Linear_Elasticity_05_Presure_Vessels

Linear_Elasticity_05_Presure_Vessels - Section 4.5 4.5 The...

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Section 4.5 Solid Mechanics Part I Kelly 127 4.5 The Thin-walled Pressure Vessel Theory An important practical problem is that of a container subjected to an internal pressure p . Such a container is called a pressure vessel , Fig. 4.5.1. In many applications it is convenient and valid to assume that (i) the material is isotropic (ii) the strains resulting from the pressures are small (iii) the wall thickness t of the pressure vessel is much smaller than some characteristic radius: i o i o r r r r t , << = Figure 4.5.1: A pressurised container Because of (i,ii), the isotropic linear Elastic model is used. Because of (iii), it will be assumed that there is negligible variation in the stress field across the thickness of the vessel, Fig. 4.5.2. Figure 4.5.2: Approximation to the stress arising in a pressure vessel As a rule of thumb, if the thickness is less than a tenth of the vessel radius, then the actual stress will vary by less than about 5% through the thickness, and in these cases the constant stress assumption is valid. Note that a pressure i zz yy xx p = = = σ means that the stress on any plane drawn inside the vessel is subjected to a normal stress i p and zero shear stress. i r 2 o r 2 p t p t t actual stress approximate stress p
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Section 4.5 Solid Mechanics Part I Kelly 128 4.5.1 Thin Walled Spheres A thin-walled spherical shell is shown in Fig. 4.5.3. Because of the symmetry of the sphere and of the pressure loading, the circumferential (or tangential or hoop) stress t σ at any location and in any tangential orientation must be the same. Figure 4.5.3: a thin-walled spherical pressure vessel Considering a free-body diagram of one half of the sphere, Fig. 4.5.4, force equilibrium requires that ( ) 0 2 2 2 = + t i o i r r p r π (4.5.1) and so () t r r p r i i t + = 0 2 ( 4 . 5 . 2 ) Figure 4.5.4: a free body diagram of one half of the spherical pressure vessel One can now take as a characteristic radius the dimension r . This could be the inner radius, the outer radius, or the average of the two – results for all three should be close: t pr t 2 = Tangential stress in a thin-walled spherical pressure vessel (4.5.3) This tangential stress accounts for the stress in the plane of the surface of the sphere. The stress normal to the walls of the sphere is called the radial stress , r . The radial stress is zero on the outer wall since that is a free surface. On the inner wall, the normal stress is p r = , Fig. 4.5.5. From Eqn. 4.5.3, since 1 / << r t , t p << , and it is reasonable to t p t t
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Section 4.5 Solid Mechanics Part I
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Linear_Elasticity_05_Presure_Vessels - Section 4.5 4.5 The...

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