Chapter 3.37: Girkmann Verification Problem

3.37 Girkmann Verification Problem

Summary 354

Detailed Description 355

Results 359

Modeling Tips 361

Input Files 365

Marc User’s GuideSummary

3.37-

Summary

Title Girkmann Verification Problem

Problem features The Girkmann problem is a numerical verification exercise in solid mechanics proposed by Juhani Pitkäranta, Ivo Babuška, and Barna Szabó in 2008. The importance of verification is self-evident.

Geometry

Material properties E = 2.059x1010 N/m2, = 0.0, = 3335.71 kg/m3

Analysis type Static with elastic material behavior

Boundary conditions Axial displacement at centerline = 0, pressure and gravity loads indicated

Element type Axisymmetric shell and solid elements are connected with displacement and slope continuity.

FE results Bending moment and shear force at shell-ring interface; meridional angle at maximum bending moment. The authors invited their readers to solve this problem and verify the results are accurate to within 5 percent. The results herein are within 0.50 percent.

Z

X

Y

p = 27,283.1 N/m2

CL

shell ρ = 3335.71 kg/m 3

ring

α

Rc

Mα Mα

Qα

Qα

Rm = Rc/sin(α)h

A Ba

b

Rc

h = 0.06 mRc = 15.00 mα = 2π/9a = 0.60 mb = 0.50 m

p = 27,283.1 N/m2CL

ρ = 3335.71 kg/m 3

10 20 30 40

-50

0

50

100

150

200

250

300Bending Moment (Nm/m)

Meridional Angle (degrees)

38.137o

3.37-CHAPTER 3.37Girkmann Verification Problem

The Girkmann problem consists of a spherical shell connected to a stiffening ring at the crown radius. The objective of the analysis is to accurately estimate: a) the shear force and bending moment acting at the junction between the spherical shell and the stiffening ring b) determine the location (meridional angle) and the magnitude of the maximum bending moment in the shell. The model problem was first discussed by Girkmann in 1956, subsequently by Timoshenko and Woinowski-Krieger in 1959. The results are compared to the solutions by the classical methods to demonstrate the accuracy.

Detailed DescriptionElement type 1, an axisymmetric, straight, thick-shell element is used for modeling the spherical shell and element type 10, an axisymmetric, four-node, quadrilateral element is used to model the stiffening ring. The geometry for the Girkmann problem is shown in Figure 3.37-1. The x axis is the axis of rotational symmetry. A spherical shell of thickness h and mid-surface radius Rm, is connected to a stiffening ring at the meridional angle α and a crown radius of Rc. The dimensions of the ring are a and b.

Figure 3.37-1 The Girkmann Problem Geometry

Z

X

Y

p = 27,283.1 N/m2

CL

ringshell ρ = 3335.71 kg/m 3

α

Rc

Mα Mα

Qα

Qα

Rm = Rc/sin(α)h

A Ba

b

Rc

h = 0.06 mRc = 15.00 mα = 2π/9a = 0.60 mb = 0.50 m

Marc User’s GuideDetailed Description

3.37-

A close-up of the shell-ring intersection for the Girkmann problem is shown in Figure 3.37-2. The Mesh consists of 2208 elements and 2270 nodes.

Figure 3.37-2 The Girkmann Shell - Ring Close-up

The axisymmetric solid elements for the stiffening ring are generated by *add_elements and re-meshed with *subdivide_elements (Figure 3.37-3). The axisymmetric shell elements for the spherical shell are generated using *expand_nodes.

Figure 3.37-3 Building the Ring using Subdivide

Z

X

Y

3.37-CHAPTER 3.37Girkmann Verification Problem

A local Cartesian coordinate system (*new_coord_system) is created with the shell solid intersection

node as origin, the local X axis along the 400 inclined edge of the ring and the local Y axis normal to

that. All the nodes on the 400 inclined edge of the ring and the end node of the shell at the intersection are transformed into this co-ordinate system (Figure 3.37-4).

Figure 3.37-4 Coordinate Transformation (colored arrows) for Joining the Shell and Ring

Servo links constrain the translation and rotation displacements of the end node of the shell joining the inclined edge of the ring. The local Y displacement of the nodes on the ring edge is the sum (with appropriate sign) of the local Y displacement of the end node of the shell at the intersection and Z rotation times the distance of that node from the end node of the shell (see Figure 3.37-8). The local X and local Y displacement of the coincident nodes of solid and shell at the intersection are constrained to be equal.

Local Y Direction

Z

X

Y

Loca

l X D

irecti

on

Marc User’s GuideDetailed Description

3.37-

The material for all elements is linear elastic, isotropic with Young’s modulus of 2.059e10 N/m2 and

density of 3335.71 Kg/m3. Pressure of 27283.14706 N/m2 is applied is applied to the bottom face (left)

of the ring as an edge load (Figure 3.37-5). An acceleration of -9.81 m/s2 is applied in the X direction (although not necessary the Y and Z components of acceleration is set to zero as well) only to the shell elements, whose mass times this acceleration will determine the weight or gravity load of the shell structure. The stiffening ring is assumed to be weightless. The displacement of the node on the axis of symmetry is constrained in the axial direction.

Figure 3.37-5 Loads and Boundary Conditions

By design, the axial (vertical in Figure 3.37-5) force on the ring equilibrates the weight of the shell.

p = 27,283.1 N/m2

CL

shell ρ = 3335.71 kg/m 3

Rc

apply1 -> Axial Disp = 0

apply2 -> Pressure Load

apply3 -> Gravity LoadZ

X

Y

3.37-CHAPTER 3.37Girkmann Verification Problem

ResultsThe internal forces from the force balance file (girkmann_job1.grd) are listed below for the node (2270) on shell at the intersection.

The bending moment at the shell-ring interface becomes, .

The axial force at the shell-ring interface becomes, .

The radial force at the shell-ring interface becomes, .

The shear stress at the shell-ring interface (Figure 3.37-6) is -15658.2 N/m2, and when multiplied by the thickness gives a shear force of -939.492 N/m.

Figure 3.37-6 Shear Stress (Component 12) at the Shell-Ring Interface

node 2270 internal force from element 2208 -0.1571E+07 0.1735E+07 0.0000E+00 -0.3475E+04 0.0000E+00 0.0000E+00 node 2270 externally applied forces -0.4710E+03 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 node 2270 tying/mpc forces 0.1572E+07 -0.1735E+07 0.0000E+00 0.3475E+04 0.0000E+00 0.0000E+00 node 2270 reaction - residual forces 0.6319E-05 -0.7750E-05 0.0000E+00 0.1676E-07 0.0000E+00 0.0000E+00

M3575D

------------ 3575 30 -------------- 36.871 Nm/m= = =

Q1571000–D

------------------------ 16668.828 N/m–= =

Qr1735000D

--------------------- 18408.922 N/m= =

80888.5

71077

62333.3

63572.1

61789.1

43299.7

-32766.4

31703.8

13389.1

29697.9

42517.6

83265.8

84066.9

77680

93494.7

73937.8

66222.7

70742.5

140071

-15236.7

-15404.7

83583.3-15573.5

-15658.2

-3.277e+004

7.668e+005

job1

Comp 12 of Stress Layer 1

Inc: 0Time: 0.000e+000

Z

X

Y

Marc User’s GuideResults

3.37-

The bending moment is estimated from the shell stresses as follows:

Using the above, the bending moment can be plotted versus the meridional angle as shown in. The

maximum bending moment is 255.103 Nm/m at a meridional angle of 38.137o.

Figure 3.37-7 Bending Moment versus Meridional Angle

The results are summarized below and compared to the reference values.

(1) The Problem of Verification with Reference to the Girkmann Problem by Barna Szabó, Ivo Babuška, Juhani Pitkäranta, and Sebastian Nervi. The Institute for Computational Engineering and Sciences Report 09-17, 2009. See www.ices.utexas.edu/research/reports/2009/0917.pdf.

Editorial Comment: The above reference is well worth reading; the authors received 15 solutions and among their comments the following is worth repeating, namely: “Another respondent wrote: “Regarding verification tasks for structural analysis software that has adequate quality for use in our safety critical profession of structural engineering, the solution of problems such as the Girkmann problem represents a minuscule fraction of what is necessary to assure quality.” We [the authors] agree with this statement. That is why we find it very surprising that the answers received had such a large dispersion. For example, the reported values of the moment at the shell-ring interface ranged between -205 and 17977 Nm/m. Solution of the Girkmann problem should be a very short exercise to persons having expertise in FEA, yet many of the answers were wildly off.”

Result Marc Reference (1) % Error

Bending Moment (Nm/m) 36.871 36.81 0.17%

Axial Force (N/m) -16668.828 -16700 -0.19%

Radial Force (N/m) 18408.922 18400 0.05%

Shear Force (N/m) -939.492 -943.6 -0.44%

Max. Bending Moment in the shell (Nm/m) 255.103 253.97 0.45%

Meridional Angle of Max. BM (degree) 38.137 38.08 0.15%

B Comp 11 of Stress at Layer 1 - Comp 11 of Stress at Layer 5 2=

MB B bd2

6 B 1 0.06 2 6 = =

10 20 30 40

-50

0

50

100

150

200

250

300Bending Moment (Nm/m)

Meridional Angle (degrees)

38.137o

3.37-CHAPTER 3.37Girkmann Verification Problem

Modeling TipsTo review the model, read in the Marc input file girkmann.dat into Mentat. All of the modeling information will be present. Axisymmetric models in Marc use the global x axis as the axis of rotation. The meshing is relatively straight forward and is not repeated here. However, an important feature in this model are the transformations and constraints between the end shell node (n:2770) where it joins the inclined plane of the ring (n:11). To review the transformations and constraints let’s read in the input file and go to modeling tools.

FILESMARC INPUT FILE READ

girkmann.dat, OKSAVE AS

girkmann, OKFILLMAIN

MODELING TOOLSTRANSFORMATONS

TRANSFORMATION PLOT SETTINGSTRANSFORMATIONS (turn on)

DRAW (should look like Figure 3.37-4)TRANSFORMATIONS (turn off)

DRAWMAIN

LINKSSERVO LINKS (see Figure 3.37-8)MAIN

We see that servo link 1 in Figure 3.37-8, constrains the ring node 110 to have its second degree of freedom related to the second (translation normal to incline) and third (rotation) of the end shell node 2770 by the moment arm of length 0.03m. This is repeated 15 more times for all nodes along the ring incline edge. Servo link 17 and 18, simply equate the first and second degrees of freedom to the coincident nodes 2270 of the shell and 11 of the ring. These servo links are automatically generated with the N to 1 SERVOS button, where you need only select the proper nodes and all of the coefficients (aka moment arms) are computed automatically by Mentat.

Furthermore, since the shell elements have three degrees of freedom per node, while the ring elements have only two degrees of freedom per node, node 11 and node 2270 should never be the same node number, but constrained together as shown here. Also since the nodes are separate, the results will not be nodal averaged across the shell and solid axisymmetric elements. Finally, getting this step wrong will give incorrect results that may not be obvious.

Marc User’s GuideModeling Tips

3.37-

Figure 3.37-8 Servo Link 1

You may wish to run the model; to do so simply go to Jobs, run and submit the simulation, for example

JOBSRUN

SUBMIT

After the simulation completes, let’s examine how we can produce check the validity of the servo links, the bending moment versus meridional angle shown in Figure 3.37-7, and some other ways to help visualize the results.

OPEN POST FILE (RESULTS MENU) (opens results and jumps to results menu)DEFORMED SHAPE SETTINGS

AUTOMATIC (turn on)RETURN

DEF & ORIG (should look like Figure 3.37-9)

The servo links must keep the angle (a right angle in this case) between the shell and ring edge the same before and after deformation. Since the deformations are very small, the scaling of the deformed shape was set to automatic and the magnification factor is over 400 in Figure 3.37-9.

88

77

66

55

44

33

22

11

2269

99

2268

108 110

2265

2264

2263

2262

187

176

165

154

143

132

121

109

2267

2266

2265

77

66

55

44

33

22

11

88

99

108

2264

2263

2262

187

176

2266

165

143

132

121

110 109

154 2267

2269

2270

2268

Z

X

Y Servo 1: dof 2 n:110 =1*dof 2 n: 2270 + 0.03*dof 3 n:2270

Z, dof 3

Local X, dof 1Local Y, dof 2

0.03

3.37-CHAPTER 3.37Girkmann Verification Problem

Figure 3.37-9 Shell-Ring Edge Originally Perpendicular must remain Perpendicular - Displacements Automatically Magnified over 400 times.

The strategy to computing the bending moment in the shell is simple; we will just collect bending stress along a path from the centerline to the end shell node.

PATH PLOTNODE PATH

n:803 n:2270 #ADD CURVES

ADD CURVEArc LengthComp 11 of Stress Layer 1Arc LengthComp 11 of Stress Layer 5SHOW ID 100FIT (should look like Figure 3.37-10)RETURN

CLIPBOARD COPY TO

The xy data is now in the clipboard and can be exported to Microsoft Excel for additional processing to compute the bending moment from the bending stresses at the top and bottom layers of the shell element that was used to produce the plot in Figure 3.37-7.

Inc: 0Time: 0.000e+000

Z

X

Y

Marc User’s GuideModeling Tips

3.37-

Figure 3.37-10 Bending Stress of Top and Bottom Shell Layers along Arc Length of Shell Elements from Center Line to Shell-ring Intersection

Also we can use the expand feature to expand (rotated 40o) our results about the axis of rotation.

Figure 3.37-11 Axisymmetric Shell Element Results Expanded about the Rotational Symmetry Axis

Y (x1e5)

1030 1130 1230 1330 1430153016301730

1830

1930

2130

2230

Comp 11 of Stress Layer 1

-0.025

-8.5291.6290 Arc Length (x10)

930830

2030

2230

Comp 11 of Stress Layer 5

830 930 1030 1130 1230 1330 1430153016301730

1830

1930

2030

2130

1

-4.712e+005

-4.243e+005

-3.775e+005

-3.306e+005

-2.837e+005

-2.369e+005

-1.900e+005

-1.431e+005

-9.625e+004

-4.939e+004

-2.520e+003

job1Comp 11 of Stress Layer 1

Inc: 0Time: 0.000e+000

ZX

Y

4

3.37-CHAPTER 3.37Girkmann Verification Problem

Finally we can visualize the shell-ring intersection by closing the post file and adjusting the plot settings as follows:

MAINRESULTS

CLOSEMAIN

GEOMETRIC PROPERTIESPLOT SETTINGS SHELL

PLOT EXPANDEDDEFAULT THICKNESS = 0.06

DRAW (should look like Figure 3.37-12)

Figure 3.37-12 Expanded Shell Plot Showing the Shell’s Thickness at the Shell-Ring Intersection

Hence we can see that the thickness of the shell is identical to the length of the inclined ring edge that has all of the servo links illustrated in Figure 3.37-8.

Input FilesThe files below are on your delivery media or they can be downloaded by your web browser by clicking the links (file names) below.

Z

X

Y

Ring

Servo LinksShell

0.06

File Description

girkmann.dat Marc input file to run the above problem

Marc User’s GuideInput Files

3.37-