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Numerical simulation study on the maximum permissible geometry deviation values for cylinders under external pressure
Thin-Walled Structures 144 (2019) 106305
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Numerical simulation study on the maximum permissible geometry deviation values for cylinders under external pressure
T
Bingbing Chen, Lei Jin, Pengfei Wang*, Peng Cao, Yi Zhou, Zengliang Gao Institute of Process Equipment and Control Engineering, Zhejiang University of Technology, Hangzhou, 310023, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Cylinders under external pressure Shape reduced coefficient Maximum permissible geometry deviation Statistic imperfections model Yield strength
According to the surface imperfections of rolling–welding cylinders, a finite element analysis (FEA) model for a cylinder with surface imperfections is proposed, namely, a statistic imperfections model. The maximum permissible geometry deviation values e of statistic imperfections models are computed based on the shape reduced coefficient βcr = 0.8. Compared with the limited value of ASME Code sections Ⅷ-1 and Ⅷ-2, it is clear that the calculated value e/t is positively correlated with the length–diameter ratio L/Do and diameter–thickness ratio Do/t and that the yield strength has a substantial influence on the maximum permissible geometry deviation values, which is not taken into account by Code Ⅷ-2.
1. Introduction Cylinders used in the petrochemical industry are usually manufactured by rolling and welding. The load-bearing capacity of a cylindrical shell under external pressure significantly depends on the characteristics of imperfections caused by various manufacturing processes. Initially, an equation for the critical buckling pressure of cylinders subjected to uniform external pressure without imperfections was proposed [1]. However, several studies found that the theoretical model of critical instability pressure cannot reasonably describe the experimental results. Koiter [2] noted that these discrepancies were caused by the high sensitivity of shells with geometric imperfections. Calladine [3] claimed that the initial stresses locked in the imperfect initial geometry had significant effects on the buckling performance. It was proved that the initial geometry imperfections of the cylindrical shells were the main reason for the inconsistencies between the experimental and theoretical values [4]. Recently, using the asymptotic analysis method, Yang et al. [5] have investigated the effects of three types of imperfections with different thicknesses on the buckling of laterally pressured cylindrical shells. Hornung and Saal numerically investigated the buckling behaviors of a shell with geometric imperfections with different shapes and [6] found that only a shell model with measured imperfections can reasonably predict the buckling phenomena observed in the experiments. Therefore, the design standard of cylinders subjected to external pressure is intended to limit the maximum permissible geometry deviation values e. For rolling–welding cylinders with initial geometry deviation, it is
*
assumed that the buckling pressures would not be less than 80% of the theoretical buckling pressures of cylinders without imperfections. The values of e from the true circular form are employed in ASME Code section Ⅷ-1 [7] to regulate the maximum permissible deviation values for cylinders subjected to external pressure according to Fig. UG-80.1, which is based on an empirical formula, Eqs. (1)–(4), by Windenburg [8].
0.0180(D / t ) e = 0.015n + t n
(1)
where d
R/t ⎞ n = c ⎜⎛ ⎟ ⎝ L/R ⎠
(2)
c = 2.28(R/ t )0.054 ≤ 2.80
(3)
d = 0.38(R/ t )0.044 ≤ 0.485
(4)
In Eqs. (1)–(4), D and t are the centerline cylinder diameter and the cylindrical shell thickness, respectively. L and R represent the length of a cylinder and the radius to midpoint of a cylindrical shell, respectively. e is the maximum permissible geometry deviation from true circular form measured over a half wave length of cylindrical shells. n is the number of circumferential waves at cylinder buckling within the domain (2 ≤ n ≤ 1.41 R/ t ). Miller [9] noted that the value of e determined by Windenburg cannot guarantee that the actual instability pressure be 80% higher than the theoretical instability pressure. The critical pressure reduction
Corresponding author. E-mail address: [email protected] (P. Wang).
https://doi.org/10.1016/j.tws.2019.106305 Received 9 January 2019; Received in revised form 17 April 2019; Accepted 6 July 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.
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μ σs σ ε σtrue εtrue _ σtrue _ εtrue PTest
Nomenclature D Do R t L Lec e
centerline diameter of cylinder, mm outside diameter of cylinder, mm radius to midpoint of cylinder, mm thickness of cylindrical shell, mm length of cylinder between lines of support, mm chord length of the arc template of the bow, mm maximum permissible geometry deviation from a true circular form, mm minimum permissible geometry deviation from a true circular form, mm number of waves with which a cylinder will buckle in the circumferential direction geometric radius on the outer wall of the steel tube, mm defect of multiple steel tubes, mm random phase for Monte Carlo simulation, mm axial coordinates of cylindrical shell, mm relative axial coordinates of cylindrical shell circumferential angle of cylindrical shell, ° relative circumferential angle of cylindrical shell dimensionless imperfection of cylindrical shell average dimensionless imperfection of cylindrical shell modulus of elasticity of material, GPa
ec n r w f z Z θ θ* W W E
PFEA PMiller Pcr
Subscripts o cr
FEA
1.069
⎜
⎟
valid for 0.2t ≤ ec ≤ 0.0242R
finite element analysis
subjected to external pressure was employed by Wang and Koizumi [15]. It was found that the existence of longitudinal joints and initial imperfections in the cylinder would have an influence on the critical buckling pressure. Schenk and Schueller [16] investigated the effects of random geometric imperfections on the limit loads of thin-walled cylindrical shells under deterministic axial compression through the FEA method with the direct Monte Carlo simulation technique, and the values of the ultimate load predicted were consistent with the actual observation. Paor et al. [17] used the FEA method to analyze the geometrical nonlinear buckling of thin-walled shell models under external pressure with imperfections, and the buckling pressure and post-buckling mode shape were in good agreement with those of the experiment. In this study, the FEA method is used to investigate the effects of the geometrical parameters (L/Do and Do/t) and the yield strength on the maximum permissible geometry deviation values e. The FEA model, which is called the statistic imperfections model, is consistent with the initial geometry imperfections of the real cylinder. Through the statistical method, this model is proposed to deal with the three-dimensional scanning data of the actual cylinders appearance. In addition, when the external pressure on the cylinder examples reaches 80% of the theoretical buckling critical pressures without imperfections, the values of e for the statistic imperfections models are acquired through the FEA simulation, and the obtained values of e are the maximum permissible geometry deviation values of the actual cylinder. Finally, the simulation e curves are compared with those specified by ASME Code sections Ⅷ-1 and Ⅷ-2. Finally, the influence of the L/Do, Do/t, and yield strength of a cylinder under external pressure on the values of e is also investigated.
(5)
L ec = 0.0165t ⎛ ec + 3.25⎞ ⎝ Rt ⎠
outside critical buckling
Abbreviations
factor βcr caused by geometry imperfections may be less than the value of ASME Code Ⅷ-1, that is, βcr < 80%. Therefore, based on the experiment and the critical pressure reduction factor βcr < 80%, the values of e can be determined as follows:
e = min[ec, 2t ]
Poisson's ratio yield strength of material, MPa stress, MPa strain true stress, MPa true strain dimensionless true stress dimensionless true strain buckling pressure of a cylinder from experimental observation, MPa buckling pressure of a cylinder from FEA simulation, MPa buckling pressure of a cylinder from Miller's experiment, MPa buckling pressure of a cylinder with the value e = 0.001 from FEA, MPa
(6)
where Lec represents the chord length of the arc template of the bow, and ec represents the minimum inward deviation from a true circular form. Eqs. (5) and (6) are applied in the current ASME Code Ⅷ-2 [10]. Eq. (6) is used to limit the range of the value of e. According to aforementioned equations, it is clear that the maximum permissible geometry deviation values e provided by ASME Code sections Ⅷ-1 and Ⅷ-2 only consider the geometrical parameters (diameter Do, length L, and thickness t) of the cylinders, except for the effects of yield strength. However, Timoshenko [11] presented a critical pressure quantitative description formula of a circle and tube under uniform buckling external pressure, taking material yield strength and initial geometry imperfections into account. A critical pressure formula given by the pressure vessel design by Annaratone [12] also showed the relationship of material yield strength with initial geometry imperfections. In recent years, Fatemi, Showkati, et al. [13] conducted external pressure experiments applying different degrees of initial geometry imperfections to cylinders with different L/Do and Do/t ratios and found that although the initial geometry imperfections applied to some cylinders were much higher than those specified in the EN 1993-1-6 ECCSDIN18800 code, their critical pressure actually was higher. Therefore, the limit of the maximum permissible geometry deviation values e enforced by existing standards still needs to be investigated. With the development of numerical simulation technology, the FEA method has been applied to investigate the instability of cylindrical shells by many researchers. Rotter and Teng [14] studied the effects of imperfections caused by the weld joint on the buckling of cylindrical shells under external pressure through the finite element method. The results showed that the critical instability pressure of a cylinder under external pressure decreased significantly owing to the initial geometrical imperfections. The FEA method for rolling–welding cylinders
2. FEA model 2.1. Establishment of the FEA model The geometry shape of a random steel tube based on the theory provided by Vryzidis and Stefanou [18] is described as follows:
r (θ , z ) = R + w (θ , z ) + f (θ , z )
(7)
where θ and z are the circumferential and axial coordinates of the tube, respectively, r (θ, z) represents the initial radius at any point of a tube, 2
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imperfections of rolling–welding cylinders proposed by Pircher [20]. In addition, the distribution of the weld joint imperfections outside the zone presents circumferential and axial waveforms, which represent the fluctuation of local potholes and oblique waveforms of the real cylindrical shells. Therefore, the imperfection distribution of the statistic imperfections models are mostly consistent with the imperfection distribution of the actual rolling–welding cylinders. According to the aforementioned method, Eqs. (7)–(12) coupled with the ABAQUS modeling module are used to compile the model of statistic imperfections by using PYTHON. For cylinders with different geometric and material parameters (L/Do, Do/t, e, E, μ, and σs), the corresponding statistic imperfections models can be established in ABAQUS efficiently.
R represents the outer radius, w (θ, z) represents the values of imperfections across any point of the tube, and f (θ, z) is a random phase for the Monte Carlo simulation. This method can present the real shape of the imperfections to the greatest extent. Given that each cylinder specimen in this study is fabricated by rolling–welding, the geometric shapes of the eleven rolling–welding cylinder samples are obtained using a 3D scanner. Further details of the specimens are presented in Fig. 1 and Table 1. In this work, eleven geometry models obtained by scanning the samples were imported into the FEA software to draw meshes. Subsequently, the column coordinates of each node on the surface of the cylinders would be established. The weld joint of cylinder was set up at 90°. Then, the dimensionless coordinate values of each node were derived according to Eqs. (8)–(12), as follows:
θ* = θ /360°
(8)
2.2. Experimental verification of the FEA imperfection model
Z = z /L
(9)
2.2.1. Comparison of the FEA and external pressure experiment results An external pressure experiment was conducted on each rolling–welding cylinder specimen (in total eleven), and an external pressure equipment device was designed as the pressure chamber structure shown in Fig. 5. The cylinder specimen 4, bottom plate 2, top plate 5, and bolt 3 were fixed and sealed in the chamber through the hand hole 1 and an O-ring. The bolt 3 connected the bottom plate 2, top plate 5, and flat head, compensating for the effects of axial force to ensure that the cylindrical shell was pressurized through the pressure nozzle with uniform circumferential pressure. The bottom plate 2 and top plate 5 were processed into three ladder shapes to make the device suitable for three types of cylinder samples with different diameters. Before the experiment, the water injection nozzle and blow-off nozzle were opened to fill water, and then, the valve was closed. At the beginning of the experiment, the pressure nozzle was opened to pressurize. At the same time, the critical instability pressure (PTest) of the specimens is obtained by observing the deformation of the cylinder body through the viewport and recording the pressure change of the pressure gauge nozzle. At the end of the experiment, the water drain nozzle and blowoff nozzle were opened to drain the internal water. Cylinder specimens are made of S30408 stainless steel, and the tensile stress–strain (σ–ε) curves of five S30408 stainless steel tensile samples are obtained by an INSTRON tensile test machine, which are converted into the curves of true stress and true strain through Eqs. (13) and (14), respectively, as follows:
w (θ*, Z ) = r (θ*, Z )/ R
(10)
W (θ*, Z ) = w (θ*, Z )/ e
(11)
W (θ*, Z ) =
1
∑θ*=0 W (θ*, Z )
(12)
where θ* and Z represent the relative circumferential coordinate and relative axial coordinate, respectively; w (θ*, z) represents the imperfection values of any point on the cylinders; W (θ, 0) represents the dimensionless imperfection values of any point on the cylinders; and W (θ , 0) represents the average dimensionless imperfection values of any point on the cylinders. Obviously, the average dimensionless imperfection distribution at the bottom of the S3 cylinder (Z = 0) and the whole cylinder S3 model can be obtained, as shown in Fig. 2 and Fig. 3, respectively. A perfect cylindrical shell is built in ABAQUS [19] based on the size parameters of cylinder S3 presented in Table 1, and the average dimensionless imperfections attained were applied as radial load to the grid node of the perfect cylinder. Then, an FEA imperfection model, namely, the statistic imperfections model, is acquired after the static analysis step, as shown in Fig. 4. It can be seen from Figs. 3 and 4 that the weld joint is located in a statistic imperfections model θ* = 0.2–0.3. The distribution characteristics of the weld joint imperfections are inward concave along the axis because the stiffening ring at the ends of the rolling–welding cylinder has an outward strengthening effect on the end. It is consistent with the characteristics of the weld joint
εtrue = ln(1 + ε )
Fig. 1. Photograph and structure of the cylinder specimens. 3
(13)
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Table 1 Dimensions and material parameters of the cylinder specimens. Cylinder number
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11
σtrue = σ (1 + ε )
Material
S30408 S30408 S30408 S30408 S30408 S30408 S30408 S30408 S30408 S30408 S30408
E (GPa)
195 195 195 195 195 195 195 195 195 195 195
μ
0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
t (mm)
Do (mm)
1.810 1.810 1.810 1.810 1.810 1.810 1.810 1.810 1.810 1.810 1.810
199.737 199.721 199.757 199.438 200.035 299.523 299.740 299.822 399.715 399.797 399.379
(14)
L (mm)
195.215 393.129 595.837 795.689 994.581 294.373 444.570 594.371 314.962 394.703 474.931
L /Do
0.977 1.968 2.983 3.990 4.972 0.983 1.483 1.982 0.788 0.987 1.189
e (mm) Min
Max
−1.297 −0.846 −1.521 −1.875 −1.990 −1.212 −1.963 −1.818 −1.370 −1.238 −1.943
2.098 1.361 2.283 1.898 1.939 2.139 1.288 1.678 1.784 0.847 1.480
Similarly, the statistic imperfections models corresponding to the other Miller's test cylinders are established in ABAQUS, and nonlinear buckling FEA simulations are conducted to acquire the buckling pressure (PFEA). According to Fig. 8, compared with Miller's experimental buckling pressure PMiller, the simulation results are still in good agreement with those of the experiment. Therefore, the statistic imperfections model of the rolling–welding cylinder is reliable for the FEA simulation of external pressure instability.
where σtrue and εtrue represent the true stress and true strain, respectively. Fig. 6 presents the material curve employed in FEA simulations, which is derived from the average of five S30408 σtrue-εtrue curves. The yield strength of the material is σs = 267.4 MPa; modulus of elasticity is E = 195 GPa; and Poisson's ratio is μ = 0.3. A fixed support boundary condition is imposed in the simulations according to the experimental device. The details of the FEA simulation parameters are presented in Table 2. Moreover, the aforementioned Eqs. (7)–(12) of the statistic imperfections models are used to establish the FEA imperfection models according to Table 1. Then, nonlinear buckling simulations are conducted with arc-length methods to obtain the simulated buckling pressure (PFEA). All the experimental results of critical instability pressure PTest are in good agreement with the simulation results, as shown in Fig. 7.
3. FEA simulation of maximum permissible geometry deviation of cylinder subjected to external pressure In ASME Code Ⅷ-2, subsection 5.4 of the analysis and design portion, the influence of the critical buckling pressure for cylinders subjected to external pressure with initial geometry imperfections was explained by the shape reduced coefficient βcr = 0.8 [10]. In other words, the buckling pressures of cylinders with initial geometry imperfections subjected to external pressure are equivalent to approximately 80% of the theoretical buckling pressures of a perfect cylinder. Similarly, the statistic imperfections models of rolling–welding cylinders with different yield strengths and dimensions are used in the simulations to calculate the value of e when the external pressure is 80% of the instability pressure of a perfect cylinder.
2.2.2. Comparison of FEA simulation results with Miller's experiment According to the aforementioned method of statistic imperfections model established in Section 2.1, the original parameters of the cylinders are processed into the variable form of the FEA imperfection model in Miller experiment [9]. Then, geometric and material parameters are compiled into the scripting language to set up the statistic imperfections models. Finally, the boundary conditions are simply set up based on Miller's experimental settings, and the setting of the other FEA simulation parameters are the same as those in Table 2. The material and geometry nonlinear simulations of the buckling external pressure PFEA are conducted by the FEA method coupled with the statistic imperfections models. The detailed parameters and buckling pressure of Miller cylinder No. 1 are presented in Table 3.
3.1. FEA simulation models and parameter setting In this section, 98 samples of the statistic imperfections models subjected to external pressure with L/Do = 0.5–7.0, and Do/t = 50, 100, and 200, are calculated by the FEA simulation. The S30408 material curve with yield strength σs = 267.4 MPa, E = 195 GPa, and μ = 0.3 mentioned in Section 2.2 is transferred into a dimensionless
Fig. 2. Average dimensionless imperfections at the bottom of cylinder S3. 4
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Fig. 3. Average dimensionless imperfections of the whole cylinder S3.
Fig. 6. Curve of true stress σtrue and true strain, εtrue.
Fig. 4. Model of statistic imperfections of cylinder S3 (30 times larger).
_ _ where σtrue and εtrue represent the dimensionless true stress and dimensionless true strain, respectively. The above material constitutive relation is used for the FEA calculation. The specified FEA simulation parameters are the same as those in Table 2.
parameter by Eqs. (15) and (16), which is employed to fit two material curves with yield strength σs = 367.4 MPa and 167.4 MPa, as seen in Fig. 9. _ σtrue = σtrue/ σs (15) _ εtrue = Eεtrue/ σs (16)
Fig. 5. Device for the external pressure test. 5
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Table 2 Parameters of the FEA simulation. Solver
ABAQUS Standard (implicit) version 6.13
Non-linear algorithm
Arc-length methods with adaptive arc length increments Fixed support S4 (with 4 nodes and 6 degrees-of-freedom per node) Depending on the model size 10–2 10–5 10–2 300 1
Boundary conditions Element type Mesh size Initial arc length increment size Min. arc length increment size Max. arc length increment size Max. number of increments Estimated total arc length
Fig. 8. Results of the FEA simulation and Miller's experiment.
Fig. 7. Results of the FEA simulation and experiments.
3.2. FEA procedure and results
Fig. 9. The σtrue–εtrue curves of S30408 material with different yield strength.
The finite element software ABAQUS coupled with the script language compiled by PYTHON is used to model the geometry and subsequently conduct material nonlinear buckling analyses. As shown in Fig. 10, the cylinder with the geometry deviation value e = 0.001 is considered to be a perfect cylinder, and its critical buckling pressure was obtained from the FEA simulation. Then, the corresponding value of e is the maximum permissible geometry deviation, until the critical pressure by the FEA iterative simulation decreases to 80% Pcr as the value of e increases gradually. The FEA simulation flowchart is included in Fig. 10. The simulation results of the critical buckling pressure Pcr and the values of e for the statistic imperfections models are presented in Table 4.
4.1. Simulation results of permissible geometry deviation compared with ASME code specified value The e/t–L/Do curves of ASME Code sections Ⅷ-1 and Ⅷ-2 are shown in Figs. 11–13. It can be clearly seen that the e/t of the two ASME curves is positively correlated with the L/Do at the same Do/t. Moreover, at the same L/Do, the two ASME curves shifted to the right as the Do/t increased. An intersection point for these two ASME curves with different Do/t values can be obtained as the allowed geometry deviation e is differently limited by ASME Code sections Ⅷ-1 and Ⅷ-2. The value of e in the ASME Code Ⅷ-1 curve is less than that in the ASME Code Ⅷ-2 curve when it is above the intersection point and L/Do increases, whereas below the intersection, the value of e in the ASME Code Ⅷ-1 curve is greater than that in the ASME Ⅷ-2 curve. On the one hand, the e/t–L/Do curves calculated by the FEA simulation in Figs. 11–13 show that in the case of the same and large L/Do value, the trend of the FEA curve is generally consistent with that of the ASME curves. In other words, the value e/t is positively correlated with the Do/t value. On the other hand, at the same Do/t, the relationship of the value e/t with L/Do is quite different with different Do/t values.
4. Discussion The results of the maximum permissible geometry deviation values of e simulated by the FEA are compared with those specified by ASME Code sections Ⅷ-1 and Ⅷ-2. The influence factors of e were discussed. Both the FEA simulation results and theoretical ones based on Eqs. (1)–(6) from the ASME Code are drawn as the e/t curves, respectively, where e/t is the X-axis and L/Do is the Y-axis. the e/t curves of the statistic imperfections models were marked as FEA, and the e/t curves of the ASME Code standards are marked as ASME Code sections Ⅷ-1 and Ⅷ-2, respectively.
(1) For the thick cylinder with Do/t = 50 shown in Fig. 11: when L/Do is large, the FEA value e/t is positively correlated with L/Do. As the L/Do value becomes larger, the e/t value of the FEA becomes smaller than that of the two ASME curves, and it is closer to the
Table 3 Detailed parameters and buckling pressure of Miller cylinder No. 1. Test No.
Do / t
L/ Do
t (mm)
e (mm)
μ
E (GPa)
σS (MPa)
PMiller (MPa)
PFEA (MPa)
1
32.36
1.97
12.537
4.114
0.3
204
272.35
15.1
13.1
6
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Fig. 11. e/t– L/Do curves of ASME Code and FEA simulation with σs = 367.4 MPa, Do/t = 50.
buckling external pressure of the cylinder increases as the value of e increases. (2) As shown in Fig. 12, when Do/t = 100, L/Do < 3.5 and L/ Do > 5.0, the FEA value e/t increases with the increase of L/Do, but when 3.5 < L/Do < 5.0, the value e/t is negatively correlated with L/Do. It is caused by the special form of the random defect distribution in the defect model and the waveform conversion when the cylinder is unstable. In most cases, as the L/Do value becomes larger, the FEA value e/t is larger than that of the ASME Ⅷ-1 curve, and it is closer to the ASME Ⅷ-2 curve. (3) The thin cylinder with Do/t = 200 is shown in Fig. 13: the change in the trend of the FEA curve is basically consistent with that of the ASME curves, which means hat the value e/t is linearly positive with L/Do. In general, the increase in the e/t value of the FEA is
Fig. 10. Flow diagram of FEA simulation.
value of the ASME Ⅷ-1 curve than to that of the ASME Ⅷ-2 curve. However, when L/Do is small, the e/t value of the FEA is negatively correlated with L/Do. It is caused by the strengthening effect of the weld height on the cylinders. Therefore, the critical Table 4 Critical buckling pressures Pcr and values of e for statistic imperfections models. Do (mm)
t (mm)
L (mm)
σs = 167.4 MPa·
Pcr (MPa) 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
50 100 200 300 400 500 600 700 50 80 100 130 160 200 250 300 350 400 450 500 550 600 650 700 50 100 200 300 400 500 600 700
1.409 0.938 0.516 0.325 0.221 0.179 0.164 0.159 3.762 3.216 3.038 2.874 2.665 2.593 2.032 1.642 1.480 1.378 1.320 1.194 0.966 0.821 0.724 0.658 7.277 7.232 6.446 5.908 5.315 4.595 4.207 3.957
σs = 267.4 MPa
σs = 367.4 MPa·
e (mm)
Pcr (MPa)
e (mm)
Pcr (MPa)
e (mm)
0.112 0.150 0.475 1.050 1.700 2.125 1.950 1.130 0.185 0.243 0.266 0.273 0.319 0.285 0.586 0.834 0.831 0.816 0.688 0.577 0.741 0.825 0.890 0.940 1.196 0.756 0.854 0.952 0.616 0.432 0.362 0.370
1.377 0.936 0.516 0.325 0.220 0.180 0.163 0.155 5.437 4.534 4.641 3.732 3.175 2.695 2.141 1.686 1.490 1.387 1.329 1.200 0.971 0.823 0.725 0.659 11.01 11.03 9.205 9.068 6.931 5.231 4.580 4.240
0.140 0.186 0.497 1.098 1.825 2.050 2.950 1.110 0.193 0.245 0.151 0.347 0.505 0.702 0.988 1.415 1.540 1.481 1.211 0.978 1.217 1.363 1.458 1.550 1.092 0.682 0.918 0.502 0.478 0.582 0.590 0.630
1.980 0.968 0.520 0.325 0.221 0.179 0.164 0.155 6.821 5.594 5.217 4.269 3.334 2.795 2.142 1.698 1.493 1.389 1.331 1.201 0.971 0.823 0.725 0.659 14.49 14.50 13.67 10.61 7.230 5.352 4.586 4.251
0.120 0.389 0.950 1.900 3.100 4.200 3.600 1.950 0.199 0.241 0.243 0.415 0.735 0.972 1.931 2.163 2.086 0.680 1.675 1.310 1.611 1.817 1.980 2.134 1.202 0.744 0.996 0.540 0.696 0.874 0.996 1.022
7
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has inapparent influence on the value of e. For example, for a thick cylinder with Do/t = 50, the e/t curves with different yield strengths basically coincide, which may be caused by the fact that the fixed support ends of the cylinder have a strengthening effect on the rigidity of the cylinders. (2) In most cases, with the same L/Do value, the bigger Do/t is, the greater the impact on the value e yield strength. Statistic imperfections models are adopted to model some of the Miller's experimental models and eleven external rolling–welding cylinder samples. The FEA simulated critical buckling pressures are in good agreement with the experimental results. To some extent, the curves of e/t versus L/Do obtained by the FEA simulation using statistic imperfections models represent the values of e for real cylinders. Furthermore, it shows that as the statistic imperfections models take the irregular initial geometry imperfections of the actual weld joint, the nonlinear distribution of the e calculated by the FEA simulation represents the influence of actual weld joint.
Fig. 12. e/t–L/Do curves of ASME Code and FEA simulation with σs = 367.4 MPa, Do/t = 100.
5. Conclusions In this study, the statistic imperfections models of cylinders with different yield strengths and sizes are subjected to external pressure and used in nonlinear FEA simulations by ABAQUS. The curve of the maximum permissible geometry deviation values e is obtained when the external pressure is 80% of the critical external pressure for the instability of perfect cylinders. Comparing and analyzing the value of e specified by ASME Code sections Ⅷ-1 and Ⅷ-2, the following conclusions can be drawn: (1) The finite element model is in good agreement with the distribution characteristics of the actual cylinder imperfections, namely, with the proposed statistic imperfections model. The values of e obtained by the FEA simulation show a nonlinear relationship with the L/Do, Do/t, and σs values of the cylinder. To some extent, it reflects that the weld joint substantially affects the maximum permissible geometry deviation of the cylinders subjected to external pressure. (2) The simulation values of e/t and L/Do with different Do/t are different from those of ASME Code Ⅷ. However, in the case of the same and large L/Do value, the trend of the FEA simulation results of the statistic imperfections models is generally consistent with those of ASME Code Ⅷ. On the one hand, when the L/Do value is the same, the e/t value moves to the right as Do/t increases, that is, the values of e for the cylinders under external pressure increase with the thickness of the cylinders. On the other hand, when the Do/ t value is the same, the e/t value is positively correlated with L/Do. That is, at the same L/Do, the e values increase with the increase in cylinder length. (3) The e/t curves obtained by the FEA simulation using the statistic imperfections models indicate that the yield strength has a certain effect on the e value of the cylinders subjected to external pressure. For cylinders with the same Do/t value, the effects of yield strength increase with an increase in the cylinder length. For cylinders with the same L/Do value, the effects of yield strength increase with a decrease in the cylinder thickness. In conclusion, the e values of a cylinder are closely related to the material yield strength σs. However, the maximum permissible geometry deviation e and the material yield strength σs are not taken into account in the current ASME Code sections Ⅷ-1 and Ⅷ-2, which will be investigated in a future work.
Fig. 13. e/t–L/Do curves of ASME Code and FEA simulation with σs = 367.4 MPa, Do/t = 200.
Fig. 14. FEA simulation curve of e/t and L/Do based on the statistic imperfections models with different yield strengths.
significantly higher than that of the two ASME curves with the same increase in L/Do.
4.2. Effect of material plasticity on maximum permissible geometry deviation
Acknowledgement
The e/t–L/Do curves of the FEA simulation results are shown in Fig. 14: (1) on the one hand, in the case of greater L/Do with the same Do/t, the e/t has a significant increase with an increase in the material yield strength σs, which has an obvious impact on the value of e; on the other hand, at the smaller L/Do with the same Do/t, the yield strength
The authors would like to acknowledge the financial support provided by The National Key Research and Development Program of China (2016YFC0801902). 8
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References [11]
[1] C.T.F. Ross, Pressure Vessels: External Pressure Technology, Woodhead Publishing Limited, UK, 2011. [2] W.T. Koiter, Over de stabiliteit van het elastisch evenwicht, (1945). [3] C.R. Calladine, Understanding imperfection-sensitivity in the buckling of thinwalled shells, Thin-Walled Struct. 23 (1995) 215–235 https://doi.org/10.1016/ 0263-8231(95)00013-4. [4] W.T. Koiter, The effect of axisymmetric imperfections on the buckling of cylindrical shells under axial compression, Proceedings Koninklijke Nederlandse Akademie Van Wetenschappen, 1963, pp. 265–279. [5] Licai Yang, Zhiping Chen, Fucai Chen, et al., Buckling of cylindrical shells with general axisymmetric thickness imperfections under external pressure, Eur. J. Mech. A Solid. 38 (2013) 90–99 https://doi.org/10.1016/j.euromechsol.2012.09. 006. [6] U. Hornung, H. Saal, Buckling loads of tank shells with imperfections, Int. J. Nonlin. Mech. 37 (2002) 605–621 https://doi.org/10.1016/S0020-7462(01)00087-7. [7] ASME Boiler and Pressure Vessel Code, Section Ⅷ, Alternative Rules for Construction of Pressure Vessel, Division 1, The American Society of Mechanical Engineers, 2013. [8] D.F. Windenburg, Theoretical and empirical equations represented in rules for the construction of unfired pressure vessels subjected to external pressure, 1927-1959, ASME, Int. J Pres Ves Pip. Collected papers (1960) 625–632. [9] C.D. Miller, The Effect of Initial Imperfections on the Buckling of Cylinders Subjected to External Pressure, Pressure Vessel Research Council, 1994. [10] ASME Boiler and Pressure Vessel Code, Section Ⅷ, Alternative Rules for
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[16]
[17]
[18]
[19] [20]
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Construction of Pressure Vessel, Division 2, The American Society of Mechanical Engineers, 2013. S.P. Timoshenko, J.M. Gere, Theory of Elastic Stability, McGraw-Hill, New York, 1961. D. Annaratone, Pressure Vessel Design, Springer-Verlag, Berlin, 2007. S.M. Fatemi, H. Showkati, M. Maali, Experiments on imperfect cylindrical shells under uniform external pressure, Thin-Walled Struct. 65 (2013) 14–25 https://doi. org/10.1016/j.tws.2013.01.004. J.M. Rotter, J.G. Teng, Elastic stability of cylindrical shells with weld depressions, J. Struct. Eng. 115 (1989) 1244–1263. J.H. Wang, A. Koizumi, Buckling of cylindrical shells with longitudinal joints under external pressure, Thin-Walled Struct. 48 (12) (2010) 897–904 https://doi.org/10. 1016/j.tws.2010.05.004. C.A. Schenk, G.I. Schuëller, Buckling analysis of cylindrical shells with random geometric imperfections, Int. J. Nonlin. Mech. 38 (2003) 1119–1132 https://doi. org/10.1016/S0020-7462(02)00057-4. C. de Paor, D. Kelliher, K. Cronin, et al., Prediction of vacuum-induced buckling pressures of thin-walled cylinders, Thin-Walled Struct. 55 (2012) 1–10 https://doi. org/10.1016/j.tws.2012.03.001. Isaak Vryzidis, Stefanou George, Stochastic stability analysis of steel tubes with random initial imperfections, Finite Elem. Anal. Des. 77 (2013) 31–39 https://doi. org/10.1016/j.finel.2013.09.002. ABAQUS, Abaqus 6.13 Analysis User's Guide, Dassault Systèmes, 2013. M. Pircher, P.A. Berry, X. Ding, et al., The shape of circumferential weld-induced imperfections in thin-walled steel silos and tanks, Thin-Walled Struct. 39 (2001) 999–1014 https://doi.org/10.1016/S0263-8231(01)00047-7.
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Full length article
Numerical simulation study on the maximum permissible geometry deviation values for cylinders under external pressure
T
Bingbing Chen, Lei Jin, Pengfei Wang*, Peng Cao, Yi Zhou, Zengliang Gao Institute of Process Equipment and Control Engineering, Zhejiang University of Technology, Hangzhou, 310023, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Cylinders under external pressure Shape reduced coefficient Maximum permissible geometry deviation Statistic imperfections model Yield strength
According to the surface imperfections of rolling–welding cylinders, a finite element analysis (FEA) model for a cylinder with surface imperfections is proposed, namely, a statistic imperfections model. The maximum permissible geometry deviation values e of statistic imperfections models are computed based on the shape reduced coefficient βcr = 0.8. Compared with the limited value of ASME Code sections Ⅷ-1 and Ⅷ-2, it is clear that the calculated value e/t is positively correlated with the length–diameter ratio L/Do and diameter–thickness ratio Do/t and that the yield strength has a substantial influence on the maximum permissible geometry deviation values, which is not taken into account by Code Ⅷ-2.
1. Introduction Cylinders used in the petrochemical industry are usually manufactured by rolling and welding. The load-bearing capacity of a cylindrical shell under external pressure significantly depends on the characteristics of imperfections caused by various manufacturing processes. Initially, an equation for the critical buckling pressure of cylinders subjected to uniform external pressure without imperfections was proposed [1]. However, several studies found that the theoretical model of critical instability pressure cannot reasonably describe the experimental results. Koiter [2] noted that these discrepancies were caused by the high sensitivity of shells with geometric imperfections. Calladine [3] claimed that the initial stresses locked in the imperfect initial geometry had significant effects on the buckling performance. It was proved that the initial geometry imperfections of the cylindrical shells were the main reason for the inconsistencies between the experimental and theoretical values [4]. Recently, using the asymptotic analysis method, Yang et al. [5] have investigated the effects of three types of imperfections with different thicknesses on the buckling of laterally pressured cylindrical shells. Hornung and Saal numerically investigated the buckling behaviors of a shell with geometric imperfections with different shapes and [6] found that only a shell model with measured imperfections can reasonably predict the buckling phenomena observed in the experiments. Therefore, the design standard of cylinders subjected to external pressure is intended to limit the maximum permissible geometry deviation values e. For rolling–welding cylinders with initial geometry deviation, it is
*
assumed that the buckling pressures would not be less than 80% of the theoretical buckling pressures of cylinders without imperfections. The values of e from the true circular form are employed in ASME Code section Ⅷ-1 [7] to regulate the maximum permissible deviation values for cylinders subjected to external pressure according to Fig. UG-80.1, which is based on an empirical formula, Eqs. (1)–(4), by Windenburg [8].
0.0180(D / t ) e = 0.015n + t n
(1)
where d
R/t ⎞ n = c ⎜⎛ ⎟ ⎝ L/R ⎠
(2)
c = 2.28(R/ t )0.054 ≤ 2.80
(3)
d = 0.38(R/ t )0.044 ≤ 0.485
(4)
In Eqs. (1)–(4), D and t are the centerline cylinder diameter and the cylindrical shell thickness, respectively. L and R represent the length of a cylinder and the radius to midpoint of a cylindrical shell, respectively. e is the maximum permissible geometry deviation from true circular form measured over a half wave length of cylindrical shells. n is the number of circumferential waves at cylinder buckling within the domain (2 ≤ n ≤ 1.41 R/ t ). Miller [9] noted that the value of e determined by Windenburg cannot guarantee that the actual instability pressure be 80% higher than the theoretical instability pressure. The critical pressure reduction
Corresponding author. E-mail address: [email protected] (P. Wang).
https://doi.org/10.1016/j.tws.2019.106305 Received 9 January 2019; Received in revised form 17 April 2019; Accepted 6 July 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.
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μ σs σ ε σtrue εtrue _ σtrue _ εtrue PTest
Nomenclature D Do R t L Lec e
centerline diameter of cylinder, mm outside diameter of cylinder, mm radius to midpoint of cylinder, mm thickness of cylindrical shell, mm length of cylinder between lines of support, mm chord length of the arc template of the bow, mm maximum permissible geometry deviation from a true circular form, mm minimum permissible geometry deviation from a true circular form, mm number of waves with which a cylinder will buckle in the circumferential direction geometric radius on the outer wall of the steel tube, mm defect of multiple steel tubes, mm random phase for Monte Carlo simulation, mm axial coordinates of cylindrical shell, mm relative axial coordinates of cylindrical shell circumferential angle of cylindrical shell, ° relative circumferential angle of cylindrical shell dimensionless imperfection of cylindrical shell average dimensionless imperfection of cylindrical shell modulus of elasticity of material, GPa
ec n r w f z Z θ θ* W W E
PFEA PMiller Pcr
Subscripts o cr
FEA
1.069
⎜
⎟
valid for 0.2t ≤ ec ≤ 0.0242R
finite element analysis
subjected to external pressure was employed by Wang and Koizumi [15]. It was found that the existence of longitudinal joints and initial imperfections in the cylinder would have an influence on the critical buckling pressure. Schenk and Schueller [16] investigated the effects of random geometric imperfections on the limit loads of thin-walled cylindrical shells under deterministic axial compression through the FEA method with the direct Monte Carlo simulation technique, and the values of the ultimate load predicted were consistent with the actual observation. Paor et al. [17] used the FEA method to analyze the geometrical nonlinear buckling of thin-walled shell models under external pressure with imperfections, and the buckling pressure and post-buckling mode shape were in good agreement with those of the experiment. In this study, the FEA method is used to investigate the effects of the geometrical parameters (L/Do and Do/t) and the yield strength on the maximum permissible geometry deviation values e. The FEA model, which is called the statistic imperfections model, is consistent with the initial geometry imperfections of the real cylinder. Through the statistical method, this model is proposed to deal with the three-dimensional scanning data of the actual cylinders appearance. In addition, when the external pressure on the cylinder examples reaches 80% of the theoretical buckling critical pressures without imperfections, the values of e for the statistic imperfections models are acquired through the FEA simulation, and the obtained values of e are the maximum permissible geometry deviation values of the actual cylinder. Finally, the simulation e curves are compared with those specified by ASME Code sections Ⅷ-1 and Ⅷ-2. Finally, the influence of the L/Do, Do/t, and yield strength of a cylinder under external pressure on the values of e is also investigated.
(5)
L ec = 0.0165t ⎛ ec + 3.25⎞ ⎝ Rt ⎠
outside critical buckling
Abbreviations
factor βcr caused by geometry imperfections may be less than the value of ASME Code Ⅷ-1, that is, βcr < 80%. Therefore, based on the experiment and the critical pressure reduction factor βcr < 80%, the values of e can be determined as follows:
e = min[ec, 2t ]
Poisson's ratio yield strength of material, MPa stress, MPa strain true stress, MPa true strain dimensionless true stress dimensionless true strain buckling pressure of a cylinder from experimental observation, MPa buckling pressure of a cylinder from FEA simulation, MPa buckling pressure of a cylinder from Miller's experiment, MPa buckling pressure of a cylinder with the value e = 0.001 from FEA, MPa
(6)
where Lec represents the chord length of the arc template of the bow, and ec represents the minimum inward deviation from a true circular form. Eqs. (5) and (6) are applied in the current ASME Code Ⅷ-2 [10]. Eq. (6) is used to limit the range of the value of e. According to aforementioned equations, it is clear that the maximum permissible geometry deviation values e provided by ASME Code sections Ⅷ-1 and Ⅷ-2 only consider the geometrical parameters (diameter Do, length L, and thickness t) of the cylinders, except for the effects of yield strength. However, Timoshenko [11] presented a critical pressure quantitative description formula of a circle and tube under uniform buckling external pressure, taking material yield strength and initial geometry imperfections into account. A critical pressure formula given by the pressure vessel design by Annaratone [12] also showed the relationship of material yield strength with initial geometry imperfections. In recent years, Fatemi, Showkati, et al. [13] conducted external pressure experiments applying different degrees of initial geometry imperfections to cylinders with different L/Do and Do/t ratios and found that although the initial geometry imperfections applied to some cylinders were much higher than those specified in the EN 1993-1-6 ECCSDIN18800 code, their critical pressure actually was higher. Therefore, the limit of the maximum permissible geometry deviation values e enforced by existing standards still needs to be investigated. With the development of numerical simulation technology, the FEA method has been applied to investigate the instability of cylindrical shells by many researchers. Rotter and Teng [14] studied the effects of imperfections caused by the weld joint on the buckling of cylindrical shells under external pressure through the finite element method. The results showed that the critical instability pressure of a cylinder under external pressure decreased significantly owing to the initial geometrical imperfections. The FEA method for rolling–welding cylinders
2. FEA model 2.1. Establishment of the FEA model The geometry shape of a random steel tube based on the theory provided by Vryzidis and Stefanou [18] is described as follows:
r (θ , z ) = R + w (θ , z ) + f (θ , z )
(7)
where θ and z are the circumferential and axial coordinates of the tube, respectively, r (θ, z) represents the initial radius at any point of a tube, 2
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imperfections of rolling–welding cylinders proposed by Pircher [20]. In addition, the distribution of the weld joint imperfections outside the zone presents circumferential and axial waveforms, which represent the fluctuation of local potholes and oblique waveforms of the real cylindrical shells. Therefore, the imperfection distribution of the statistic imperfections models are mostly consistent with the imperfection distribution of the actual rolling–welding cylinders. According to the aforementioned method, Eqs. (7)–(12) coupled with the ABAQUS modeling module are used to compile the model of statistic imperfections by using PYTHON. For cylinders with different geometric and material parameters (L/Do, Do/t, e, E, μ, and σs), the corresponding statistic imperfections models can be established in ABAQUS efficiently.
R represents the outer radius, w (θ, z) represents the values of imperfections across any point of the tube, and f (θ, z) is a random phase for the Monte Carlo simulation. This method can present the real shape of the imperfections to the greatest extent. Given that each cylinder specimen in this study is fabricated by rolling–welding, the geometric shapes of the eleven rolling–welding cylinder samples are obtained using a 3D scanner. Further details of the specimens are presented in Fig. 1 and Table 1. In this work, eleven geometry models obtained by scanning the samples were imported into the FEA software to draw meshes. Subsequently, the column coordinates of each node on the surface of the cylinders would be established. The weld joint of cylinder was set up at 90°. Then, the dimensionless coordinate values of each node were derived according to Eqs. (8)–(12), as follows:
θ* = θ /360°
(8)
2.2. Experimental verification of the FEA imperfection model
Z = z /L
(9)
2.2.1. Comparison of the FEA and external pressure experiment results An external pressure experiment was conducted on each rolling–welding cylinder specimen (in total eleven), and an external pressure equipment device was designed as the pressure chamber structure shown in Fig. 5. The cylinder specimen 4, bottom plate 2, top plate 5, and bolt 3 were fixed and sealed in the chamber through the hand hole 1 and an O-ring. The bolt 3 connected the bottom plate 2, top plate 5, and flat head, compensating for the effects of axial force to ensure that the cylindrical shell was pressurized through the pressure nozzle with uniform circumferential pressure. The bottom plate 2 and top plate 5 were processed into three ladder shapes to make the device suitable for three types of cylinder samples with different diameters. Before the experiment, the water injection nozzle and blow-off nozzle were opened to fill water, and then, the valve was closed. At the beginning of the experiment, the pressure nozzle was opened to pressurize. At the same time, the critical instability pressure (PTest) of the specimens is obtained by observing the deformation of the cylinder body through the viewport and recording the pressure change of the pressure gauge nozzle. At the end of the experiment, the water drain nozzle and blowoff nozzle were opened to drain the internal water. Cylinder specimens are made of S30408 stainless steel, and the tensile stress–strain (σ–ε) curves of five S30408 stainless steel tensile samples are obtained by an INSTRON tensile test machine, which are converted into the curves of true stress and true strain through Eqs. (13) and (14), respectively, as follows:
w (θ*, Z ) = r (θ*, Z )/ R
(10)
W (θ*, Z ) = w (θ*, Z )/ e
(11)
W (θ*, Z ) =
1
∑θ*=0 W (θ*, Z )
(12)
where θ* and Z represent the relative circumferential coordinate and relative axial coordinate, respectively; w (θ*, z) represents the imperfection values of any point on the cylinders; W (θ, 0) represents the dimensionless imperfection values of any point on the cylinders; and W (θ , 0) represents the average dimensionless imperfection values of any point on the cylinders. Obviously, the average dimensionless imperfection distribution at the bottom of the S3 cylinder (Z = 0) and the whole cylinder S3 model can be obtained, as shown in Fig. 2 and Fig. 3, respectively. A perfect cylindrical shell is built in ABAQUS [19] based on the size parameters of cylinder S3 presented in Table 1, and the average dimensionless imperfections attained were applied as radial load to the grid node of the perfect cylinder. Then, an FEA imperfection model, namely, the statistic imperfections model, is acquired after the static analysis step, as shown in Fig. 4. It can be seen from Figs. 3 and 4 that the weld joint is located in a statistic imperfections model θ* = 0.2–0.3. The distribution characteristics of the weld joint imperfections are inward concave along the axis because the stiffening ring at the ends of the rolling–welding cylinder has an outward strengthening effect on the end. It is consistent with the characteristics of the weld joint
εtrue = ln(1 + ε )
Fig. 1. Photograph and structure of the cylinder specimens. 3
(13)
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Table 1 Dimensions and material parameters of the cylinder specimens. Cylinder number
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11
σtrue = σ (1 + ε )
Material
S30408 S30408 S30408 S30408 S30408 S30408 S30408 S30408 S30408 S30408 S30408
E (GPa)
195 195 195 195 195 195 195 195 195 195 195
μ
0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
t (mm)
Do (mm)
1.810 1.810 1.810 1.810 1.810 1.810 1.810 1.810 1.810 1.810 1.810
199.737 199.721 199.757 199.438 200.035 299.523 299.740 299.822 399.715 399.797 399.379
(14)
L (mm)
195.215 393.129 595.837 795.689 994.581 294.373 444.570 594.371 314.962 394.703 474.931
L /Do
0.977 1.968 2.983 3.990 4.972 0.983 1.483 1.982 0.788 0.987 1.189
e (mm) Min
Max
−1.297 −0.846 −1.521 −1.875 −1.990 −1.212 −1.963 −1.818 −1.370 −1.238 −1.943
2.098 1.361 2.283 1.898 1.939 2.139 1.288 1.678 1.784 0.847 1.480
Similarly, the statistic imperfections models corresponding to the other Miller's test cylinders are established in ABAQUS, and nonlinear buckling FEA simulations are conducted to acquire the buckling pressure (PFEA). According to Fig. 8, compared with Miller's experimental buckling pressure PMiller, the simulation results are still in good agreement with those of the experiment. Therefore, the statistic imperfections model of the rolling–welding cylinder is reliable for the FEA simulation of external pressure instability.
where σtrue and εtrue represent the true stress and true strain, respectively. Fig. 6 presents the material curve employed in FEA simulations, which is derived from the average of five S30408 σtrue-εtrue curves. The yield strength of the material is σs = 267.4 MPa; modulus of elasticity is E = 195 GPa; and Poisson's ratio is μ = 0.3. A fixed support boundary condition is imposed in the simulations according to the experimental device. The details of the FEA simulation parameters are presented in Table 2. Moreover, the aforementioned Eqs. (7)–(12) of the statistic imperfections models are used to establish the FEA imperfection models according to Table 1. Then, nonlinear buckling simulations are conducted with arc-length methods to obtain the simulated buckling pressure (PFEA). All the experimental results of critical instability pressure PTest are in good agreement with the simulation results, as shown in Fig. 7.
3. FEA simulation of maximum permissible geometry deviation of cylinder subjected to external pressure In ASME Code Ⅷ-2, subsection 5.4 of the analysis and design portion, the influence of the critical buckling pressure for cylinders subjected to external pressure with initial geometry imperfections was explained by the shape reduced coefficient βcr = 0.8 [10]. In other words, the buckling pressures of cylinders with initial geometry imperfections subjected to external pressure are equivalent to approximately 80% of the theoretical buckling pressures of a perfect cylinder. Similarly, the statistic imperfections models of rolling–welding cylinders with different yield strengths and dimensions are used in the simulations to calculate the value of e when the external pressure is 80% of the instability pressure of a perfect cylinder.
2.2.2. Comparison of FEA simulation results with Miller's experiment According to the aforementioned method of statistic imperfections model established in Section 2.1, the original parameters of the cylinders are processed into the variable form of the FEA imperfection model in Miller experiment [9]. Then, geometric and material parameters are compiled into the scripting language to set up the statistic imperfections models. Finally, the boundary conditions are simply set up based on Miller's experimental settings, and the setting of the other FEA simulation parameters are the same as those in Table 2. The material and geometry nonlinear simulations of the buckling external pressure PFEA are conducted by the FEA method coupled with the statistic imperfections models. The detailed parameters and buckling pressure of Miller cylinder No. 1 are presented in Table 3.
3.1. FEA simulation models and parameter setting In this section, 98 samples of the statistic imperfections models subjected to external pressure with L/Do = 0.5–7.0, and Do/t = 50, 100, and 200, are calculated by the FEA simulation. The S30408 material curve with yield strength σs = 267.4 MPa, E = 195 GPa, and μ = 0.3 mentioned in Section 2.2 is transferred into a dimensionless
Fig. 2. Average dimensionless imperfections at the bottom of cylinder S3. 4
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Fig. 3. Average dimensionless imperfections of the whole cylinder S3.
Fig. 6. Curve of true stress σtrue and true strain, εtrue.
Fig. 4. Model of statistic imperfections of cylinder S3 (30 times larger).
_ _ where σtrue and εtrue represent the dimensionless true stress and dimensionless true strain, respectively. The above material constitutive relation is used for the FEA calculation. The specified FEA simulation parameters are the same as those in Table 2.
parameter by Eqs. (15) and (16), which is employed to fit two material curves with yield strength σs = 367.4 MPa and 167.4 MPa, as seen in Fig. 9. _ σtrue = σtrue/ σs (15) _ εtrue = Eεtrue/ σs (16)
Fig. 5. Device for the external pressure test. 5
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Table 2 Parameters of the FEA simulation. Solver
ABAQUS Standard (implicit) version 6.13
Non-linear algorithm
Arc-length methods with adaptive arc length increments Fixed support S4 (with 4 nodes and 6 degrees-of-freedom per node) Depending on the model size 10–2 10–5 10–2 300 1
Boundary conditions Element type Mesh size Initial arc length increment size Min. arc length increment size Max. arc length increment size Max. number of increments Estimated total arc length
Fig. 8. Results of the FEA simulation and Miller's experiment.
Fig. 7. Results of the FEA simulation and experiments.
3.2. FEA procedure and results
Fig. 9. The σtrue–εtrue curves of S30408 material with different yield strength.
The finite element software ABAQUS coupled with the script language compiled by PYTHON is used to model the geometry and subsequently conduct material nonlinear buckling analyses. As shown in Fig. 10, the cylinder with the geometry deviation value e = 0.001 is considered to be a perfect cylinder, and its critical buckling pressure was obtained from the FEA simulation. Then, the corresponding value of e is the maximum permissible geometry deviation, until the critical pressure by the FEA iterative simulation decreases to 80% Pcr as the value of e increases gradually. The FEA simulation flowchart is included in Fig. 10. The simulation results of the critical buckling pressure Pcr and the values of e for the statistic imperfections models are presented in Table 4.
4.1. Simulation results of permissible geometry deviation compared with ASME code specified value The e/t–L/Do curves of ASME Code sections Ⅷ-1 and Ⅷ-2 are shown in Figs. 11–13. It can be clearly seen that the e/t of the two ASME curves is positively correlated with the L/Do at the same Do/t. Moreover, at the same L/Do, the two ASME curves shifted to the right as the Do/t increased. An intersection point for these two ASME curves with different Do/t values can be obtained as the allowed geometry deviation e is differently limited by ASME Code sections Ⅷ-1 and Ⅷ-2. The value of e in the ASME Code Ⅷ-1 curve is less than that in the ASME Code Ⅷ-2 curve when it is above the intersection point and L/Do increases, whereas below the intersection, the value of e in the ASME Code Ⅷ-1 curve is greater than that in the ASME Ⅷ-2 curve. On the one hand, the e/t–L/Do curves calculated by the FEA simulation in Figs. 11–13 show that in the case of the same and large L/Do value, the trend of the FEA curve is generally consistent with that of the ASME curves. In other words, the value e/t is positively correlated with the Do/t value. On the other hand, at the same Do/t, the relationship of the value e/t with L/Do is quite different with different Do/t values.
4. Discussion The results of the maximum permissible geometry deviation values of e simulated by the FEA are compared with those specified by ASME Code sections Ⅷ-1 and Ⅷ-2. The influence factors of e were discussed. Both the FEA simulation results and theoretical ones based on Eqs. (1)–(6) from the ASME Code are drawn as the e/t curves, respectively, where e/t is the X-axis and L/Do is the Y-axis. the e/t curves of the statistic imperfections models were marked as FEA, and the e/t curves of the ASME Code standards are marked as ASME Code sections Ⅷ-1 and Ⅷ-2, respectively.
(1) For the thick cylinder with Do/t = 50 shown in Fig. 11: when L/Do is large, the FEA value e/t is positively correlated with L/Do. As the L/Do value becomes larger, the e/t value of the FEA becomes smaller than that of the two ASME curves, and it is closer to the
Table 3 Detailed parameters and buckling pressure of Miller cylinder No. 1. Test No.
Do / t
L/ Do
t (mm)
e (mm)
μ
E (GPa)
σS (MPa)
PMiller (MPa)
PFEA (MPa)
1
32.36
1.97
12.537
4.114
0.3
204
272.35
15.1
13.1
6
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Fig. 11. e/t– L/Do curves of ASME Code and FEA simulation with σs = 367.4 MPa, Do/t = 50.
buckling external pressure of the cylinder increases as the value of e increases. (2) As shown in Fig. 12, when Do/t = 100, L/Do < 3.5 and L/ Do > 5.0, the FEA value e/t increases with the increase of L/Do, but when 3.5 < L/Do < 5.0, the value e/t is negatively correlated with L/Do. It is caused by the special form of the random defect distribution in the defect model and the waveform conversion when the cylinder is unstable. In most cases, as the L/Do value becomes larger, the FEA value e/t is larger than that of the ASME Ⅷ-1 curve, and it is closer to the ASME Ⅷ-2 curve. (3) The thin cylinder with Do/t = 200 is shown in Fig. 13: the change in the trend of the FEA curve is basically consistent with that of the ASME curves, which means hat the value e/t is linearly positive with L/Do. In general, the increase in the e/t value of the FEA is
Fig. 10. Flow diagram of FEA simulation.
value of the ASME Ⅷ-1 curve than to that of the ASME Ⅷ-2 curve. However, when L/Do is small, the e/t value of the FEA is negatively correlated with L/Do. It is caused by the strengthening effect of the weld height on the cylinders. Therefore, the critical Table 4 Critical buckling pressures Pcr and values of e for statistic imperfections models. Do (mm)
t (mm)
L (mm)
σs = 167.4 MPa·
Pcr (MPa) 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
50 100 200 300 400 500 600 700 50 80 100 130 160 200 250 300 350 400 450 500 550 600 650 700 50 100 200 300 400 500 600 700
1.409 0.938 0.516 0.325 0.221 0.179 0.164 0.159 3.762 3.216 3.038 2.874 2.665 2.593 2.032 1.642 1.480 1.378 1.320 1.194 0.966 0.821 0.724 0.658 7.277 7.232 6.446 5.908 5.315 4.595 4.207 3.957
σs = 267.4 MPa
σs = 367.4 MPa·
e (mm)
Pcr (MPa)
e (mm)
Pcr (MPa)
e (mm)
0.112 0.150 0.475 1.050 1.700 2.125 1.950 1.130 0.185 0.243 0.266 0.273 0.319 0.285 0.586 0.834 0.831 0.816 0.688 0.577 0.741 0.825 0.890 0.940 1.196 0.756 0.854 0.952 0.616 0.432 0.362 0.370
1.377 0.936 0.516 0.325 0.220 0.180 0.163 0.155 5.437 4.534 4.641 3.732 3.175 2.695 2.141 1.686 1.490 1.387 1.329 1.200 0.971 0.823 0.725 0.659 11.01 11.03 9.205 9.068 6.931 5.231 4.580 4.240
0.140 0.186 0.497 1.098 1.825 2.050 2.950 1.110 0.193 0.245 0.151 0.347 0.505 0.702 0.988 1.415 1.540 1.481 1.211 0.978 1.217 1.363 1.458 1.550 1.092 0.682 0.918 0.502 0.478 0.582 0.590 0.630
1.980 0.968 0.520 0.325 0.221 0.179 0.164 0.155 6.821 5.594 5.217 4.269 3.334 2.795 2.142 1.698 1.493 1.389 1.331 1.201 0.971 0.823 0.725 0.659 14.49 14.50 13.67 10.61 7.230 5.352 4.586 4.251
0.120 0.389 0.950 1.900 3.100 4.200 3.600 1.950 0.199 0.241 0.243 0.415 0.735 0.972 1.931 2.163 2.086 0.680 1.675 1.310 1.611 1.817 1.980 2.134 1.202 0.744 0.996 0.540 0.696 0.874 0.996 1.022
7
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has inapparent influence on the value of e. For example, for a thick cylinder with Do/t = 50, the e/t curves with different yield strengths basically coincide, which may be caused by the fact that the fixed support ends of the cylinder have a strengthening effect on the rigidity of the cylinders. (2) In most cases, with the same L/Do value, the bigger Do/t is, the greater the impact on the value e yield strength. Statistic imperfections models are adopted to model some of the Miller's experimental models and eleven external rolling–welding cylinder samples. The FEA simulated critical buckling pressures are in good agreement with the experimental results. To some extent, the curves of e/t versus L/Do obtained by the FEA simulation using statistic imperfections models represent the values of e for real cylinders. Furthermore, it shows that as the statistic imperfections models take the irregular initial geometry imperfections of the actual weld joint, the nonlinear distribution of the e calculated by the FEA simulation represents the influence of actual weld joint.
Fig. 12. e/t–L/Do curves of ASME Code and FEA simulation with σs = 367.4 MPa, Do/t = 100.
5. Conclusions In this study, the statistic imperfections models of cylinders with different yield strengths and sizes are subjected to external pressure and used in nonlinear FEA simulations by ABAQUS. The curve of the maximum permissible geometry deviation values e is obtained when the external pressure is 80% of the critical external pressure for the instability of perfect cylinders. Comparing and analyzing the value of e specified by ASME Code sections Ⅷ-1 and Ⅷ-2, the following conclusions can be drawn: (1) The finite element model is in good agreement with the distribution characteristics of the actual cylinder imperfections, namely, with the proposed statistic imperfections model. The values of e obtained by the FEA simulation show a nonlinear relationship with the L/Do, Do/t, and σs values of the cylinder. To some extent, it reflects that the weld joint substantially affects the maximum permissible geometry deviation of the cylinders subjected to external pressure. (2) The simulation values of e/t and L/Do with different Do/t are different from those of ASME Code Ⅷ. However, in the case of the same and large L/Do value, the trend of the FEA simulation results of the statistic imperfections models is generally consistent with those of ASME Code Ⅷ. On the one hand, when the L/Do value is the same, the e/t value moves to the right as Do/t increases, that is, the values of e for the cylinders under external pressure increase with the thickness of the cylinders. On the other hand, when the Do/ t value is the same, the e/t value is positively correlated with L/Do. That is, at the same L/Do, the e values increase with the increase in cylinder length. (3) The e/t curves obtained by the FEA simulation using the statistic imperfections models indicate that the yield strength has a certain effect on the e value of the cylinders subjected to external pressure. For cylinders with the same Do/t value, the effects of yield strength increase with an increase in the cylinder length. For cylinders with the same L/Do value, the effects of yield strength increase with a decrease in the cylinder thickness. In conclusion, the e values of a cylinder are closely related to the material yield strength σs. However, the maximum permissible geometry deviation e and the material yield strength σs are not taken into account in the current ASME Code sections Ⅷ-1 and Ⅷ-2, which will be investigated in a future work.
Fig. 13. e/t–L/Do curves of ASME Code and FEA simulation with σs = 367.4 MPa, Do/t = 200.
Fig. 14. FEA simulation curve of e/t and L/Do based on the statistic imperfections models with different yield strengths.
significantly higher than that of the two ASME curves with the same increase in L/Do.
4.2. Effect of material plasticity on maximum permissible geometry deviation
Acknowledgement
The e/t–L/Do curves of the FEA simulation results are shown in Fig. 14: (1) on the one hand, in the case of greater L/Do with the same Do/t, the e/t has a significant increase with an increase in the material yield strength σs, which has an obvious impact on the value of e; on the other hand, at the smaller L/Do with the same Do/t, the yield strength
The authors would like to acknowledge the financial support provided by The National Key Research and Development Program of China (2016YFC0801902). 8
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