Cylindrical shells, stress

Equations 1 to 3 enable the stresses which exist at any point across the wall thickness of a cylindrical shell to be calculated when the material is stressed elastically by applying an internal pressure. The principal stresses cannot be used to determine how thick a shell must be to withstand a particular pressure until a criterion of elastic failure is defined in terms of some limiting combination of the principal stresses. 

For a cylindrical shell the minimum thickness required to resist internal pressure can be determined from equation 13.7 the cylindrical stress will be the greater of the two principal stresses. 

Figure 13.17. Stresses in a cylindrical shell under combined loading...

Determine the shear stress distribution and velocity profile for steady, fully developed, laminar flow of an incompressible Newtonian fluid in a horizontal pipe. Use a cylindrical shell element and consider both sign conventions. How should the analysis be modified for flow in an annulus ... 

Figure 8.7 Stresses in a thin cylinder snbjected to an internal pressure, P (a) cylindrical shell nnder internal flnid pressure (b) longitudinal stress development (c) hoop stress development. Reprinted, by permission, from G. Lewis, Selection of Engineering Materials, p. 139. Copyright 1990 by Prentice-Hill, Inc.

Similarly, the total downward pressure on the semicircular portion of the cylindrical shell below the diametral plane XX is also 2prL. These two equal and opposite pressures act to burst the cylinder longitudinally at the plane XX. The resisting force comes from the hoop stress. Thus... 

Fig. 3.11 The cylinder on the left is filled with a gas at pressure p and bounded by two pistons that can move with velocity u. The long cylindrical annulus on the left is filled with a fluid. The center rod is fixed, but the outer cylindrical shell moves upward at a constant velocity. Under these circumstances a steady state-velocity distribution will develop in the fluid as illustrated u(r), with the zero velocity at the inner-rod wall and the wall velocity at the shell surface. A cylindrical control volume with its zrz shear stresses is illustrated.

Fig. 4.12 A long cylinder rotates within an outer cylindrical shell. The inner cylinder suddenly begins rotating with angular velocity Q, with the fluid in the annulus initially at rest. Also shown is a control volume illustrating the pressure and shear stresses...

The problem of definition of modulus applies to all tests. However there is a second problem which applies to those tests where the state of stress (or strain) is not uniform across the material cross-section during the test (i.e. to all beam tests and all torsion tests - except those for thin walled cylinders). In the derivation of the equations to determine moduli it is assumed that the relation between stress and strain is the same everywhere, this is no longer true for a non-linear material. In the beam test one half of the beam is in tension and one half in compression with maximum strains on the surfaces, so that there will be different relations between stress and strain depending on the distance from the neutral plane. For the torsion experiments the strain is zero at the centre of the specimen and increases toward the outside, thus there will be different torque-shear modulus relations for each thin cylindrical shell. Unless the precise variation of all the elastic constants with strain is known it will not be possible to obtain reliable values from beam tests or torsion tests (except for thin walled cylinders). 

The cylindrical shell is frequently used in pressure vessel design. For initial designs, it is useful to calculate the stresses in a thin-walled cylindrical shell that is uniformly loaded with internal pressure. For thin-walled pressure vessel calculations to be valid, the radial stresses in the shell need to be negligible. This is usually taken to be a valid assumption when the ratio of the vessel inner radius to the wall thickness (R/t) is greater than 10. "... 

The ASME code formula for the thickness of a cylindrical shell is listed in UG-27, as t = PR/(SE — 0.6P)S In this formula, t is the minimum thickness of the shell (in.), P is the maximum allowable working pressure (MAWP) (psi), R is the internal radius of the vessel (in.), S is the allowable stress in the material listed in ASME Section II, and E is the weld joint efficiency. 

The ellipsoidal shape is usual for closures of vessels that are six feet in diameter or greater. The hemispherical shape is preferred for vessels of a lesser diameter. The basic relationships for thin cylindrical shells under internal pressure assume that circumferential stress is dependent on the pressure and vessel diameter, but independent of the shell thickness. 

From basic theoretical considerations for thin-walled vessels, the minimum thickness of a closure for a cylindrical shell is obtained when the closure has a hemispherical shape. In this case, the membrane theory predicts that, using the same construction materials and allowable stresses, a hemispherical closure needs to be only half as thick as the cylindrical shell to which it is attached. 

The ellipsoidal dished head with a major to minor axis ratio of 2 1 is popular for economic reasons, even though the theory for thin-walled vessels predicts that the head of this shape should have twice the thickness of a hemispherical head where the major and minor axes are equal. Such an ellipsoidal head used for vessels under internal pressure has the same thickness as the cylindrical shell if the same allowable stresses and joint efficiencies are applied to both parts. The 1962 ASME Code Section VIII, Division 1 gives the following equation for the thin-walled ellipsoidal dished heads with a 2 1 major to minor axis ratio ... 

From the consideration of induced compressive (or tensile) stress in thin cylindrical shells, it was shown by equation 4-1 that f = pdl2t. 

Local Stresses in Spherical and Cylindrical Shells Due to External Loadings, WRC Bulletin 107, 3rd revised printing, April 1972,... 

STRESSES IN CYLINDRICAL SHELLS FROM EXTERNAL LOCAL LOADS [7, 9, 10, 11]... 

For Stress Concentration Factors see Stresses in Cylindrical Shells from External Local Loads, Procedure 5-4. 

Compact bone is found in the cylindrical shells of most long bones in vertebrates. It often contains osteons which consist of lamellae that are cylindrically wrapped around a central blood vessel (Haversian system or secondary osteon). These secondary osteons form during bone remodeling. Studies have shown that the orientation of the collagen fibrils within individual lamellas is a function of external stresses on the bone and adapted to the specific function." " ... 

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