Asme_stress Classification Lines Straight Through Singularities

Proceeding s o f PVP2008 PVP2008 2008 2008 ASME ASME Pressu Pressu re Vessels Vessels and Piping Divisio n Conferenc e Jul y 27-31, 27-31, 2008, 2008, Chicago, Illin ois USA USA

PVP2008-61746 STRESS STRESS CLASSIFICATION CL ASSIFICATION L INES STRAIGHT THROUGH THROUGH SINGULARITIE SINGULA RITIES S  Ar tu rs Kalni Kal ni ns Professor Emeritus of Mechanics Lehigh University Bethlehem, Pennsylvania E-mail: ak01@Lehig ak01@Lehig h.edu

 AB STRACT The paper considers geometries of pressure vessels and components for which the theoretical models contain sharp corners, representing singularities. The idea is proposed that a stress classification line passed straight through the singularity can yield linearized stresses that are applicable in pressure vessel design. Using elastic finite element analysis, details of the procedures by which this result can be achieved are given for two examples. One is a sharp corner at the toe of a fillet weld. Membrane and bending stresses are calculated directly in the toe plane, showing little or no dependence on mesh size. The other is an axisymmetric shell with a flat head and a sharp corner at the joint. The objective is to determine the primary plus-secondary stress intensity on a Stress Classification Line (SCL) through the joint. Two methods are used. One is by determining the zone of valid SCLs and extrapolating the linearized stresses to the joint. The other is by calculating the linearized stresses directly on the SCL through the joint. Conditions for the use of the SCL through the joint for the shell/flat head model are established. NOMENCLATURE S11, S22, S12 = in-plane stresses in FEA model of 2-D solid elements in X,Y,Z coordinate system, as shown in Figure 4 S33 = out-of-plane stress in FEA model, in Z direction, hoop stress for axisymmetric 2-D elements  NFORC1, NFORC2 = nodal forces of 2-D 2-D solid elements elements σ  

m

, σ  b  = membrane and bending stress, respectively

P+Q = primary plus secondary Tresca stress m, b = suffixes for membrane and bending stresses F = force developed by stresses on SCP M = moment developed by stresses on SCP SCP = Stress Classification Plane SCL = Stress Classification Line, an SCP of infinitesimal width

1 INTRODUCTION For design-by-analysis of pressure vessels, the leap from shell analysis to finite element analysis (FEA) about forty years ago brought much benefit but left some details in a more difficult position. One of such details is a sharp corner *  at a local structural discontinuity. It plays no role in shell analysis  but influences the stresses in FEA. If the corner radius is unspecified at the design stage, it is common practice to model the corner with a zero radius. In that case, these corners represent a singularity. The problem with singularities is that FEA-calculated elastic stresses at the singularity increase without bound as the mesh is refined. These stress values have no physical meaning for any mesh size. The question is whether the linearized stresses acting on planes straight through the singularity are still meaningful for design. This is addressed in the paper.

2 PREVIOUS WORK Previous work on the classification of FEA-calculated stresses goes back to more than three decades. The early papers are by Kroenke1, Kroenke, Addicott, and Hinton 2, Hinton and Hechmer 3, and Gordon4. Hechmer and Hollinger 5 extended the concept to three-dimensional geometries. Hollinger and Hechmer 6 gave a summary of this work. More recent papers are  by Ming-Wan Lu, Yong Chen, Jian-Guo Li7, and by Strzelczyk and Ho8. The focus of these papers is on the membrane and membrane-plus-bending stress intensities that are appropriate for assessing the stress intensity limits involving primary and secondary stresses.

*

  For the purposes of this paper, a “sharp corner” is one for which the angle through the material between two intersecting boundary planes is greater than π. In a continuum analysis, the stresses at such corners are infinite. For angles less than π, the stresses at the corners are zero.

1

Copyright © 2008 by ASME

The question posed in section 1 of this paper is also relevant to fatigue assessment of welded joints that is based on the assumption that the controlling stress for fatigue is a function of the elastic membrane plus bending stress normal to a hypothetical crack plane. This concept has been developed by Dong9 and his colleagues at the Battelle Memorial Institute. It is now included in 2007 Section VIII-Division 2 of the ASME B&PV Code10.

Abaqus). FEA models for two mesh sizes were b uilt, consisting of 1 and 4 elements through thickness, shown in Figure 2. The deformed shape that results from this loading and boundary conditions is shown in Figure 3 for the 4-element model.

336 120 12

3 SCOPE AND OBJECTIVE The paper is concerned with linearized stresses that are derived from the resultant action (forces and moments) on a selected plane of a structural element, commonly called stress classification plane (SCP), or, for 2-D and axisymmetric models, the stress classification line (SCL) †. The objective of the paper is to show that an SCL passed straight through the singularity can yield results that are not influenced by the numerical disturbance of the singularity. Two examples are selected to show how to edit elastic FEA-calculated stresses obtained from standard FEA output (no macros) to obtain the stresses needed in design. The edits are based on the work of Gordon4. In the first example, Gordon’s Force and Moment Edit  is used to determine the membrane and bending stresses of a stress component that is applied to a specified SCP. Two methods are given by Gordon for this edit, the stress method and the force method. Both methods are used. In the second example, Gordon’s Stress Intensity Edit  is used to determine the primary-plus-secondary (P+Q) Tresca stress intensity. Examples of performing these two edits are given next.

P SCL Figure 1: Geometry of the example

Figure 2: Finite element meshes for the example

4 FORCE AND MOMENT EDIT 4.1

MODEL

This is meant to be a simple example to illustrate a case in which an SCL passes straight through a singularity. Gordon’s 4 edits are performed to calculate the membrane and bending stresses for the model shown in Figure 1. The dimensions are in mm. Both the nodal stress and nodal force methods are used. For the stress method, the nodal stresses are integrated over the SCL to obtain the forces and moments, while, for the nodal force method, the internal or reaction forces are summed over the nodes of the SCL The results are also checked with Abaqus/Standard 11  version 6.7-1 CAE postprocessor’s stress linearization option. The middle node of the left end of the model is restrained from vertical and horizontal displacement. The middle node at the right end is restrained from vertical displacement and subjected to a force of P=1200 Newtons (270 lb). Solutions are obtained by the Abaqus11  finite element program using 8noded, quadratic, 2-D solid, plane strain elements (CPE8 in

Figure 3: Deformed shape According to Dong’s approach9, only the membrane and  bending stresses produced by the force and moment that are applied to the structural element to the right of the SCL are considered. The reason for choosing this example is that the exact values are known. They can be calculated from static equilibrium and are given by equations (1) and (2).

F = 1 2 0 0 N    1200 ×12  M = ×120 = 5,143N ⋅ mm   336

(1) (2)

†  The term “SCL” refers to a line that is used to represent an SCP in a 2-D model. This justifies the term “singularity”, which refers to a point on the SCL. It is understood that stresses are applied to a plane (SCP), not to a line (SCL).

2

Copyright © 2008 by ASME

The corresponding membrane and bending stresses, assuming a unit out-of-plane length of the plane strain structural element, are then given by equations (3) and (4).

σ  

m

σ  

b

=

1200

= 100 Mpa   12 6 = 2 5,143 = 214.3 Mpa   12

(3)

SCL

(4)

The objective for this example is to compare the values in equations (1) to (4) with those calculated by FEA. This objective is achieved based on the FEA results discussed next. 4.2

RESULTS FROM FEA

Abaqus11  finite element program is used to calculate the stresses (variable name S) and nodal forces (variable name  NFORC) for the element set (ELSET=STACK) on the righthand side of the SCL shown in Figure 4. An important point is that the nodal stresses not be averaged with those on the lefthand side of the SCL. The STACK is defined for elements 29, 97, 165, and 233, as shown in Figure 4. The print command includes the  parameter POSITION=NODES, which does not average the stresses at the nodes. The resulting output is shown in Table 1. The node numbers are identified in Figure 5. The results are shown for the 4-element model. Similar results were obtained for a 1-element model.

STACK Figure 4: Zoom of singularity region

SCL

Table 1: Nodal stresses and forces from Abaqus output Element

Node

S11

NFORC1

29

57

-89.34

-43.17

29

257

-40.93

-80.79

29

457

-0.26

-6.97

97

457

-0.18

9.54

97

657

42.09

83.39

97

857

83.75

34.62

165

857

83.49

44.18

165

1057

112.00

244.30

165

1257

155.60

77.79

233

1257

163.60

90.60

233

1457

272.10

465.00

233

1657

477.70

281.50

yc y

The stress S11 and force NFORC1, both per unit out-of plane length, are normal to the SCL. In the table, the nodes 457, 857, and 1257 have two values, one received from each adjoining element. The fact that they are not the same indicates that they have not been averaged. Even though the stresses could have been averaged between the elements of the STACK, the integration scheme for not-averaged stresses, given in subsection 4.4, is convenient for biased nodal distances.

Figure 5: ELSET STACK revealing nodes on SCL

3

Copyright © 2008 by ASME

4.3

 y 2

OBJECTIVE

∫ ( y − y ) S11dy ≈

 M k =

Two singularity effects are investigated in this subsection. The first is the behavior of stresses at a singularity. This is shown in Figure 6 for the stress component normal to the SCL. For the 4-element model, the stresses for Figure 6 are taken from Table 1. It is seen that S11 at node 1657 increases from 333 MPa for 1 element mesh to 478 MPs for a 4 element mesh. This illustrates the main problem addressed in this paper that stresses at singularity diverge with refined mesh.

c

 y1

hk 

6

(6)

[( y − yc )1 S111 + ( y − yc ) 2 4 S112 + ( y − yc )3 S 113 )

where  y  is the coordinate with origin at bottom of SCL (see Figure 5), k   denotes the k-th   element, y1 and y2 are the y coordinates at bottom and top of k-th element, hk   is its height, subscripts 1, 2, and 3 refer to the three nodes on the SCL of the

k-th element, and  yc  is the  y value to the centroid of the SCP represented by the SCL (node 857 in Figure 5). For the 4element model, the membrane and bending stresses acting on the SCL are then obtained from equations (7) and (8).

500 400   a 300    P    M 200  ,   s   s 100   e   r    t    S 0

σ  

m

σ  

b

=

=

1 k = 4

∑ F   

t  k =1

6 2

t 

(7)

k 

k = 4

∑ M   

(8)

k 

k =1

-100

4.5

-200 0

2

4

6

8

10

12

For plane stress and strain 2-D solid elements, standard Abaqus output gives two internal nodal forces in X and Y directions. For the 4-element model, the forces in X direction that are applied to the nine nodes of the elements of the STACK are listed in the last column of Table 1. Just as for the nodal stresses, there are 12 forces that represent 3 nodal forces of 4 elements. The membrane and bending stresses acting on the SCL are calculated from equations (9) and (10)

Distance across thickness, mm 1-element

4-elements

Figure 6: Distribution of stress normal to SCL Upon broader view, it follows that nodal stresses and strains obtained by FEA at a singularity have no physical meaning in any part of pressure vessel design. The second effect investigated in this subsection is the dependence of the membrane and bending stresses on mesh refinement. For the purpose of this paper, the key test is whether or not these stresses also diverge with refined mesh. To check on that test, the force and moment that are acting on the SCL, and the corresponding stresses, are calculated from an Excel spreadsheet. The basis for the calculation comes from Gordon’s4 Effective Force and Moment Edit. 4.4

σ  

m

σ  

b

∫ S11dy ≈  y1

hk 

6

( S111 + 4 S112 + S113 )  

=

1 m =12  NFORC 1m   t  m =1

∑

6 2

t 

m =12

∑ ( y

m

(9)

− yc ) NFORC 1m  

(10)

m =1

 ym  are the nodal force and the  y coordinate of the m-th node, respectively, on the SCL. 4.6

The integration for the force and moment acting on the SCL portion of the k-th element was performed by Simpson’s quadratic rule for each element separately using equations (5) and (6).

Fk  =

=

where t is the thickness (length of SCL), and  NFORC m  and

NODAL STRESS METHOD

 y 2

NODAL FORCE METHOD

A BAQUS CAE  LINEARIZATION

Abaqus/Standard 11  version 6.7-1 CAE postprocessor’s stress linearization option is used to check the Excel results. However, the following procedure was found to be necessary to obtain consistent results. First, when using the CAE, the ELSET=STACK was selected so that it alone appeared on the screen, like that shown in Figure 5. Then a PATH was created over the nodes on the SCL. Finally, the stress averaging box was switched off on the menu Results/Options. Wrong results can be obtained if this procedure is not followed.

(5)

4

Copyright © 2008 by ASME

4.7

RESULTS

5 STRESS INTENSITY EDIT

The results are shown in the tables below.

The heading refers to the term used in Gordon’s paper 4. The objective is to investigate the use of an SCL straight through the singularity for the calculation of the primary-plussecondary Tresca stress intensity, referred to as P+Q stress. The general and local primary stress limits are left out of this edit  because they can be easily dealt with by limit analysis (see, for example the WRC Bulletin #464 by Kalnins 12). The edit is applied to the model discussed next, and conclusions are drawn for this model. Extensions to other models may be possible but are left out of the scope of this paper.

Table 2: Membrane stress on SCL Method

Elements

σm

Error %

Nodal Stress Method

1

103.2

3.2

4

100.6

0.6

Nodal Force Method

1

100.0

0.0

4

100.0

0.0

 Abaqus CAE Postprocessor

1

103.2

3.2

4

100.7

0.7

Exact

n/a

100.0

0.0

5.1

The edit is performed for the axisymmetric vessel shown in Figure 7. To assess divergence with refined mesh, FEA models with 1, 2, 4, and 8 elements across the shell wall were built. The 4-element model is shown in Figure 7. The axis of symmetry is marked by the dot-dashed line and the SCL of interest by the red line at the joint of the shell and the head. For the cylindrical shell, the inside and outside radii are 10 and 11.5 inches (254 and 292 mm), respectively, and the length is 15 inches (381 mm). The flat head is 3 inches (76.2 mm) thick. Uniform internal pressure is applied. The lower end of the shell is subjected to a symmetry boundary condition in Y direction. The nodes of the shell and head are defined separately and tied together at the joint. The modulus of elasticity is 30,000 ksi (207 GPa), Poisson’s ratio is 0.3, and the

Table 3: Bending stress on SCL Method

Elements

σb

Error %

Nodal Stress Method

1

214.2

0.0

4

217.0

1.3

Nodal Force Method

1

214.3

0.0

4

214.3

0.0

 Abaqus CAE Postprocessor

1

214.4

0.1

4

217.5

1.5

Exact

n/a

214.3

0.0

4.8

MODEL

design stress intensity S m  is 17.5 ksi (120.7 MPa).

DISCUSSION OF EXAMPL E

The results of Table 2 and Table 3 illustrate the key point made in this paper that while stresses at singularity diverge with refined mesh, the membrane and bending stresses do not. Comparison of the results of the nodal stress and nodal force method support Gordon’s4  finding that “… the nodal force method is the more accurate of the two”, matching, in fact, the exact results for both the 1 and 4 element models in this case. These results support the claim that an SCL passed straight through the singularity of a finite element model can yield linearized stresses that do not diverge with refined mesh, at least for the 2-D solid elements used in the example. The close agreement of the nodal stress method and the Abaqus11  CAE postprocessor’s stress linearization option is expected because the stress integration rules are essentially the same as those given by equations (5) to (8). However, the  procedure stated in subsection 4.6 had to be followed to achieve that result. For the primary and secondary stress categories, the calculation of stress intensities is required. This is what Gordon’s4 Stress Intensity Edit  is meant for. It will be discussed next. Figure 7: Model for Shell/Flat Head Example

5

Copyright © 2008 by ASME

5.2

These results will now be used to investigate which of the stress components may prevent the calculating the P+Q stress  by passing the SCL straight through the joint. This will be  performed by extrapolating the stresses to the joint.

P+Q INGREDIENTS

In common practice, the following stresses are included in the P+Q stress for the axisymmetric model shown of Figure 7: 1. 2. 3. 4.

Meridional membrane plus bending stress, S22(m+b) Hoop membrane plus bending stress, S33(m+b) Average in-plane shear stress, S12(m) Average through-thickness normal stress, S11(m)

S11, S22, S12 are the in-plane stresses in the X,Y directions shown in Figure 7, and S33 is the out-of-plane hoop stress in Z direction. The suffixes “m” and “b” denote membrane and  bending stresses, except in S11(m) and S12(m), where the “m” has nothing to do with physical membrane action. 5.3

Critical SCL

VALID STRESS CLA SSIFICATION LINES

Valid SCLs

When discussing criteria for the placement of SCLs, Hechmer and Hollinger 5 recommended certain conditions that are necessary for a “valid” SCL. Similar recommendations are stated in the Informative Annex 5.A of 2007 Section VIIIDivision 2 of the ASME B&PV Code 10. The important conditions for the purposes of this paper are: 1. The through-thickness normal stress (S11) distribution over the SCL should be linear, with the surface stresses close to the applied pressure 2. The in-plane shear stress (S12) distribution over the SCL should be parabolic, with the surface stresses close to zero

Figure 8: Valid SCLs for the shell

To assess these conditions, the stresses were calculated by Abaqus11  for the 8-element model by passing SCLs at 15 consecutive element boundaries of the shell, at and below that of the critical SCL in Figure 8, including those in both the valid and invalid zones. Figure 9 and Figure 10 show the S11 and S12 stresses up to the distance of 1.3125 inches from the joint. Table 4 lists their values on the ID of the shell. Table 4, Figure 9, and Figure 10 show that the above conditions are satisfied within a zone beginning somewhere between 0.75 and 1.125 inches and extending further away from the joint. Figure 8 shows the zone of the valid SCLs along the shell.

   i 30   s    k  ,    1 25    1    S   s 20   s   e   r    t    S 15  .   n    k 10   c    i    h    t   - 5    h   g   u 0   o   r    h    T -5

Table 4: S11 and S12 stresses on shell ID, ksi Distance from  joint, inches 0.000

S11

S12

25.97

-19.13

0.1875

-1.20

1.72

0.375

-0.26

-0.08

0.5625

-0.87

-0.66

0.75

-0.99

-0.01

0.9375

-1.04

-0.01

1.125

-1.01

-0.02

1.3125

-1.01

-0.02

0.00

0.25

0.50

0.75

1.00

1.25

1.50

X-coordinate Across Wall, inches From Joint 0

1

2

3

4

5

6

7

Figure 9: Through-thickness normal stress across wall on SCLs at 3/16 inch (4.8 mm) intervals away from joint

6

Copyright © 2008 by ASME

5.4

The P+Q stress at the joint can be obtained by extrapolating the stresses from the valid zone to the joint. This  procedure was used in Appendix IV of Hechmer and Hollinger’s5 WRC Bulletin No. 429 for a similar geometry. To see how this would work for the current example, the linearized stresses on the ID of the shell, which were determined in subsection 5.3, are plotted in Figure 11 and Figure 12. In these figures, a valid-SCL limit is assumed at Y=0.75 from the joint. If the limit were assumed at Y=1.125 inches from the joint, two more points would move from the valid to the invalid zone. Figure 11 indicates no problem of extrapolating S22(m+b) and S12(m) from the valid into the invalid zone and arriving at the same value at the joint (at Y=0) as that obtained from  placing an SCL straight through the joint. Thus, for the current example, the validity requirement that the in-plane shear stress distribution over the SCL appear parabolic and its surface stresses be close to zero is unnecessary. While the S12 and S22 stresses at the singularity diverge with refined mesh and are unusable, this is not true for the linearized stresses over the SCL through the joint. This behavior of S22(m+b) and S12(m) is attributed to the fact that these stress components arise from the meridional force and moment and shear force that are necessary to satisfy the continuity and equilibrium of the structure at the joint. This is in line with the results obtained in the Force and Moment  Edit  of section 4, confirming the expectation that finite element stresses satisfy equilibrium of structural elements. On the other hand, Figure 12 indicates a more uncertain extrapolation of S33(m+b) and S11(m) to the joint.

5    i   s 0    k  ,    2    1    S -5   s   s   e   r    t -10    S   r   a   e    h -15    S

-20

0.00

0.25

0.50

0.75

1.00

1.25

1.50

X-coordinate Across Wall, inches From joint 0

1

2

3

4

5

6

7

Figure 10: In-plane shear stress across the wall on SCLs at 3/16 inch (4.8 mm) intervals away from joint

20 S22(m+b)-valid 15

EXTRAPOLATION TO JOINT

S22(m+b)-invalid S12(m)-valid

   i   s    k 10  ,   s   s   e   r    t 5    S

S12(m)-invalid 8 7 6    i   s    k  ,   s   s   e   r    t    S

0

-5 0.000 0.375 0.750 1.125 1.500 1.875 2.250 2.625 3.000

5

S33(m+b)-valid S33(m+b)-invalid S11(m)-valid S11(m)-invalid

4 3 2 1

Y-coordinate from Joint, inches

0 -1

Figure 11: Meridional membrane-plus-bending stress on shell ID and average in-plane shear stress

0.000 0.375 0.750 1.125 1.500 1.875 2.250 2.625 3.000 Y-coordinate f rom Joint, inches

Figure 12: Hoop membrane-plus-bending stress on shell ID and average through-thickness normal stress

7

Copyright © 2008 by ASME

This is attributed to the fact that these stresses do not  participate in establishing continuity and equilibrium of the structure at the joint. By any reasonable extrapolation technique, it is unlikely that it will arrive at the same value at the joint (Y=0) as that obtained from placing an SCL straight through the joint. Regarding the S33(m+b) hoop stress, examination of the stress values in Figure 11 and Figure 12 reveals that it plays no role in the P+Q stress for the current example. The average in plane shear stress (Figure 11) is small (about 2 ksi), and the highest principal stress remains close to S22(m+b). Since S33(m+b) is greater than S11(m), as shown in Figure 12, the lowest principal stress remains close to S11(m). Thus, whatever S33(m+b) is extrapolated to the joint (about 3 to 4 ksi in Figure 12), or calculated directly on the SCL through the joint (7.6 ksi), it will not affect the maximum Tresca P+Q stress. However, this is not the case for the average throughthickness normal stress, S11(m), at the joint. If included in P+Q, it does affect the maximum Tresca P+Q stress for the vessel. The consequence of that is considered next. 5.5

curve is -0.43. The S11(m) over the joint SCL is expected to be higher. But how much higher?

0.2 0    i -0.2   s    k  ,   s -0.4   s   e   r    t    S -0.6    1    1    S -0.8

0.75 in. from Joint

-1

1.125 in. from Joint

-1.2 0.00

0.25

0.50

0.75

1.00

1.25

1.50

X-coordinate Across Wall, inches

THROUGH-THICKNESS NORMAL STRESS

5.5.1 S11(m) fro m S11 of FEA Soluti on The fact that the average through-thickness normal stress, S11(m), on the SCL through the joint is 4.4 ksi (Figure 12), while a rough extrapolated value appears less than 1.0 ksi, indicates that the former is affected by the purely numerical disturbance of the singularity. It is easy to see why. The reason is that it is obtained from the average of the FEA-calculated S11 stress on the SCL at the joint, which is the curve marked with red squares in Figure 9. While the curve should go down to a known value of -1.0 ksi to match the applied pressure, it goes up to the value of 26 ksi for the 8element model and will diverge with a more refined mesh. Since all the red squares indicate positive numbers, it is obvious that the average stress, S11(m), will also diverge with refined mesh. Therefore, the FEA-calculated value of 4.4 ksi is wrong. The conclusion at this point is that if the FEA-calculated S11(m) stress on the SCL through the joint were used in the calculation of the maximum Tresca P+Q stress, it would also be affected by the singularity. This, of course, is the reason why the SCL through the joint has been marked as “invalid”. So, what is the right   value of S11(m), extrapolated or determined otherwise? This will be discussed next.

Figure 13: Through-thickness normal stress distributions on SCLs at 0.75 (19) and 1.125 inches (33 mm) away from joint There does not appear to be a direct way to answer that question. The answer could be estimated by the extrapolation from the valid-SCL zone, with some uncertainty. According to Figure 12, the S11(m) value at the joint is estimated at about 0.7 ksi if the SCL validity limit is assumed at 0.75 inches and at about 0.0 ksi if the SCL validity limit is assumed at 1.125 inches from the joint. The P+Q stresses for these estimates are listed in Table 5.

Table 5: Tresca P+Q stress as a function of S11(m), ksi Method

S11(m)

Extrapolation Extrapolation FEA-calculated

0.7 0.0 4.4

Tresca P+Q Error % 17.5 18.1 14.0

0.0 3.4 -20.0

The error of the P+Q stresses in the table is calculated with respect to the extrapolated S11(m) value of 0.7. The positive error indicates a conservative value and the negative error an unconservative value with respect to that obtained for S11(m)=0.7 ksi. The FEA-calculated value, which has already  been established as wrong in subsection 5.5.1, is included for comparison. When assessing the errors, the potential uncertainties of the extrapolation should also be considered. The selection of the estimated value of 0.7 ksi was made to show that the estimated 0.0 value leads to a higher, thus more conservative, P+Q stress.

5.5.2 “ Right” value of S11(m) The question now is: How can S11(m) be calculated to escape the numerical disturbance of the singularity that is apparent from the FEA-calculated S11 distribution in Figure 9? It is observed that instead of that in Figure 9, the S11 stress distribution over the joint SCL ought to be like that shown in Figure 13, in which the two curves shown represent the S11 stress over the SCLs at the two estimated SCL validity limits. The S11(m) of the 0.75 inch curve is -0.20 and that of the 1.125

8

Copyright © 2008 by ASME

5.5.3

Abaqus str ess li nearization

6 CONCLUSIONS

Abaqus/Standard 11  version 6.7-1 CAE postprocessor’s linearization routine includes not only the FEA-calculated average through-thickness normal stress but also its “bending” component. The definition of the bending component includes the through-thickness normal stress values at the endpoints of the SCL. If the selected SCL passes through a singularity, the  bending component will diverge with a refined mesh. The resulting values of what are output as “Tresca Stress” and “Mises Stress” (i.e., P+Q stresses) on the CAE report file will also diverge with refined mesh. This linearization routine should not be used for the calculation of the P+Q stresses over an SCL that contains a singularity at one of its end points. These stresses are only valid remote from SCLs passing through singularities. 5.6

1. Passing an SCL straight through a singularity may offer a simpler alternative to other methods for the determination of linearized and Tresca P+Q stress intensity at discontinuities modeled by sharp corners. 2. When an SCL is passed through the weld toe that is modeled by a sharp corner, the stresses at the singularity diverge with refined mesh but the membrane and bending stresses on the toe plane do not and show little dependence on mesh size. 3. For the weld toe example, Gordon’s 4 nodal force method and the nodal stress method both give results that agree closely with available exact results, with the nodal force method being the more accurate of the two, confirming Gordon’s finding.

DISCUSSION

The two methods considered for calculating P+Q stresses at a joint with a discontinuity were the extrapolation and the direct calculation over an SCL through the joint. The former involved calculation of stresses over many SCLs, a search for the valid SCL zone, and then extrapolation of the stresses to the  joint. The latter involved the stress calculation over a single SCL. By any measure, the latter appears preferable. The only obstacle in its application was the uncertainty in the calculation of the average through-thickness normal stress. The question can be raised whether this obstacle is real. It is not immediately obvious what role, if any, the throughthickness normal stress plays in the P+Q stress. It is not  primary because it does not equilibrate applied loading on structural elements, and it is not secondary because it does not  participate in satisfying continuity of the shell/flat head structure. It is neither P nor Q. Even if this argument is not accepted and convincing support for its inclusion can be formulated, the results of subsection 5.5.2 showed minimal effect on the P+Q stress. Most important, the FEA-calculated S11(m) gave a P+Q stress 20% less than that obtained from the extrapolated values, rendering it unusable. It also showed that its extrapolated magnitude, depending on the extrapolation technique, could vary from zero to 3.4 % of the meridional membrane plus  bending stress, which is the main actor in ensuring continuity and equilibrium of the joint. Two other options could be considered for the throughthickness normal stress contribution to the P+Q stress at the  joint, which use its value in the valid SCL zone. Gordon’s  paper 4  sets it equal to the applied pressure on the shell I.D., which for this case would be -1.0 ksi. The other is to use the FEA-calculated -0.5 ksi for its average that is shown in Figure 12. Referring to Table 5, this would increase P+Q to 19.1 ksi and 18.6 ksi, respectively. As shown in subsection 5.5.2, the average stress is bound to be greater than these values, which does not justify the increased P+Q stress. Setting it to zero seems like a valid compromise for using an SCL straight through the singularity.

4. For the axisymmetric shell/flat head example, an FEAcalculated average through-thickness normal stress over an SCL containing a singularity is affected by a numerical disturbance of the singularity. If used in the Tresca P+Q stress intensity, it is shown to be unconservative. 5. The results suggest that neither the average throughthickness normal stress nor its “bending” component (i.e., the linear part) should be included in the P+Q stress at a joint with a discontinuity. If accepted, this could result in a simple, conservative, and justifiable compromise for using an SCL straight through the singularity. 6. The values of Tresca and Mises P+Q stresses, output by the Abaqus/Standard 11  version 6.7-1 CAE postprocessor linearization routine should not be used on an SCL passing through a singularity.

REFERENCES 1

  Kroenke, W. C., 1974, “Classification of Finite Element Stresses According to ASME Section III Stress Categories,” Pressure Vessels and Piping, Analysis and Computers , ASME,  New York, NY. 2  Kroenke, W. C., Addicott, G. W., and Hinton, B. M., 1975, “Interpretation of Finite Element Stresses According to ASME Section III,” ASME Paper 75-PVP-Vol. 63. 3  Hinton, B. M., and Hechmer, J. L., 1976, "Secondary Stress Evaluation at a Singularity," ASME Technical Paper 76-PVP67, New York, The American Society of Mechanical Engineers. 4   Gordon, J. L., 1976, “Outcur: An Automated Evaluation of Two-Dimensional Finite Element Stresses According to ASME Section III Stress Requirements”, 1976 ASME Winter Annual Meeting, Paper 76-WA/PVP-16.

9

Copyright © 2008 by ASME

5

  Hechmer, J. L., and Hollinger, G. L., 1998, ‘‘3D Stress Criteria Guidelines for Application’’, Welding Research Council Bulletin No. 429, Feb., The Pressure Vessel Research Council, New York, NY. 6   Hollinger, G. L., and Hechmer, J. L., 2000, “ThreeDimensional Stress Criteria—Summary of the PVRC Project”, ASME Journal of Pressure Vessel Technology, vol. 122, pp. 105–109. 7   Ming-Wan Lu, Yong Chen, and Jian-Guo Li, 2000, "TwoStep Approach of Stress Classification and Primary Structure Method", Journal of Pressure Vessel Technology, February 2000, Vol. 122, pp. 2-8. 8   Strzelczyk, A. T., and Ho, S. S., 2007, “Evaluation of Linearized Stresses Without Linearization”, 2008 ASME PVP Conference Proceedings, Paper PVP2007-26357. 9  Dong, P., 2001, “A Structural Stress Definition and Numerical Implementation for Fatigue Analysis of Welded Joints”, International Journal of Fatigue, vol. 23, pp 865–876. 10   ASME Boiler and Pressure Vessel Code, 2007, American Society of Mechanical Engineers, Three Park Avenue, New York, NY 10016-5990. 11   Abaqus Finite Element Program, Version 6.7-1, Hibbitt, Karlsson, and Sorensen, Inc., Pawtucket, R.I., by Educational License to Lehigh University 12  Kalnins, A., 2001, “Guidelines for Sizing of Vessels by Limit analysis”, Welding Research Council Bulletin No. 464, August 2001, The Pressure Vessel Research Council, New York, NY.

10

Copyright © 2008 by ASME