Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

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824 ELIA EFRAIM AND MOSHE EISENBERGER ⎧ ⎪⎨ ⎪⎩ σφ σθ σφθ σφz σθz ⎫ ⎪⎬ = ⎪⎭ E 1 − µ 2 ⎡ ⎤ ⎧ 1 µ 0 0 0 ⎢ µ 1 0 0 0 ⎥ ⎪⎨ ⎥ ⎢ 0 0 (1−µ)/2 0 0 ⎥ ⎣ 0 0 0 (1−µ)/2 0 ⎦ ⎪⎩ 0 0 0 0 (1−µ)/2 ɛφ ɛθ γφθ γφz γθz ⎫ ⎪⎬ , (6) where E is the modulus of elasticity and µ is Poisson’s ratio. For isotropic materials the force and moment resultants are obtained by integrating the stresses through the shell thickness, which in this case is variable along the meridian: ⎧ ⎪⎨ ⎪⎩ ⎧ ⎪⎨ ⎪⎩ Nφ(φ, θ) Nθ(φ, θ) Nφθ(φ, θ) Nθφ(φ, θ) ⎫ ⎪⎬ = ⎪⎭ � � Qφ(φ, θ) = κ Qθ(φ, θ) Mφ(φ, θ) Mθ(φ, θ) Mφθ(φ, θ) Mθφ(φ, θ) ⎫ ⎪⎬ = ⎪⎭ � +h(φ)/2 −h(φ)/2 � +h(φ)/2 −h(φ)/2 � +h(φ)/2 −h(φ)/2 ⎧ ⎫ ⎪⎨ σφ(z) ⎪⎬ � σθ(z) 1 + ⎪⎩ σφθ(z) ⎪⎭ σθφ(z) z � dz, (7) R0 � � � σφz(z) 1 + σθz(z) z � dz, (8) R0 ⎧ ⎫ ⎪⎨ σφ(z) ⎪⎬ � σθ(z) 1 + ⎪⎩ σφθ(z) ⎪⎭ σθφ(z) z R0 ⎪⎭ � z dz, (9) where κ is a shear correction factor. Various derivations of the shear correction factor have been proposed. Mindlin [1951] gave an implicit result for the shear correction factor for isotropic elastic plates that depends on Poisson ratio µ. Hutchinson [1984] determined the shear coefficient in a Mindlin plate equation based on matching a mode of the Mindlin plate theory to the exact Rayleigh–Lamb frequency equation for the flexural wave response at long wavelengths and proposed the value κ = 5/(6 − µ). Later, Stephen [1997] reexamined this solution, and called this the “best” shear coefficient. In the present work this value of Hutchinson’s shear coefficient is used in the calculations. Considering the stress-strain relations, the kinematical relations for shells with variable thickness the constitutive relations become Nφ(φ, θ) = E 1 − µ 2 h(φ)(ɛ0φ + µɛ0θ), Nθ(φ, θ) = E 1 − µ 2 h(φ)(µɛ0φ + ɛ0θ), κ E Qφ(φ, θ) = 2(1 + µ) h(φ)γ0φn, κ E Qθ(φ, θ) = 2(1 + µ) h(φ)γ0φn, Nφθ(φ, θ) = E 2(1 + µ) h(φ)(γ0φθ + γ0θφ), Nθφ(φ, θ) = Nφθ(φ, θ), Mφ(φ, θ) = E 1 − µ 2 h3 (φ) 12 (kφ + µkθ), Mθ(φ, θ) = E 1 − µ 2 h3 (φ) 12 (µkφ + kθ), Mφθ(φ, θ) = E h 2(1 + µ) 3 (φ) 12 (τφθ + τθφ), Mθφ(φ, θ) = Mφθ(φ, θ). (10)
STIFFNESS VIBRATION ANALYSIS OF SPHERICAL SHELL SEGMENTS WITH VARIABLE THICKNESS 825 3. Solution procedure We introduce a nondimensional coordinate ξ = (φ − φb)/φ0 that vary from 0 to 1. The variation of the geometric parameters h and R p is taken in a polynomial form as follows: h(φ) �⇒ h(ξ) = nh� i=0 hiξ i � , R p(φ) �⇒ R p(ξ) = R piξ i , (11) Spherical shell segments with a wide range of meridian opening angles, concave or convex thickness variation can be described in this way, up to any desired accuracy. In case of shell with wavy or corrugated surface, it could be represented by segmented shell with sequential shell segments with convex and concave thickness variation. When the force and moment resultants are substituted into the equations of motion (4), assuming harmonic vibrations, and using the assumed displacement field with the notation U0(φ, θ, t) = u(φ) cos nθ sin ωt, V0(φ, θ, t) = v(φ) sin nθ sin ωt, W0(φ, θ, t) = w(φ) cos nθ sin ωt, �φ(φ, θ, t) = ψφ(φ) cos nθ sin ωt, �θ(φ, θ, t) = ψθ(φ) sin nθ sin ωt, U(φ) = {u(φ), v(φ), w(φ), ψφ(φ), ψθ(φ)} T S10 S6 S8 φ0 S 3 = Qφ S 1 = Nφ S 2 = Nφθ S 5 = Mφθ S 4 = Mφ Figure 2. Dynamic stiffnesses defined by resultant forces along a unit angle segment of the perimeter of the shell edges (ξ = 0, ξ = 1). S7 cL S9 θ =1 n R p i=0 (12) (13)
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Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

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