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Theory of elasticity and plasticity
Equations sheet, Part 3
Circular and annular plates
Equilibrium equation
∂2
∂r2
+
1
r2...
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
Nβ...
Constitutive equations








Nα
Nβ
Nαβ
Mα
Mβ
Mαβ








=








Dm νDm 0 0 0 0
νDm Dm 0 0 0 0...
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
Circ...
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Theory of elasticity and plasticity (Equations sheet, Part 3) Att 9035

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. 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. 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. 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. 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

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