150 CHAPTER 9 INFLUENCE OF RESIDUAL STRESSES ON THE FAILURE PRESSURE OF CYLINDRICAL VESSELS In the design of pressure vessels, evaluation of failure pressure that a cylindrical pressure vessel can withstand is an important consideration. While prediction of failure pressure of pressure vessels it is also necessary to consider the residual stresses already present in the pressure vessels. The knowledge of the residual stresses present in the pressure vessel due to welding is necessary. Of several formulae for calculating bursting pressure of vessels, the Faupel s formula is the most popular one. However there is no unique failure theory applicable to all materials. 9.1 FAILURE PRESSURE ESTIMATES OF UNFLAWED PRESSURE VESSELS WITHOUT RESIDUAL STRESS Various methods are being used to estimate the failure pressure (Faupel 1956, Svensson 1958, Beena et al 1995 and Christopher et al 2002). Finite element techniques based on the Global Plastic Deformation (GPD) are also used to evaluate the failure pressure and its results were found to be in good agreement with test results (Aseer et al 2009). Experimental methods are useful for verifying the correctness of analytical or computational analysis. Usually stress cannot be measured directly and hence most of the experimental methods serve to measure strains by bonding the gauges to the surface of the structure under test. If nothing is known in advance with the
151 strain field, a three element rosette is the best choice for finding the elements of the small strain tensor. Most of the existing solutions to elasto-plastic problems are based on Tresca s yield criterion and in associated flow rule leading to analytical or quasi analytical solution. Under the application of the internal pressure, the material deforms both elastically and plastically. Inelasticity is always present under all types of loading. Hence elasto-plastic deformations cannot be treated independently. Therefore elasto-plastic analysis has to be performed while analyzing a pressure vessel. Failure pressure estimates from FEA based on GPD are found to be in good agreement with test results of thin as well as thick walled vessels made of ductile materials (Aseer et al 2009). The pressure corresponding to GPD will be the failure pressure. Applicability of the present FEA procedure for failure prediction is examined with the existing test results (Faupel et al 1953). The geometrical model and FEA model of the ASTM 36 carbon steel pressure vessel similar to Figures 6.1 and 6.2 is considered for analysis. Axial displacement is suppressed at both ends of the cylindrical shell to arrest the axial growth under internal pressure. = 1+ (9.1) Equation (9.1) is Inverse Romberg-Osgood relationship (Beena et al 1995), is a constitutive relationship and gives the stress as an explicit function of strain. Where =, is the ultimate strength of the material and is the parameter defining the shape of the non-linear stress-strain relationship. = 0.002143, = 1.798 for ASTM 36 carbon steel. The stress-strain curve (see Figure 9.1) generated using Equation (9.1) is given as input for FEA to take care of material behavior during the application of internal pressure. ANSYS has the provision for checking the (GPD).
152 It indicates the pressure level to cause complete plastic flow through the cylinder walls (ie. bursting pressure). Bursting pressure of pressure vessel (Faupel et al 1953), = ln (9.2) Where = Ultimate strength of the material = = Yield strength of the material Wall ratio of hollow cylinder (Ro/Ri) For ASTM 36 steel, = 450 MPa and = 380 MPa 9.1.1 Results and Discussion Figure 9.2 shows the effective stress at failure of the pressure vessel. The analysis results indicate the failure pressure as 48 MPa while it is 46 MPa by using Equation (9.2), having good agreement. Table 9.1 gives the cylinder dimensions, material properties and Table 9.2 gives the material constants in Equation (9.1) for the stress-strain curve and failure pressure of cylindrical vessels analyzed in Faupel (1953). Figure 9.3 gives the stressstrain curve of Cr-Ni-Mo-V steel generated by Equation (9.1). Figure 9.4 gives effective, hoop and meridional stress plot of Cr-Ni-Mo-V steel pressure vessel No.2 in Table 9.1 with the applied internal pressure up to global plastic deformation. The failure pressure evaluated through the present analysis (FEA) is 1358 MPa, where the experimental and calculated values are 1379 MPa and 1325.2 MPa respectively (Faupel et al 1953), and found in good agreement.
153 Figure 9.1 Stress-strain curve of ASTM 36 carbon steel generated by Equation (9.1) 500 Effective stress, MPa 400 300 200 100 0 At Inner wall At Middle At outer wall 0 10 20 30 40 50 Applied pressure (MPa) Figure 9.2 Effective stress plot of ASTM 36 carbon steel pressure vessel up to global plastic deformation
154 1400 1200 Stress - MPa 1000 800 600 400 Cr-Ni-Mo-V steel 200 0 0 5 10 15 20 25 30 Strain x 10 3 Figure 9.3 Stress-strain curve of Cr-Ni-Mo-V steel generated by Equation (9.1) Figure 9.4 Effective, hoop and meridional stress plot of Cr-Ni-Mo-V steel pressure vessel (Faupel et al 1953) (Cylinder No.2) with the applied internal pressure up to global plastic deformation
155 Table 9.1 Details of cylindrical pressure vessels, materials and Cylinder No strength properties (Faupel et al 1953) Cylindrical shell dimensions (mm) Outer diameter Thickness Material Tensile strength properties (MPa) Yield strength () Ultimate strength ( ) 1 68.00 21.00 Cr-Ni-Mo-V 329.6 641.9 steel 2 68.00 21.00 Cr-Ni-Mo-V 1101.8 1223.8 steel 3 101.60 31.68 SAE 3320 548.1 726.7 4 31.75 9.5 SAE 4340 716.4 855.6 5 31.75 9.5 SAE 4340 596.4 795.0 6 31.75 9.5 SAE 4340 797.0 859.0 7 101.60 31.68 SAE 1045 419.9 701.9 8 101.60 31.68 SAE 1045 562.6 842.5 Table 9.2 Material constants in Equation (9.1) for the stress-strain curve and failure pressure of cylindrical vessels in Table 9.1 Material constants in Eqn (9.1) Failure pressure, (MPa) Cylinder No Young s modulus, E (GPa) Observed Calculated (Faupel et al 1953) FEA 1 207 0.00310 0.945 689.5 561.9 711.0 2 207 0.00592 3.600 1379.0 1325.2 1358.0 3 207 0.00351 1.710 730.8 770.8 820.0 4 207 0.00414 2.340 917.0 954.9 901.0 5 207 0.00384 1.754 827.4 856.3 836.0 6 207 0.00415 3.512 806.7 892.2 904.0 7 207 0.00339 1.164 737.7 662.6 791.0 8 207 0.00407 1.464 820.5 846.0 950.0
156 9.2 INFLUENCE OF RESIDUAL STRESSES ON FAILURE PRESSURE OF UNFLAWED PRESSURE VESSELS The ASTM 36 carbon steel pressure vessel as in Figure 6.1 is considered for failure pressure analysis of butt-welded cylindrical pressure vessels having residual stresses. Axial displacement is suppressed at both ends of the cylindrical shell to arrest the axial growth under internal pressure. Failure pressure of the pressure vessel before welding (ie., without residual stress) is initially obtained by an elasto-plastic analysis. Then a thermomechanical FEA with the similar procedure and same welding parameters in the earlier analysis is carried out to assess the weld-induced residual stress and again failure pressure is obtained by another elasto-plastic analysis for this pressure vessel having residual stress to assess the effect of residual stresses in failure pressure. For failure pressure analysis for this pressure vessel along with the residual stresses present, the analysis is restarted from the terminating (final) load step of thermal stress analysis along with applied internal pressure and performed up to GPD. The pressure which is corresponding to GPD will be the failure pressure. 9.2.1 Results and Discussion 2D Finite element analysis with axisymmetric model has been carried out using ANSYS software package to access the failure pressure of cylindrical pressure vessel made of ASTM 36 carbon steel having weldinduced residual stresses. An elasto-plastic analysis is performed to find out the failure pressure of the pressure vessel not having residual stresses. The analysis results indicate the failure pressure as 48 MPa while it is 46 MPa by using Equation (9.2), having good agreement. Figure 9.5 shows the effective stress up to failure with and without residual stress. The failure analysis of the pressure vessel having residual stresses shows that the pressure vessel fails at
157 a pressure of 15 MPa, while the failure pressure without residual stresses is 48 MPa. Faupel et al (1953) have analyzed eight number of pressure vessels shown in Tables 9.1 and 9.2, in which three cylinders of each material were heat treated under identical conditions. One cylinder was then used for the residual stress test, the second for the static internal pressure test, and the third one was used to determine the mechanical properties resulting from the heat treatment. They have concluded that the residual stresses in pressure vessels due to heat treatment do not appear to influence overstrain or bursting pressure. It should be noted that the material properties were measured after the heat treatment process. Now while calculating the failure pressure analytically, in which if these material properties were used then the effect of residual stresses on failure pressure will not be felt. If the properties of the material before the heat treatment were used, then the effect would have been felt. 500 Effective stress, (MPa) 400 300 200 100 0 Inner without RS Middle without RS Outer without RS Inner with RS Outer with RS GPD 0 10 20 30 40 50 Applied pressure (MPa) Figure 9.5 Effective stress plot of ASTM 36 carbon steel pressure vessel with and without residual stress up to global plastic deformation
158 9.3 INFLUENCE OF RESIDUAL STRESSES ON FAILURE PRESSURE OF FLAWED PRESSURE VESSELS Christopher et al (2002) applied the three parameter fracture criterion to correlate the fracture data on aluminium, titanium and steel materials from test results on flawed cylindrical tanks/pressure vessels without considering residual stresses. In this work to quantify the influence of residual stresses on failure pressure of flawed pressure vessels, the point stress criterion with two parameter fracture criterion is used in AA2014-T6. Figure 9.6 shows the cracked configuration of specimen and cylindrical pressure vessel. First a failure assessment diagram is drawn for tensile specimens having through thickness cracks with the data given in Table 9.3 used in Christopher et al (2002). Figure 9.7 shows the failure assessment diagram for AA2014-T6 specimen. For cylindrical pressure vessels with axial surface crack (see Figure 9.6) the value of is calculated using Equations (9.3 to 9.12) from the stress intensity factor expressions (Newman 1985). = ( ) / (9.3) = (9.4) = (9.5) (9.6) = 1+ 1.464. for a c, (9.7) = 1 + 1.464. for a > c, = + ( - ) ( ), (9.8)
159 =2+8( ) (9.9) = (1 + 0.52 + 1.29-0.074 ) for 0 10 (9.10) =, q = 2+ 8 (9.11) = ln 1+ (9.12) where = crack depth, c = half the crack length,, : outer and inner diameter of cylindrical vessel, = stress intensity factor at failure, = failure pressure of unflawed cylindrical vessel, = failure pressure of flawed pressure vessel, = internal pressure, = inner radius of cylinder, = thickness of cylinder, = failure stress, = nominal stress required to produce a fully plastic region on the net section, = ultimate tensile strength and =yield strength or 0.2 % proof stress. Table 9.4 shows the failure pressure of pressure vessels with different crack sizes without considering residual stresses. Figure 9.8 shows the failure assessment diagram for AA2014-T6 center crack tension specimen in which design points of specimen and cylindrical pressure vessel were plotted. Figure 9.9 shows the FAD for AA2014-T6 center crack tensile specimen and design points for four cylindrical pressure vessels of dimensions and crack sizes given in Table 9.4. The residual stress values are calculated through SINTAP procedure used in Chapter-8 are: = =3, =12, = =560 MPa, =680 MPa,
160 Welding speed = 6 mm/sec and plate thickness =1.52 mm R = 71.1 mm, a = t = 1.52 mm, = 554 Mpa, = 687 MPa 9.3.1 Results and Discussion In SINTAP procedure the welding residual stresses values are given as trapezoidal profile, in which the residual stress values are available up to yield zone only ( ). Beyond yield zone the residual stress values are taken as zero. Hence from Table 9.4 the crack sizes less than yield zone is considered for analysis. Figure 9.9 also shows that three of the four pressure vessels are critical with the available residual stresses. Therefore it is necessary for the pressure vessel to undergo stress relieving process. Figure 9.10 shows the failure assessment diagram for AA2014-T6 cylindrical pressure vessel with 70% residual stress relieved. Figure 9.10 and Table 9.5 show that the point D is unsafe with the existing residual stress and an applied internal pressure of 3.25 MPa. When the applied internal pressure is increased to 3.75 MPa the point C also becomes unsafe. When the applied internal pressure is increased to 5 MPa the point B also becomes unsafe. When the applied internal pressure is increased to 7.5 MPa the point A also becomes unsafe. This analysis brings out the reduced internal pressure withstanding capacity in the presence of residual stress as indicated in Table 9.6.
161 Table 9.3 Comparison of analytical and test results of center crack tension specimens made of AA2014-T6 ( = MPa, = MPa, K F = 44.38 MPa m, m = 0.3022) Specimen dimensions (mm) Crack Width, W Thickness, t length, 2c Fracture strength, (MPa) Test (MCIC 1975) Analysis Relative error (%) 76.2 1.56 7.04 408.9 410.97-0.51 76.2 1.58 7.06 405.4 410.62-1.29 76.2 1.57 14.15 320.6 320.07 0.16 76.2 1.52 19.68 288.9 277.28 4.02 76.2 1.56 21.49 272.4 265.75 2.44 76.2 1.55 26.21 235.8 239.44-1.54 76.2 1.53 30.15 222.0 220.31 0.76 76.2 1.53 30.58 217.2 218.37-0.54 76.2 1.54 35.53 195.8 196.94-0.58 76.2 1.55 36.37 189.6 193.51-2.06 Standard error (SE) = 0.018
162 Table 9.4 Comparison of analytical and test results of cylindrical pressure vessels made of AA2014-T6 ( = MPa, =, =., =. ) Points in FAD Crack length, 2c(mm) Failure pressure, (MPa)-Test (MCIC 1975), ( ) A 2.64 12.15 38.14 0.83 B 6.35 9.37 49.45 0.64 C 12.7 5.85 51.58 0.40 D 19.05 4.73 60.53 0.32 E 25.4 3.1 53.59 0.21 F 31.75 2.93 65.19 0.20 G 44.45 1.95 65.04 0.13 H 50.8 1.76 69.33 0.12 Table 9.5 Fracture parameters of AA2014-T6 pressure vessels with 70% residual stress relieved Points in FAD Residual stress 100% 30% For 30% RS only 30% RS+ 3.25 MPa 30% RS+ 3.75 MPa 30% RS+ 5 MPa 30% RS+ 7.5 MPa A 560 168 0.25 11.3 0.45 20.7 0.48 22.3 0.55 25.4 0.72 33.3 B 540 162 0.24 18.3 0.44 34.1 0.47 36.8 0.54 42.0 0.71 55.2 C 340 102 0.15 19.2 0.35 45.7 0.39 50.1 0.45 58.9 0.63 80.9 D 160 48 0.07 13.1 0.27 51.5 0.31 57.9 0.38 70.7 0.55 102.7
163 Table 9.6 Failure pressure of AA2014-T6 pressure vessels with and without residual stress Points c (MPa) (MPa) in FAD (mm) without RS with 30% RS A 1.32 12.15 7.50 B 3.18 9.37 5.00 C 6.35 5.85 3.75 D 9.53 4.73 3.25 Figure 9.6 Cracked configurations
164 Figure 9.7 Failure assessment diagram for AA2014-T6 center crack tension specimen Figure 9.8 Failure assessment diagram for AA2014-T6 center crack tension specimen and cylindrical pressure vessel
165 Figure 9.9 Failure assessment diagram for AA2014-T6 showing residual stress levels without any internal pressure Figure 9.10 Failure assessment diagram for AA2014-T6 cylindrical pressure vessel with 70% residual stress relieved