Membrane stress analysis

In conclusion, membrane stress analysis is not completely accurate but allows certain simplifying assumptions to be made while maintaining a fair degree of accuracy. The main simplifying assumptions are that tire stress is biaxial and that the stresses are uniform across the shell wall. For thin-walled vessels these assumptions have proven themselves to be reliable. No vessel meets the criteria of being a true membrane, but we can use this tool with a reasonable degree of accuracy. 

The analysis of the membrane stresses induced in shells of revolution by internal pressure gives a basis for determining the minimum wall thickness required for vessel shells. The actual thickness required will also depend on the stresses arising from the other loads to which the vessel is subjected. 

In the stress analysis of pressure vessels and pressure vessel components stresses are classified as primary or secondary. Primary stresses can be defined as those stresses that are necessary to satisfy the conditions of static equilibrium. The membrane stresses induced by the applied pressure and the bending stresses due to wind loads are examples of primary stresses. Primary stresses are not self-limiting if they exceed the yield point of the material, gross distortion, and in the extreme situation, failure of the vessel will occur. 

The thin-wall bellows element should be designed for membrane stresses to conform to code-allowable stresses. The sum of membrane and secondary bending stresses should not exceed 1.5 times the yield stress in order to prevent the collapse of the corrugations caused by pressure. Bellows subjected to external pressure can be analyzed in a manner similar to a cylinder, utilizing an equivalent moment of inertia. The fatigue life can be estimated based on the sum of deflections and pressure stresses as compared to S/N curves based on bellows test data or using the curves in B31.3 Appendix X, Metal Bellows Expansion Joints. Formulas for the stress analysis of bellows are available in the Expansion Joints Manufacturing Association (EJMA) Standards (37). 

The Code establishes allowable stresses by stating in Para. UG-23(c) that the maximum general primary membrane stress must be less than allowable stresses outlined in material sections. Further, it states that the maximum primary membrane stress plus primary bending stress may not exceed 1.5 times the allowable stress of the material sections. In other sections, specifically Paras. l-5(e) and 2-8, higher allowable stresses are permitted if appropriate analysis is made. These higher allowable stresses clearly indicate that different stress levels for different stress categories are acceptable. 

Primary local membrane stresses are a combination of membrane stresses only. Thus only the membrane stresses from a local load are combined with primary general membrane stresses, not the bending stresses. The bending stresses associated with a local loading are secondary stresses. Therefore, the membrane stresses from a WRC-lOT-ri pe analysis must be broken out separately and combined with primary general stresses. The same is true for discontinuity membrane stresses at head-shell junctures, cone-cylinder junctures, and nozzle-shell junctures. The bending stresses would be secondary stresses. 

This analysis combines the primary membrane stress due to pressure with the secondary bending stress resulting from the flexure of the nozzle about the hard axis. 

For the cooldown analysis, only the l/4t location needs to be examined. During cooldown, the thermal stresses are tensile at the inside surface and compressive at the outside. Again, membrane stresses due to pressure are always positive. As a result, the total stress will always be greater at the inside than the outside surface. Since the maximum allowable stresses, taking into account irradiation effects, will decrease more at the inside than outside surface, it is clear that the total stress at the inside surface will approach the maximum allowable before those at the outside surface. Therefore, only the l/4t location needs to be examined for the cooldown transient. 

The bending stresses associated with a local loading are almost always classified as secondary stresses. Therefore, the membrane stresses from a WRC-107-type analysis must be broken out separately and combined with general primary stresses due to internal pressure, for example. 

Khadhraoui, M., Di Vona, M.L, and Knauth, P. (2010) Mechanical properties of proton-conducting sul-fonated aromatic polymer membranes stress—strain tests and dynamical analysis. J. Power Sources, 195, TTJO-TnS. 

Another rapid loading condition in underwater applications is the application of external hydrostatic stress to plastic structures (also steel, etc.). Internal pressure applications such as those encountered in pipe and tubing or in pressure vessels such as aerosol containers are easily treated using tensile stress and creep properties of the plastic with the appropriate relationships for hoop and membrane stresses. The application of external pressure, especially high static pressure, has a rather unique effect on plastics. The stress analysis for thick walled spherical and tubular structures under external pressure is available. 

However, it is a general practice to provide detailed stress analysis for the vessel components outside the Code approved details using either the maximum-stress or the maximum-shear theory of failure and to select allowable stresses for design conditions other than normal operations or for computed stresses other than direct membrane or direct membrane plus primary bending Code stresses. CEGB R6 method based on fracture mechanics is highly recommended. This method later on has been thoroughly discussed in this chapter. 

Kwok, K Frandsen, H.L., Sogaard, M and Hendriksen, P.V. (2014) Stress analysis and fail-safe design of bi-layered tubular supported ceramic membranes. /. Membr. 

As mentioned previously, it is necessary to calculate stresses at various points in order to determine which combination controls. Certain analysis methods, such as that in the Swedish Pressure Vessel Code, combine the membrane stress and the bending stress in the same equation. It may be necessary to separate them for evaluation when different efficiencies apply to the membrane stress rather than to the bending stress. 

For analysis purposes, the noncircular cross section of the vessel is considered as a structural frame. Each component of the rectangular or obround frame contains a load that causes a membrane stress and a moment that causes a bending stress. Hie total stress at any point is the summation of these two... 

This means that the cylinder will contract axially at a rate that is directly proportional to the interfacial energy and inversely proportional to the viscosity and the radius. The same result could have been obtained from a conventional stress analysis, using Laplace s equation for the radial stress (<7r = -y v o) and recognizing that there is an axial membrane stress of... 

When significant uplift takes place in large-diameter tanks, the state of stresses in the uplifted part of the base plate at the ultimate limit state is dominated by plate bending (including the effect of the pressure acting on the tank base), not by membrane stresses. In such cases, a finite element analysis method should be used for the calculation of the state of stresses. 

From this stress analysis, we can conclude that for exhibiting a good stability, a composite membrane has to contain a thin metal layer. On the contrary, to achieve... 

Further progress in understanding membrane instability and nonlocality requires development of microscopic theory and modeling. Analysis of membrane thickness fluctuations derived from molecular dynamics simulations can serve such a purpose. A possible difficulty with such analysis must be mentioned. In a natural environment isolated membranes assume a stressless state. However, MD modeling requires imposition of special boundary conditions corresponding to a stressed state of the membrane (see Refs. 84,87,112). This stress can interfere with the fluctuations of membrane shape and thickness, an effect that must be accounted for in analyzing data extracted from computer experiments. 

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