By A.S. Kompaneyets
A process Theoretical Physics, quantity 2. Statistical legislation: Statistical Physics, Hydrodynamics and fuel Dynamics, Electrodynamics of continuing Media, actual Kinetics. writer: A. S. Kompaneyets. MIR Publishers, Moscow.
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Additional resources for A course of theoretical physics, vol.2: statistical laws
13 Cylindrical pressure vessel with thick end plates Consider a cylindrical pressure vessel of radius a, length L and uniform thickness t which is rigidly welded at its ends to thick end plates, as shown in Fig. 10. If the vessel is subjected to an internal L/2 L/2 Fig. 10. Cylindrical pressure vessel. pressure p9 then the applied components of load X9 Y and Z will be X=0 7=0 Z = -p. 28) is therefore dV -+4ß*w=-plD. 33), then the complete solution will be w = exp ßx[Ci cos ßx-\-C2 sin ßx] + exp-ßx[C 3 cos ßx+Ct sin ßx]-pa2/Et.
E. at x = 0. 40), C 3 = qa^LjEt. 43) which is the complete solution for the radial deflexion w. The stress resultants may then be obtained by using the relevant differentials of w. However, in differentiating w it is expedient to use the following substitutions: Θ = exp (—ßx) cos ßx and Φ = exp (—ßx) sin ßx. 45) ά3Φ 4dx^3 - = 2β*[θ+Φ]. 35) Mx= ρ £ [ - Φ + (1-1/£Ζ,)Θ]/2£ 2 , Μφ = rMo, Νφ = QaL[l-xlL-0-(l-l/ßL^l Qx = ρΕ[-2Θ+(Θ+Φ)/βΣ]/2β. 7 gave the following stress resultants: Νφ = Qa(L-x), Νχφ = 0 and Nx = 0.
This differential equation is termed an equation of compatibility. When solved the remaining unknowns are readily determined from the equilibrium and stress-resultant displacement equations. In this particular case the radial deflexion w will be selected as the dependent variable and the differential equation is derived in the following section. 4) to this we eliminate dQJdx and obtain d2Mx Νφ _ _ + _ + , = 0. 22) we have Νφ =Et(-w/a + tfiw/a)/(l-v2) = -Etw/a. 28) ß* = Et/4Da2 = 3(1 - v2)/a2t2.
A course of theoretical physics, vol.2: statistical laws by A.S. Kompaneyets