Accuracy of MSC/NASTRAN First- and Second-Order Tetrahedral Elements inSolid Modeling for Stress Analysis

A. Entrekin – Engineer, Production Airframe StructuresBell Helicopter Textron Inc.

P.O. Box 482Fort Worth, Texas 76101

(817) 280-4220 AEntrekin@bellhelicopter.textron.com

ABSTRACT

Solid element FEM models are often used in the analysis of complex structural components to applycomplex loading conditions and predict stress levels. The simplest solid element available to the finiteelement modeler is the first-order four-node tetrahedron, or TET4. The tetrahedral element is often usedbecause of its ability to mesh almost any solid, regardless of complexity. However, the formulation of theTET4 element makes it necessary to use a very large number of elements to accurately model areas aroundstress concentrations. In general, the elements’ edge lengths must all be a fraction of the size of thesmallest feature to get accurate results. This density of mesh produces models that are often toocumbersome to be analyzed. In these situations, second-order tetrahedral elements, or TET10 elements, arevery useful. Since second-order elements are not restricted to straight-line edges, they can model complexsolids more accurately with fewer elements. In addition, the stress levels predicted by TET10 elements areslightly conservative in areas of stress concentration, while those predicted by the first-order TET4 areconsiderably unconservative using meshes of comparable size. The TET10 element produces better resultsin bending applications with fewer elements required through the thickness of the part. However,MSC/NASTRAN does not support second-order elements in nonlinear analyses.

These conclusions are based on two solid models that were created and analyzed using Unigraphics ,MSC/PATRAN and MSC/NASTRAN . The results from these models suggest that TET10 elementsproduce slightly conservative results in areas of stress concentration, while TET4 models of similar sizeproduce unconservative and less accurate results. The results also indicate that the TET10 element modelsbending behavior in structures with fewer elements than are required in a TET4 model to produce similarresults. For these reasons, when modeling parts that have areas of stress concentration or that may besubjected to bending, second-order TET10 elements perform more accurately and more reliably than first-order TET4 elements. Since MSC/NASTRAN does not support second-order elements in nonlinearanalyses, the use of solid elements in nonlinear applications should be avoided.

IntroductionThe accuracy (and, therefore, the usefulness) of MSC/NASTRAN tetrahedral elements for stress analysishas been called into question recently. It has been suggested that the first-order tetrahedral element (TET4)is not reliable in the stress levels it predicts. More accurate stress level predictions are attainable using thesecond-order tetrahedral element (TET10). However, the modification of a TET4 mesh to a TET10 meshwith the same number of elements results in a significant increase in model size and analysis run time. Aninvestigation was conducted in order to verify this phenomenon and its implications for analysis ofcomplex parts subjected to complex loading conditions.

Investigation IA solid stepped shaft with a blend/shoulder radius was modeled in Unigraphics . The shaft varied indiameter from 4.0 inches at the large end to 2.0 inches at the small end. Each section was 10.0 inches inlength, and a radius of 0.25 inches was used to blend the step between sections. This shaft was cantileveredon its large end and a 1000-pound load was applied to the small end of the shaft, offset from the end by adistance of 1.0 inches. This offset from the end of the shaft made viewing the model easier, since theloaded node was not in the same plane as the end of the shaft. The model and applied load are illustrated inFigure 1.

The solid from Unigraphics was saved as a parasolid file for further model development and analysis usingMSC/PATRAN and MSC/NASTRAN .

The imported solid was meshed using a variety of global edge lengths and element topologies.Specifically, meshes were created using TET4 elements with global edge lengths of 0.25, 0.5, 0.75, and 1.0inches, and TET10 elements with global edge lengths of 0.25, 0.5, and 1.0 inches. The nodes on the largeend face of the mesh were constrained against translation in all directions to model the cantilever support,and all of the nodes in the solid mesh were constrained against rotation about all three axes. The rotationalconstraint of all solid-element nodes is necessary because solid elements have no stiffness in rotation. Anode was generated 1.0 inches from the center of the small end face of the mesh. This node was connectedto the nodes of the small end face with a rigid body element (RBE2) in all translational degrees of freedom.A 1000-pound load was applied to this node. Each model was then written out as an MSC/NASTRAN

input file for analysis. Representative plots of the models are presented in Figures 2 and 3.

A linear static analysis of each model was performed using MSC/NASTRAN , and the results wereevaluated using MSC/PATRAN . Three stresses were investigated: Von Mises, axial (X-direction, alongthe length of the shaft), and maximum principal. These stresses were averaged over all elements in each ofthe models. Although unaveraged results are known to typically show higher peak stress levels thanaveraged results, this investigation was conducted to show the differences between the models rather thanbetween any one of them and theoretical values. The peak values of each stress were observed andrecorded.

As can be seen in these figures, the size of the model varies with element topology. Table I shows thevariation in model size due to element topology for several global edge lengths.

It was noted after running the models that the theoretical stress concentration factor (Kt) had beendetermined for a shaft with an applied moment rather than a moment induced by an applied shear. In themodels with applied forces, the moment varies along the length of the shaft. This makes the choice of thesection at which the stress is calculated a factor in the comparison of the theoretical results to those fromMSC/NASTRAN . To avoid introducing this additional variable, two new models were created. TheTET4 model with a global edge length of 0.25 inches and the TET10 model with a global edge length of0.5 inches were modified to use a moment as the applied load rather than a force. This ensured that theresults obtained from MSC/NASTRAN were comparable to those calculated from theory. These modifiedmodels were identical to the original models with the exception of the applied load. The magnitude of theapplied moment in the revised models was 1000 inch-pounds, generated by a 500-pound force couple witha 2-inch couple distance. Using the properties of the smaller end of the shaft, the stress concentrationfactor (Kt) at the root of the blend radius on the smaller shaft was determined to be approximately 1.625.

The stress level calculated using classical methods was 2,069 psi. MSC/NASTRAN results for the TET4mesh predicted a peak axial stress of 1600 psi, which was 23% lower than the stress calculated by classicaltechniques. TET10 results predicted a peak axial stress of 2210 psi, which was 7% greater than thepredicted analytical stress.

Conclusions of Investigation IBased on the data produced in this investigation, it can be concluded that the TET4 mesh does indeedproduce misleading stress results in areas where stress concentrations are of interest. In a very fine TET4mesh (relative to the features being modeled), results may be closer to those predicted by classical methods.The primary disadvantage of this approach is that such fine meshes produce very cumbersome models thatgreatly increase computing resource demands. In this investigation, with a 0.25-inch radius, the finestmesh generated used a global edge length of 0.25 inches. It is possible that a global edge length of 0.0625or less may result in a significant increase in accuracy, but such a model would greatly increase computingresource demands. Even as the mesh density doubles, the stress levels in the TET4 models fails toapproach those of the TET10 models. At best, the TET4 results show stresses about twenty percent lowerthan the TET10 results. At worst, the error is approximately fifty percent. However, the TET10 stressesconverged toward the theoretical value rather quickly, with the Von Mises stress increasing by only 1.27percent when the mesh density was doubled by reducing the global edge length of 1.0 inches to 0.5 inches.

The data generated in this study, however, suggests that it is not necessary to keep the same number ofelements when moving from TET4 elements to TET10 elements. Fewer TET10 elements can be used toget more accurate stress predictions.

Another finding of this investigation was that the stress levels predicted in areas away from stressconcentrations are conservative for both TET4 and TET10 meshes. At the extreme fiber of the small shaftin the modeled problem, the TET4 mesh predicted an axial stress of 1,340 psi. The TET10 mesh predicted1,400 psi. The stress level computed by classical methods was 1,273 psi. In this situation, the TET4 modelwas slightly more than 5% conservative and the TET10 model was conservative by nearly 10%. It wouldtherefore appear that although TET4 elements tend to underestimate stress levels in areas of stressconcentrations, they appear to be reasonably accurate in areas away from stress concentrations.

Investigation IIA second investigation was made to determine the number of elements required through the thickness of asolid model to ensure accurate displacement and stress predictions in bending applications. In generalFEM solid modeling, at least three elements are required through the thickness of a member in bending toadequately model the bending behavior of the member. A more accurate statement may be that thereshould be more than one integration point through the thickness of a member in bending. This would meanthat a single second-order element would yield more accurate results than multiple first-order elements.

A bar of rectangular cross section was modeled in Unigraphics and transferred as a parasolid for analysisusing MSC/PATRAN and MSC/NASTRAN . Its height was 2.0 inches and its width was 1.0 inch. Thebar was 10.0 inches in length. Material properties for 2024-T3 were given to the solid elements of the bar.One end of the bar was constrained against all translation to model a cantilever support. A rigid bodyelement (RBE2) was used at the free end to introduce the applied moment. A grid was created one inchfrom the end of the bar, along the centerline of the bar. This grid was defined as the independent grid ofthe RBE2, and the grids on the end face of the bar were defined as the dependent grids in their translationaldegrees of freedom (UX, UY, UZ). Attached to the independent grid were two stiff bars, extending oneinch up and one inch down, whose ends were loaded with a force couple to create the applied moment. Themagnitude of the force couple was 500,000 pounds, which resulted in a moment of 1,000,000 inch-pounds.The geometry and load application for this model are shown in Figure 4.

The solid was then meshed with four TET10 mesh densities, using global edge lengths to obtain meshesthat had one, two, three, and then four elements through the bar’s thickness. The four models are shown inFigure 5. The models were all submitted to MSC/NASTRAN for linear static analysis.

The results of this second investigation were read in to MSC/PATRAN and stress contour plots weregenerated. The displacement of the loaded point was also compared. The vertical downward displacementof the loaded point and the extreme fiber axial stress are shown in Table II. Classical analysis predicts anextreme fiber stress of 1.5 million PSI. Simple cantilever analysis predicts a downward end deflection of7.1429 inches.

As the results in Table II show, the deflection at the load application point does not vary significantly (lessthan 4% variation) as the mesh density increases. The stress levels are all quite close to the theoreticalvalue of 1,500,000 psi. The maximum error occurs in the coarsest mesh, and that error is less than 7%conservative. This indicates that the bending stress levels predicted by coarser meshes are slightly moresevere than those predicted by finer meshes. The penalty for using finer meshes, however, is the rapidincrease in the size of the model as measured by the number of degrees of freedom.

This same rectangular beam was then meshed with six different densities of TET4 elements to examinehow the accuracy of stresses predicted by first-order elements improves with mesh density. The deflectionand stress results obtained from this investigation are presented in Table III.

The variation of stress level predicted as the number of elements increases through the thickness of thebeam is plotted in Figure 6, along with theoretical results.

When compared to the TET10 results in Table II, it is obvious that the second-order element mesh isconsiderably more accurate in predicting stress levels for this geometry. The TET4 model with fourelements through the thickness of the beam is more conservative in its stress estimate than the TET10model with only two elements through the beam thickness. Thus, a part analyzed using the first-ordermodel would probably be heavier than if it were analyzed using the second-order model with a similarnumber of degrees of freedom.

Conclusions of Investigation IIBased on this investigation, it can be concluded that second-order solid elements give satisfactory stresspredictions with as few as two elements through the thickness of a structure. This increase in accuracy overa first-order element (TET4) is primarily due to the use of multiple integration points in the higher-orderelement formulation. Additionally, although the coarser meshes are less accurate in their prediction ofstress levels, they are conservative. This indicates that even coarse solid models are acceptable for thedesign of structural components, as long as second- or higher-order elements are used. The mesh size isthen determined by the accuracy required in modeling the geometry of the component in question. Anotheradvantage of second- and higher-order elements is that they are not restricted to straight edges, but canmodel curved surfaces and solids with reasonable accuracy. It is therefore not necessary to use very smallelements, as is the case with first-order elements.

It can also be concluded that as mesh density increases, first-order TET4 elements converge towards thetheoretical stress from lower initial predictions. Stress levels predicted using coarse meshes areunconservative. Second-order TET10 elements converge from higher stress levels as mesh density isincreased. Stress levels predicted using coarse TET10 meshes are overly conservative.

One limitation of second- and higher-order elements is that they are not supported in MSC/NASTRAN asnonlinear elements. Only first-order elements are supported in nonlinear analyses. Since these elementsproduce unreliable stress predictions in areas of stress concentration, it is not advisable to useMSC/NASTRAN solid elements to analyze structures that behave in a nonlinear manner.

Figures and Tables

Figure 1. Stepped shaft geometry and applied load.

Figure 2. TET4 Mesh

Figure 3. TET10 Mesh

Figure 4. Rectangular bar geometry and applied load.

1000 lbs4 in

2 in

10 in10 in 1 in

R = 0.25 in

500,000 lbs2 in2 in

10 in 1 in

500,000 lbs

1 in

XY

Z

XY

Z

XY

Z

XY

Z

XY

Z

XY

Z

XY

Z

XY

Z

Figure 5. TET10 models of rectangular bar.

0

200

400

600

800

1000

1200

1400

1600

1800

1 2 3 4 5 6

Number of Elements Through Beam Thickness

Pea

k A

xial

Str

ess,

ksi

TheoryTET4TET10

1500 ksi

Figure 6. Peak stress convergence, rectangular beam model.

Global EdgeLength

TET4Nodes

TET10Nodes

TET4Elements

TET10Elements

TET4D.O.F.

TET10D.O.F.

0.250 in 15,723 119,432 84,531 84,533 46,287 355,0110.500 in 1,962 13,912 9,157 9,169 5,646 40,9020.750 in 647 - 2,593 - 1,827 -1.000 in 321 1,948 1,065 1,069 888 5,619

Table I. Variation of model size with element topology.

Number of ElementsThrough Beam

Thickness

DegreesOf

Freedom

DownwardDeflection ofLoaded Point

Maximum AxialStress (Upper &Lower Surfaces)

1 282 8.2320” 1,600,000 psi2 918 8.4573” 1,510,000 psi3 2,406 8.4807” 1,500,000 psi4 4,932 8.4939” 1,500,000 psi

Table II. Deflection and stress results for rectangular bar model (TET10).

Number of ElementsThrough Beam

Thickness

DegreesOf

Freedom

DownwardDeflection ofLoaded Point

Maximum AxialStress (Upper &Lower Surfaces)

1 69 1.6098” 537,000 psi2 186 5.5371” 1,160,000 psi3 441 6.2822” 1,300,000 psi4 828 6.8980” 1,570,000 psi5 1,728 7.5786” 1,610,000 psi6 2,433 7.8700” 1,570,000 psi

Table III. Deflection and stress results for rectangular bar model (TET4).

Trademark AcknowledgementsNASTRAN is a registered trademark of NASA. MSC/NASTRAN is an enhanced, proprietary versiondeveloped and maintained by the MacNeal Schwendler Corporation.MSC/PATRAN is a registered trademark of the MacNeal Schwendler Corporation.Unigraphics is a registered trademark of Electronic Data Systems.