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Investigation of the effects of perturbation forces to buckling in internally pressurized torispherical pressure vessel heads
Advances in Engineering Software 45 (2012) 232–238
Contents lists available at SciVerse ScienceDirect
Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft
Investigation of the effects of perturbation forces to buckling in internally pressurized torispherical pressure vessel heads Ahmet Zafer Sß enalp ⇑ Department of Mechanical Engineering, Gebze Institute of Technology, 41400 Gebze, Kocaeli, Turkey
a r t i c l e
i n f o
Article history: Received 27 April 2010 Received in revised form 18 May 2011 Accepted 26 September 2011 Available online 21 October 2011 Keywords: Perturbation Torispherical pressure vessel heads Buckling Instability Eigenvalue Internal pressure
a b s t r a c t The object of this paper is to investigate the effects of perturbation forces to buckling in pressure vessel heads. The pressure vessel heads in concern are confined to torispherical geometry with thin walls. Perturbation forces can alter not only the critical load for buckling but the buckled shape as well. In this paper in addition to previously used perturbation model, three more different perturbation force configurations are applied to the knuckle of the vessel. Internally pressurized three-dimensional torispherical pressure vessel head model that is previously used in literature is constructed and finite element program ANSYS Workbench is used for the solutions. First of all eigenvalue solutions are performed for each model. Then nonlinear instability solutions are conducted to obtain more realistic instability pressure values. For the nonlinear analyses not only large deformation static analyses but also large deformation transient analyses are conducted. For nonlinear analyses, perfectly plastic material model is used. It is concluded that instability pressures obtained by transient analyses are closer to plastic pressure values by PWC and ASME TES criteria and perturbation models increase instability pressure and equivalent plastic strain values. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Thin-walled torispherical pressure vessel heads are widely used in industry. Buckling is one of the major problems that a designer has to deal with during the design process of these thin-walled structures in which structural members collapse under compressive loads greater than the material can withstand. Due to the existence of unstable post-buckling behavior, torispherical pressure vessel heads are sensitive to small geometric or load imperfections. Finite element analysis is widely used in the design of these structures [1,2]. Extensive studies are presented on the buckling of pressure vessels. Khan et al. presented an experimental technique for the buckling test of shells under external pressure to determine buckling load [3]. Miller worked on buckling criteria for torispherical heads under internal pressure which are especially outside the limits of ASME codes [4]. Some of the previous works deal with elastic analysis of pressure vessel components. According to limit analysis theorems, the elastic compensation method (ECM) can obtain both upper bound and lower bound limit loads. Though the upper bound limit load given by the ECM is more accurate than the lower bound limit
⇑ Tel.: +90 262 605 18 06; fax: +90 262 653 06 75. E-mail address: [email protected] 0965-9978/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2011.09.027
load [5], the lower bound limit load is safer. Yang et al., proposed modified elastic compensation method (MECM) to improve the precision of the elastic compensation method (ECM) [6]. MECM can provide a good estimation of plastic limit loads for complex structures. Athiannan and Palaninathan’s study concerns experimental studies on buckling of thin-walled circular cylindrical shells under transverse shear. The buckling loads are also obtained from finite element models and empirical formulas and codes are compared [7]. Li et al. made an investigation of structures to identify and characterize the condition of gross plastic deformation in pressure vessel design by analysis. Limit analysis and bilinear hardening plastic analysis is performed. A criterion of plastic collapse based on the curvature of the load–plastic work history is proposed [8]. Muscat et al. proposed a criterion for evaluating the critical limit values and determining the plastic loads in pressure vessel design. The proposed criterion is based on the plastic work dissipated in the structure as loading progresses and can be used for structures subject to a single load or a combination of multiple loads. The limit and plastic loads are determined purely by the inelastic response of the structure and are not influenced by the initial elastic response [9]. Blachut’s study provides results of a numerical and experimental investigation into static stability of externally pressurized layered hemispherical and torispherical domes. Buckling/collapse tests are also conducted on domes from various materials [10].
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In pressure vessel design, it is required to satisfy certain criteria related to failure modes. In general, the fundamental failure mechanism related to static loading is denoted as gross plastic deformation (GPD). PD5500 Unfired fusion welded pressure vessels [11], ASME Boiler and pressure vessel code Sections III and VIII [12] and EN13445-3:2002 Unfired pressure vessels [13] codes specify two different approaches to the designer. The most commonly used method is based on linear elastic stress analysis of the vessel. GPD failure is related to the primary stress category, which is yieldlimited to preclude failure due to this mechanism. The second method requires an inelastic analysis concerning post yield behavior to simulate the GPD mechanism. The GPD load is calculated directly from the inelastic analysis. In EN13445 regulations, this method is called as ‘‘the direct route’’. Mackenzie et al. made an extensive review on the descriptions of the code contents [14]. The ASME Twice Elastic Slope (TES), criterion uses an empirical procedure for calculating collapse loads in experimental stress analysis of pressure vessels [12]. In Mackenzie et al.’ study, plastic collapse or gross plastic deformation loads are evaluated for two sample torispherical heads by 2D and 3D FEA based on an elastic-perfectly plastic material model [14]. They considered small and large deformation effects and the geometry and load perturbations. Their study contains the formation of the gross plastic deformation mechanism in the models in relation to the elastic–plastic buckling response of the vessels. In their study both ASME TES and plastic work criteria (PWC) are considered. The PWC criterion requires a plot of load against normalized load–plastic work curvature. The object of this paper is to investigate the effects of perturbation forces to buckling in pressure vessel heads. Internally pressurized three-dimensional torispherical pressure vessel head model that is previously used in literature is constructed and finite element program ANSYS Workbench is used for the solutions. Four different perturbation models are set up to investigate the effects of perturbation forces to buckling. As a first step, linear buckling analyses are conducted prior to solving the nonlinear buckling shapes to understand the effect of perturbation models to the deformation modes of the geometry in concern. Consequently, nonlinear instability analyses are performed for each perturbation model. Two types of nonlinear analyses are conducted. These are large deformation static analysis and transient analyses. Elastic perfectly plastic material model is used for all nonlinear analyses. The pressure vessel head model is assumed to have no initial shape imperfections. 2. Finite element model
Fig. 1. Torisperical head geometry.
Fig. 2. Finite element model.
Table 1 Material properties of torispherical pressure vessel head. Young’s modulus (GPa)
Yield strength (MPa)
200
353
2.1. Geometry For the analyses of a thin wall torispherical head, the same geometry investigated previously by Miller et al. [15] and Galletly and Blachut [16] and Mackenzie et al. [14] is considered. The geometry of torispherical head is given in Fig. 1.
Each complete model is 3D and consists of 8672 elements as shown in Fig. 2.
2.2. Finite element mesh
Material properties used in static analyses are given in Table 1. For large deformation static and transient analyses elastic perfectly plastic material model is used.
Ansys Workbench version 12 is used for the finite element analyses [17]. In the finite element mesh 4-noded SOLSH190 solid shell element with shell option is used for simulating torispherical head. SOLSH190 element can be used for a wide range of thickness from thin to moderately thick geometry (supports Mindlin-Reissner shell theory). The element has plasticity, hyperelasticity, stress stiffening, creep, large deflection, and large strain capabilities. The element formulation is based on logarithmic strain and true stress measures.
2.3. Material properties
2.4. The loading and boundary conditions The bottom free face of torispherical head is subjected to frictionless support boundary condition. As loading condition different values of pressures are applied to crown, knuckle and cylinder regions. The graph of these pressures is presented in Fig. 6.
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(a) Full model
(c) Quarter model 1
(b) Half symmetric model
(d) Quarter model 2 (different force directions)
Fig. 3. Perturbation force models.
(a) Full model
(b) Half symmetric model
(c) Quarter model 1
(d) Quarter model 2
Fig. 4. First buckling mode shapes for four models (perturbation force = 2000 N).
A.Z. Sßenalp / Advances in Engineering Software 45 (2012) 232–238
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Fig. 5. Total deformation at first critical mode versus perturbation force.
3. Perturbation force models Mackenzie et al. [14] used 2 kN perturbation forces which are applied normal to the mid-section of the knuckle region of each quadrant, In order to simulate different perturbation states, four different perturbation force models are used in this paper as shown in Fig. 3. Each perturbation force is applied normal to the knuckle surface. 4. Eigenvalue solution As a first step, conducting a eigen-buckling analysis prior to solving a nonlinear buckling problem to understand the frequency content of the system is useful [18,19]. Eigenvalue analysis provides a ‘classical’ solution to a buckling problem. Although the critical load determined by eigenvalue analysis is unconservative, eigenvalue analysis is essential as the eigenvalue buckling yields an estimate of the critical load to induce buckling. In spite of the fact that the calculated value is higher than the actual critical load, it provides a good starting point to see the possible buckling mode shapes. The solution time for linear buckling analysis is comparatively
Fig. 6. Pressure deformation graph [14].
Fig. 7. Equivalent plastic strain distribution at the onset of buckling for no perturbation model (instability pressure 0.906 MPa-large deformation static analysis).
much less than the solution time for nonlinear post-buckling analysis. First of all, first buckling mode shapes for four perturbation models with 2 kN perturbation forces are obtained using eigenvalue analysis as seen in Fig. 4. For all the eigenvalue solutions the internal pressure is kept constant at 0.1 MPa. Then, several analysis are performed with different perturbation force values in order to achieve the effect of perturbation forces to the total deformation at first buckling mode. The results obtained are presented in Fig. 5. From these results it is important to remark that full model yields the same mode shape for positive and negative perturbation force application. Only critical pressure values change. Models apart from the full model show asymmetrically irregular distributions.
Fig. 8. Equivalent plastic strain distribution at the onset of buckling for no perturbation model (instability pressure 0.881 MPa-transient analysis).
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A.Z. Sßenalp / Advances in Engineering Software 45 (2012) 232–238
5. Nonlinear instability solutions Pressure vessel materials exhibit linear elastic behavior up to yield and thereafter the stress and strain increase in a non-proportional manner (work hardening). In such cases, post-buckling behavior must be studied carefully. Since nonlinear buckling analysis is usually the more accurate and realistic approach, it is highly recommended to use in design or evaluation of actual structures. This technique employs a nonlinear analysis with gradually increasing loads in order to seek the load level at which the structure becomes unstable. In this problem both material and load nonlinearity are considered. Large deformation effect also yields to nonlinear problem. For nonlinear solutions, it is common to apply both quasi-static and transient analysis. ANSYS employs the ‘‘Newton–Raphson’’ approach to solve nonlinear problems. In this approach, the load is subdivided into a series of load increments that can be applied over several load steps.
In this paper both large deformation static and large deformation transient analysis are applied to four perturbation models and no perturbation model. The pressure deformation graph for no perturbation model which is also presented by Mackenzie et al. [14] is given in Fig. 6. In transient analysis internal body loads are updated in each time step and iteration which yields more accurate solution than incremental large displacement static analysis. Solution time for transient analysis is almost 3.75 times the solution time needed for large displacement static analysis. The equivalent plastic strain distributions at the onset of buckling for no perturbation models for large deformation static and transient analyses are shown in Figs. 7 and 8 respectively. The geometries obtained at the onset of buckling for all the presented models are almost the same with the no perturbation model. However maximum equivalent plastic strain value and location of maximum value change from one model to another. The results for full and full with reversed loads are given in Figs. 9 and 10 respectively. Full model results higher instability pressure and equivalent strain values compared to full model with reversed load. The equivalent plastic strain distributions for quarter model
Fig. 9. Equivalent plastic strain distribution at the onset of buckling for full model (instability pressure 0.987 MPa-large deformation static analysis).
Fig. 11. Equivalent plastic strain distribution at the onset of buckling for quarter model 1 (instability pressure 0.968 MPa-transient analysis).
Fig. 10. Equivalent plastic strain distribution at the onset of buckling for full modelload reversed (instability pressure 0.968 MPa-large deformation static analysis).
Fig. 12. Equivalent plastic strain distribution at the onset of buckling for quarter model 2 (instability pressure 1.007 MPa-large deformation static analysis).
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A.Z. Sßenalp / Advances in Engineering Software 45 (2012) 232–238 Table 2 Instability pressures and corresponding equivalent plastic strains. Model
Large deformation static solver
No perturbation Full model Full model (load reversed) Half symmetric model Quarter model 1 Quarter model 2
Large deformation transient solver
Instability pressure (MPa)
Eq. plastic strain
Instability pressure (MPa)
Eq. plastic strain
0.906 0.987 0.968 1.001 1.008 1.007
0.005586 0.009063 0.006068 0.005664 0.007196 0.007070
0.881 0.960 0.946 0.965 0.968 0.967
0.004490 0.005150 0.005583 0.005156 0.005298 0.005109
Table 3 Results of Mackenzie et al. [14]. Model
No perturbation Full model (load reversed)
Instability
PWC
ASME TES
Pressure (MPa)
Eq. plastic strain
Plastic pressure (MPa)
Eq. plastic strain
Plastic pressure (Knuckle) (MPa)
0.91 0.96
0.012547 n/a
0.87 0.84
0.002533 0.002343
0.87 0.84
1 for transient analysis and for quarter model 2 large deformation static analyses are given in Figs. 11 and 12 respectively. The perturbation forces for these analyses are shown on the figures. The point of maximum equivalent strain occurrence can be observed from the figures. The equivalent strain values differ from model to model and these values are given in Table 2 and the results presented by Mackenzie et al. [14] are given in Table 3. Full model (load reversed) case used in this paper corresponds to load perturbation model of Mackenzie et al. [14]. For no perturbation model, instability pressure obtained by large deformation static analysis is 0.906 MPa and the result of large deformation transient analysis is 0.881. These values can be compared with the results of Mackenzie et al. [14] from Table 3. Mackenzie et al. obtained instability pressure as 0.91 MPa and plastic pressures as 0.87 MPa. with principle work criterion (PWC) and ASME TES (knuckle region) criterion. Large deformation transient results obtained in this paper are closer to plastic pressures obtained by PWC and ASME TES. However, when equivalent plastic strains are in concern it can be stated that the results achieved in this work are not in good accordance with the results of Mackenzie et al. [14].
6. Discussion and conclusions The effects of four different perturbation load models to buckling are investigated through this study. As a first step, eigenvalue analyses are conducted. The results of eigenvalue study yield the buckling shapes given in Fig. 4. By the altering the perturbation loads from 0 to 2 kN the magnitude effect of these forces to the total deformation at first buckling mode are obtained as seen in Fig. 5. It is important to note that these deformations given in Fig. 5 are not real deformations. They are used just for comparison. As 2 kN is relatively small load value compared with the geometry, the variations in the obtained results are also small in magnitude. However obtained distribution is important to observe the effect of magnitude of perturbation force and perturbation mode. For full model, a linear variation is obtained. The rest of the models show nonlinear deformation distribution for perturbation forces ranging from 0 to 2 kN. Higher force magnitudes are not applied not to violate perturbation logic. As a second step, nonlinear instability analyses are conducted for each model. Both large deformation static and large deformation transient analyses are solved. When the results are compared with the work of Mackenzie et al. [14], it can be observed that instability pressures obtained by transient analyses are closer to
plastic pressures obtained with plastic work criterion (PWC) and to ASME TES (knuckle region) criterion (Tables 2 and 3). It is also obtained that all the perturbation models result instability pressures higher than the model with no perturbation. Among all the perturbation models, full model with load reversed has the lowest and quarter model 1 has the highest instability pressure values. The equivalent plastic strain values obtained at the onset of buckling increase with the perturbation application and transient solution results have lower equivalent plastic strain values than large deformation static results. It is also clear from the results that full model has the highest strain value. The effect of perturbation force direction can be observed by comparing the results of full model and full model with load reversed. When the eigenvalue results are compared with nonlinear solutions it can be stated that nonlinear solutions do not yield the same deformation modes with eigenvalue solutions. However in nonlinear solutions, the points where the maximums occur show a good match with the perturbation mode applied as can be observed in Figs. 11 and 12. It can also be stated that transient analysis yields more accurate solution than incremental large displacement static analysis as internal body loads are updated in each time step and iteration. References [1] Mackerle J. Finite elements in the analysis of pressure vessels and piping, an addendum (1996–1998). Int J Press Ves Piping 1999;76:461–85. [2] Mackerle J. Finite elements in the analysis of pressure vessels and piping, an addendum: a bibliography (1988–2001). Int J Press Ves Piping 2002;79:1–26. [3] Khan R, Akanda S, Uddin W. A new approach to instability testing of shells. Int J Press Ves Piping 1998;75:75–80. [4] Miller CD. Buckling criteria for torispherical heads under internal pressure. J Press Ves Technol 2001;123(3):318–23. [5] Mackenzie D, Boyle JT, Boyle JT, Hamilton R. The elastic compensation method for limit and shakedown analysis: a review. J Strain Anal 2000;35(3):171–87. [6] Yang P, Liu Y, Ohtake Y, Yuan H, Cen Z. Limit analysis based on a modified elastic compensation method for nozzle-to-cylinder junctions. Int J Press Ves Piping 2005;82:770–6. [7] Athiannan K, Palaninathan R. Buckling of cylindrical shells under transverse shear. Thin Wall Struct 2004;42:1307–28. [8] Li H, Mackenzie D. A characterising gross plastic deformation in design by analysis. Int J Press Ves Piping 2005;82:777–86. [9] Muscat M, Mackenzie D, Hamilton R. A work criterion for plastic collapse. Int J Press Ves Piping 2003;80:49–58. [10] Blachut J. Buckling of multilayered metal domes. Thin Wall Struct 2009;47:1429–38. [11] BSI. PD5500 Unfired fusion welded pressure vessels. London: British Standards Institution; 1999. [12] ASME. ASME Boiler and pressure vessel code sections III and VIII. New York, NY: The American Society of Mechanical Engineers; 2003. [13] CES. EN13445-3:2002 Unfired pressure vessels. Brussels: European Committee for Standardisation; April 2002.
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[14] Mackenzie D, Camilleri D, Hamilton R. Design by analysis of ductile failure and buckling in torispherical pressure vessel heads. Thin Wall Struct 2008;46:963–74. [15] Miller CD, Grove RB, Bennett JG. Pressure testing of large-scale torispherical heads subject to knuckle buckling. Int J Pres Ves Piping 1986;22:147–59. [16] Galletly GD, Blachut J. Torispherical shells under internal pressure—failure due to asymmetric plastic buckling or axisymmetric yielding. Proc Inst Mech Eng 1985;199:225–38.
[17] ANSYS Workbench version 12; 2009. [18] Cisloiu R, Wang J. Understanding nonlinear stabilization features. ANSYS Solutions 2006;7(5):32–5. [19] Imaoka S. Analyzing buckling in Ansys workbench simulation. ANSYS Advantage 2008;2(1):41–3.
Contents lists available at SciVerse ScienceDirect
Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft
Investigation of the effects of perturbation forces to buckling in internally pressurized torispherical pressure vessel heads Ahmet Zafer Sß enalp ⇑ Department of Mechanical Engineering, Gebze Institute of Technology, 41400 Gebze, Kocaeli, Turkey
a r t i c l e
i n f o
Article history: Received 27 April 2010 Received in revised form 18 May 2011 Accepted 26 September 2011 Available online 21 October 2011 Keywords: Perturbation Torispherical pressure vessel heads Buckling Instability Eigenvalue Internal pressure
a b s t r a c t The object of this paper is to investigate the effects of perturbation forces to buckling in pressure vessel heads. The pressure vessel heads in concern are confined to torispherical geometry with thin walls. Perturbation forces can alter not only the critical load for buckling but the buckled shape as well. In this paper in addition to previously used perturbation model, three more different perturbation force configurations are applied to the knuckle of the vessel. Internally pressurized three-dimensional torispherical pressure vessel head model that is previously used in literature is constructed and finite element program ANSYS Workbench is used for the solutions. First of all eigenvalue solutions are performed for each model. Then nonlinear instability solutions are conducted to obtain more realistic instability pressure values. For the nonlinear analyses not only large deformation static analyses but also large deformation transient analyses are conducted. For nonlinear analyses, perfectly plastic material model is used. It is concluded that instability pressures obtained by transient analyses are closer to plastic pressure values by PWC and ASME TES criteria and perturbation models increase instability pressure and equivalent plastic strain values. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Thin-walled torispherical pressure vessel heads are widely used in industry. Buckling is one of the major problems that a designer has to deal with during the design process of these thin-walled structures in which structural members collapse under compressive loads greater than the material can withstand. Due to the existence of unstable post-buckling behavior, torispherical pressure vessel heads are sensitive to small geometric or load imperfections. Finite element analysis is widely used in the design of these structures [1,2]. Extensive studies are presented on the buckling of pressure vessels. Khan et al. presented an experimental technique for the buckling test of shells under external pressure to determine buckling load [3]. Miller worked on buckling criteria for torispherical heads under internal pressure which are especially outside the limits of ASME codes [4]. Some of the previous works deal with elastic analysis of pressure vessel components. According to limit analysis theorems, the elastic compensation method (ECM) can obtain both upper bound and lower bound limit loads. Though the upper bound limit load given by the ECM is more accurate than the lower bound limit
⇑ Tel.: +90 262 605 18 06; fax: +90 262 653 06 75. E-mail address: [email protected] 0965-9978/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2011.09.027
load [5], the lower bound limit load is safer. Yang et al., proposed modified elastic compensation method (MECM) to improve the precision of the elastic compensation method (ECM) [6]. MECM can provide a good estimation of plastic limit loads for complex structures. Athiannan and Palaninathan’s study concerns experimental studies on buckling of thin-walled circular cylindrical shells under transverse shear. The buckling loads are also obtained from finite element models and empirical formulas and codes are compared [7]. Li et al. made an investigation of structures to identify and characterize the condition of gross plastic deformation in pressure vessel design by analysis. Limit analysis and bilinear hardening plastic analysis is performed. A criterion of plastic collapse based on the curvature of the load–plastic work history is proposed [8]. Muscat et al. proposed a criterion for evaluating the critical limit values and determining the plastic loads in pressure vessel design. The proposed criterion is based on the plastic work dissipated in the structure as loading progresses and can be used for structures subject to a single load or a combination of multiple loads. The limit and plastic loads are determined purely by the inelastic response of the structure and are not influenced by the initial elastic response [9]. Blachut’s study provides results of a numerical and experimental investigation into static stability of externally pressurized layered hemispherical and torispherical domes. Buckling/collapse tests are also conducted on domes from various materials [10].
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A.Z. Sßenalp / Advances in Engineering Software 45 (2012) 232–238
In pressure vessel design, it is required to satisfy certain criteria related to failure modes. In general, the fundamental failure mechanism related to static loading is denoted as gross plastic deformation (GPD). PD5500 Unfired fusion welded pressure vessels [11], ASME Boiler and pressure vessel code Sections III and VIII [12] and EN13445-3:2002 Unfired pressure vessels [13] codes specify two different approaches to the designer. The most commonly used method is based on linear elastic stress analysis of the vessel. GPD failure is related to the primary stress category, which is yieldlimited to preclude failure due to this mechanism. The second method requires an inelastic analysis concerning post yield behavior to simulate the GPD mechanism. The GPD load is calculated directly from the inelastic analysis. In EN13445 regulations, this method is called as ‘‘the direct route’’. Mackenzie et al. made an extensive review on the descriptions of the code contents [14]. The ASME Twice Elastic Slope (TES), criterion uses an empirical procedure for calculating collapse loads in experimental stress analysis of pressure vessels [12]. In Mackenzie et al.’ study, plastic collapse or gross plastic deformation loads are evaluated for two sample torispherical heads by 2D and 3D FEA based on an elastic-perfectly plastic material model [14]. They considered small and large deformation effects and the geometry and load perturbations. Their study contains the formation of the gross plastic deformation mechanism in the models in relation to the elastic–plastic buckling response of the vessels. In their study both ASME TES and plastic work criteria (PWC) are considered. The PWC criterion requires a plot of load against normalized load–plastic work curvature. The object of this paper is to investigate the effects of perturbation forces to buckling in pressure vessel heads. Internally pressurized three-dimensional torispherical pressure vessel head model that is previously used in literature is constructed and finite element program ANSYS Workbench is used for the solutions. Four different perturbation models are set up to investigate the effects of perturbation forces to buckling. As a first step, linear buckling analyses are conducted prior to solving the nonlinear buckling shapes to understand the effect of perturbation models to the deformation modes of the geometry in concern. Consequently, nonlinear instability analyses are performed for each perturbation model. Two types of nonlinear analyses are conducted. These are large deformation static analysis and transient analyses. Elastic perfectly plastic material model is used for all nonlinear analyses. The pressure vessel head model is assumed to have no initial shape imperfections. 2. Finite element model
Fig. 1. Torisperical head geometry.
Fig. 2. Finite element model.
Table 1 Material properties of torispherical pressure vessel head. Young’s modulus (GPa)
Yield strength (MPa)
200
353
2.1. Geometry For the analyses of a thin wall torispherical head, the same geometry investigated previously by Miller et al. [15] and Galletly and Blachut [16] and Mackenzie et al. [14] is considered. The geometry of torispherical head is given in Fig. 1.
Each complete model is 3D and consists of 8672 elements as shown in Fig. 2.
2.2. Finite element mesh
Material properties used in static analyses are given in Table 1. For large deformation static and transient analyses elastic perfectly plastic material model is used.
Ansys Workbench version 12 is used for the finite element analyses [17]. In the finite element mesh 4-noded SOLSH190 solid shell element with shell option is used for simulating torispherical head. SOLSH190 element can be used for a wide range of thickness from thin to moderately thick geometry (supports Mindlin-Reissner shell theory). The element has plasticity, hyperelasticity, stress stiffening, creep, large deflection, and large strain capabilities. The element formulation is based on logarithmic strain and true stress measures.
2.3. Material properties
2.4. The loading and boundary conditions The bottom free face of torispherical head is subjected to frictionless support boundary condition. As loading condition different values of pressures are applied to crown, knuckle and cylinder regions. The graph of these pressures is presented in Fig. 6.
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(a) Full model
(c) Quarter model 1
(b) Half symmetric model
(d) Quarter model 2 (different force directions)
Fig. 3. Perturbation force models.
(a) Full model
(b) Half symmetric model
(c) Quarter model 1
(d) Quarter model 2
Fig. 4. First buckling mode shapes for four models (perturbation force = 2000 N).
A.Z. Sßenalp / Advances in Engineering Software 45 (2012) 232–238
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Fig. 5. Total deformation at first critical mode versus perturbation force.
3. Perturbation force models Mackenzie et al. [14] used 2 kN perturbation forces which are applied normal to the mid-section of the knuckle region of each quadrant, In order to simulate different perturbation states, four different perturbation force models are used in this paper as shown in Fig. 3. Each perturbation force is applied normal to the knuckle surface. 4. Eigenvalue solution As a first step, conducting a eigen-buckling analysis prior to solving a nonlinear buckling problem to understand the frequency content of the system is useful [18,19]. Eigenvalue analysis provides a ‘classical’ solution to a buckling problem. Although the critical load determined by eigenvalue analysis is unconservative, eigenvalue analysis is essential as the eigenvalue buckling yields an estimate of the critical load to induce buckling. In spite of the fact that the calculated value is higher than the actual critical load, it provides a good starting point to see the possible buckling mode shapes. The solution time for linear buckling analysis is comparatively
Fig. 6. Pressure deformation graph [14].
Fig. 7. Equivalent plastic strain distribution at the onset of buckling for no perturbation model (instability pressure 0.906 MPa-large deformation static analysis).
much less than the solution time for nonlinear post-buckling analysis. First of all, first buckling mode shapes for four perturbation models with 2 kN perturbation forces are obtained using eigenvalue analysis as seen in Fig. 4. For all the eigenvalue solutions the internal pressure is kept constant at 0.1 MPa. Then, several analysis are performed with different perturbation force values in order to achieve the effect of perturbation forces to the total deformation at first buckling mode. The results obtained are presented in Fig. 5. From these results it is important to remark that full model yields the same mode shape for positive and negative perturbation force application. Only critical pressure values change. Models apart from the full model show asymmetrically irregular distributions.
Fig. 8. Equivalent plastic strain distribution at the onset of buckling for no perturbation model (instability pressure 0.881 MPa-transient analysis).
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A.Z. Sßenalp / Advances in Engineering Software 45 (2012) 232–238
5. Nonlinear instability solutions Pressure vessel materials exhibit linear elastic behavior up to yield and thereafter the stress and strain increase in a non-proportional manner (work hardening). In such cases, post-buckling behavior must be studied carefully. Since nonlinear buckling analysis is usually the more accurate and realistic approach, it is highly recommended to use in design or evaluation of actual structures. This technique employs a nonlinear analysis with gradually increasing loads in order to seek the load level at which the structure becomes unstable. In this problem both material and load nonlinearity are considered. Large deformation effect also yields to nonlinear problem. For nonlinear solutions, it is common to apply both quasi-static and transient analysis. ANSYS employs the ‘‘Newton–Raphson’’ approach to solve nonlinear problems. In this approach, the load is subdivided into a series of load increments that can be applied over several load steps.
In this paper both large deformation static and large deformation transient analysis are applied to four perturbation models and no perturbation model. The pressure deformation graph for no perturbation model which is also presented by Mackenzie et al. [14] is given in Fig. 6. In transient analysis internal body loads are updated in each time step and iteration which yields more accurate solution than incremental large displacement static analysis. Solution time for transient analysis is almost 3.75 times the solution time needed for large displacement static analysis. The equivalent plastic strain distributions at the onset of buckling for no perturbation models for large deformation static and transient analyses are shown in Figs. 7 and 8 respectively. The geometries obtained at the onset of buckling for all the presented models are almost the same with the no perturbation model. However maximum equivalent plastic strain value and location of maximum value change from one model to another. The results for full and full with reversed loads are given in Figs. 9 and 10 respectively. Full model results higher instability pressure and equivalent strain values compared to full model with reversed load. The equivalent plastic strain distributions for quarter model
Fig. 9. Equivalent plastic strain distribution at the onset of buckling for full model (instability pressure 0.987 MPa-large deformation static analysis).
Fig. 11. Equivalent plastic strain distribution at the onset of buckling for quarter model 1 (instability pressure 0.968 MPa-transient analysis).
Fig. 10. Equivalent plastic strain distribution at the onset of buckling for full modelload reversed (instability pressure 0.968 MPa-large deformation static analysis).
Fig. 12. Equivalent plastic strain distribution at the onset of buckling for quarter model 2 (instability pressure 1.007 MPa-large deformation static analysis).
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A.Z. Sßenalp / Advances in Engineering Software 45 (2012) 232–238 Table 2 Instability pressures and corresponding equivalent plastic strains. Model
Large deformation static solver
No perturbation Full model Full model (load reversed) Half symmetric model Quarter model 1 Quarter model 2
Large deformation transient solver
Instability pressure (MPa)
Eq. plastic strain
Instability pressure (MPa)
Eq. plastic strain
0.906 0.987 0.968 1.001 1.008 1.007
0.005586 0.009063 0.006068 0.005664 0.007196 0.007070
0.881 0.960 0.946 0.965 0.968 0.967
0.004490 0.005150 0.005583 0.005156 0.005298 0.005109
Table 3 Results of Mackenzie et al. [14]. Model
No perturbation Full model (load reversed)
Instability
PWC
ASME TES
Pressure (MPa)
Eq. plastic strain
Plastic pressure (MPa)
Eq. plastic strain
Plastic pressure (Knuckle) (MPa)
0.91 0.96
0.012547 n/a
0.87 0.84
0.002533 0.002343
0.87 0.84
1 for transient analysis and for quarter model 2 large deformation static analyses are given in Figs. 11 and 12 respectively. The perturbation forces for these analyses are shown on the figures. The point of maximum equivalent strain occurrence can be observed from the figures. The equivalent strain values differ from model to model and these values are given in Table 2 and the results presented by Mackenzie et al. [14] are given in Table 3. Full model (load reversed) case used in this paper corresponds to load perturbation model of Mackenzie et al. [14]. For no perturbation model, instability pressure obtained by large deformation static analysis is 0.906 MPa and the result of large deformation transient analysis is 0.881. These values can be compared with the results of Mackenzie et al. [14] from Table 3. Mackenzie et al. obtained instability pressure as 0.91 MPa and plastic pressures as 0.87 MPa. with principle work criterion (PWC) and ASME TES (knuckle region) criterion. Large deformation transient results obtained in this paper are closer to plastic pressures obtained by PWC and ASME TES. However, when equivalent plastic strains are in concern it can be stated that the results achieved in this work are not in good accordance with the results of Mackenzie et al. [14].
6. Discussion and conclusions The effects of four different perturbation load models to buckling are investigated through this study. As a first step, eigenvalue analyses are conducted. The results of eigenvalue study yield the buckling shapes given in Fig. 4. By the altering the perturbation loads from 0 to 2 kN the magnitude effect of these forces to the total deformation at first buckling mode are obtained as seen in Fig. 5. It is important to note that these deformations given in Fig. 5 are not real deformations. They are used just for comparison. As 2 kN is relatively small load value compared with the geometry, the variations in the obtained results are also small in magnitude. However obtained distribution is important to observe the effect of magnitude of perturbation force and perturbation mode. For full model, a linear variation is obtained. The rest of the models show nonlinear deformation distribution for perturbation forces ranging from 0 to 2 kN. Higher force magnitudes are not applied not to violate perturbation logic. As a second step, nonlinear instability analyses are conducted for each model. Both large deformation static and large deformation transient analyses are solved. When the results are compared with the work of Mackenzie et al. [14], it can be observed that instability pressures obtained by transient analyses are closer to
plastic pressures obtained with plastic work criterion (PWC) and to ASME TES (knuckle region) criterion (Tables 2 and 3). It is also obtained that all the perturbation models result instability pressures higher than the model with no perturbation. Among all the perturbation models, full model with load reversed has the lowest and quarter model 1 has the highest instability pressure values. The equivalent plastic strain values obtained at the onset of buckling increase with the perturbation application and transient solution results have lower equivalent plastic strain values than large deformation static results. It is also clear from the results that full model has the highest strain value. The effect of perturbation force direction can be observed by comparing the results of full model and full model with load reversed. When the eigenvalue results are compared with nonlinear solutions it can be stated that nonlinear solutions do not yield the same deformation modes with eigenvalue solutions. However in nonlinear solutions, the points where the maximums occur show a good match with the perturbation mode applied as can be observed in Figs. 11 and 12. It can also be stated that transient analysis yields more accurate solution than incremental large displacement static analysis as internal body loads are updated in each time step and iteration. References [1] Mackerle J. Finite elements in the analysis of pressure vessels and piping, an addendum (1996–1998). Int J Press Ves Piping 1999;76:461–85. [2] Mackerle J. Finite elements in the analysis of pressure vessels and piping, an addendum: a bibliography (1988–2001). Int J Press Ves Piping 2002;79:1–26. [3] Khan R, Akanda S, Uddin W. A new approach to instability testing of shells. Int J Press Ves Piping 1998;75:75–80. [4] Miller CD. Buckling criteria for torispherical heads under internal pressure. J Press Ves Technol 2001;123(3):318–23. [5] Mackenzie D, Boyle JT, Boyle JT, Hamilton R. The elastic compensation method for limit and shakedown analysis: a review. J Strain Anal 2000;35(3):171–87. [6] Yang P, Liu Y, Ohtake Y, Yuan H, Cen Z. Limit analysis based on a modified elastic compensation method for nozzle-to-cylinder junctions. Int J Press Ves Piping 2005;82:770–6. [7] Athiannan K, Palaninathan R. Buckling of cylindrical shells under transverse shear. Thin Wall Struct 2004;42:1307–28. [8] Li H, Mackenzie D. A characterising gross plastic deformation in design by analysis. Int J Press Ves Piping 2005;82:777–86. [9] Muscat M, Mackenzie D, Hamilton R. A work criterion for plastic collapse. Int J Press Ves Piping 2003;80:49–58. [10] Blachut J. Buckling of multilayered metal domes. Thin Wall Struct 2009;47:1429–38. [11] BSI. PD5500 Unfired fusion welded pressure vessels. London: British Standards Institution; 1999. [12] ASME. ASME Boiler and pressure vessel code sections III and VIII. New York, NY: The American Society of Mechanical Engineers; 2003. [13] CES. EN13445-3:2002 Unfired pressure vessels. Brussels: European Committee for Standardisation; April 2002.
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