Stress calculation

The stress theorem relies upon the variational principle applied together with a strain-scaling of the quantum system, as discussed in detail by the present authors elsewhere (Nielsen and Martin, to be published). The strain scales particle positions as x- (1 + e)x, and by definition the macroscopic stress per volume n (a and B denote cartesian coordinates) is derived from the total energy by 

In the case of plane-wave basis sets the scaling proceeds on the reciprocal-space vectors as G- (1+ ) G, which is seen by the definition aj b = 6 4, where and b are real- and reciprocal-lattice primitive translation vectors, respectively. Thus one finds the derivative of reciprocal-space vectors given by 

It is noted that structure factors, exp(iG x), are unchanged since aj -bj = 6j. . Volume SI times charge density p(G) (or S2 times 

Here e denotes a convergence parameter (and not the strain jjg) which may be chosen for computational performance. T denotes the real-lattice translation vectors, and x the atomic positions in the unit cell. The function H (x) is 

The terms i) through vii) add up to the total stress per unit volume, and are calculated in exactly the same way as total energy and forces(Wendel and Martin, 1979 Ihm et al., 1979). 

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The outlined scheme is shown to yield stable solutions for non-zero Weissenberg number flows in a number of benchmark problems (Swarbric and Nassehi, 1992b). However, the extension of this scheme to more complex problems may involve modifications such as increasing of elemental subdivisions for stress calculations from 3 x 3 to 9 x 9 and/or the discretization of the stress field by biquadratic rather than bi-linear sub-elements. It should also be noted that satisfaction of the BB condition in viscoelastic flow simulations that use mixed formulations is not as clear as the case of purely viscous regimes. 

Stress calculations, fault tree analysis, failure modes analysis, and worst case analysis... 

For stress calculations, the pressure loss in the annulus may be ignored.)... 

The sections to follow discuss determining keyway depth and width, keyway manufacturing tolerances, key stress calculations, and shaft stress calculations. 

Tensile testing is susceptible to invalid runs due to sample flaws, poor mounting, or many other other sources of error. It is therefore essential that outliers be identified and removed. After all runs have been analyzed, STRESS calculates means and standard deviations for each parameter. It also performs Student t-tests on... 

Sm is the maximum allowable operating stress, calculated as specified minimum yield strength x Hf, where Hf is the material performance factor from Mandatory Appendix IX, Table IX-5A or IX-5B. Material performance factors account for the adverse effects of hydrogen gas on the mechanical properties of carbon steels used in the construction of pipelines. 

Figures 4E, 5E, 6E and 7E depict the individual coil hoop stresses along the radial direction at the middle plane of each coil. Particularly for the high-field 11.75T magnet, the stress calculation indicates that the most inner coils are the ones that are most crucial in the design, because they are exposed to the greatest magnetic fields and sfresses. If is possible to use different superconductors (i.e. cheaper) to build the outer superconducting coils, since these are well within the superconductivity limit.

Creeping of the material under pressure and temperature must be additionally taken into account. The time shield limit of the material must be known for a correct stress calculation. The results influence the life span of the high pressure vessels and other components. Time shield values are listed in the pressure vessel codes. 

Fig. 16a, b. Temperature dependence of swelling stress calculated from Figs. 14 and 15... 

It should be noted that the stresses usually used are engineering stresses calculated from the ratio of force and original cross section area whereas true stress is the ratio of the force and the actual cross sectional area at that deformation. Clearly, the relationship between stress and strain depends on the definition of stress used and taking the case of tensile strain, for example, the true stress is equal to the engineering stress multiplied by the extension ratio. 

It is usual in rubber testing to calculate tensile stresses, including that at break, on the initial cross-sectional area of the test piece. Strictly, the stress should be the force per unit area of the actual deformed section but this is rather more difficult to calculate and in any case, it is the force that a given piece of rubber will withstand which is of interest. The stress calculated on initial cross-section is sometimes called nominal stress. ... 

Jordan et al. (100) refined the thermoelastic stress calculation for the analysis of the spatial distribution of dislocations in GaAs grown with the LEC method. Their analysis was based on a two-dimensional model for the... 

Figure 15. Comparison of measured dislocation density in a GaAs wafer grown by the LEC method (top) with the thermoelastic stress calculation by Jordan et al. (bottom) (100). The high dislocation density around the periphery is predicted by the calculations.

Thermal stress calculations in the five cell stack for the temperature distribution presented above were performed by Vallum (2005) using the solid modeling software ANSYS . The stack is modeled to be consisting of five cells with one air channel and gas channel in each cell. Two dimensional stress modeling was performed at six different cross-sections of the cell. The temperature in each layer obtained from the above model of Burt et al. (2005) is used as the nodal value at a single point in the corresponding layer of the model developed in ANSYS and steady state thermal analysis is done in ANSYS to re-construct a two-dimensional temperature distribution in each of the cross-sections. The reconstructed two dimensional temperature is then used for thermal stress analysis. The boundary conditions applied for calculations presented here are the bottom of the cell is fixed in v-dircction (stack direction), the node on the bottom left is fixed in x-direction (cross flow direction) and y-direction and the top part is left free to... 

Stresses are induced by piling the cells or binding each bundle. Mechanical loads or pressure are also considered in the stress simulations. During cell operation, thermally induced stresses are superposed on the stresses induced by the mechanical loads. This state can be treated by the stress calculations. 

Stress calculations are carried out by the finite element method. Here, the commercial finite method code ABAQUS (Hibbit, Karlsson, and Sorensen, Inc.) is used. Other codes such as MARC, ANSYS are also available. To calculate the stresses precisely, appropriate meshes and elements have to be used. 2D and shell meshes are not enough to figure out stress states of SOFC cells precisely, and thus 3D meshes is suitable for the stress calculation. Since the division of a model into individual tetrahedral sometimes faces difficulties of visualization and could easily lead to errors in numbering, eight-comered brick elements are convenient for the use. The element type used for the stress simulation here is three-dimensional solid elements of an 8-node linear brick. In the coupled calculation between the thermo-fluid calculation and the stress calculation a same mesh model have to be used. Consequently same discrete 3D meshes used for the thermo-fluid analysis are employed for the stress calculation. Using ABAQUS, the deformations and stresses in a material under a load are calculated. Besides this treatment, the initial and final conditions of models can be set as the boundary conditions and the structural change can thus be treated. 

Thus Equation (10.33) is solved as the new equilibrium equation. To calculate the thermal expansion behavior of the model, the thermal expansion coefficient is necessary as the calculating parameter. If the temperature of the model is even, the initial temperature and the final temperature are used as just a calculating parameter. If a temperature distribution exists in the model, the temperature distribution data is dispensable for the stress calculation. The temperature is firstly calculated by a CFD and the calculated data is used as the boundary condition in the stress calculation. If the thermal expansion coefficient is temperature dependent, the temperature dependence must be considered in the calculation. Here the temperature data at the nodes is transferred from STAR-CD to the ABAQUS. 

Numerical calculations for the residual stresses in the anode-supported cells are carried out using ABAQUS. After modeling the geometry of the cell of the electro-lyte/anode bi-layer, the residual thermal stresses at room temperature are calculated. The cell model is divided into 10 by 10 meshes in the in-plane direction and 20 submeshes in the out-plane direction. In the calculation, it is assumed that both the electrolyte and anode are constrained each other below 1400°C and that the origin of the residual stresses in the cell is only due to the mismatch of TEC between the electrolyte and anode. The model geometry is 50 mm x 50 mm x 2 mm. The mechanical properties and cell size used for the stress calculation are listed in Table 10.5. 

All the parameters used for the stress calculation on the doped-LaCr03 are listed in Table 10.6. 

Most of the time, the three-point bending test (Fig. 12.2c) induces fracture without exhibiting yielding. The stress calculated here is a maximum stress due to the inhomogeneity of the stress field along the thickness ... 

Table 7.4 Properties of ceramics used in the stress calculation...

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