dr2 + ν r dw dr , Mθ = −D ν d2 w dr2 + 1 r dw dr , Mrθ = 0 Shells General membrane theory Strain-displacement equations εαα εββ γαβ = ∂ ∂α 0 −κα 0 ∂ ∂β −κβ ∂ ∂β ∂ ∂α 0 uα uβ uz 1
- 2. Stresses and membrane forces σαα σββ σαβ = E 1 − ν2 1 ν 0 ν 1 0 0 0 1−ν 2 εαα εββ γαβ , Nαα Nββ Nαβ = σαα σββ σαβ t Equilibrium equations − ∂ ∂α 0 − ∂ ∂β 0 − ∂ ∂β − ∂ ∂α −κα −κβ 0 Nαα Nββ Nαβ = pα pβ pz Membrane theory of shells of revolution Strain-displacement equations and membrane forces εφ εθ = 1 r1 ∂ ∂φ 1 r1 1 r3 1 r2 uφ uz , Nφ Nθ = Et 1 − ν2 1 ν ν 1 εφ εθ Equilibrium equations − 1 r1 ∂ ∂φ 1 r3 1 r1 1 r2 Nφr Nθr = pφr pzr Solution of the equilibrium equation Nφ = 1 r sin φ F(φ), Nθ = pzr2 − 1 r1 sin2 φ F(φ) where F(φ) = (pz cos φ − pφ sin φ)rr1dφ General bending theory Strain-displacement equations εα εβ γαβ ρα ρβ ραβ = ∂ ∂α 0 −κα 0 ∂ ∂β −κβ ∂ ∂β ∂ ∂α −2καβ 0 0 − ∂2 ∂α2 0 0 − ∂2 ∂β2 0 0 − ∂2 ∂α∂β uα uβ uz 2
- 3. Constitutive equations Nα Nβ Nαβ Mα Mβ Mαβ = Dm νDm 0 0 0 0 νDm Dm 0 0 0 0 0 0 Dm 1−ν 2 0 0 0 0 0 0 Db νDb 0 0 0 0 νDb Db 0 0 0 0 0 0 Db 1−ν 2 εα εβ γαβ ρα ρβ ραβ where Dm = Et 1−ν2 and Db = Et3 12(1−ν2) Equilibrium equations − ∂ ∂α 0 − ∂ ∂β 0 0 0 0 − ∂ ∂β − ∂ ∂α 0 0 0 −κα −κβ −2καβ − ∂2 ∂α2 − ∂2 ∂β2 − ∂2 ∂α∂β Nα Nβ Nαβ Mα Mβ Mαβ = pα pβ pz Displacement differential equation Db 2 2 2 2 uz + Dm(1 − ν2 )Γ2 uz = 2 2 pz − κα pαdα − κβ pβdβ + Γ ∂ ∂β2 pαdα + ∂ ∂α2 pβdβ − ν ∂pα ∂α − ν ∂pβ ∂β Bending theory of shells of revolution Strain-displacement and constitutive equations εα εβ ρα ρβ = ∂ ∂α −κα 0 −κβ 0 − ∂2 ∂α2 0 0 uα uz , Nα Nβ Mα Mβ = Dm νDm 0 0 νDm Dm 0 0 0 0 Db νDb 0 0 νDb Db εα εβ ρα ρβ Equilibrium equations − ∂ ∂α 0 0 0 −κα −κβ − ∂2 ∂α2 0 Nα Nβ Mα Mβ = pα pz Displacement equilibrium equation Db d4 uz dα4 + Dm(1 − ν2 )κ2 βuz = pz − (κα + νκβ) pαdα 3
- 4. Homogeneous solution uz(α) = e−µα (C1 cos µα + C2 sin µα) + eµα (C3 cos µα + C4 sin µα) where µ4 = Dm(1−ν2 ) 4Db κ2 β Circular cylindrical shells Kinematic and constitutive equations εx εθ ρx ρθ = ∂ ∂x 0 0 −κθ 0 − ∂2 ∂x2 0 0 ux uz , Nx Nθ Mx Mθ = Dm νDm 0 0 νDm Dm 0 0 0 0 Db νDb 0 0 νDb Db εx εθ ρx ρθ Equilibrium equations − ∂ ∂x 0 0 0 0 −κθ − ∂2 ∂x2 0 Nx Nθ Mx Mθ = px pz Displacement equilibrium equation Db d4 uz dx4 + Dm(1 − ν2 )κ2 θuz = pz − νκθ pxdx Solution of the equilibrium equation uz(x) = e−µx (C1 cos µx + C2 sin µx) + eµx (C3 cos µx + C4 sin µx) + r2 Et pz − νr Et Nx Yield line theory of slabs Virtual work principle Qu∆ = Muxθxy0 + Muyθyx0 where Qu is load resultant on the segment; ∆ is the displacement of its centroid; Mux and Muy are the components of the ultimate moment normal to the yield line; θx and θy are the components of the rotation and x0 and y0 are the components of the yield line length Minimum load principle ∂Qu ∂x1 = 0, . . . ∂Qu ∂xn = 0 where Qu = f(x1, x2, . . . , xn) 4