Chapter 62: Girkmann Problem using Axisymmetric Shell Elements

62 Girkmann Problem using Axisymmetric Shell Elements

Summary 1178

Introduction 1179

Problem Statement 1179

Modeling Details 1179

Results and Discussions 1182

Conclusion 1184

Input File(s) 1184

Video 1185

References 1185

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SummaryTitle Chapter 62: Girkmann Problem using Axisymmetric Shell Elements

Contact Features • Use of axisymmetric shell elements• Use of MPCs

Geometry

Material Properties , , for sperical shell subjected to gravity loading. Same material properties except density is used for the stiffening ring not subjected to gravity loading.

Analysis Type • Use of axisymmetric shell element (CAXISYM)• Use of MPCs

Boundary Condition Boundary conditions of axisymmetry, at the free end of the sperical shell.

Applied loads • Gravity loading for the axisymmetric shell• Uniform normal pressure of 27.306 KPa is applied at the base AB of the stiffening ring

Element Type • 2-node axisymmetric shell element CAXISYM• 4-node axisymmetric solid quad element CQUADX

FE Results 1. Interface shear force per unit circumference length between spherical shell and solid stiffener.

2. Interface bending moment per unit circumference length.3. Meridional angle at which maximum bending moment occurs and the value of maximum

bending moment per unit circumference length.

E 20.59 GPa= 0= 3335.71 kg\m3=

Uy 0=

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Girkmann Problem using Axisymmetric Shell Elements

IntroductionThis example showcases the capability of CAXISYM elements. The problem demonstrated here was first discussed by PitkÄaranta et al. (References). The FEA solution of the Girkmann problem was obtained and compared with the solutions obtained by classical methods.

Problem StatementA spherical shell of thickness , crown radius is connected to a stiffening ring at the meridional angle . The dimensions of the ring are: and . The radius of the midsurface of the sperical shell is .

The notation is shown in Figure 62-1. The z axis is the axis of rotational symmetry.

Figure 62-1 Girkmann Problem

Consider gravity loading only. The equivalent (homogenized) unit weight of the material comprised of the shell and the cladding is . Assume that uniform normal pressure PAB is acting at the base AB of the stiffening ring. The resultant of PAB equals the weight of the structure. Assume that the stiffening ring is weightless. The goals of the computations are as follows:

• To find the shear force at the interface between spherical shell and stiffening ring.

• To find the maximum bending moment.

• To find the meridional angle at which the maximum bending moment occurs.

Modeling DetailsThe spherical shell was modeled with axisymmetric shell element and stiffening ring with axisymmetric solid elements as shown in Figure 62-2.

h 0.06m= Rc 15.00m=

2 9 40 = a 0.60m= b 0.50m=

Rm Rc sin=

32.69 KN\m3

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Figure 62-2 FEA Model

Element ModelingThe spherical shell was modeled with 2-node axisymmetric shell elements CAXISYM. The properties for CAXISYM were entered as given below:

PAXISYM 2 1 0.06 prop_axi

The CAXISYM elements were defined as given below:

CAXISYM 769 2 830 831 CAXISYM 770 2 831 832 CAXISYM 771 2 832 833 CAXISYM 772 2 833 834 CAXISYM 773 2 834 835

The solid stiffener was modeled with axisymmetric solid element CQUADX. First, the properties PLPLANE for the CQUADX were defined:

PLPLANE 1 2 prop_axiPSHLN2 1 2 + + C4 AXSOLID L prop_axiThe CQUADX elements were defined as given below,CQUADX 1 1 1 10 11 2CQUADX 2 1 2 11 12 3CQUADX 3 1 3 12 13 4…

Material PropertiesThe material properties were modeled by using MAT1 entry as given below:

MAT1 12.059+10 0.0 3335.71 0.0 mat_axi_MAT1 22.059+10 0.0 mat_axi_

As Nastran applies gravity over the entire model by default, two material cards were defined. The material without density was used for solid stiffener modeled with CQUADX so that the gravity load does not get applied on it.

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Girkmann Problem using Axisymmetric Shell Elements

Transformations and MPCsTo ensure the compatibility of rotational degrees of freedom between axisymmetric solid and shell elements, MPCs were used. For the convenience of defining MPCs, transformations using local coordinate systems were defined for the node at the junction between axisymmetric shell and axisymmetric solid elements,

Figure 62-3 Local Coordinate System

CORD2R 1 0 15. 17.8763 0.0 15. 17.8763 15.+ + 15.0193017.89930 0.0

Then MPCs were defined between the node at the junction and the other nodes at the interface as given below:

MPC 1 187 2 1. 1185 2 -1. + + 1185 6 0.03 MPC 3 186 2 1. 1185 2 -1. + + 1185 6 0.02625MPC 17 91 2 1. 1185 2 -1. + + 1185 6 -0.03…

Loads and Boundary ConditionsFor CAXISYM and CQUADX elements, the axis conventions for axisymmetry are: Y-axis is the axis of symmetry and X-axis defines the radial direction. The symmetry boundary conditions of the axisymmetric problem was defined constraining the axial degree of freedom of the node which lies on the axis of symmetry.

SPC 1 830 2 0.0

Gravity load was applied on the axisymmetric shell elements as given below:

GRAV 3 1. 0.0 -9.81 0.0

Pressure loading was applied at the bottom face of the solid stiffener. Pressure value was multiplied by as per the Nastran convention for the application of the pressure edge load on the axisymmetric problem.

PLOADX1 4 353 171425. 386 387 0.0PLOADX1 4 354 171425. 387 388 0.0

Uy 0=

2

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Analysis ParametersSubcase corresponding to the loadcase that contains above loads and the boundary conditions were defined and analysis was carried out by SOL 400 by using following parameters:

$# NASTRAN Control Section$# File Management SectionNASTRAN SYSTEM (316)=19ASSIGN OUTPUT2 = 'girk_fo_s400.op2', UNIT = 12, FORM = FORMATTED$# Executive Control SectionSOL 400CENDECHO = NONE$# Case Control SectionSUBCASE 1 STEP 1$ Subcase name: DefaultLoadCase$LBCSET SUBCASE1 DefaultLbcSet TITLE=DefaultLoadCase SET 10 = 769 THRU 1123 SET 20 = 830 THRU 1185 MPC = 33 SPC = 1 LOAD = 6 DISPLACEMENT (SORT1, PLOT, PRINT, REAL) =ALL NLSTRESS (SORT1, PLOT, PRINT, REAL, VONMISES, CENTRE) =ALL GPFORCE (PLOT, PRINT) =20 ANALYSIS = LNSTAT

Results and Discussions

Figure 62-4 Results

The f06 file generated out of the analysis was studied carefully and the STRESS-XY for the axisymmetric shell element CAXISYM 1123 were noted as

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Girkmann Problem using Axisymmetric Shell Elements

Shear Stress-(sig-xy) for element CAXISYM 1123 is -1.577151E+04

The grid point force balance for the interface element CAXISYM 1123 is as given below,:

So the bending moment/circumference at the interface i:

= 3475.662*2*π*r

= 3475.662*2*3.14159*15

= 36.878 N.m/m

In order to calculate the maximum bending moment and its location, we need to find the maximum bending stress and its location. Now in order to calculate the bending stress one must know the stress distribution for the shell element. The f06 file reports the stresses for upper layer and lower layer (i.e., LAYER I and LAYER 2) for the CAXISYM elements. For example, the bending stress for the CAXISYM element 1123 is calculated as given below:

The bending stress can also be calculated as

Interface shear force/circumference length = sig-xy * shell thickness

Interface shear force/ circumference length = -1.577151E+04 * 0.06

= 946.3 KN/m

Bending stress = [Total stress (LAYER 1) – Total stress (LAYER 5)]/2

= [(-4.711647E+05) – (-3.562409E+05)]/2

= -57461.9 N/m2

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The bending stress values for all CAXISYM elements are calculated like this and the node where the max. bending stress occurs was noted. At that particular node, the bending moment would be maximum. Figure 62-5 shows the variation of bending moment/thickness (N.m/m) with meridional angle.

Figure 62-5 Bending Moment vs. Meridional Angle

The maximum bending moment of 255.126 N.m/m occurs at the meridional angle of 38.15° as shown in Figure 62-5.

ConclusionThe following table shows the comparison of the results obtained from FEA modeling with that of theoretical one,

Input File(s)

Results MD Nastran Theory % Error

Moment (Nm/m) 36.878 36.81 0.18%

Q1 (N/m) -946.291 -943.6 0.29%

Max. Moment (Nm/m) 255.126 253.97 0.46%

Angle (degrees) 38.150 38.08 0.18%

Files Description

nug_62.bdf MD Nastran input for Girkmann problem using axisymmetric shell and solid elements.

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Girkmann Problem using Axisymmetric Shell Elements

VideoClick on the image or caption below to view a streaming video of this problem; it lasts approximately 20 minutes and explains how the steps are performed.

Figure 62-6 Video of the Above Steps

ReferencesPitkÄaranta, J., Babuska, I. and Szabo, B. The Problem of Verification with Reference to the Girkmann Problem. IACM Expressions. 24, January 2009, 14-15.