Thin-walled cylinders

For a detailed description of the reflection at grazing incidence of X-rays, see e.g. [4.37]. X-ray astronomy began after World War II with rockets. The 

In 1951 Walter analyzed mirrors having concentric figures of revolution, i.e., paraboloids, hyperboloids, and ellipsoids [4.38]. He showed that in order to achieve a true image over an extended field of view the X-rays have to undergo two successive reflections from either a paraboloid/hyperboloid or paraboloid/ellipsoid combination which are mounted in a coaxial and confocal arrangement (see Fig. 4.52). 

The index of refraction or the optical constants can be computed from dispersion theory. 5 in (4.2) reads as follows  

An example of a very successful X-ray telescope with Zerodur mirrors is ROSAT. The resolution is 3 arc seconds. The X-ray detectors work in the energy range of 0.1-2.5 KeV (corresponding to 0.5-12.4 nm). ROSAT was sent into space in June 1990. 

ROSAT (an abbreviation of Roentgen satellite ) is a German X-ray telescope having four nested tubular mirrors constructed with Zerodur mirror substrates of 881mm maximum diameter and 580 mm height. This system should have been sent up with the Space Shuttle in 1987. The loss of the 

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Thin wall cylinders under pressure OyjpC -... 

If the drill string is thought to be a thin wall cylinder with closed ends, then the drill pipe pressure produces the axial stress and tangential stress given by the following formulas ... 

For thin-walled cylinders subject to in-plane (axial and circumferential) loading and axial torsion, Whitney and Halpin [8] have developed an analytic solution for strains. Their analysis is valid in the central region of the cylinder, end support effects are neglected. 

Now most of /2E data that are based on cylinder expansion velocities (Table 4) are derived from LRL measurements of velocities that appear to approach an asymptote at about seven-fold cylinder expansion. Hershkowitz Rigdon (Ref 15) used larger cylinders than LRL and obtained velocity measurements of up to 10- to 13-fold expansions. Comparison of these velocities with those obtained for approximately 7-fold expansion is as follows for Comp B, vt 0/v7 = 1.029 for thick-walled cylinders and 1.027 for thin-walled cylinders for 40/40/20 RDX/TNT/AN, v10/v7 = 1.025 for thick-walled cylinders and 1.023 for thin-walled cylinders and for 20/40/40 RDX/TNT/AN, v10/v7 = 1.030 for thick-walled cylinders and 1.022 for thin-walled cylinders... 

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 bursting pressure of a thin-walled cylinder may be estimated from the following equation ... 

Cambium uaually (in Dicotyledons) a thin-walled cylinder (or... 

If a thin-walled cylinder is subjected to a given pressure,/, Eqs. (5.3) and (5.4) can be used to calculate the stress in the longitudinal and peripheral direction. 

Pressure drop in competitive countercurrent air-water flow in a porous medium, which consisted of raschig rings, which are thin-walled cylinders 1 in long and 1 in in diameter, with both ends open. -[From B. J. Lerner and E. S. Grove, Jr., Critical conditions of two-phase flow in packed columns, Ind. Eng. Chem. 53 216 (1951), Copyright 1951 by the American Chemical Society. Reproduced by permission of the publisher.]... 

Let a CNT bundle contains N infinitely long metallic CNTs, closely packed together, with surface conductivity aQ. The bundle radius Rb is much less than the wavelength A. Since the incident field is almost homogeneous over the bundle cross-section, a symmetrical surface wave is excited in the bundle. In order to take into account the symmetrical local field distribution inside and outside the bundle we model one as a system of n coaxial thin-walled cylinders with the radii R, (l = l,2.,.n, Rb = Rn> Rn x> > Rx) and the surface conductivity cr, /(2 R,), where cr, is equal to the sum of linear conductivities of CNTs placed between the surfaces of cylinders with radii R, and R,, . Boundary conditions for electric Hertz potential on the surface of I -th cylinder in the cylindrical coordinate system (p, q>, z) is as follows [2] ... 

We have proposed a simple electromagnetic model of the CNT bundle, in which the bundle is considered as a system of coaxial thin-walled cylinders with effective conductivities. The bundle surface wave phase velocity was found increasing with the bundle radius. This shifts the frequency of geometric resonances into infrared range as compared with a single-wall CNT. 

Figure I 3.9 Specific torsional stiffness of beams a) the same cross-sectional area b) cross-ribbed, Dimensions in mm, all wall thicknesses 2mm wall thickness for torsional stiffness, relative to the thin-walled cylinder.

Figure 8.27 Fibrous glass-reinforced plastic thin-wall cylinder (a) internal pressure alone and (b) internal pressure plus twisting moment...

A technique suggested to fabricate complex shaped articles is the matrix transfer molding technique [69]. Woven stmctures used as reinforcement are arranged inside a mold cavity. Fluid matrix is transferred at high temperature into the mold cavity to fill the void space around the reinforcement stmcture. In this way, for example, thin-walled cylinders can be fabricated. Another proposed method for obtaining structures of complex shapes uses a... 

The hoop stress in the wall of a thin-walled cylinder of diameter D and wall thickness W is given by... 

The test method involves obtaining a relatively intact soil sample by driving a thin-walled cylinder and the subsequent activities for the determination of in-place density. When sampling or in situ density is required at depth, test Method D 1587 should be used. 

For applications below either of these parameters, the standard equation (UG-27) for thin walled cylinders will yield the same results. 

An estimate of the relative amounts of materials used in the two types of construction may be obtained using the classical equation for the stress in a thin-walled cylinder ... 

The slide valve is a thin-walled cylinder through which gas flows from the work chamber, through the open inlet valve, through the cylindrical slide valve, and exiting into the pump cavity through the inlet sleeve. The inlet sleeve is a slit in the sidewall of the slide valve. This flow path for the fill stroke is shown in Figure 2.13. 

A thin walled cylinder with closed ends of radius r and wall thickness is fabricated from a polymer with a yield stress in pure shear of k. Calculate the internal pressure required to produce yielding of the cylinder walls if the yield criterion under appropriate conditions of temperature and strain rate may be written as... 

With thin walled cylinders, as are generally seen in aircraft components, a diametrical stretch occurs due to high fluid pressures. In the design of rubber O rings or seals, establishing the dimensions of the rings, selection of the base rubber and its compounding with additives should all be considered carefully. 

Example 5 In the case of a heavy-walled cylinder under internal pressure (see Fig. 3-61). Using methods similar to those used for the thin-walled cylinder, the creep type of radial expansion for a heavy-walled cylinder can be shown to be... 

Figure 6-47. A fiberglass-reinforced plastic thin-walled cylinder showing internal pressure alone.

C-i Standards for Welding on Thin Walled Cylinders. Standards applicable to DOT welded or brazed compressed gas cylinders designed for 500 psig service pressure or less and having 1000 lb water capacity maximum also minimum wall thickness under inch. Covers procedure and operator qualification, radiographic inspection, and container repair. (24 pages)... 

Strum, R. G., A Study of the Collapsing Pressure of Thin Walled Cylinders, University of Illinois Bulletin, Vol. XXXIX, No. 12, 1941. 

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