Reasoning – How exactly should equipment flanges be entered into calculations?

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This post was last edited by Freestyle-sky on 2022-4-3 at 19:38. Regarding the calculation of equipment flanges, in the previous article the author used simple comparative calculations to illustrate the problems present in such calculations and to show how to carry out them more effectively. This comparative approach is actually a simple yet highly effective method for studying issues; rather than relying on numerous theoretical explanations, a few simple comparisons can reveal the facts more clearly, making them easier to understand and accept. Of course, the premise for comparative verification must be based on correct theoretical foundations; if the premise is incorrect, then comparative verification loses all meaning. Therefore, it ultimately still has to be grounded in accurate theoretical understanding. In the terms of the standards, an accurate flange stress analysis is quite complex, mainly for the following two reasons: (1) it is difficult to determine the compressive resilience of the gasket as well as the minimum compressive force per unit area on the gasket required to ensure sealing ； (2) If it is difficult to determine the gasket compression force, then the bolt load cannot be accurately determined either. The bolt load directly affects the magnitude of the flange torque, which in turn determines the accuracy of the calculated stress values. Therefore, it seems pointless to conduct a highly detailed stress analysis of the equipment’s flanges using an imprecise bolt load – after all, the input data is inaccurate, no matter how precise the finite element model is, how fine the mesh division is, or how high the calculation accuracy is. To conduct an accurate finite element stress analysis of the equipment flange, it is necessary to have the rebound performance curve of the gasket used; this curve is used to determine the gasket compression force as well as the bolt load required to calculate the flange stresses. But the fact is that determining the rebound curve of the performance of each gasket material can only be done through experiments, which involves significant costs in terms of labor and time. As a result, there is a severe shortage of data on gasket performance curves; such data are rarely found in standards. Occasionally, some curve data for individual gasket materials can be found in research papers, but their accuracy cannot be guaranteed. The mechanical model calculated by the Waters method employs an approximate calculation approach based on significant simplifications – the Waters method. This method makes a series of assumptions regarding flanges, thereby enabling the calculation of flanges to be carried out as an approximate process in which differential equations are established based on shell theory and boundary conditions, and these equations are simplified into solvable systems of linear algebraic equations so that an analytical solution can be obtained. The calculated model for the flange, after assumptions and simplifications, is as follows: Assumption 1: The discontinuous stresses at the boundary joints of various flange components, caused by the radial effect of pressure, are much smaller than the stresses induced by the flange torque. Similarly, the membrane stresses in the cylinder and cone necks, resulting from the axial action of internal pressure, are also much smaller than the stresses caused by the flange torque; in other words, they are negligible on a quantitative scale. Therefore, when using the Waters method to calculate flanges, the effect of internal pressure on the stresses at the edges where the cylinder meets the small end of the cone neck, as well as at the edges where the large end of the cone neck meets the flange ring, is indeed not taken into account. Hypothesis 2: Based on the plate-shell theory, the flange is considered to consist of three main components. Cylindrical shell: considered as an infinite cylindrical shell of a plate subjected to edge moments and transverse shear forces ； Neck: Considered as a linearly thickened cylindrical shell subjected to edge moments and transverse shear forces at both the major and minor ends. Flange ring: Considered as a thin annular plate, it is subjected to a torque resulting from uniformly distributed forces on its inner and outer edges, in addition to a bending moment that is also uniformly distributed along the inner circumference. In addition to the assumptions regarding the aforementioned model, there are also assumptions about the materials and loads, such as the material remaining in a purely elastic state without undergoing plastic deformation or creep, and the bolt load and lever arm being determined according to the assumed conditions. Based on these assumptions and the conditions of internal force equilibrium as well as boundary conditions, 8 equilibrium equations with 8 unknowns are established, which enable the determination of the deflection, rotation, and stress values at any point on the flange. For a more detailed analysis of the assumptions and calculation processes, those interested can refer to the relevant standards. The conclusions of the Waters method The results obtained using the Waters method provide only the calculation formulas for the three main stresses that control the strength of the flange, namely axial stress, circumferential stress, and radial stress, along with certain control conditions. It is important to understand that: (1) axial stress refers to the maximum axial bending stress in the neck of the cone. Analysis shows that this maximum axial bending stress always occurs at the two ends of the neck – it may be at the smaller end or at the larger end – so one should not assume that the axial stress calculated by the software corresponds to the stress at the smaller end of the neck. (2) Both the hoop stress and the radial stress refer to the stresses in the flange ring; that is, the maximum hoop stress and the maximum radial stress occur in the flange ring, not on the cone neck. These two stresses are each composed of two components: film stress and bending stress. The membrane stress is calculated by treating the flange ring as a thick-walled cylinder subjected to an equivalent internal pressure, while the bending stress is determined using a mechanical model of an annular plate subjected to bending moments resulting from uniform loads acting on its inner and outer edges. By adding the membrane stress and the bending stress together, the maximum hoop stress and maximum radial stress of the flange ring can be obtained. (3) The analysis results show that the maximum radial stress in the flange ring is located at the inner edge of the ring and at the junction with the larger end of the cone neck, namely point A in the figure above ； The maximum circumferential stress in the flange ring occurs at the inner edge of the ring, near the gasket sealing surface, that is, at point B in the diagram above. (4) As can be seen from the above analysis, the maximum axial stress, maximum radial stress, and maximum circumferential stress do not occur at the same point. The axial stress checking formula provided by the Waters method may apply to either the larger end or the smaller end; both the radial stress and circumferential stress formulas are based on point A. In other words, the point chosen for calculating the circumferential stress is not the point where the maximum circumferential stress occurs. As a result, the value of circumferential stress calculated using this formula is lower than the actual maximum value, though the difference is not significant. Therefore, point A, which is under triaxial stress conditions, was selected as the critical point for verification. Based on the Waters method for analyzing standard equipment flanges: In the previous article, calculations were carried out and results were compared for different ways of entering data related to actual standard equipment flanges in SW6, and the comparison results were clear at a glance. Here, we further understand and analyze this using the theoretical calculation model of the Waters method. Still taking the standard flange shown in the figure below as an example: (a) the actual standard flange model. If calculations are performed in SW6 using the formula thickness of the smaller end of the flange’s cone – corrosion allowance = 13 mm, the results will be as follows. Then, the maximum axial stress value calculated using Waters’ method in SW6 should correspond to the stress value at the section indicated by the blue arrow or the red arrow in the figure below. No further discussion will be held regarding the fact that the calculated values of tangential and radial stresses are quite low, far below the allowable values. According to the theoretical analysis of the Waters method, the maximum axial bending stress may occur at the larger end section or the smaller end section of the neck, but for most flanges, the maximum axial bending stress still occurs at the smaller end of the neck. Based on the calculations above, the axial stress at the smaller end of the conical neck is 177.66 Mpa, which is less than or equal to 1.5f = 1.5*169.2 = 253.8 Mpa. Thus, the axial stress at the joint interface between the straight section and the shell cylinder was not checked. If we ignore the stress reduction effect in the straight section and calculate the axial stress caused by internal pressure using σ = PD/4δ = 0.7*900/(4*7) = 22.5 Mpa, then the axial stress at the jointed cylinder interface would be 177.66 + 22.5 = 200.16 Mpa, which is still less than or equal to 1.5f = 253.8 Mpa. If we also take into account the edge stress generated by internal pressure on the cylinder surface, theory suggests that this edge stress is much smaller compared to the bending stress caused by the flange torque. Although the exact value cannot be determined, theoretically, a margin of 53.8 Mpa is more than sufficient to cover this stress. From the above analysis, it can be seen that even when considering the most severe scenarios (axial bending stress caused by flange torque + axial membrane stress generated by internal pressure + axial bending stress resulting from structural discontinuities), the axial stress on the butt jointed cylinder section is not necessarily exceeding the allowable values. Moreover, our standard flanges feature a straight edge section that not only helps to reduce stress but also minimizes discontinuities at the junction with the cylinder. Although this straight edge section does not meet the requirements regarding the length necessary for reducing bending stress, the significant reduction in edge stress indicates that its length plays a substantial role in lowering stress levels. If the calculation is performed in SW6 by entering the value of thickness of the butted cylinder minus corrosion allowance equal to 7 mm, the results are as follows: According to Waters’ theory, the axial bending stress is related to the edge moment Mh0 at the smaller end of the cone-shaped neck, as well as the square of δ0. If the thickness of the butted cylinder is used for the calculation, then the edge moment Mh0 at that end will increase, while δ0 will decrease. As a result, the axial stress δh ends up being higher than actual values, making the calculation too conservative. Based on the above analysis and theoretical considerations, it can be seen that China’s standard equipment flanges take into account various factors regarding stiffness and strength, and already possess sufficient safety margins. Finally, let’s discuss a few personal opinions: (1) The formulation of all standards is actually based on widely accepted theoretical foundations, as well as data derived from extensive practice and over time; therefore, there are generally no major issues. Even if errors are found in the standards we can see, they are likely to be mere typos, and certainly not fundamental errors that go against those theoretical foundations. One cannot simply use one’s own calculation results to question the reliability of standards, because it is possible that our calculations have produced incorrect or overly conservative results due to an incomplete understanding of the theoretical principles behind them. The software itself is not at fault; what happens is that the calculations for the pressure-bearing elements of pressure vessels are used to determine the required thickness. Increasing this thickness seems to be a universal solution in design – whenever the calculations do not meet the requirements or the stress levels are insufficient, the solution is simply to increase the thickness, regardless of the underlying cause. However, everything has two sides, and not all problems can be solved by simply increasing the thickness or using more material. To take a well-known example, in the case of heat exchanger tube sheets, it’s not true that the thicker the sheet, the better; a greater thickness can lead to higher temperature difference stresses, resulting in an increase rather than a decrease in stress. Applied to the example of equipment flange calculation in this article, it is not true that the more material used for the flange, the better – greater flange stiffness will result, while the stiffness of the cylinder section connected to the flange will be lower. A large difference in their stiffness ratios can cause most of the deformation to be borne by the cylinder, leading to an increase in local stresses. Therefore, reasonable design involves identifying the causes of problems and their potential consequences, and then making appropriate judgments based on these different causes and levels of risk, in order to optimize the design accordingly. This is also the fundamental principle behind analytical design. (3) Regarding the use of finite element software to analyze equipment flanges, it has been mentioned earlier that it is difficult to conduct an accurate analysis of these flanges, due to the lack of precise data on bolt preload and loads, as well as the absence of curves showing the springback behavior of gaskets. Currently, most approaches do not take into account the effects of gasket performance, and the bolt loads are determined using the formulas specified in standards. It is indeed difficult to determine whether it is more reliable to use the results from finite element calculations or the analytical solutions derived from Waters’ plate-shell theory. Since the Waters method is itself a stress analysis technique based on the practical elastic mechanics theory of plates and shells, it calculates the stress at the inspection points and imposes certain constraints; however, it does not classify the stresses and limits them using a uniform value of 1.5 times the allowable flange stress. Therefore, the Waters method is also considered an incomplete stress analysis method. On the other hand, if the finite element method is used to calculate stresses, the values obtained may be higher than those calculated using the Waters method, as the Waters method involves simplifications and ignores the effect of internal pressure on edge stresses. If stress values calculated by the finite element method are used for classification and evaluation, assessing the stress at the conical neck section using primary + secondary stress SⅣ might result in an underestimation of safety, whereas assessment using primary membrane + primary bending stress SⅢ might be too conservative. It’s truly difficult to determine whether the finite element calculations or the analytical solution provided by the Waters method is more accurate, more conservative, or more aggressive. Therefore, the idea of relying solely on finite element calculations to solve problems should also be abandoned. Finite element calculations can yield vastly different results depending on an individual’s theoretical and practical skills; resorting to these methods without proper consideration is not a good strategy. Apart from human factors, various other subjective elements can also affect the reliability of the results obtained through finite element calculations, making them less reliable compared to conventional calculation methods. Principled theoretical knowledge is the foundation of all design; software is merely a tool. We should be designers who use theory to guide these tools, rather than becoming slaves to them and letting the tools dictate our design decisions. In reality, however, many designers rely too heavily on software, depending entirely on its automated calculation results, and thus become prisoners of that software. They ignore the most basic theoretical principles, and even when the software’s calculations are incorrect, they prefer to follow those erroneous results rather than making better judgments based on theory and facts. As a result, many designs turn out to be ridiculous. In fact, the software isn’t at fault; it’s the users who lack a proper understanding and fail to apply basic theoretical and empirical judgment to the calculation results. Well, as I wrote it, I felt it started to deviate from the topic and went a bit too far. This article represents only the author’s personal views. Due to the author’s limited expertise, any shortcomings are kindly requested to be overlooked and guidance is highly appreciated. Welcome to search for and follow the WeChat official account \"ANSYS Analysis and Design Professionals\" – a platform dedicated to the analysis and design of pressure vessels. It has already attracted over 4,000 professionals in the field of stress analysis from various design institutes, engineering companies, manufacturing units, and universities across the country.