Finite Element Analysis and Optimization IntroductionPeter BudgellE-mail Address: Please see main page. Return to Home Page FEA Modeling Issues Page ANSYS Tips Page Finite Element Analysis A few years ago, Boeing released a few television episodes on the development of their 777 aircraft. I found them fascinating. They included a destructive test of a prototype wing. The wing was loaded in a test rig in a manner that simulated the application of severe aerodynamic pressure during flight. As I recall it, the wing was specified to meet a load of 150% of the design load. The design load was the maximum g-load that the aircraft was permitted to experience. (If I remember correctly, the design load was 4 g's, although it would never be intentionally operated at that level.) In the test, the wing failed at 153%. Since there is a severe weight penalty in aricraft design and significant overdesign would be unwanted, I consider this test result to be a fantastic accomplishment. The precision of this result must have required superb Finite Element Analysis, as well as material characterization, dimensional tolerancing, exact manufacturing methods, and a precise test rig. Needless to say, the deformation of the wing was "Large Displacement", so nonlinear analysis must have been used. I imagine that it was an example of FEA at its very best -- a variation on The Wonderful One-Hoss Shay in which no component is over- or under-designed. What is Finite Element Analysis? Finite Element Analysis (FEA) is a computer-based numerical technique for calculating the strength and behavior of engineering structures. It can be used to calculate deflection, stress, vibration, buckling behavior and many other phenomena. It can be used to analyze either small or large-scale deflection under loading or applied displacement. It can analyze elastic deformation, or "permanently bent out of shape" plastic deformation. The computer is required because of the astronomical number of calculations needed to analyze a large structure. The power and low cost of modern computers has made Finite Element Analysis available to many disciplines and companies. In the finite element method, a structure is broken down into many small simple blocks or elements. The behavior of an individual element can be described with a relatively simple set of equations. Just as the set of elements would be joined together to build the whole structure, the equations describing the behaviors of the individual elements are

joined into an extremely large set of equations that describe the behavior of the whole structure. The computer can solve this large set of simultaneous equations. From the solution, the computer extracts the behavior of the individual elements. From this, it can get the stress and deflection of all the parts of the structure. The stresses will be compared to allowed values of stress for the materials to be used, to see if the structure is strong enough. The term "finite element" distinguishes the technique from the use of infinitesimal "differential elements" used in calculus, differential equations, and partial differential equations. The method is also distinguished from finite difference equations, for which although the steps into which space is divided are finite in size, there is little freedom in the shapes that the discreet steps can take. Finite element analysis is a way to deal with structures that are more complex than can be dealt with analytically using partial differential equations. FEA deals with complex boundaries better than finite difference equations will, and gives answers to "real world" structural problems. It has been substantially extended in scope during the roughly 40 years of its use. How is Finite Element Analysis Useful? Finite Element Analysis makes it possible to evaluate a detailed and complex structure, in a computer, during the planning of the structure. The demonstration in the computer of the adequate strength of the structure and the possibility of improving the design during planning can justify the cost of this analysis work. FEA has also been known to increase the rating of structures that were significantly overdesigned and built many decades ago. In the absence of Finite Element Analysis (or other numerical analysis), development of structures must be based on hand calculations only. For complex structures, the simplifying assumptions required to make any calculations possible can lead to a conservative and heavy design. A considerable factor of ignorance can remain as to whether the structure will be adequate for all design loads. Significant changes in designs involve risk. Designs will require prototypes to be built and field tested. The field tests may involve expensive strain gauging to evaluate strength and deformation. With Finite Element Analysis, the weight of a design can be minimized, and there can be a reduction in the number of prototypes built. Field testing will be used to establish loading on structures, which can be used to do future design improvements via Finite Element Analysis. For an interesting note on the very beginnings of Finite Element Analysis, see this web page on Courant. What Is Required To Do Finite Element Analysis? Finite Element Analysis is done principally with commercially purchased software. These commercial software programs can cost roughly $1,000 to $50,000 or more. Software at the high end of the price scale features extensive capabilities -- plastic deformation, and

specialized work such as metal forming or crash and impact analysis. Finite element packages may include pre-processors that can be used to create the geometry of the structure, or to import it from CAD files generated by other software. The FEA software includes modules to create the element mesh, to analyze the defined problem, and to review the results of the analysis. Output can be in printed form, and plotted results such as contour maps of stress, deflection plots, and graphs of output parameters. The choice of a computer is based principally on the kind of structure to be analyzed, the detail required of the model, the type of analysis (e.g. linear versus nonlinear), the economics of the value of timely analysis, and the analyst's salary and overhead. An analysis can take minutes, hours, or days. Extremely complex models will be run on supercomputers. Usually, "Faster and Bigger is Better!" if you can afford it -- this might be called "The fundamental theorem of finite element analysis." People who analyze large structures, or run nonlinear models, will tell you that you can never get a machine that is as fast as you would like -- hopefully, good business sense will prevail. Many things can be analyzed in good detail on computers costing from roughly $2,000 to $20,000. Budget for a large monitor, fast graphics card, large hard drive, large memory, fast processor(s), and an appropriate printer -- faster printers are to be preferred. Color is usually desirable, although you can sometimes live without it by doing gray-scale plots where the palette for contour maps has been changed to a progressive gray-scale. Higher prices will let you consider computers with more memory, larger hard drives, and one or more high-speed processors. Depending on the complexity of the structures to be studied and the volume of manufacturing, the expense for FEA hardware can be small in comparison with the savings in weight and construction cost that can result from design improvements, and speed of analysis. The expense can be very small in comparison to the cost of a failure. Problem is, when there is no failure, how do you "prove" that the investment was warranted? It helps if senior management trusts engineering staff and is at least somewhat familiar with the nature of their work. The background of a finite element analyst includes an understanding of engineering mechanics (strength of materials & solid mechanics) as well as the fundamentals of the theory underlying the finite element method. The analyst must appreciate the basics of numerical methods. An engineering degree is typical, though not an absolute requirement. Use of a particular finite element program requires familiarity with the interface of the program in order to create and load the models, and to review the results. To do the work well requires experience, comprehension of structures and their classical (manual analytical) analysis, an understanding of a variety of FEA modeling issues (see link below), and an appreciation of the specialized field in which the design work is taking place. Click for Information onFEA Modeling Issues

Types of Analysis on Structures Structures can be analyzed for small deflection and elastic material properties (linear analysis), small deflection and plastic material properties (material nonlinearity), large

deflection and elastic material properties (geometric nonlinearity), and for simultaneous large deflection and plastic material properties. By plastic material properties, we mean that the structure is deformed beyond yield of the material, and the structure will not return to its initial shape when the applied loads are removed. The amount of permanent deformation may be slight and inconsequential, or substantial and disastrous. In metal forming, deformation is substantial and intentional (consider the shaping of a fender for an automobile). In some structures, "shakedown" producing residual stress due to local permanent deformation, may in some circumstances reduce fatigue problems in zones that will remain in compressive stress as a consequence. An example is the hydrostatic pressure test on a new, post weld heat treated, steel pressure vessel (opinions on this may vary). In this test the pressure may be taken to 1.5 times the design pressure. Local yielding means that some zones will usually be in compressive stress during conventional use of the pressure vessel, and may be less prone to fatigue crack development. By large deflection, we mean that the shape of the structure has changed enough that the relationship between applied load and deflection is no longer a simple straight-line relationship. This means that doubling the loading will not double the deflection. The material properties can still be elastic. In addition to analyzing structures for their stress and deflection, other typical analyses are an evaluation of the natural frequency of vibration, and calculation of buckling loads. Steady state, transient, and random vibration behavior can be analyzed, too. Loads on structures can be represented by using the force of gravity on the mass of the structure, by applying distributed pressure over surfaces of the structure, or by applying forces directly to positions in the structure. Centrifugal load can be entered by indicating the axis for the motion, and the rate of rotation. Displacements of the structure can be specified at positions in the structure. This can include boundary conditions that imply symmetric structures where only a portion of the structure is modeled. Other boundary conditions will indicate where the structure is supported against movement, by the outside world. Temperature distribution that causes thermal expansion and stress can be applied directly to nodes or to elements with appropriate commands. Uniform temperatures and reference temperatures can also be applied to full models. Typical Modeling Difficulties Certain modeling problems can be considered "typical". (Note: The Toolbox in ANSYS has, by default, a save model button. This saves the database under the jobname. The button can be used before employing any "dangerous" operation, such as modifying the position of a group of keypoints or a group of nodes. Then, a single press to the restore button will restore the state of the model when it was last saved. Regular use of the save button can prevent the loss of time-consuming work. There is no automatic save feature.)

A typical user modeling problem is the case of keypoints, lines, areas, volumes, nodes, and elements that are identical and occupy the same space. This can lead to erroneous models. Proper use of the merge command can eliminate many instances of these problems. The merge can fail if, for example, two elements share the same space, but were defined via alternative sequences of nodes (e.g. elements in the same place, one numbered by nodes selected clockwise, the other counterclockwise). Another problem is failure of keypoints, or lines, or areas to be shared by higher geometric modeling entities. When this happens, the higher entities are not "fused" or "welded" together as intended. Consequently, the elements will not share nodes along what should have been the common boundary. The analyst must always use caution and double-check everything while developing a model. A problem most users will encounter is to make a change to a model in error, long after the database was saved. The user will have to learn to use a text editor on the log file, to extract that portion of the log file after the last time the database was saved, or retrieved (whichever was most recent). Remove the offending command. That portion of the log file will have to be run on the model as it was the last time it was saved or retrieved. Make sure you are in the correct part of ANSYS (usually /PREP7) when you read in the instructions with /INPUT. The same method can apply if your computer is subject to a power failure, or if ANSYS crashes without leaving an "ansabort.db" file. After restarting, take a text editor to the log file, and re-run the appropriate instructions on the model database file as it was when last saved or retrieved. I've done this more times than I care to remember, though less often with more recent ANSYS revisions and more experience. The most common of all errors in Finite Element Modeling is the incorrect application of loads and boundary conditions. This must be thought about very carefully. Most models (not all) are prevented from undergoing free body motion in 2-D or 3-D space, by eliminating at least a minimal number of degrees of freedom (2 translations plus 1 rotation in 2-D, and 3 translations plus 3 rotations in 3-D). Rotations can be prevented either by having constraints on translations at enough distinct nodes in space, or by directly constraining a rotational degree of freedom at a node. A common check on results is to see whether the sums of the reaction forces at the constrained nodes equal the sums of the applied forces and gravity loads. On rare occasions, the Finite Element Analysis model may not be bug-free as a result of imperfect programming of the FEA software, not the user's mistakes. An FEA software package has to keep track of the relationships between keypoints, lines, areas and volumes, including the changes that result from Boolean operations. In addition it must keep track of the relationships between those geometric entities and the nodes and elements that result from meshing, even when geometric entities are cleared, and the model is modified. The process for coordinating all the pointers and tables that are generated and changed is not perfect. There are times when the relationships will be erroneous. An example will be nodes that cannot be deleted because ANSYS claims they are attached to elements, yet the ESLN command will not select any elements from them.

The same thing can happen with geometric entities. I have seen an area for which the software thought one of the bounding lines was in a distant part of the model. The area would not mesh. If the user cannot step back to an earlier "good" saved database, then the offending parts may have to be deleted, and generated from scratch. The process of archiving and restoring a model via CDWRITE and IGES can sometimes leave the misbehaving portions of a model behind, if only the "good" portions of a model are selected. The good news is that each release of the FEA software eliminates more of these bugs. ERRORS ! If you have worked with inferential statistics, you will have heard of Type I and Type II errors. There are also major types of errors in FEA work. They could be classified something like this: "Type 0" -- the model contains fundamental flaws: parts missing, not connected, or details are inappropriate. "Type I" -- The model may not properly represent the structure as built, or recorded in the engineering drawings. "Type II" -- The loads or boundary conditions may not represent the real world or the customer specification. "Type III" -- failure to consider that a particular type of analysis was needed. "Type IV" -- experience of the analyst inappropriate to the task at hand plus inadequate training and supervision. "Type V" -- wrong element type and associated option settings. "Type VI" -- errors in applying design codes. "Type VII: -- computer is too small and slow to use fine meshing, run nonlinear analysis, and review results in sufficient detail. The list goes on... The "Type III" problems can be "unsettling". Several years and a few employers ago, I saw troubles when heat treating deformations were not considered soon enough during design (not my design!). It is not impossible to imagine that failure to consider buckling, or bending-compression on a beam with nonlinear analysis, or reduction of bending cross section in a hollow tube subjected to external pressure, or a prying load, or an unusual load configuration, or a fatigue detail, or overloading or fracture-inducing loading of a weld, or flow induced vibrations, or missing stiffeners, can lead to "unhappy events". Element Connectivity Error, 8-Node Curved Shell Elements In this image, the red stiffener was intended to be welded to the purple pipe. Note that the elements of the red stiffener do not match up with those on the pipe. There is no connection, and the meshing was done independently. This is due to a geometric modeling error by the user (me). There are superimposed curved lines where the interface is located. There should have been a shared line for the connection to have worked. I found this only because of careful examination of the model -I had already run a stress analysis. What to do about these error concerns? Read and think. Share and listen to ideas and concerns with others. Review your own work, and the work of your co-workers. (Recently an experienced co-worker who does not even do FEA work asked me if I had eliminated the added mass of water in pipes when evaluating shipping loads on a product.

I hadn't. Eliminating the added mass got rid of a high-stress problem. These errors are very easy to make.) Be friendly. Communicate with other departments. Have a check list and design reviews. Never use FEA blindly, or believe the results of an analysis without some critical review. Accept a critical review without taking it personally. Develop a good understanding of the intent of the design codes that regulate your work. Consult an expert when it is appropriate. Pay attention to the ethics and standards of your professional association. Choose your employer wisely. (Some of these things you were supposed to have learned in Kindergarten, but life isn't always that simple.) Go Up to Top of Page

Engineering Optimization What is Optimization? Optimization of an engineering design is an improvement of a proposed design that results in the best properties for minimum cost. One of the simplest examples is determining the shape of a fence that will enclose the most area. If the fence can be any shape, but only a certain amount of fencing is available, then a circle will enclose the most area with the given amount of fencing. In order to minimize the amount of steel used in manufacturing a cylindrical tin can, a certain relationship between the diameter of the can and the height of the can is found. This will enclose a volume with the least amount of steel used for the surface area. In each these simple optimization examples, there have been two criteria -- one was a criterion to be made best. In the fence, it was the enclosed area. In the tin can, it was the amount of steel in the body. The other criterion was a constraint on the design. In the fence, it was the amount of available of fencing material. In the can it was the specified volume to be enclosed. In more elaborate problems encountered in engineering, there will be a property to be made best (optimized) such as weight or cost of a structure. Then there will be constraints, such as the load to be handled, and the strength of the steel that is available. Constraints on the design are of two types. One is Equality Constraints. An equality constraint specifies a property of the design that must hit a specified value. In our fence example, the length of the fencing (perimeter of the enclosure) was a certain number. This is an equality constraint. In a structure, the steel throughout the structure will need to be kept below the yield strength (localized stress concentration regions excepted). In many parts of the structure, the stress will be below yield stress. Consequently, there is an Inequality Constraint. In an inequality (or one-sided) constraint, a property of the design will be required to be kept above or below some number. Once a preliminary design has been developed, variations in some of the dimensions of the design can be evaluated. The particular dimensions that will be permitted to be

changed are the degrees of freedom, known simply as variables. Some of the resulting properties of the design will be required not to exceed certain boundary values, or constraints. There may be constraints on the degrees of freedom, as well as on derived properties, such as the stress in a structure. If the initial design was feasible it did not violate any constraints. Variations on the design may result in properties that are an improvement. When the degrees of freedom have been set to values that give the best possible properties for the design, the design is said to have been optimized. In the case of the fence above, if we started out by trying a rectangular shape, and eventually arrived at the circle, we would have optimized the design. This would require that there was no constraint on the permitted shapes, such as requiring that the fence be rectangular! What is Required to Do Optimization? Classical optimization is done manually with algebra, calculus, and the calculus of variations. Problems with a variety of constraints may be handled symbolically using Lagrangian multipliers. Many modern design problems are too complex to be handled with purely algebraic symbolic methods. Computers are used for numerical assessment of variations in a design. Computer codes to optimize design have been developed ever since the inception of modern digital computer use. Today, codes for optimization can be acquired for free, or purchased as part of mathematical subroutine libraries. Some coding of the problem to be solved is required, as are calls to the optimization subroutines to be employed. A faster method suitable for many optimization problems is to use the optimization engine bundled into spreadsheet programs, such as Microsoft Excel. (This is an option that you must intentionally install.) Then the only significant work required is to put the problem in spreadsheet form. If the problem is difficult to program, Excel spreadsheet cells can reflect the results of code written in Visual Basic. The spreadsheet program then does most of the work, and the user interface is easy to construct. In Finite Element Analysis, optimization is more difficult, because each variation in the design takes a significant amount of time to evaluate. This can make brute-force iterative optimization excessively time consuming. The analyst will usually attempt nonlinear optimization under both equality and inequality constraints, when optimization is used with FEA. One approach is to run a small set of variations on the design, then fit curves to the relationship between degrees of freedom, and the properties of the optimization

function and properties to be constrained. Software can be used to search this design space, and suggest good starting points for the next set of design checks. Finite Element Programs are beginning to have tools of this type built in. Alternatively, external tools can be used to evaluate the design that is being tested in a small set of analyses. For further information, see the Optimization section of the ANSYS Analysis Guides (included in the Help Files). It should be noted that substantial new optimization tools have been added to the ANSYS product line in recent releases. Optimization versus Innovation Optimization, as I have used the term above, implies changing the setting of independent variables in a continuous manner, to get best possible structure properties. An approach like this will look at variations in an existing configuration, but not invent significantly new configurations for a structure. When trying to improve a structure, or respond to a defined need to support a set of loads with a newly created structure, there is far more involved than the rather narrow definition of optimization that I have pursued above. The analyst should keep in mind that creativity in finding a structural configuration should not be sidetracked by a narrow approach to optimizing an existing shape. Innovation is often more important in the early stages of a new job.

Finite Element Analysis Modeling Issues and Ideas

Peter Budgell

E-mail Address: Please see my main page. 1998 & 2004 by Peter C. Budgell -- You are welcome to print and photocopy these pages. These tips and comments are intended for user education purposes only. They are to be used at your own risk. The contents are based on my experience with ANSYS 5.3 -- more recent versions may change things. The contents do not attempt to discuss all the concepts of the finite element method that are required to obtain successful solutions. It is your responsibility to determine if you have sufficient knowlege and understanding of finite element theory to apply the software appropriately. I have attempted to give accurate information, but cannot accept liability for any consequences or damages which may result from errors in this discussion. Accordingly, I disclaim any liability for any damages including, but not limited to, injury to person or property, lost profit, data recovery charges, attorney's fees, or any other costs or expenses.

Return to Main Page FEA and Optimization Introduction Page A Quick Overview of FEA ANSYS Tips Page My Collection of Tips on ANSYS Use. Modeling issues include a host of topics. I will mention some that have been relevant to my experience. After almost six years of continuous use of the ANSYS program, I continue to learn new features of the software, discover more ways to represent or approximate features, and develop new ways to get useful output information from the models.Example of Approximation: I wrote a macro to give the surface area (one side) of a previously selected set of shell elements. A force divided by this area can be applied as pressure over these shell elements, for smooth force application, if the elements are flat. Writing the macro required a few lines of code that: determine the number of elements, get the first element identity, create an array of correct size to hold data, put the areas of the elements into the array, sum the array entries, report the result, and delete the variables and array. NOTE: The user must be careful to apply the pressure to the CORRECT FACE of the set of shell elements. Force or pressure on a flat shell may require Large Displacement (geometrically nonlinear) analysis.

CONTENTS: 1. FEA is Approximate 2. Meshing 3. Shell versus Solid versus Beam Elements 4. Reduction of the model to a shell structure 5. Pressure on Shell Elements 6. Reflecting Part of a Model 7. Representation of Bolted Connections 8. Warning about Nodal Coupling 9. Development of geometry in which surfaces cut each other with shared lines 10. Application of boundary conditions 11. Application of loading 12. Pressure loading of a wall containing granular material 13. Deformation of thin flat panels by pressure loading 14. Use of Units 15. Buckling analysis and failure 16. Ramping Loads in ANSYS

17. Plotting results 18. Coping with Design Changes 19. Computer Aided Engineering Environment 20. FEA versus Hand Calculations 21. Choosing an Appropriate Shell Element 22. Using P-Elements 23. Harmonic Response 24. Failure Modes to Consider 25. Stress Limits and Margin of Safety 26. Representation of a group of bolts (or rivets) 27. Adequate Computer Hardware for FEA MODELING ISSUES that are faced include (but are by no means limited to): FEA is Approximate. The first issue to understand in Finite Element Analysis is that it is fundamentally an approximation. The underlying mathematical model may be an approximation of the real physical system (for example, the Euler-Bernoulli beam ignoring shear deformation). The finite element itself approximates what happens in its interior with interpolation formulas. The interior of a 2-D or 3-D finite element has been mapped to the interior of an element with a perfect shape, so a severely distorted element can not deform in a manner that has an accurate match to the real physical response. Integration over the body of the element is often approximated by Gaussian Quadrature (depending on the element, an analytical integral can be either impractical or exceedingly difficult -- I've done a few with the computer algebra system MACSYMA and the number of terms can explode unless constants are extracted during the derivation and the integrand is kept factored; some elements are said to be more accurate with numerical integration at a limited number of points). The continuity of deformation between connected elements is interrupted at some level. Badly shaped (by distortion, warping or extreme aspect ratio) elements can give less accurate results. Elements approximate the local shape of the real body. Numerical analysis difficulties such as ill-conditioned matrices may reduce the accuracy of calculated results. A linear analysis is an approximation of the real behavior. The loading of the model is an approximation of what happens in the real world. The boundary conditions approximate how the structure is supported by the outside world. The material properties assumed are approximate. Flaws are not represented unless the analyst incorporates a model of a flaw. The overall dimensions of the model approximate real structures that are manufactured within a tolerance. Many details are idealized, simplified, or ignored. Element results may be reported at integration points or nodes, not continuously evaluated with the interpolation functions over the whole element interior. Stress and strain results are based on the derivatives of the displacement solution, amplifying the errors. The result of an analysis contains the accumulated errors due to all of the contributing approximations. Good analysis and interpretation of results requires knowing what is an acceptable approximation, development of a complete list of what should be evaluated, appreciation of the need for margin of safety, and comprehension of what remains unknown after an analysis.

Meshing. Production of a good quality mesh is a major topic. The mesh should be fine enough for good detail where information is needed, but not too fine, or the analysis will require considerable time and space in the computer. A mesh should have well-shaped elements -- only mild distortion and moderate aspect ratios. This can require considerable user intervention, despite FEA software promotional claims of automatic good meshing. The user should put considerable effort into the generation of well-shaped meshes. This will include setting element densities, gradients in element size, concatenation of lines or areas to permit mapped meshing, playing with automatic meshing controls, and remeshing individual areas and volumes until the result looks "just right". In ANSYS, the command "LSEL,S,NDIV,,0" will select all the lines that have not had mesh density assigned. This can help find missed lines when setting mesh densities manually. On a curved surface, quadrilateral shell elements should not be generated with a warped form. (The theory manual discusses shell element warping, but I suspect that the discussion is more relevant to element deformation under load, than to the initial undeformed element shape. ANSYS will give warnings if there is more than very slight warping of the original un-deformed quad shell element shape.) Quad shell elements can sometimes be fitted to a cylindrical curve so that they are rectangular in shape and not warped. On other curved surfaces, finely meshed triangular 3-node or four-sided curved 8-node shell elements may be needed. Mid-side node elements can follow complex curved surfaces, so if they are capable of any nonlinearity that will be needed, they may be acceptable and preferred. The 8-node Shell93 shell element of ANSYS has mid-side nodes, follows curved surfaces, and supports nonlinearity. Remember that most finite elements are stiffer than the real structure. For these elements, a coarse mesh generally results in a structure that underpredicts deflection, and overpredicts buckling load and vibration frequency. A coarse mesh is less sensitive to and "hides" stress concentrations. A fine mesh generally gives an answer closer to the exact solution. A fine mesh also results in larger models, more data storage, and longer model solution and display times. Shell versus Solid versus Beam Elements. Ideally, structures would be represented for Finite Element Analysis by solid elements, for this would eliminate the problem of positioning the mid-plane of shell elements, exactly represent the sectional properties of components, and position welds in their design location. Unfortunately, there would have to be several solid elements through the thickness of sheets of steel or aluminum to capture local bending effects with any accuracy, and the other dimensions of the elements would have to be kept small so that the aspect ratios of the elements were acceptable. Consequently, the number of elements would be unbelievably large. It is not feasible to model many thin-wall structures with solid elements. Shell elements were originally developed to efficiently represent thin sheets or plates of steel or aluminum, both flat and curved surfaces. They include out-of-plane bending effects in their fundamental formulation, as well as transferring shear, tension, and

compression in the plane. Developing an interface between a shell portion and a solid element portion of a model has a difficulty: Most solid elements do not include rotational degrees of freedom at the nodes, and this results in a rotational "joint" if shell elements are connected to a solid. Even if a solid element with rotational degrees of freedom is used, the rotational stiffness at a solid's edge node is not appropriate for connection to shell elements -- these solid elements were intended to be connected to each other. In addition, high order solid elements like these are not usually capable of nonlinear analysis. A modeling trick that is often used is to overlap one shell element with the first element in a solid, and join the nodes in two locations in order to imply continuity of rotations, as well as deflections. This is not a perfect fix. Rigid regions with node pairs (rigid links with CERIG) may be used to enforce connection, although high local stresses will result. Some finite element software may have tools to address this problem. Of course, beam elements are even simpler and more efficient, when structures employ beam-like details. There are occasions in FEA work when structural beams (including I, wide-flange, channels and angles) will be more fully represented as shells or solids, in order to examine in detail how they are behaving, or interacting with the structure where they are connected to other parts. Structural steel tubing and rolled sections can sometimes be simplified as beam elements. NOTE: Remember that when shapes are simplified as beam elements, we lose the possibility of predicting flange buckling, web buckling, and concentrated stresses, so caution must be used. Link elements will not show bending stress or Euler buckling of a link. On the XANSYS listserver, I have seen the opinion that the ANSYS PCG solver is not significantly faster than the frontal solver with shell elements, because of the great stiffness difference between in-plane deflections of shell elements, and out-of-plane deflections. In the ANSYS manuals the PCG solver is not recommended where significant numbers of coupled nodes (CP) and rigid regions (CERIG) have been defined. Gap and contact elements may introduce the same problem. This has usually been my experience. However, when modeling a perforated flat plate with shell elements that were roughly square, all about the same shape and size, and as thick as they were wide, using about 200,000 degrees of freedom, I achieved good convergence with the PCG solver. The frontal solver could not fit this problem into my computer because of the size and large wavefront. Of course, you can speed up the "solution" of the PCG solver by accepting a larger convergence error. You know you are having PCG convergence trouble when the convergence error is not decreasing monotonically (when it goes up and down instead of dropping smoothly). The PCG solver is not recommended for use with nonlinear solutions. One time I tried it I got a negative on the diagonal, which would have resulted in bisection with the Frontal solver and adaptive time stepping, but crashed ANSYS with the PCG solver. However, for better behaved models, I have achieved apparently good results with the PCG solver, with shell elements, in nonlinear Large Displacement runs. Reduction of the model to a shell structure. Shell elements are appropriate for many steel structures, since the plates of steel are thin in comparison with their other dimensions. (This applies to aluminum and other materials, too.) The ideal position for

the shell element is on the mid-plane position of the sheet of steel. Consequently, a variety of approximations are needed to link parts of the model together, so that the surfaces act as if they are welded together. ANSYS supports shell elements for which the element thickness varies within the element. This could require a REAL value for every element in order to input a different shell thickness at each node. Input from external programs such as CAD packages sometimes generates such elements and information. User-written macros are sometimes employed to generate elements with varying thickness, or to set up REAL values for existing elements, with the thickness that is assigned being based on node position. There is a helpful if less-than-ideal fix for the case when somewhat thick shells overlap each other, and are welded together. Place the shell mid-surfaces correctly in space, and mesh them so that nodes where welds are used are positioned directly "above" one another on the two surfaces. Join those nodes in pairs with rigid regions (CERIG) or with massless high-stiffness beam elements. The beam elements have the advantage of working properly in large displacement (geometrically nonlinear) solutions. The problem with this technique is that it requires proper mesh control if the user wants to automate generation of the model, and it is tedious to implement manually. In some cases it will be desired to place gap or surface contact elements (with the gap set closed) between the nodes or elements in the interior of the pair of shells, requiring more work. The gap elements keep their original orientation in a large displacement solution, so they will not be applicable in large displacement analyses (unless you can live with the error), and surface contact elements will be needed. Surface contact elements on shell elements must be applied to the correct face of the shell elements. The following figure shows two areas that are offset with one above the other. Lines have been created so that the CERIG command can be used to join them as if they were welded together. Mesh densities have been set so that the rigid region pairs can be created.

The next figure shows the same two areas after meshing and the creation of the rigid region pairs with CERIG. The shell elements have been plotted with the shell thickness

shown, so that the positioning of the nodes in the center of the shell elements is visible, and the touching of the plates is implied. Remember that rigid regions only apply accurately with Small Displacement analysis.

Automating the creation of these CERIG pairs could be done with a macro that: 1. Has the user identify the set of lines on one surface, and the set on the other surface. 2. Steps through the nodes on the first set of lines. 3. For each node on the first set of lines, uses a *GET command to select the closest node from the nodes on the other set of lines. 4. Create a rigid region from the pair of nodes. The macro would work as long as the nodes for the sets of lines are located "above" one another by appropriate mesh control on the lines. A similar macro could join the node pairs with the massless high stiffness beam elements mentioned. Alternatives to using a macro include applying the CEINTF or the EINTF commands with appropriate tolerance values. The reader is cautioned that this technique tells us little about the stresses in the weld, or about fatigue, crack growth and fracture. A prying load applied to the above example could tear the weld apart if the weld was small in comparison to the shell thickness. The example does not illustrate good design practice for handling certain loads. The FEA evaluation of loading of welds in shell structure models is a whole separate topic. Pressure on Shell Elements. In ANSYS, shell elements have two sides. These are known as the TOP and the BOTTOM faces. They are also known as FACE 1 (the BOTTOM) and

FACE 2 (the TOP). The nodes I,J,K,L form a path around the element. If the "right hand rule" is used on this path, the fingers of the right hand following the path, then the thumb points out of the TOP surface (FACE 2). If positive (into the element) pressure is to be applied to FACE 2, a positive pressure vector points into FACE 2, the TOP. If positive pressure is to be applied to FACE 1, a positive pressure vector points into FACE 1, the BOTTOM. Areas act similarly. If a simple primitive solid (for example a cube) is created in ANSYS, it is bounded by areas. The areas will have FACE 1 on the inside surface, while FACE 2 is on the outside of the solid. If the volume was deleted, and the areas that bounded the solid were to be pressurized on the interior of the box that was formed, the pressure should be applied to FACE 1 on all sides. In other models, where Boolean operations have been performed, the FACE 1 and FACE 2 orientations get very scrambled.

For the user to apply pressure, careful checking must be used to assure that the correct faces of shell element have been pressurized. ANSYS can plot elements or areas for which the positive vector points out of the screen (when coming out of FACE 2), or when it points into the screen. This lets the user plot only those areas or elements for which the user sees FACE 2, or for which the user sees only FACE 1. This helps in choosing whether to apply pressure to FACE 1 or FACE 2 when using picking to select areas or elements. Alternatively, ANSYS 5.3 (and presumably later) plots shell elements with different colors for FACE1 and FACE2 under PowerGraphics when the numbering options are set with "No Numbering" and with "Colors" or "Colors and Numbers". To add to the challenge, the direction of the pressure arrows (choose arrows to be shown to indicate pressures under the SYMBOLS choice under PlotCtrls on the Utility Menu) for areas may differ from the direction of the arrows shown for the elements attached to those areas, depending on surfaces visible and sides to which the pressure was applied. The arrow plots for the elements are the ones to believe. Pressures have to be transferred from geometric entities to elements in order for these plots to take place. You have to activate plotting of arrows with the /PSF command -- by default surface symbols are used. ANSYS only plots pressure arrows on shell elements when the arrows point into the screen, so you have to look at a model from all directions when inspecting a shell model. Have fun!

Final notes on pressures: ANSYS can include a gradient in the applied pressure to show the effect of, for example, pressure increasing as a depth of water increases. "Suction" can also be applied by using a minus sign. Remember that "suction" in physically realistic models cannnot be applied beyond the point at which a liquid boils, or below zero absolute pressure. ANSYS, however, does not limit the negative pressure values that a user enters. The hydrostatic pressure of oil floating on water might be modeled by setting the "zero" position of the water pressure gradient above the position where the water starts, in order to include the pressure of the oil. A variety of other tricks can be applied. Reflecting Part of a Model. Where symmetry in the design exists, only a partial model need be built; the rest can be created by reflecting (mirror imaging) the geometry. Where structure is repeated (e.g. a set of posts) multiple copies can be made. Reflection in ANSYS can be done across the XY, YZ, or ZX planes of any ACTIVE Cartesian coordinate system. Since the active coordinate system can be any local system that the user has defined, any kind of reflection in 3D Cartesian space can be accomplished. If the reflection included geometry, nodes, or elements that were on the XY, YZ, or ZX plane about which the reflection took place, copies of those entities will overlay the original copy on the plane of reflection. Entity appropriate merge (NUMMRG) commands will be needed to connect the original and reflected entities. Warning: As discussed elsewhere, elements lying in the plane of reflection get copied with the node order reversed, and will NOT merge with the element from which they were generated. These elements may have to be deleted, depending on your intentions. Representation of Bolted Connections. This non-trivial item can be tackled at a simplified level, or with detailed 3-D representation. The simplest approximation is to represent the bolted (or riveted) connection of overlapping shell structures by locating a node of each surface at the location of the bolt. The nodes have to be located at the same X,Y,Z location in space. This means offsetting one or both shells from its nominal position so that the nodes and shells can touch. One then uses nodal coupling (the CP command in ANSYS) to tie the X, Y, and Z locations in space. It will generally be desirable to tie two of the three rotations as well. The only rotation that is free is that about an axis perpendicular to the planes of elements (about the axis of the bolt). When any rotations in a 3-D analysis are coupled (a result of the bolt clamping surfaces together) the rotation coupling is generally valid only in a small displacement (geometrically linear) analysis. Large displacement (geometrically nonlinear) analysis introduces an error based on the difference between "sin(theta) and theta" (expressed in radians). If contact surfaces are added between the shells that are bolted together, the coupling of rotation is not needed, but the solution becomes a nonlinear iterative process, taking several times longer. NOTE: Contact surfaces on shell elements have to be defined carefully, so that the correct surfaces (Face 1 or Face 2) of the shell elements are the ones in contact -- shell element orientation may need to be doctored to get this to work.

Another bolt representation is to use a rigid region to link pairs of nodes. Rigid regions in ANSYS assume small displacement (geometrically linear) analysis. The degree of freedom for rotation about the axis of the bolt must be free at one end of the rigid region node pair, for bolt representation. This representation has the advantage that the shells can be positioned properly in space. However, contact surfaces may become desirable, depending on the dimensions of the clamped parts. The ANSYS rigid region (CERIG) couples rotations about global axes, so the axis of the bolt would have to be along one of the global axes for the rotational degree of freedom to be correct. The analyst may do better to use a very stiff beam element, with incomplete nodal DOF coupling at one beam end and shell, and the other beam end attached to the other shell; the rotational degree of freedom about the beam axis is free at the end with the nodal coupling. A beam with arbitrary orientation may require the nodes at the coupled end to have their coordinate system rotated to have the rotational degree of freedom oriented properly (I haven't tried this). The problem of contact surfaces remains. It can be partially addressed by using gap elements at nearby nodes, for which the nodes of the two shell surfaces must be aligned "above" one another, so the gap elements are perpendicular to the two shell surfaces. Note: Gap elements keep their original orientation in a large displacement analysis, and will not be applicable where there is significant rotation. Contact surfaces (with the gap closed) may be needed where there will be large displacement. The previous warning about applying a contact surface to the correct side of a shell element applies. Note that nodal coupling acts in the coordinate system of the nodes coupled. The nodal coordinates systems of the coupled nodes should, in general, be identical. The ability of nodal coupling to act in the nodal coordinate system means that the user is not restricted to coupling in global coordinate system directions. Two of the previous bolt representation methods (nodal coupling and rigid region CERIG) are missing the possibility of representing bolt preload. Preload can be implied if a bolted connection is represented with a link or beam element that is capable of "initial strain". In ANSYS these include: Link1 (2-D Spar), Beam3 (2-D Elastic Beam), Beam4 (3-D Elastic Beam), Link8 (3-D Spar), and Link10 (Tension or Compression Only Spar). They must be squeezing surfaces together, which means that either nodal contact elements (gap elements) or surface contact elements must be in use between separated shell element surfaces, or that surface contact elements must be used on the interface between touching 3-D solid elements or touching shell element surfaces. Other ways to represent bolt preload include: 1. Use all 3-D representations of bolts and parts, use contact elements, and apply a temperature difference to the bolt to cause it to "shrink" an intended amount. 2. Use 3-D elements and contact surface elements, with an initial interference between bolt and parts, such that the initial interference results in the intended preload. Temperature setting, interference setting, and setting the "surface normal stiffness" value of surface contact elements in ANSYS must be carefully done to result in the intended

preload. Setting the surface normal stiffness value appropriately is nontrivial. The intended preload must exist BEFORE the structure is loaded. An iterative process may help, but be time-consuming. If the bolts are not overloaded when the structure is loaded, the bolt preload will be nearly unchanged when the structure is loaded. Whether any gap or contact element friction coefficient should be included in the model needs to be considered carefully for it can hide or prevent shear loading on the bolts. For conservatism and safety, friction coefficients may need to be zero, so that the bolts take all the load. When postprocessing, loading on bolts should be assessed using established criteria. My experience has been that if a full 3-D model of a bolted connection (bolt and materials represented with 3-D elements, and contact elements on the surfaces) starts out with the bolt loose and none of the contact elements touching, convergence may be difficult when the solver begins work. Various analyst "cheats" may help, such as moving the bolt or parts so that there is some contact, and/or using some very soft spring stiffness combination elements to keep the model from "flying off into space", when the solver is working to converge. Warning about Nodal Coupling. Nodal coupling has its uses: one is a quick-and-dirty representation of a bolted or riveted connection with shell elements (see above). More exotic applications can be invented. When nodal coupling is used to represent a bolted connection of 3D shells, the nodes that are coupled must occupy the same position in space. Otherwise, body rotation at that part of the structure will result in an artificial mechanism acting on the structure. If the nodes were tied in the X,Y,Z directions, structure rotation would not result in the necessary change in the relative X,Y,Z positions of the two nodes. High local stresses, and an external couple would result if the coupled nodes were not located at the same position. This is not good! Development of geometry in which surfaces cut each other with shared lines. The lines must be shared between different areas if the finite elements are to act as if the surfaces are welded together, when meshing takes place. Considerable care and checking is always necessary as a model is built, to see that connectedness is complete. I can still make errors of this type, for they sneak in even when being careful. Hopefully, a beginning ANSYS user will have had some training in the development of ANSYS solid geometry within /PREP7. New revisions of ANSYS improve the capability of /PREP7, with not all improvements being publicized. I lived with ANSYS 5.0 and 5.1, and much prefer the more recent ANSYS versions. The solid modeling engine does not like singularities, e.g. you can't have a line that cuts half way through an area, the way that you can cut half way into a sheet of paper with a pair of scissors. It is necessary to cut an original area into two areas, in order to get a line that extends into the interior of the original area. Recent ANSYS versions appear to be more tolerant of cusps and some other difficulties. Development of complex structure solid-model geometry with /PREP7 calls on analyst creativity, intelligence, and puzzle-solving skills, as well as a good dose of patience. This tends not to be understood by those who have never done the work.

ANSYS does not assign the attributes (REAL, MAT, TYPE, and ESYS) of a parent geometric entity (Line, Area, or Volume) to the entities that are formed by a Boolean operation such as dividing the original entity into parts. I consider this unfortunate, since it increases the work required of the analyst who is developing the model. It is an easy way to forget to assign attributes. Application of boundary conditions. Structural FEA displacement boundary conditions are the limitations on movement of the structure at places such as anchor locations. The boundary conditions in a finite element model must limit translation or rotation in a manner appropriate to the case at hand. Boundary conditions can be used to imply symmetric behavior in a structure that has symmetry, so that the model size can be halved, quartered, or similarly reduced, if the loading of the structure is also symmetrical. Boundary conditions can also be used to imply anti-symmetry, for example, where a warping displacement is applied to a symmetric structure (envision twisting a shoebox about the long axis -- a quarter model could be sufficient). There are occasions when a displacement boundary condition needs to be applied to a single node so that the structure can rotate around the support point. This single node support, however, can result in a serious local stress spike. Depending on the model, the elements where the single node support will be applied might be artificially stiffened. Alternatively, if there is a surrounding "pad", an even pressure could be applied to the pad, that generates a force equal to the reaction otherwise found at the constrained node. Two stress runs could be used: (1) Run without the pressure on the "pad" and find the reaction at the constrained node. (2) Take the reaction, spread it smoothly over the pad as a pressure, and run again. The reaction could be spread over nearby nodes at stiffeners, instead of applied as a pressure, depending on the nature of the model and structure. The goal here is to approximate reality in an acceptable way, while avoiding the timeconsuming use of contact and other non-linear elements. (Of course, in some cases, it will be necessary to exactly model a support complete with many non-linear complexities.) NOTE: If you do this, the reaction forces will no longer equal the previous applied load plus gravity load on the structure, because of the new load that has been introduced. Application of loading in a manner that is of satisfactory accuracy, without becoming overly complex. It is often sufficient to apply forces directly to a small set of nodes. However, better representation of loading can be needed to avoid local stress spikes in some analyses. As discussed above, application of pressure over a region of elements, producing the desired force, can help avoid a local stress spike. Artificially stiffening a local region where a point force is applied can help, if this is acceptable. The load to consider may need to be increased because of the possibility of dynamic effects, if you are doing only a static analysis. Your industry may have standards for this. Consider road vehicle design -- you wouldn't want the tires to blow out from the increased force due to a vehicle roll-over. (If they did, how would you prove that tire failure did not cause the accident?) This would call for the tires to stand at least twice the "normal max rating" without immediate failure. I once sighted a non-professional driver pulling a simple trailer grossly overloaded with crushed stone. It appeared that the wheel

bearings failed before the tires let go (there was a lot of smoke so it was hard to tell). Somebody did good tire design! (Some transportation structures have to be limited in size under the knowlege that users will fill them to the maximum possible volume, without regard to the density and total weight of the material loaded.) For structures that do not have a severe weight penalty (e.g. those that do not have to fly), getting a conservative result is often satisfactory. An analyst will develop a feel for this as the result of experience in a particular industry. However, where there are high material costs, or large volumes manufactured, extra modeling detail to reduce unjustified conservatism may be economically sound. Pressure loading of a wall containing granular material is particularly challenging. Earth, sand, grain, coal, or other granular material pressure is a civil engineering topic. Because of internal friction in the material, the lateral pressure on walls is usually less than simple hydrostatic pressure would be for a liquid of the same average density. For some dry materials, the pressure would be roughly 40 to 60 percent of hydrostatic pressure (look up a proper value) on a vertical wall. The pressure loading varies with the depth of the material, and varies if the slope of a wall changes (a horizontal surface could see hydrostatic pressure). On the other hand, in a long column filled with granular material, the pressure may be constant past a certain depth -- this affects the function of an hourglass. The ANSYS Finite Element program is capable of applying a pressure with a gradient, so pressure can ramp up smoothly as the depth increases. The pressure load must be applied to the correct face of a shell finite element. Considerable FEA checking is needed to assure that the whole structure model is properly loaded. Extra analyst work is needed to apply a series of gradient loads that increase smoothly in intensity if curvature of a wall or container surface causes change of slope. The Rankine formula describes granular material pressure on a vertical wall. Non-vertical sides might require the Coulomb formula to give a higher accuracy representation of how non-vertical slope affects granular material pressure on a wall (go visit a library, plus talk to a civil engineer). Take a look at EJGE/Magazine Feature for more information. After creating loads that represent a granular material in a container, under a 1.0 g vertical load, the vertical component of the applied pressure should result in a total force that equals the weight of the granular material. It may be desired to scale the granular material pressures so that the total vertical force component under 1.0 g equals the weight of the contained material. This should be checked in reviewing the results of the analysis. A perfect FEA model of containers (bin, hopper, hold, box, trailer, etc.) loaded by granular material may be impossible. The pressure required to push inward and deform a surface of a granular material is greater than the load with which the granular material pushes outward. This is because of the internal friction in the material. A finite element model of a loaded wall can include pressure on the inside surface that would result from contained material. However, that pressure will not be adjusted according to whether the wall moves inward, or expands outward, as the container deforms under various loads. Since an FEA analysis results in deformation of the walls, exact representation of the pressure loading will be unachievable. I have not been able to find an expert who would

say that a granular material nonlinear solid element finite element model can be included inside a shell structure container model in a successful manner, using contact elements on the interface between the solid elements and shell elements (geotechnical engineers should know far more about this than I do). Material properties such as Drucker-Prager are included in ANSYS and some other FEA packages, but I don't know if they are applicable to this type of structure and granular material modeling. ANSYS manuals discuss this material option briefly. An engineer often settles for a model and design thought to be conservative or adequate, given industry experience. The worrying starts when a design departs significantly from previous practice. Deformation of thin flat panels by pressure loading causes the panels to curve. When flat panels are loaded on one of their surfaces, the panels curve, then start to carry applied loading with membrane forces. The only way in which this can be represented is to activate large displacement (geometrically nonlinear) analysis. A rule of thumb is that membrane forces begin to be significant when the out-of-plane deflection exceeds half the thickness of the panel. Nonlinear analysis requires considerable experience, because of the difficulty in achieving converged solutions. Failure to use nonlinear analysis where it is appropriate can result in considerable ignorance of the real structural mechanics involved. Nonlinear analysis becomes very time consuming because of the iterative solutions needed. Fast computers are very desirable when doing this kind of work with a large model. Failure to consider that significant out-of-plane deflection can result during nonlinear analysis can, in some cases, lead to inadequate designs. In other cases, the curvature can lead to significant increases in strength of the structure. The designer needs to be aware of the need to include nonlinear effects in some work. Use of Units. Vibration and transient analysis require that the mass of the structure be entered in units consistent with the other units in the model. Some North American industries normally work in inches-pounds-seconds. This requires that mass be represented as pounds/in/sec^2. Pounds here means "pounds force", the force with which 1.0 g of gravity pulls on the mass. This means dividing the weight in "pounds force", or the density in pounds/in^3, by the number 386.1 (more accurate than 32.2*12=386.4), which is the acceleration due to gravity expressed in inches per second squared (in/sec^2). In consequence, when mass and mass density have been defined this way (the density of steel, which depends on the alloy, if given as 0.2836 lb/in^3 would be entered into ANSYS as 0.0007345) it is necessary to enter 1.0 g of gravity as 386.1 in/sec^2 to let ANSYS apply the correct force due to gravity on the structure. Loads will be entered in pounds. Pressures and stresses will be referred to as pounds per square inch. ANSYS refers to these units as "BIN" (see the /UNITS command for "British system using inches", noting that the /UNITS command is for annotation of the database, and has no effect on the analysis or data). In the metric world, fundamental units are meters-kilograms-seconds. However, in engineering work, analysts often use millimeters-kilograms-seconds. Forces are expressed in Newtons (1 Newton accelerates 1 kilogram at 1 meter per second squared). Pressure is Newtons per square meter (1 Newton/Meter^2 = 1 Pascal). A pressure of 1 Newton per square millimeter is referred to as 1 megapascal. When working in

millimeters-kilograms-seconds, it is common to refer to pressures, stresses, and Young's modulus in megapascals or kilopascals. Acceleration due to gravity is 9.807 meters/sec^2, or 9807 mm/sec^2. ANSYS does not care what units are used, nor does it issue warnings. The analyst must be consistent in the set of units in one model, to avoid errors. Getting the mass and mass density into the correct units is particularly important if any form of vibration, transient, or transient heat transfer work will be done. Tip: Check the values for typical materials in the ANSYS material library as a guide, even if you do not use these exact materials. A comparison will indicate if your values are in the right range. The ANSYS materials library includes material values in various systems of units. Many design codes will, for example, give densities in lb/in^3, where pounds is actually the weight expressed as "pounds force". This Imperial value cannot be used directly for vibration and transient work, and must be converted. (When I try to explain this to non-North American people, and even recent Canadian graduates, they think the whole Imperial units business is insane -- I can't blame them.) The usual question on Imperial units is, "Why can't I enter density for steel as 0.2836 and 1.0 g of gravity as 1.0 ?" The answer is, "This would work for gravity loading on a structure, but if you ever do vibration or transient analysis on the same model in the future, your answer will be garbage." My own policy is to always use the "correct" units, similar to those that the ANSYS material library supplies for the BIN system, in case vibration or other work is done in future. If densities have been entered "correctly" in Imperial units (e.g. 0.2836/386.1=0.0007345 for steel), then when ANSYS reports the "mass" of the model during the SOLVE process, that mass will have to be multiplied by "g" (386.1 in this example) to recover the weight of the model in "pounds force". Buckling analysis and failure can be pursued in two ways: Linear eigenvalue buckling, and geometrically nonlinear (Large Displacement) buckling analysis. Eigenvalue buckling (also known as Euler buckling or classical buckling) will be sufficient for some structures, but much greater detail about stress amplification and margin of safety can be found with geometrically nonlinear analysis. Note that margin of safety is not a simple concept in a nonlinear analysis. The margin of safety will be based on the difference between the intended design load and either the load that reaches failure conditions or the load that exceeds allowables set by design codes. The relationship between loading and consequent stress and deflection cannot be extrapolated linearly when a nonlinear analysis is used, or when it is needed. Design codes may address this concept with reference to combined compression and bending of beams, but many codes were written before the availability of nonlinear finite element analysis, so the analyst will need to comprehend the intent of the design code and interpret it, if this is permissible. A difficulty here is to establish what level of loading has reached "failure" conditions. If the structure starts to buckle in a Large Displacement analysis, solution convergence will become slow, as the load is ramped up. The fact that the FEA solution stops converging at

some level does not guarantee that the failure load has been reached -- it could be just a numerical analysis difficulty. The Arc-length method is useful here, since it will follow the load up and back down as the load/deflection curve first rises and then falls. An advantage to Large Displacement, Plastic material property analysis is that the failure can be followed in detail (if the model is small, or the computer is very fast). Defining margin of safety still requires a human decision as to what load "reaches" unacceptable stress and deflection, before complete collapse happens. Simply basing margin of safety on the highest load reached in a plastic, Large Deflection, Arc-length analysis would not satisfy the rules in most design codes, and usually not make good engineering sense. A problem with eigenvalue analysis of some structures is that localized "popping" of panels or other components happens long before the whole structure begins to fail via buckling induced deformation. The problem with geometrically nonlinear analysis of the same structure and loading is that convergence troubles may make analysis exceedingly difficult and/or time consuming. This is particularly true when applied force is ramped up. Convergence of applied displacement is more successful in nonlinear studies, but applied displacement is not the most common way in which loads are analyzed. A possible advantage of a geometrically nonlinear Large Displacement run is that if convergence of the model is achieved, it may sometimes be shown that the structure will handle a load considerably greater than the first several eigenvalue buckling loads, without exceeding yield, or allowable stress, or undergoing deflection significant enough to merit concern. A geometrically nonlinear analysis with loads that exceed the eigenvalue buckling level should have loading ramped up, with substep information saved in fine detail. The substep results should be examined carefully to see whether sudden changes in the stress or deflection patterns develop. With a shell model, this should be done for both mid-plane and surface (use Powergraphics) stresses and for deflection plots. The ability of the ANSYS program to generate an animation file from the set of substep results is helpful here. Deflection can be set 1:1 or exaggerated using the /DSCALE command. Ramping Loads in ANSYS. Loads are ramped up if the appropriate settings are used for time stepping. The fun starts when the user tries to ramp the loads back down (as when wanting to find the permanent deformation that results from plastic deformation). If the loads are deleted, there is nothing to ramp down to, the force drops immediately to zero, and convergence may be a problem. One solution is to reduce forces and pressures to an extremely small number. Another problem is that if the loading has been applied to geometric entities, it cannot be scaled down directly, for ANSYS lacks commands to do this. An unsatisfactory but adequate fix is to transfer the loading to the nodes and elements, then delete the relationship between geometric entities using the MODMSH,DETA command from /PREP7 (Warning: make sure your model is saved before doing this -MODMSH ruins the connection between your geometry and your FEA mesh), then scale down the loading on the nodes and elements. If you merely scale down the loading on the nodes and elements, it will be replaced by the loading on the geometric entities when the SOLVE command is executed.

A more satisfactory way to ramp loads that were originally applied to geometric entities will be to write and read load step files. The full loading on the geometric entities can be transferred to the elements, then a load step file written. The load step file includes pressures on elements, not information about loading on geometric entities. Then, the loading on geometric entities can be deleted. Next, the load step file can be read, bringing back in the pressures on the elements. Finally, that loading can be scaled down to an extremely small number. This method works in general for keeping the loading that geometric entities transferred to elements and nodes, while discarding the original assignment of loading to geometry, and so can be quite convenient. Plotting results can show the stresses in the structure with colored contour maps. Plotting with stresses averaged at nodes (PLNSOL) results in smoother cleaner contours that are easier to study, and that tend to average out stress fluctuations due to local variations in element shape. However, such plots have the disadvantage that they average stresses at shell intersections (at corners, "Tee" intersections, thickness discontinuities, and material changes, for example). This results in considerable loss of information, and masking of high stress areas in some models. Either element stress plots with no nodal averaging must be used when this matters (PLESOL), or element selection must be limited to continuous panels of material, so that the averaging is not performed where it is not appropriate. This is a very common error in the reporting of results from shell models (and solid models with material type changes). I have seen stresses hidden that would cause fatigue troubles, because of nodal stress averaging with shell FEA models. In addition, fatigue-causing stresses often need to be shown at shell surfaces, not just at the mid-plane, so both mid-plane and surface stress plotting will often be required for complete model evaluation. In a complex model, components may need to be examined from a number of viewing angles, and with cutting planes, in order to inspect the stresses everywhere. ANSYS has introduced its "Powergraphics" setting that can show VISIBLE SURFACE shell stresses with discontinuity at intersections, and changes in REAL and MATerial (see the AVRES command). However, a user often wants stress at the shell mid-plane. ANSYS keeps track of the surface stresses in its database, and calculates the mid-plane average when needed. I have written a macro that will move the mid-plane stress for each node of each shell element, element-by-element, to the top and bottom surfaces, so that the Powergraphics setting can show mid-plane shell stress with discontinuities and intersections. The problem with the macro is that it executes VERY slowly -- it was about two seconds per SHELL 63 element on a Pentium-Pro 180 under Windows NT in a 70,000 DOF model, taking 7 hours to process one load case. Surrounding macro executable lines with /NOPR and /GOPR speeded up the process by roughly a factor of 3. The database is permanently modified by this macro, so the analysis results database must be stored on disk BEFORE this macro is used. It must be used with caution. The ANSYS contour map colors can be customized. I set them to shades of gray when I want to plot to a black-and-white laser printer (directly from ANSYS, not the DISPLAY program). The contour levels can be set automated to be evenly applied (default), or can be set by the user. I sometimes set all levels but the "red" contour to be evenly spread out

up to the material yield, or the allowable stress, and let red color the region above. I wrote a macro to automate this, using the *GET command to find the max and min stresses, in order to calculate the custom levels. The macro has to be re-applied every time stresses are plotted for new elements, or for a different stress plot type. The automatic contour level mode should be returned to when done. Shell mid-plane stresses are often preferred for review of structures. There are also good reasons to review shell surface stresses. They include checks on: direct shell bending, torque causing torsion stress in open sections, plastic hinge development and the onset of plastic failure, local stress concentrations, locations for possible fatigue or fracture, nonlinear buckling, stresses from design errors or modeling errors, and prying loads. Torsion on an open section can cause substantial shell surface stresses at shell intersections such as corners -- an invitation to fatigue failure, fracture, or possible structure collapse. This phenomenon will be completely overlooked if only mid-plane stresses are plotted. In limited testing I did, ANSYS gave me surprisingly good values for surface stress caused by torque applied to open sections modeled with shell elements. (I created equivalent solid models with a few solid elements through the wall thickness for the comparison runs that gave the "real" answer.) Mid-plane stress plots don't hint that torsional load is causing high shell stress on the surfaces of open sections. I wouldn't extrapolate my test result to any structure, but it suggests that shell surface stress plots will help to detect a class of design problems (shortcomings) that mid-plane stress plots will miss. ANSYS PowerGraphics plotting helps considerably. Coping with Design Changes. A fun topic! The analyst must be able to modify existing models. The ability to do this can be enhanced if the model has been planned for later modification (see parametric design comments below). The commands that move keypoints can help a little... the keypoint moves will destroy curved lines, and only work if affected areas are not severely distorted, and topology does not try to change. KEEPING THE GEOMETRY on which the mesh was based is an important part of being able to do significant future modifications of models. It is easier to move a set of nodes than a set of keypoints, so under rare circumstances the elimination of geometry may be desired (nodes cannot be moved while they attached to underlying geometry; see the MODMSH command but do not use it without knowing exactly what you are doing). However, any substantial model changes become very difficult when only elements and nodes are available. Computer Aided Engineering Environment. I often develop finite element models the "hard" way: Generate all the geometry from scratch in the pre-processor of ANSYS. For existing designs, I may get copies of a few dozen drawings, sometimes scaling dimensions off the drawings (I did say finite element analysis is approximate) when the dimensions are not explicit on the drawings (I don't like this). I adjust the position of parts in space to achieve a good mid-plane representation of steel sheets for shell element development. Adjustments and modeling tricks are used to approximate some connections of thick parts and of bolted parts. For a complex model it can become very time consuming to modify a model's fundamental dimensions after model development

has progressed significantly. This makes exploration of cost-saving alternatives difficult on a tight time schedule (what other kind of time schedule is there?), even though significant money might be at stake. Significant money is involved with expensive structures, weight penalties, high-volume production, and with failures. There exist CAD systems that can link the 3-dimensional CAD model to a complex shell finite element model (e.g. Pro/Engineer and SDRC IDEAS, probably others as progress is made). The CAD models can be parametrically defined so that overall dimensions can be updated quickly with all associated part and assembly prints, and the bill of materials being automatically updated, as well as the finite element model. This can make exploration of design alternatives much more sophisticated. Otherwise, the analyst may be limited to exploring shell thickness alternatives, and development of ANSYS models parametrically, so that the ANSYS log files can be re-run with different fundamental dimensions. Such a finite element model "program" requires careful planning and experience. FEA versus Hand Calculations. This issue comes up when a new design needs to be configured. The "first cut" at a design must start with the invention of a configuration that supports the applied loads, and carries these loads to the support points of a structure. A variety of loads usually need to be supported, and structural details must be present that will handle each kind of load in a manner that is acceptable for the type of structure being considered (e.g. welded steel structures, bolted, pipes and pressure vessels, and others). The initial layout of the components of the structure, and the initial sizing of the parts has to begin with manual calculations. Several concerns arise in the initial configuration, such as:

Adequate section properties and crossectional area to handle applied loading. The presence of bracing and stiffeners that prevent structure instability. Sufficient wall and beam thickness and stiffening to avoid detrimental buckling of local regions. Avoidance of unacceptable stress concentrations by methods such as stiffeners, shapes, finishing, or other details. Adequate weld and bolt size to handle all applied loads. Design for manufacturing. Development of geometry that respects dimensional constraints on the overall structure. Minimizing cost: material cost, uncut raw material size, material availability, fabrication expense, delivery dates and penalties, and risk. Use of standard thicknesses, hot rolled sections, bolt sizes, available maximum dimensions, and affordable material choices. Discussions with suppliers regarding not just supply and cost, but possible costsaving customization of the scope of supplied material and parts. Remember that suppliers are familiar with the practices of others, including your competitors -they can be a valuable resource to a designer (and to a job-hunter, so treat them well, but don't give away secrets).

Given an initial structural concept, an FEA model can be created. If the model is only of moderate complexity, the geometry for the FEA model can be created parametrically, so that the log file can be reused in the future to regenerate the design with different dimensional values. This will require that there be no changes in the topology of the structure (e.g. varying the number of stiffeners, or shortening a part until it no longer meets another part) or else the parametric approach must include means to accommodate these changes. If the model is complex, it may not be feasible to create the geometry parametrically, and the finite element model will be created with exact dimensions entered numerically. During the finite element analyses that follow, the thicknesses of shells or beams can be varied in order to investigate the possibility of weight savings and cost reduction. The FEA package can be used to investigate stress, deflection, buckling, vibration, and nonlinear effects if these matter. Properly interpreted results will show where the structure is overdesigned, underdesigned, or if it has significant inadequate design details (e.g. complete lack of stiffeners where they are needed) and needs modification. Design sensitivity can be assessed with respect to variations in some dimensions. Optimization may be possible if time and sufficient skill are available. Given modern CAD software, a parametric model can be built in the CAD system. An FEA model can be derived from the CAD model such that updating the CAD model leads to updating of the FEA model. This makes the modify-and-assess design loop much more effective and can lead to significant cost savings. Progress with development and deployment of these CAD systems continues. Choosing an Appropriate Shell Element. There are several shell elements types available under ANSYS. The usual workhorse shell element is Shell63, a 4-node shell element. This element supports large displacement, but not plastic material properties. (If plastic material properties have been entered, they will be ignored by Shell63.) If your element type 1 was Shell63, you can directly enter (by hand) a command like "ET,1,181" to convert the elements to Shell181, which has plastic capability. You may want to modify the KEYOPT values after this command. Note that the effect of stress stiffening is activated with shell elements like Shell63 by adjusting one of the KEYOPT values for this element. Other 4-node elements that are capable of plasticity include Shell43, Shell143, and Shell181. I have recently found Shell93, an 8-node shell element, to give satisfactory results for a problem I ran. This element is capable of plasticity (ANSYS manuals note that lower order elements (4-node in this case) may be preferred for nonlinear and plastic analysis), in addition to large displacement, so it gives "one size fits all" service. The advantage to this element is that mesh density does not have to be as great, and it follows curved surfaces very well, since it is a curved element. (4-node shell elements are flat, and any significant warping of their shape during meshing will cause the FEA program to complain, and presumably give degraded results.) Some user work is required with midside node elements, because they do not want to curve too much. Meshing an area fillet has to be carefully controlled. To change a model with 4-node elements, to 8-node elements with mid-side nodes, the usual thing to do would be to clear elements and re-

mesh, after possibly modifying mesh density. Stress stiffening is activated for Shell93 in the Solution part of ANSYS, not by setting a KEYOPT value as with Shell63. Using P-Elements. The use of P-elements can reduce the effort required to mesh models. The user is cautioned that the P-elements do not support large displacement or plasticity. Harmonic Response. This is what ANSYS calls Steady State Frequency Response to constant harmonic input (an input forcing frequency that is sinusoidal steady state). There are three ways available in ANSYS: full, reduced, and modal. A damping ratio can be input using the DMPRAT command. The output is complex numbers that imply amplitude and phase. The phase differs from the phase of the input if the input is not at an eigenfrequency. Only the reduced and modal methods can handle stress stiffening. The /POST26 Time History postprocessor can plot amplitude for a node versus frequency (see the PLCPLX key value); the /POST1 postprocessor can use the SET command to load either the Real or the Imaginary component, but not both. The manuals say that the /POST26 postprocessor can do things with the components. As with all vibration and transient analyses, the units of mass must be input appropriately. Failure Modes to Consider. Textbooks are written on this topic. There are many things an analyst may overlook. Just a few of the many things to think and worry about include: 1. Static loads lead to stresses exceeding yield (or allowable stress) over a significant region. Dynamic loads exceed anything considered in load factors for static loading. 2. Loads on bolts, rivets, spot welds, plug welds, stitch welds, fillet welds, bevel welds, full-penetration welds, adhesives, nails, tie-rods, links, or other connection devices are too high. Prying loads are not considered or properly assessed, and are too high. Moments tear a bolt circle apart because it was represented as a pinned (one bolt) joint. Compression of tie-rods or links reaches buckling levels (FEA will not detect this for link elements). 3. Loads on bolt holes are too high. Bolt holes weaken a section. 4. Strains reach fracture levels in brittle materials. 5. Surface strains cause damage to protective coatings. 6. Deformations cause lock-up of parts that should slide or rotate. 7. Buckling of components leads to local damage, or to progressive collapse. 8. Buckling of the full structure is reached. 9. Combined bending and compression leads to excessive stress and failure. 10. Fatigue failure and/or sudden fracture is reached. If the FEA model ignores stress concentrations, and representation of details where trouble can occur, fatigue or fracture may never have been properly assessed. If cracks grow without detection, sudden fracture conditions may be reached. Growing cracks need to be of a detectable size without causing sudden fracture. (The capacity of a crack to cause sudden fracture in a structure increases with the size of the crack and with the stress level, and depends on the properties of the material. Remember that when materials are welded together there is an implicit crack formed except where good-quality full-penetration welds are used.) To paraphrase a writer whose name

I unfortunately can't recall, "A tolerable crack size needs to be large enough that it can be detected by a tired inspector on a Friday afternoon a half hour before quitting time." To keep a crack of detectable size from causing sudden fracture, the material choice, allowable str