Analysis stress

A calculated value of stress means little until it is associated with its location and distribution in the vessel and with the type of loading by which it was produced. Different types of stresses have different degrees of significance. 

The designer must be familiar with the various types of loadings and their stresses in order to accurately understand the results of the analysis. The designer must also consider the stress categories to determine the allowable stress limits. 

The following sections will provide the fundamental knowledge for determining and understanding the results of an analysis. The topics covered in Chapter 1 form the basis by which the rest of the book is to be used. A section on special problems and considerations is included to alert the designer to more complex problems that exist 

Stress analysis is the determination of the relationship between external forces applied to a vessel and the corresponding stresses. The emphasis of this book is not how to do stress analysis in particular, but rather how to analyze vessels and flieir component parts in an effort to arrive at an economical and safe design—the difference being that we analyze stresses where necessary to determine thickness of material and sizes of members. We are not so concerned wifli building mathematical models as 

The starting place for stress analysis is to determine all the design conditions for a given problem and then determine all the related external forces. We must then relate these external forces to the vessel parts which must 

There are different techniques to evaluate the quantitative stress level in prototype and production products. They can predict potential problems. Included is the use of electrical resistance strain gauges bonded on the surface of the product. This popular method identifies external and internal stresses. Their various configurations are made to identify stresses in different directions. This technique has been extensively used for over a half century on very small to very large products such as toys to airplanes. There is the optical strain measurement system that is based on the principles of optical interference. It uses Moire, laser, or holographic interferometry (2,3,20). 

Another very popular method is using solvents that actually attack the product. It 

Photoelastic measurement is a very useful method for identifying stress in transparent plastics. Quantitative stress measurement is possible with a polarimeter equipped with a calibrated compensator. It makes stresses visible (Fig. 5-2). The optical property of the index of refraction will change with the level of stress (or strain). When the photoelastic 

This photoelastic stress analysis is a technique for the nondestructive determination of stress and strain components at any point in a stressed product by viewing a transparent plastic product. If not transparent, a plastic coating is used such as certain epoxy, polycarbonate, or acrylic plastics. This test method measures residual strains using an automated electro-optical system. 

This concept has been known for over a century. Expressed as Brewster s Constant law, it states that the index of refraction in a strained material becomes directional, and the change of the index is proportional to the magnitude of the stress (or strain) present. Therefore, a polarized beam in the clear plastic splits into two wave fronts in the X and Y directions that contain vibrations oriented along the directions of principal stresses. An analyzing filter passes only vibrations parallel to its own transmitting plane (Chapter 4, TRANSPARENT AND OPTICAL PRODUCT, Polarized Lighting). 

In Chapter 8 the topic of design is described systematically, taking into account many factors such as chemical resistance, fracture, fatigue, and forming constraints, in addition to the viscoelastic limitations. We here examine briefly two matters stress analysis with a viscoelastic material and the method of design when the stress applied to the viscoelastic material is neither a step-function nor sinusoidal function of time. The following treatments rest on the assumption of linear viscoelastic behaviour. 

At time i after the imposition of the rotation 0 the shear stress is 

It is thus possible to determine r(f) for all values of / for which G(t) is known. The solution is identical to the elastic solution. 

This will be true only if the material is linear up to the maximum strain. The maximum strain occurs at r = a and is 

It can be appreciated that the problem is much more complex if part, of the cross-section (the outer part) deforms in the non-linear range. If this occurs, the strain in t he outer part is still given by eqn (4.73). The complexity arises because the stress generated at each value of r is less than that given by eqn (4.74) (see Fig. 4.6). The problem is in some respects analogous to the twisted elastic-plastic rod. 

The hydrostatic test pressure is 1.25 times the design pressure corrected for temperature, rather than the usual 1.5. Division 2 establishes upper limits for the test pressure relative to the yield strength at test temperature. The pneumatic test pressure is 1.15 times the design pressure corrected for temperature rather than 1.25 required by Division 1. Division 2 has no provision for proof tests to establish the maximum allowable working pressure. But Appendix 6, Experimental Stress Analysis, provides for the determination of the critical or governing stresses for unusual geometries for which theoretical stress analysis is inadequate. 

The Division 2 code implied from the beginning that the safety factor was being lowered from 4 1, as required by the ASME Section VIII, Division 1 code, to the 3 1 permitted by the Division 2 code. At first it seems that a vessel built to the Division 2 code will be less expensive than a Division 1 code vessel. This conclusion is correct if only the materials and welding used in the manufacture of any such vessel are considered. 

In applications where internal pressures range upward from 3,000 psi and for large diameter vessels requiring huge quantities of materials, the Division 2 code saves material costs. For most process equipment it is doubtful whether the Division 2 code should be used at all, especially where the vessels are subjected to dynamics problems in which moment of inertia and modulus of elasticity are more critical than strength values. 

Vessels with pressures low enough to require a thickness governed by fabrication minimums do not require Division 2 standards unless then-operation demands attention to pulsating pressure causing fatigue or some other peculiar safety problem. 

Computer programs for Division 1 vessels have been refined to such a point that full confidence can be placed in the calculations. The cost of producing engineering calculations is therefore lower by computer than by the old hand calculation method. 

Metallic components can usually be designed for load-bearing duties on 

28 The origin of physical ageing in an amorphous polymer. After quenching to a temperature below Tg (but above T, the temperature of the highest secondary relaxation), the volume stCM4y contracts the movement is towards the equilibrium, which is the extrapolMed V - T line for the liquid (after Struik). 

---

The eddy current method allows to evalute the state of stress in ferromagnetic material. The given method is used for determining own stress as well as that formed in effect of outside load. With regard to physical principles of own stress analysis, the dependence between the magnetic permeability and the distance between atomic surfaces is utilized. 

K. Kabe, M. Koishi, and T. Akasaka, "Stress Analysis for Twisted Cord and mbbet of FRR," presented at 6th Japan—U.S. Conference on Composite Materials, Odando, Fla., June 1992. 

Other interference-produced colors falling into this section include doubly refracting materials such as anisotropic crystals and strained isotropic media between polarizers, as in photoelastic stress analysis and in the petrological microscope. 

Table 10-56 gives values for the modulus of elasticity for nonmetals however, no specific stress-limiting criteria or methods of stress analysis are presented. Stress-strain behavior of most nonmetals differs considerably from that of metals and is less well-defined for mathematic analysis. The piping system should be designed and laid out so that flexural stresses resulting from displacement due to expansion, contraction, and other movement are minimized. This concept requires special attention to supports, terminals, and other restraints. 

Elastic Behavior The assumption that displacement strains will produce proportional stress over a sufficiently wide range to justify an elastic-stress analysis often is not valid for nonmetals. In brittle nonmetallic piping, strains initially will produce relatively large elastic stresses. The total displacement strain must be kept small, however, since overstrain results in failure rather than plastic deformation. In plastic and resin nonmetallic piping strains generally will produce stresses of the overstrained (plasfic) type even at relatively low values of total displacement strain. 

Appendix 4 gives definitions and rules for stress analysis for shells, flat and formed heads, and tube sheets, layered vessels, and nozzles including discontinuity stresses. Of particular importance are Table 4-120.1, Classification of Stresses for Some Typical Cases, and Fig. 4-130.1, Stress Categories and Limits of Stress Intensity. These are veiy useful in that they clarify a number of paragraphs and simphfy stress analysis. 

Appendix 6 contains requirements of experimental stress analysis. Appendix 8 has acceptance standards for radiographic examination. Appendix 9 covers nondestructive examination. Appendix 10 gives rules for capacity conversions for safety valves, and Appendix 18 details quahty-control-system requirements. 

General Considerations Most pressure vessels for the chemical-process industry will continue to be designed and built to the rules of Sec. T11, Division 1. While the rules of Sec. T11, Division 2, will frequently provide thinner elements, the cost of the engineering analysis, stress analysis and higher-quality construction, material control, and inspection required by these rules frequently exceeds the savings from the use of thinner walls. 

H. Tada, P. Paris and G. Irwin, The Stress Analysis of Cracks Handbook, Del Research Corporation, St Louis, 1973 (for Tabulation of Stress Intensities). 

Consider the design of a glass window for a vacuum chamber (Fig. 18.6). It is a circular glass disc of radius R and thickness f, freely supported in a rubber seal around its periphery and subjected to a uniform pressure difference Ap = 0.1 MPa (1 atmosphere). The pressure bends the disc. We shall simply quote the result of the stress analysis of such a disc it is that the peak tensile stress is on the low-pressure face of... 

Figure 27.1 summarises the methodology for designing a component which must carry load. At the start there are two parallel streams materials selection and component design. A tentative material is chosen and data for it are assembled from data sheets like the ones given in this book or from data books (referred to at the end of this chapter). At the same time, a tentative component design is drawn up, able to fill the function (which must be carefully defined at the start) and an approximate stress analysis is carried out to assess the stresses, moments, and stress concentrations to which it will be subjected. 

The next step is a detailed specification of the design and of the material. This may require a detailed stress analysis, analysis of the dynamics of the system, its response... 

The S -shaped flexible elements were required to keep the stiffness and stresses low, due to the relatively heavy rotor weight as evident by the finite element stress analysis shown in Figure 6-30. The wire EDM teehnology allows the produetion of sueh a damper deviee, whieh ean be easily designed with an offset to eompensate for the defleetion due to rotor weight. 

Engineering science problems (stress analysis errors, etc.)... 

The factor of safety had little scientific background, but had an underlying empirical and subjective nature. No one can dispute that at the time that stress analysis was in its infancy, this was the best knowledge available, but they are still being applied today Factors of safety that are recommended in recent literature range from 1.25 to 10 for various material types and loading conditions (Edwards and McKee, 1991 Haugen, 1980). 

The varianee equation provides a valuable tool with whieh to draw sensitivity inferenees to give the eontribution of eaeh variable to the overall variability of the problem. Through its use, probabilistie methods provide a more effeetive way to determine key design parameters for an optimal solution (Comer and Kjerengtroen, 1996). From this and other information in Pareto Chart form, the designer ean quiekly foeus on the dominant variables. See Appendix XI for a worked example of sensitivity analysis in determining the varianee eontribution of eaeh of the design variables in a stress analysis problem. 

The calculated loading stress, L, on a component is not only a function of applied load, but also the stress analysis technique used to find the stress, the geometry, and the failure theory used (Ullman, 1992). Using the variance equation, the parameters for the dimensional variation estimates and the applied load distribution, a statistical failure theory can then be formulated to determine the stress distribution, f L). This is then used in the SSI analysis to determine the probability of failure together with material strength distribution f S). 

Often in stress analysis we may be required to make simplified assumptions, and as a result, uneertainties or loss of aeeuraey are introdueed (Bury, 1975). The aeeuraey of ealeulation deereases as the eomplexity inereases from the simple ease, but ultimately the eomponent part will still break at its weakest seetion. Theoretieal failure formulae are devised under assumptions of ideal material homogeneity and isotropie behaviour. Homogeneous means that the materials properties are uniform throughout isotropie means that the material properties are independent of orientation or direetion. Only in the simplest of eases ean they furnish us with the eomplete solution of the stress distribution problem. In the majority of eases, engineers have to use approximate solutions and any of the real situations that arise are so eomplieated that they eannot be fully represented by a single mathematieal model (Gordon, 1991). 

The failure determining stresses are also often loeated in loeal regions of the eomponent and are not easily represented by standard stress analysis methods (Sehatz et al., 1974). Loads in two or more axes generally provide the greatest stresses, and should be resolved into prineipal stresses (Ireson et al., 1996). In statie failure theory, the error ean be represented by a eoeffieient of variation, and has been proposed as C =0.02. This margin of error inereases with dynamie models and for statie finite element analysis, the eoeffieient of variation is eited as Q = 0.05 (Smith, 1995 Ullman, 1992). 

Weber, M. A. and Penny, R. K. 1991 Probabilistic Stress Analysis Methods. Stress Analysis and Failure Prevention, DE-Vol. 30, ASME, 21-27. 

Cohen, R., Valve Stress Analysis for Fatigue Problems, ASHRAE Journal, January 1973, pp. 57-61. 

WILLIAMS, j. G., Stress Analysis of Polymers, Longman, London (1973)... 

It is becoming common practice to have the cross-section of a plastic moulding made up of several different materials. This may be done to provide a permeation barrier whilst retaining attractive economics by having a less expensive material making up the bulk of the cross-section. To perform stress analysis in such cases, it is often convenient to convert the cross-section into an equivalent section consisting of only one material. This new section will behave in exactly the same way as the multi-layer material when the loads are applied. A very common example of this type of situation is where a solid skin and a foamed core are moulded to provide a very efficient stiffness/weight ratio. This type of situation may be analysed as follows ... 

This book is intended primarily for students in the various fields of engineering but it is felt that students in other disciplines will welcome and benefit from the engineering approach. Since the book has been written as a general introduction to the quantitative aspects of the properties and processing of plastics, the depth of coverage is not as great as may be found in other texts on the physics, chemistry and stress analysis of viscoelastic materials, this has been done deliberately because it is felt that once the material described here has been studied and understood the reader will be in a better position to decide if he requires the more detailed viscoelastic analysis provided by the advanced texts. 

Analyses are types of calculations but may be comparative studies, predictions, and estimations. Examples are stress analysis, reliability analysis, hazard analysis. Analyses are often performed to detect whether the design has any inherent modes of failure and to predict the probability of occurrence. The analyses assist in design improvement and the prevention of failure, hazard, deterioration, and other adverse conditions. Analyses may need to be conducted as the end-use conditions may not be reproducible in the factory. Assumptions may need to be made about the interfaces, the environment, the actions of users, etc. and analysis of such conditions assists in determining characteristics as well as verifying the inherent characteristics. (See also in Part 2 Chapter 14 under Detecting design weaknesses.)... 

---