Spherical shells, stress

Internal-pressure design rules and formulas are given for cylindrical and spherical shells and for ellipsoidal, torispherical (often called ASME heads), hemispherical, and conical heads. The formulas given assume membrane-stress failure, although the rules for heads include consideration for buckling failure in the transition area from cylinder to head (knuckle area). 

There are four commonly occurring states of stress, shown in Fig. 3.2. The simplest is that of simple tension or compression (as in a tension member loaded by pin joints at its ends or in a pillar supporting a structure in compression). The stress is, of course, the force divided by the section area of the member or pillar. The second common state of stress is that of biaxial tension. If a spherical shell (like a balloon) contains an internal pressure, then the skin of the shell is loaded in two directions, not one, as shown in Fig. 3.2. This state of stress is called biaxial tension (unequal biaxial tension is obviously the state in which the two tensile stresses are unequal). The third common state of stress is that of hydrostatic pressure. This occurs deep in the earth s crust, or deep in the ocean, when a solid is subjected to equal compression on all sides. There is a convention that stresses are positive when they pull, as we have drawn them in earlier figures. Pressure,... 

Shen and Kaelble (29) found the same linear dependence in the region —60° and 60°C but state that below —50°C and above 80°C the temperature dependence of Kraton 101 could be described by the WLF equation with cx = 16.14, C2 = 56, and Tr — — 97°C below —50°C, and Tr — 60°C above 80°C. They ascribe the temperature dependence below —50 °C to the pure polybutadiene phase and that above 80 °C to the pure polystyrene phase. They then assume that at temperatures between —50° and 80°C the molecular mechanisms for stress relaxation are being contributed by an interfacial phase visualized as a series of spherical shells enclosing each of the pure polystyrene domains and characterized... 

Thick-walled cylindrical and spherical shells (internal pressure), minimum thickness based upon circumferential stress (longitudinal joints)... 

For most particulate composites the mismatch between the particles and the matrix is more important than the anisotropy of either component (though alumina/aluminium titanate composites provide a notable exception and are described below). The main features of the stresses can therefore be understood in terms of a simple elastic model assuming thermoelastic isotropy and consisting of a spherical particle in a concentric spherical shell of matrix with dimensions chosen to give the appropriate volume fractions. The particles are predicted to be under a uniform hydrostatic stress, ap after cooling. If the particles have a larger thermal expansion coefficient than the matrix, this stress is tensile, and vice versa. For small particle volume fractions the stress... 

This compares well with the upper craze yield stress of 10.5 MPa shown in Fig. 31 c for the modified spherical shell particle that approximates to the solid PB particle. 

The determination of the interface stress When the temperature varies, the TiC ceramic sphere and NisAl spherical shell are deformed simultaneously. The displacement of the interface between TiC sphere and NisAl spherical shell are equal, namely, it must satisfy the condition of compatibility of the interface displacement, and the radial stress is equal and opposite in the interface.It can be written as R1 r2 (14)... 

The radial stress in NisAl spherical shell is rapidly decrease and equal to zero on the surface of the sphere, the tangential tensile stress is decrease with the increase of the thickness of the thickness of NisAl ... 

Stress considerations make the spherical geometry most convenient for the construction of calorimetric bombs, chemical containers, etc. For an insulated spherical shell, the electric analogy gives, in terms of Fig. 2.7 and Eq. (2.30),... 

To calculate stress due to radial load (Pr), and/or moment (M), on a spherical shell or head ... 

The stresses found from these charts will be reduced by the effect of internal pressure, but this reduction is small and can usually be neglected in practice. Bijlaard found that for a spherical shell with R y T = 100, and internal pressure causing membrane stress of 13,000 psi, the maximum deflection was decreased by only 4%-5% and bending moment by 2%. In a cylinder with the same Rm/T ratio, these reductions were about 10 times greater. This small reduction for spherical shells is caused by the smaller and more localized curvatures caused by local loading of spherical shells. 

Procedure 5-4 Stresses in Cylindrical Shells from External Local Loads, 269 Procedure 5-5 Stresses in Spherical Shells from External Local Loads, 283 References, 290... 

The approach outlined above for a fiber in an infinite matrix can also be used for a spherical particle in an infinite matrix, using the elastic solution for a thick-walled spherical shell under the action of internal and external pressures. For this case, the stresses are given by a =agg=-P in the particle and... 

Fig. 13.3 Stress-strain curves of homo-PS and three toughened blends containing particles having KRO-1 diblock morphology, HIPS particles, and particles with concentric-spherical-shell (CSS) morphology having the highest elastic compliance (from Argon et al. (1987) courtesy of Pergamon Press).

The tensile toughness Wp exhibited by homo-PS and the three PS blends containing particles of KRO-1 resin, HIPS, and concentric spherical shells (CSSs) is presented in Fig. 13.5. It shows the very substantial improvement in toughness achievable by lowering the eraze-flow stress <7oo by incorporation of compliant particles, but demonstrates also that the peak toughness achievable eventually plateaus when the craze-flow stress becomes too low as in the case of blends with the CSS particles. 

Fig. 6 Thermal deformation and stress distribution of a thin half spherical shell with circular cutting defined in 256 x 256 x 256 grid space computed by 3D-HMG...

The first model proposed for approximated analysis of the left ventricle of the heart was a spherical shell (Pao, 1980a and Mirsky, 1974) which was adopted by Woods in 1892 so that the Laplace law could be applied for calculation of the wall stresses. When the biplane silhouettes can be obtained by the X-ray technique, the left ventricle has since been analyzed as axisymmetric thick-walled shells. The advances in computer-aided tomography in recent years make it possible to image and reconstruct the cross-sectional shapes of the heart (Ritman, 1983). As a result of this development, the true three-dimensional structural shape of the heart can be accurately formed by stacking of the reconstructed cross sections together. Various finite element models have been proposed (Figure 1) for the analyses of the ventricles as well as for the cardiac valves both natural and prothetic (Pao,... 

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