A method for judging failure mode of pressure-bearing equipment cylinder under external explosion load

Abstract

This invention relates to the field of energy transmission technology and discloses a method for determining the failure mode of a pressure vessel cylinder under external explosion load. The method includes the following steps: obtaining the design parameters of the pressure vessel and the external explosion load parameters; establishing a numerical model of the dynamic response of the pressure vessel under external explosion load; setting n data points along the axial direction of the cylinder; calculating the dimensionless distance of each data point; constructing a three-dimensional data matrix containing normalized radial displacement, time, and dimensionless distance; determining the critical dimensionless distance; drawing a contour map of the normalized radial displacement; and determining the failure mode as dynamic buckling or plastic indentation deformation through a discriminant function calculation. This method can effectively distinguish between dynamic buckling and plastic indentation deformation failure modes of the pressure vessel cylinder based on the discriminant function. The method is simple and intuitive, overcoming the complexity of traditional analysis methods. The judgment results can directly guide the optimization of the explosion-resistant structure design of the pressure vessel, providing targeted protection for extreme explosion load conditions.

Description

Technical Field

[0001] This invention relates to the field of energy transmission technology, and in particular to a method for determining the failure mode of a pressure vessel cylinder under external explosion load. Background Technology

[0002] Pressure vessels, as key equipment in energy storage and transportation systems, bear the responsibility of safely storing and transporting strategic resources such as oil, natural gas, and hydrogen. They are the core hubs for ensuring energy needs for people's livelihoods and stabilizing energy supply, and are also strategic supports for national economic and social development and national defense potential. However, during the service of pressure vessels, external explosive impact loads caused by accidental combustion and explosion of surrounding facilities or deliberate military bombing strikes can directly threaten the structural integrity of pressure vessels. The failure of pressure vessel cylinders caused by external explosive impacts mainly manifests in two typical modes: dynamic buckling and plastic indentation deformation. The former refers to the sudden dynamic instability of the structure under load disturbance, while the latter is the irreversible deformation that occurs when the material stress exceeds the yield strength. Although both appear as structural indentation deformation, their failure mechanisms are fundamentally different. Understanding the changing patterns and judgment methods of failure modes of pressure vessel cylinders under explosive loads has important guiding value for the explosion-resistant design of equipment.

[0003] Currently mature failure mode identification methods mainly target quasi-static external loads. For example, the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code provides theoretical calculation methods for the critical loads of dynamic buckling and plastic deformation of pressure equipment under radial or axial pressure. NASA's Shell Buckling Design Guidelines propose a method for calculating the critical buckling load based on a reduction factor, stating that buckling occurs when the critical buckling load is below the material's yield strength, and plastic deformation occurs when it exceeds the yield strength. However, external explosion loads are highly transient, and the material strain rate effect and structural inertia effect are significant under transient loads, limiting the limitations of identification methods based on quasi-static loads. Although there are explosion-resistant design methods for pressure equipment under explosive impact loads, such as the equivalent single-degree-of-freedom design method, the British AWE design method, and the American ASME design method, these methods have not effectively established criteria for the two failure modes of dynamic buckling and plastic indentation deformation of pressure equipment under external explosive loads. Summary of the Invention

[0004] To address the limitation of existing determination methods that only apply to pressure-bearing equipment under quasi-static loads, this invention provides a method for determining the failure mode of pressure-bearing equipment cylinders under external explosion loads. This method can effectively distinguish between two failure modes of pressure-bearing equipment cylinders: dynamic buckling and plastic indentation deformation, based on a discriminant function. The method is simple and intuitive, easy for engineers to understand, and can provide targeted guidance for engineers in designing pressure-bearing equipment under extreme loads such as explosions.

[0005] This invention provides a method for determining the failure mode of a pressure vessel cylinder under external explosion load, comprising the following steps:

[0006] S1: Obtain the design parameters and external explosion load parameters of the pressure equipment;

[0007] S2: Establish a numerical model of the dynamic response of pressure-bearing equipment under external explosion load;

[0008] S3: Set n data points along the cylinder axis in the numerical model of the dynamic response, where data point 1 is located at the explosion center projection point and data point n is located at the far end of the cylinder.

[0009] S4: Calculate the dimensionless distance between each data point. , ,in, is the actual distance from data point i to the projection point of the explosion center; D is the outer diameter of the pressure vessel.

[0010] S5: Using the aforementioned dynamic response numerical model, the radial displacement u of all data points at each time point is normalized by a time interval Δt to obtain the normalized radial displacement. The normalized radial displacement corresponding to data point i is denoted as . Construct a three-dimensional data matrix containing the normalized radial displacement, time, and dimensionless distance of each data point;

[0011] S6: Extracting the explosive impulse as a function of dimensionless distance L d The change curve is used to determine the critical dimensionless distance P through differentiation. I P I L is the point where the rate of change of the curve is maximum. d .

[0012] S7: Define the deformation propagation function Normalized radial displacement is plotted based on a three-dimensional data matrix. A contour map, where the function of any contour line in the map is denoted as... We introduce a discriminant function for the trend of contour line changes over the time interval (t1, t2). Its expression is:

[0013] ;

[0014] in, Selected contour lines Projection onto the time axis;

[0015] S8: Selected The contour lines are used for calculation and discrimination:

[0016] If the discriminant function satisfy:

[0017] ;

[0018] The failure mode is then determined to be dynamic buckling;

[0019] If the discriminant function satisfy:

[0020] ;

[0021] The failure mode is then determined to be plastic indentation deformation.

[0022] In some embodiments, in step S1, the design parameters of the pressure-bearing equipment include the material parameters, geometric dimensions, working pressure, internal medium composition and physical properties of the pressure-bearing equipment.

[0023] In some embodiments, in step S1, the external explosion load parameters include the type, equivalent, and detonation distance of the explosion load.

[0024] In some embodiments, in step S2, the dynamic response numerical model is constructed by a fluid-structure interaction numerical model.

[0025] In some embodiments, the fluid-structure interaction numerical model is constructed using at least one of the arbitrary Lagrange-Euler method, the structured arbitrary Lagrange-Euler method, or the smoothed particle fluid dynamics-finite element method.

[0026] In some embodiments, the constitutive model used to describe the mechanical properties of the pressure-bearing equipment in the solid domain portion of the fluid-structure interaction numerical model is a thermoviscoplastic constitutive model.

[0027] In some embodiments, in step S3, the data points are arranged in a manner that ensures they are evenly distributed from the projection point of the explosion center to the far end of the cylinder, and the number of data points satisfies the following: ;

[0028] Where L is the length from the projection point of the explosion center to the far end of the cylinder, δ is the thickness of the cylinder of the pressure equipment, and D is the outer diameter of the cylinder of the pressure equipment.

[0029] In some embodiments, in step S5, the normalized radial displacement The expression is:

[0030] ;

[0031] in, Let be the radial displacement of data point i.

[0032] In some embodiments, in step S5, the expression for the time interval Δt is:

[0033] ;

[0034] Where L is the length of the pressure equipment cylinder, n is the number of data points, ρ is the density of the pressure equipment cylinder material, and E is the elastic modulus of the pressure equipment cylinder material.

[0035] In some embodiments, the advantages and positive effects of the present invention compared with the prior art are:

[0036] The above-mentioned method for determining the failure mode of pressure equipment cylinder under external explosion load can effectively distinguish between two failure modes: dynamic buckling and plastic indentation deformation of pressure equipment cylinder based on the discriminant function. The method is simple and intuitive, overcomes the complexity of traditional analysis methods, is easy for engineers to understand, and the judgment results can directly guide the optimization of the explosion-proof structure design of pressure equipment, providing a targeted protection basis for extreme explosion load conditions. Attached Figure Description

[0037] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0038] Figure 1 This is a flowchart of the method for determining the failure mode of the cylinder of a pressure-bearing device under external explosion load according to the present invention;

[0039] Figure 2 In the method for determining the failure mode of a pressure vessel cylinder under external explosion load of the present invention, data point i and the actual distance from data point i to the projection point of the explosion center are given. Relationship;

[0040] Figure 3 This is a table of simulated working conditions for the wall thickness of two gas cylinders and the equivalent of the external TNT explosion they are subjected to, as well as the detonation distance, in the method for determining the failure mode of the cylinder of a pressure-bearing device under external explosion load of the present invention.

[0041] Figure 4 These are the constitutive model parameters for 4130X steel and Q235 steel materials in the method for determining the failure mode of the cylinder of a pressure-bearing device under external explosion load in this invention.

[0042] Figure 5 This is the finite element model constructed in the method for determining the failure mode of the cylinder of a pressure-bearing device under external explosion load of the present invention;

[0043] Figure 6 In the method for determining the failure mode of a pressure vessel cylinder under external explosion load in this invention, the explosion impulse of gas cylinder #1 varies with dimensionless distance. The curves showing the change in impulse and the rate of change in impulse;

[0044] Figure 7 In the method for determining the failure mode of a pressure vessel cylinder under external explosion load in this invention, the explosion impulse of gas cylinder #2 varies with dimensionless distance. The curves showing the change in impulse and the rate of change in impulse;

[0045] Figure 8 The normalized radial displacement of gas cylinder #1 in the method for determining the failure mode of pressure vessel cylinder under external explosion load in this invention Contour map;

[0046] Figure 9 The normalized radial displacement of gas cylinder #2 in the method for determining the failure mode of pressure vessel cylinder under external explosion load in this invention Contour map; Detailed Implementation

[0047] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0048] In the description of this invention, it should be understood that the terms "center", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0049] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "joining" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances. In the description of the above embodiments, specific features, structures, materials, or characteristics can be combined in any suitable manner in one or more embodiments or examples.

[0050] The terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this invention, unless otherwise stated, "a plurality of" means two or more.

[0051] Reference Figures 1-9 This is an embodiment of the method for determining the failure mode of a pressure-bearing equipment cylinder under external explosion load according to the present invention. Figure 1 As shown, the method for determining the failure mode of a pressure vessel shell under external explosion load includes the following steps:

[0052] S1: Obtain the design parameters of the pressure equipment and the external explosion load parameters.

[0053] S2: Establish a numerical model of the dynamic response of pressure-bearing equipment under external explosion load.

[0054] S3: Set n data points along the axial direction of the pressure equipment cylinder in the dynamic response numerical model, where data point 1 is located at the explosion center projection point and data point n is located at the far end of the cylinder.

[0055] S4: Calculate the dimensionless distance between each data point. , ,in, Let be the actual distance from data point i to the projection point of the explosion center; D is the outer diameter of the pressure vessel. (Data point i and the actual distance from data point i to the projection point of the explosion center are also mentioned.) Relationship such as Figure 2 As shown.

[0056] S5: Using the dynamic response numerical model, the radial displacement u of all data points at each time step is normalized by time intervals Δt (Δt, 2Δt, ...) to obtain the normalized radial displacement. The normalized radial displacement corresponding to data point i is denoted as . A three-dimensional data matrix containing the normalized radial displacement, time, and dimensionless distance of each data point is constructed.

[0057] S6: Extract the impulse curves of the explosion shock wave from data point 1 to data point n, and plot the explosion impulse as a function of dimensionless distance L. d The change curve is used to determine the critical dimensionless distance P by differentiating the curve. I Among them, the point corresponding to the maximum rate of change of the curve is L. d Defined as the dimensionless distance P that separates high impulse from low impulse. I .

[0058] S7: Define the deformation propagation function Its meaning is: size is The dimensionless distance corresponding to the normalized radial displacement at time t.

[0059] Normalized radial displacement plotted based on 3D data matrix A contour map, where the function of any contour line in the map is denoted as... We introduce a discriminant function for the trend of contour line changes over the time interval (t1, t2). Its expression is:

[0060] ;

[0061] in, Selected contour lines Projection on the timeline.

[0062] S8: Selected The contour lines are used for calculation and discrimination:

[0063] If the discriminant function satisfy:

[0064] ;

[0065] The failure mode is then determined to be dynamic buckling;

[0066] If the discriminant function satisfy:

[0067] ;

[0068] The failure mode is then determined to be plastic indentation deformation.

[0069] It should be noted that in step S8, the selected The value of 0.5, used as the basis for failure mode determination, is the result verified through systematic numerical experiments.

[0070] Extensive numerical simulation studies have revealed that when selecting When the contour line value is 0.5, the prediction results of the criterion are completely consistent with the baseline failure modes of all examples, demonstrating extremely high accuracy and stability. However, when contour lines with other values ​​are selected, misjudgments occur. Therefore, this invention determines the selection of contour lines... The contour line with a value of 0.5 is used as the empirically optimal threshold for judgment.

[0071] The above-mentioned method for determining the failure mode of pressure equipment cylinder under external explosion load can effectively distinguish between two failure modes: dynamic buckling and plastic indentation deformation of pressure equipment cylinder based on the discriminant function. The method is simple and intuitive, overcomes the complexity of traditional analysis methods, is easy for engineers to understand, and the judgment results can directly guide the optimization of the explosion-proof structure design of pressure equipment, providing a targeted protection basis for extreme explosion load conditions.

[0072] The following is a detailed description of each step.

[0073] In step S1, the design parameters of the pressure equipment include the material parameters, geometric dimensions, working pressure, internal medium composition and physical properties, and installation requirements.

[0074] External explosion load parameters include the type of explosion load, its equivalent, and the detonation distance.

[0075] In some embodiments of this application, in step S2, the dynamic response numerical model can be established using finite element software such as LS-DYNA, ABAQUS, and AUTODYN.

[0076] Considering the buffering effect of transient structural indentation deformation on the blast shock wave, the dynamic response numerical model is constructed using a fluid-structure interaction (FSI) numerical model. This FSI numerical model can be constructed using at least one of the following methods: arbitrary Lagrange-Euler method, structured arbitrary Lagrange-Euler method, or smoothed particle hydrodynamics-finite element method. Furthermore, the construction method for the FSI numerical model is not limited to these three methods; any method capable of effectively simulating the fluid-structure interaction effect is acceptable.

[0077] Furthermore, considering the strain hardening and strain rate effects of metallic materials under explosive loads, the constitutive model used to describe the mechanical properties of pressure-bearing equipment in the solid domain part of the fluid-structure interaction numerical model is a thermoviscoplastic constitutive model.

[0078] In some embodiments of this application, in step S3, the data points are arranged in a manner that ensures they are evenly distributed from the projection point of the explosion center to the far end of the cylinder, and the number of data points satisfies the following: ;

[0079] Where L is the length from the projection point of the explosion center to the far end of the cylinder, δ is the thickness of the cylinder of the pressure equipment, and D is the outer diameter of the cylinder of the pressure equipment.

[0080] In some embodiments of this application, in step S5, the normalized radial displacement is... The expression is:

[0081] ;

[0082] in, Let be the radial displacement of data point i.

[0083] In some embodiments of this application, the expression for the time interval Δt is:

[0084] ;

[0085] Where L is the length of the pressure equipment cylinder, n is the number of data points, ρ is the density of the pressure equipment cylinder material, and E is the elastic modulus of the pressure equipment cylinder material.

[0086] In some embodiments of this application, in step S7, the normalized radial displacement is... Contour maps can be drawn using data processing software such as Origin and Excel.

[0087] To further illustrate the technical solution of the present invention, a specific embodiment will be described in detail below.

[0088] This embodiment, based on the technical solution steps of the present invention, determines the failure mode of two single-layer gas cylinders (numbered #1 and #2) under external TNT explosion load.

[0089] Cylinder dimensions: Cylinder body length 1325mm, outer diameter 485mm, end cap type hemispherical end cap. The end of the cylinder body is fixed by a saddle. The wall thickness of the two cylinders, the equivalent of the external TNT explosion they can withstand, and the detonation distance are as follows: Figure 3 As shown in Table 1.

[0090] 1. Constructing a finite element model

[0091] Fluid-structure interaction (FSI) finite element analysis of a pressure-bearing device under external explosion load was performed using LS-DYNA software. A FSI numerical model was established, consisting of four parts: the external ambient air, the explosive, the pressure-bearing device, and the saddle. To simplify the calculation, the gas inside the pressure-bearing device was assumed to be air at normal pressure. The gas cylinder was made of 4130X steel, and the saddle was made of Q235 steel.

[0092] The Johnson-Cook constitutive model is used to describe the mechanical properties of gas cylinder materials:

[0093] ;

[0094] In the formula, and Equivalent stress and equivalent plastic strain; , , , These are the constitutive parameters of the material; The reference strain rate is used.

[0095] The constitutive model parameters for 4130X steel and Q235 steel obtained from relevant literature, such as... Figure 4 Table 2 in the table.

[0096] To improve solution efficiency, a quarter-finite element model is constructed. The pressure equipment and saddle are discretized using Lagrangian elements, while the ambient air and explosives are discretized using Eulerian elements. A contact element is established between the saddle and the pressure equipment. The finite element model is as follows: Figure 5 As shown.

[0097] 2. Data Processing

[0098] Elastic-plastic dynamic response analysis was performed based on the established numerical analysis model. According to the given constraints, the number of data points (n) for both gas cylinder #1 and gas cylinder #2 was set to 70, and the dimensionless distance corresponding to each data point was calculated. .

[0099] To analyze the distribution of radial displacement *u* of the pressure vessel cylinder obtained from the simulation, the time history curves of the radial displacement from data point 1 to data point 70 were extracted, covering the entire dynamic response process at that point. Based on the simulation results, the time histories for cylinders #1 and #2 were both extracted at 2000 μs. Taking a time interval Δt of 1.6 μs, the radial displacements at each time point were normalized to obtain the normalized radial displacements. Dimensionless distance A three-dimensional data matrix with time t.

[0100] Extract the impulse curves of the explosion shock wave from data point 1 to data point 70, and plot the explosion impulse as a function of dimensionless distance. The curve shows the change rate. Differentiating this curve, the dimensionless distance corresponding to the point of maximum rate of change is taken as the boundary between high and low impulse of the explosion, i.e., the critical dimensionless distance P. I .like Figure 6 and Figure 7 As shown, based on the simulation results, the critical dimensionless distance P between gas cylinder #1 and gas cylinder #2 is... I The values ​​are 0.233 and 0.13, respectively.

[0101] Based on the obtained normalized radial displacement, dimensionless distance, and time three-dimensional data matrix, using dimensionless distance... Using time t as the independent variable, normalized radial displacement was plotted using data processing software such as Origin and Excel. The contour map, and find f( The contour lines corresponding to when t) = 0.5, such as Figure 8 and Figure 9 As shown.

[0102] 3. Failure Mode Determination:

[0103] The failure mode of gas cylinder #1 under external explosion load was determined. Numerical results show that its... The contour lines corresponding to a value of 0.5 satisfy the following condition:

[0104] ;

[0105] Therefore, it was determined that when the cylinder #1 was subjected to an external explosion with an explosive equivalent of 2kg TNT and an initiation distance of 200mm, the failure mode was dynamic buckling.

[0106] The failure mode of gas cylinder #2 under external explosion load was determined. Based on the numerical results, =0.20, =0.22, calculate the discriminant function. :

[0107] ;

[0108] Right now When = 0.5, the discriminant function satisfies the following condition:

[0109] ;

[0110] Therefore, it was determined that when the cylinder #2 was subjected to an external explosion with an explosive equivalent of 4 kg TNT and an initiation distance of 250 mm, the failure mode was plastic indentation deformation.

[0111] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions claimed by the present invention.

Claims

1. A method for determining the failure mode of a pressure-bearing equipment cylinder under external explosion load, characterized in that, Includes the following steps: S1: Obtain the design parameters and external explosion load parameters of the pressure equipment; S2: Establish a numerical model of the dynamic response of pressure-bearing equipment under external explosion load; S3: Set n data points along the cylinder axis in the numerical model of the dynamic response, where data point 1 is located at the explosion center projection point and data point n is located at the far end of the cylinder. S4: Calculate the dimensionless distance between each data point. , ,in, D is the actual distance from data point i to the projection point of the explosion center; D is the outer diameter of the cylinder of the pressure equipment. S5: Using the aforementioned dynamic response numerical model, the radial displacement u of all data points at each time point is normalized by a time interval Δt to obtain the normalized radial displacement. The normalized radial displacement corresponding to data point i is denoted as . Construct a three-dimensional data matrix containing the normalized radial displacement, time, and dimensionless distance of each data point; S6: Extracting the explosive impulse as a function of dimensionless distance L d The change curve is used to determine the critical dimensionless distance P through differentiation. I P I L is the point where the rate of change of the curve is maximum. d； S7: Define the deformation propagation function Normalized radial displacement is plotted based on a three-dimensional data matrix. A contour map, where the function of any contour line in the map is denoted as... We introduce a discriminant function for the trend of contour line changes over the time interval (t1, t2). Its expression is: ； in, Selected contour lines Projection onto the time axis; S8: Selected Calculate and determine the contour lines: If the discriminant function satisfy: For any ,have ; The failure mode is then determined to be dynamic buckling; If the discriminant function satisfy: exist ,make ; The failure mode is then determined to be plastic indentation deformation.

2. The method for determining the failure mode of a pressure-bearing equipment cylinder under external explosion load according to claim 1, characterized in that, In step S1, the design parameters of the pressure-bearing equipment include the material parameters, geometric dimensions, working pressure, internal medium composition and physical properties of the pressure-bearing equipment.

3. The method for determining the failure mode of a pressure-bearing equipment cylinder under external explosion load according to claim 1, characterized in that, In step S1, the external explosion load parameters include the type, equivalent, and detonation distance of the explosion load.

4. The method for determining the failure mode of a pressure-bearing equipment cylinder under external explosion load according to claim 1, characterized in that, In step S2, the dynamic response numerical model is constructed through a fluid-structure interaction numerical model.

5. The method for determining the failure mode of a pressure-bearing equipment cylinder under external explosion load according to claim 4, characterized in that, The fluid-structure interaction numerical model is constructed using at least one of the following: arbitrary Lagrange-Euler method, structured arbitrary Lagrange-Euler method, or smoothed particle fluid dynamics-finite element method.

6. The method for determining the failure mode of a pressure-bearing equipment cylinder under external explosion load according to claim 5, characterized in that, The solid domain portion of the fluid-structure interaction numerical model, which describes the mechanical properties of the pressure-bearing equipment, is a thermoviscoplastic constitutive model.

7. The method for determining the failure mode of a pressure-bearing equipment cylinder under external explosion load according to claim 1, characterized in that, In step S3, the data points are arranged in a way that they are evenly distributed from the projection point of the explosion center to the far end of the cylinder, and the number of data points satisfies the following: ; Where L is the length from the projection point of the explosion center to the far end of the cylinder, δ is the thickness of the cylinder of the pressure equipment, and D is the outer diameter of the cylinder of the pressure equipment.

8. The method for determining the failure mode of a pressure-bearing equipment cylinder under external explosion load according to claim 1, characterized in that, In step S5, The expression is: ； in, Let be the radial displacement of data point i.

9. The method for determining the failure mode of a pressure-bearing equipment cylinder under external explosion load according to claim 1, characterized in that, In step S5, the expression for the time interval Δt is: ； Where L is the length of the pressure equipment cylinder, n is the number of data points, ρ is the density of the pressure equipment cylinder material, and E is the elastic modulus of the pressure equipment cylinder material.

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