easily made errors in fea 1

8/8/2019 Easily Made Errors in FEA 1

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THE PROBLEM WITH BUILT-IN SUPPORTS A classic problem involves finding the maximum princi-pal stress in a plate with a center hole. Two-dimensionalplane stress is assumed. Results show the highest stressequal to 377 MPa, quite close to 370 MPa predicted by theanalytical solution. But 377 MPa is not the highest maxi-mum principal stress in this model. The highest one is ac-tually infinite. Why? Because the formula that predicts370 MPa assumes tension is applied to both sides of theplate, while our model shows the left vertical edge rigidlyconstrained. The constrained edge tries to shrink side-ways under the tensile load, the effect of Poissons ratio.

The tensile strip with a hole looks innocentenough. Its 200 100 mm, and 10 mm thick.The hole is 40 mm diameter. Load is 100,000 Nin tension and the material has a modulus of200,000 MPa and Poissons ration of 0.27.Results are in following images.

Easily made errorsmar FEA results

MACHINE DESIGN SEPTEMBER 13, 2001 www.machinedesign.com 69

CADEdited by Paul Dvorak

Even before you mesh a part, you may have introducedthe potential for erroneousstresses and deflections.Paul KurowskiPresidentDesign Generator Inc.London, Ontario, Canada

E very day brings news of powerful analysissoftware for shortening development cycles.Trying to keep pace with such progressmakes it easy to forget that all those greatprograms provide accurate results only when

properly used.One of the stumbling blocks on the road to quick and

correct FEA results includes idealization errors, thosethat come from simplifying the real world. They arecommon yet often unrecognized and dangerous. In thisdiscussion of them, lets use the term finite-elementmethod (FEM), the foundation of FEA, because it em-phasizes the underlying numerical method.

Its also useful to briefly review the four steps pres-ent in any FEM project. Step one transforms boundaryconditions, material properties, and geometry intoterms acceptable for analysis. Simplifications are al-most always necessary, but they introduce idealiza-tion errors. Some are benign but others are hazardous.

And dont be too smug if youre using advanced soft-ware because idealization errors have nothing to dowith mesh, elements, or the type of solver used.

Mathematical models formed by simplifications andcontaining idealization errors then take the form of differential equations. These are usually too difficultto solve analytically, so solvers use an approximate nu-merical method. Most often we choose the FEM, whichhas dominated engineering analysis because of itsadaptability and numerical efficiency. FEM requiressplitting the continuous model into discrete regions orelements. Call this step two, another operation withopportunity for errors.

Step three, the solution, leaves little opportunity foruser intervention. The software can introduce numeri-cal round-off errors, but recent programming mini-mizes their impact.

And in step four, after solving a model, you apply re-sults to the design. At that time, its important to recallthe assumptions made during the simplifications andmeshing because they have influenced the results.

The boxes that follow show several particular ideal-ization errors that are introduced in step one. And thearticle avoids referencing FEM-specific issues, such aselements and meshing, as much as possible. Howeverwe must touch on the convergence process and theidea of degrees-of-freedom in FEM models becausewell use those as tools to expose the problems of ideal-izations. To illustrate the convergence process, the ex-amples that follow are meshed and solved with a pro-gram that uses p-elements, but the problems they il-lustrate apply to all FEM-based analysis or any otherkind of numerical analysis for that matter.

A plate in tension


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A DOMINATING FAILURE MODESometimes its easier to think of safety in terms of stress levels whileforgetting other issues. For example, while analyzing A curved I-beamit is easy, yet deadly, to forget that buckling, not the stress, will definethe structures safety.

With a little work, we could fillthis entire issue with examples of modeling errors introduced duringthe idealization process. Indeed,reducing 3D models to 2Drepresentations, beam and shellmodeling, defeaturing andgeometry clean up, all that is doneto simplify models and allowmeshing. Each process abounds in

traps awaiting an unsuspectinguser.Modeling errors originate from

incorrect mathematical models.Some modeling errors, such assingularities, can be revealed (butnot cured) using the FEM-basedconvergence process. Most remainhidden. The only defense is fullunderstanding of the analyzedproblem.

But built-in supports prevent theedge from shrinking. There-fore, the mathematical model, asshown in A closer look at the cor-

ners predicts infinite stresses inboth corners. (Infinite stress mayalso be called singular stress.)

Large elements, such as thosein Results for the test plate , letthe FEM model overlook high

stresses altogether. Even thougheach iteration adds degrees of freedom (dof) to the model, thereare still not enough dof to detect

those very localized stresses inthe two corners. A convergence of the highest principal stress inthe chart Convergence for the

plate model refers to stress at thehole, not to corner

stresses, and pro- vides another ex-ample of how cor-ner singularities goundetected.

A closer look atthe corners moreclearly shows that

placing small ele-ments in cornersand using nonadap-tive convergence,reveals high cornerstresses. And thegraph Convergencein the corner showshow stresses there

grow with each iteration. It startshigh from the beginning. This isbecause small elements can de-tect stress concentrations even at

low p (polynomial) levels. Perhapsmost interesting is that stressshows no signs of convergence.Each consecutive iteration simplyproduces higher stress.

70 MACHINE DESIGN SEPTEMBER 13, 2001 www.machinedesign.com

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Results for the test plate

Results for the plate show a high stress at thehole boundary and in the corners. The146 MPa is high enough to warrant furtherinvestigation. A closer examination will revealthat the 146 MPa found there is meaningless.

Convergence for the plate model

A plot of the highest maximum principle stresses forthe plate show stresses around the hole.

Buckling is not obvious, butits the dominating mode offailure for the curved beam.

A curved I-beam


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So you might ask: Which modelis correct, Results for the test plateor A closer look at the corners ? An-swer: Both are correct if we are in-

terested in stress around the hole,but neither one is correct if we areinterested in corner stresses be-cause those are singular, a condi-tion that causes ever increasing

stress while adding more dof.Consequently, the model cant beused for finding corner stresses.Discretizatio n (step two) just

masks singularities introduced instep one, which become visibleonly by looking for convergence one way to spot singularities. Itwould be easy to produce results

showing 1,000 or 1 millionMPa stress. Just make thecorner elements smallenough. The same wont hap-pen around the hole. Stress insingular locations (corners)returned by the discrete FEMmodels are purely discretiza-tion dependent and therefore

completely unreliable.We should emphasize thatour plate with the centerhole illustrates benign ide-alization errors introducedin by a rigid support. Wehad to hunt for singularitiesand use the t r ick of verysmall e lements and non-

adaptive convergence to revealthem. Benign singularities arecommon and pract ical ly un-

avoidable. Even the followingtest bracket is not f ree fromthem. In pract ice, we ei therdont notice pesky stress concen-trations or learn to ignore them,but it is still worthwhile to re-member about occasional trou-bles caused by Mr. Poisson andhis ratio.

The square area behind the plate shows amagnified corner element. It shows a highstress hidden in the results of the previouscontour plot of the plate. Fortunately, thissingularity is benign and, most often, we can

just ignore it.

A closer look at the corners

MACHINE DESIGN SEPTEMBER 13, 2001 www.machinedesign.com 71

CAD Convergence in the corner

The highest maximum principal stress inthe plate shows no signs of convergingto a finite value. Each subsequentiteration produces higher stress.

THE HAZARD OF MEMBRANE STIFFENINGThe next objective is to analyze a deflection in the model in A

flat, round membrane . Deflections wont be more than20 mm, which is small compared to the 1,000-mm diameter.So we run a linear-static analysis which assumes small dis-placements, another grave error. Results show a maximumdisplacement of 35.5 mm, much more than from a physicaltest. What went wrong?

Small-displacement theory assumesmembranes dont change stiffness as theydeform and so it accounts only for the ini-tial bending stiffness. However, the ma-terial under deformation acquires mem-brane stiffness, which adds to the origi-nal value for bending stiffness. As a re-sult, the overall stiffness increases as themembrane deforms. A nonlinear-geome-try analysis, also called large-deforma-tions analysis, is required even thoughthe displacement magnitude seemssmall. The difference between results re-turned by linear and nonlinear modelsappears in the red and blue graph.

Deflection per theory

The red graphassumes a large-deformationsolution while theblue graphassumes small-deformation theory.

A flat, round membrane

The membrane is supported around its diameter.The material has E = 73,000 MPa and = 0.33


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SHARP RE-ENTRANT EDGES HINTAT STRONG STRESS SINGULARITIESIdealization errors are not always mild. They often lead to hazardoussituations. For example, find the maximum principal stress in a simple

2D plane-stress model for an L-shaped bracket. The Maximum principal stresses seem acceptable. As expected, the highest stress is in thecorner. But is the maximum-principal stress really equal to 79 MPa?Examining the chart Stress convergence in the L-shaped bracket hintsthat something is wrong. Stress values climb with each iteration andthis time we dont have to hunt diverging stress, we just use a standardadaptive-convergence process. To find out what is going on, do as be-fore: place smaller elements in areas of interest. A closer look at thebracket shows a maximum stress of 415 MPa and again the conver-gence curve relentlessly keeps climbing. Can we say which result betterapproximates reality: 72 MPa or 415 MPa? No, they are the same theyre wrong. Thats because the correct FEM model created in step 2is based on the wrong math model created in step 1. The area of interestholds a singularity so no meaningful results can be produced.

As revealed by the convergence process, reducing element size andupgrading element order lets us chase infinity 415 MPa is just as faraway from infinity as 79 MPa. This time we cant ignore the singularityand still produce meaningful results because we are interested in stressin the location where its singular.

The remedy is to change the model. One way is to add a fillet, whichis always present in real parts anyway. You can also avoid stress singu-larities by using a different material model, for example the elastic-plastic model instead of a linear-elastic material. The elastic-plasticmaterial model would put an upper bound on stress, and instead of pro-ducing meaningless high stress, a plasticity zone would be formed.

Geometric details, such as a fillet, are frequently difficult to mesh. Atechnique called defeaturing removes such offending geometry to sim-plify meshing. Defeaturing, however, can be dangerous. Take the Sup-

port bracket for example. If the indicated fillet is removed, the modelcan still be used for deformation or modal analysis. It doesnt have a dis-placement singularity. Displacements are still finite despite the sharpedge at the base of the center bosses. But removing a fillet creates asharp re-entrant edge constituting a stress singularity which makes itimproper and potentially dangerous for stress analysis. Remember,stress results are meaningless around the sharp reentrant edges (thearrow). This is the area of most engineering interest inthe bracket. We can producestresses as low or high as wewant by manipulating ele-ments size and order. Try ityourself.

Convergence should deter-mine whether or not the re-sults are significantly dis-cretization dependent. Onlywhen results converge can weuse them with confidence tomake design decisions. TheL-shape bracket demon-strates the opposite: resultsare entirely discretization de-pendent and completelymeaningless.


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Stress results look plausibleand might fool us intobelieving that 79 MPa is agood answer.

With small elements placed aroundthe sharp inside corner, the solvernow finds much higher stress. But isthis result closer to the truth than thatprovided in the previous model?

Maximum principal stresses

A closer look at the bracket

It may be tempting to remove filletsat the base of the center bosses foreasier meshing, but results could beeither meaningless or dangerous iffillets are in the area of interest forstress analysis.

The highest maximum principal stress inthe L-shaped bracket shows no sign ofconverging.

Asupport bracket

Stress convergence in the L-shaped bracket


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DISPLACEMENT SINGULARITIESSingularities can also affect displacements.Imagine a beam in bending supported at oneend by two welds, as in A cantilever beam . Theobjective is to calculate stresses and deflec-

tions. Model geometry and boundary condi-tions lend themselves to a 2D plane-stress rep-resentation. Because the weld is small in com-parison to the overall beam, we decide tomodel them as point supports (a bad mistake)surrounded by small elements to more pre-cisely capture the local stress distribution.

A stress convergence curve (not shown) reveals aproblem: The curve is not converging. Clearly, thismodel is not useful for stress analysis becausestresses are discretization dependent, that is, theyare singular. Notice again that the error in defini-tion of the math model is revealed by the conver-gence curve. Looking at results of just one single run

would have been misleading. So a useful stressanalysis is out of thequestion. But can we usethe model for deflectionanalysis? Displacements

for the cantilever beamlooks okay, and the maxi-mum deflection is justabove 8 mm as predictedby beam theory. How-ever, a closer examina-tion of the graph revealsan unnerving fact: Itscurve is not converging.Instead, its slowly butsurely increasing. How isthat possible? Point sup-ports at the corner showstress that tends to infin-ity. In fact, the strainalso tends to infinity. Af-

ter all, it is proportional to stress. (The linear rela-tion between stress and strain is expressed byHookes law, or = E ) With strains tending to infin-ity, displacements have no choice but to follow.

The model in A cantilever beam is completelywrong. Point support is a cardinal sin of FEMmodeling. So why do all FEM programs include

point supports? Because they can be used to re-strict rigid body movement when a support generateszero reaction. Point supportsare also useful in beam-ele-ment models.

One can conclude that bothtypes of modeling errors(sharp re-entrant corners andpoint supports) originate froman improper definition of themathematical model uponwhich the FEM model is con-structed. But the errors havenothing to do with FEM. Wecommitted them in step onebefore FEM even entered thescene. So the problems cant befixed using FEM. Convergencestudies, however, can identifythe benign from the severe.

STEPS IN FEM PROJECTStep 1 Simplifications of reality

lead to a mathematicalmodel and introduceidealization errors.

Step 2 Replacing a continuum witha set of discrete regions(meshing) introducesdiscretization errors.

Step 3 The solution of a discretesystem introduces numericalerrors.

Step 4 Analysis of resultsintroduces interpretationerrors.

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A FINAL WORD ONCONVERGENCEThe convergence process adds degrees of freedom to the FEM model to see how re-sults change. Degrees of freedom are addedeither by using more elements (mesh refine-ment, called h convergence), or by usinghigher-element orders (p-convergence).

Convergence should demonstrate that re-sults converge to a finite value and, there-fore, are not significantly dependent on thechoice of discretization. If the process showsthat results are significantly discretizationdependent, then the results are unreliable.

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Beam displacements are diverging to infinity, anindicator of a singularity.

Displacements in the cantilever

beam

The cantilever beam has a material E = 200,000 MPa, = 0.27,and its 20 mm thick.

A cantilever beam

easily made errors in fea 1

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