Equations from a thermal handbook have been placed into   TK Solver from Universal Technical Systems, Rockford, Ill. (<a href=www.uts.com). The user is testing heat convection coming off a flat plate for several variables in the spread sheet. Charts and graphs provide more visual detail. The temperature distribution might give a thermal analyst an estimate of results before starting FEA.">

Equations from a thermal handbook have been placed into TK Solver from Universal Technical Systems, Rockford, Ill. (www.uts.com). The user is testing heat convection coming off a flat plate for several variables in the spread sheet. Charts and graphs provide more visual detail. The temperature distribution might give a thermal analyst an estimate of results before starting FEA.


MathSoft's (<a href=www.mathsoft.com) version of Roark's formulae looks much like the text. The values are variables for a beam equation. Users can cut and paste these pages into their own worksheets, change the variables, generate new figures, and even tweak the equations for special conditions.">

MathSoft's (www.mathsoft.com) version of Roark's formulae looks much like the text. The values are variables for a beam equation. Users can cut and paste these pages into their own worksheets, change the variables, generate new figures, and even tweak the equations for special conditions.


The Web page for Maple software from Waterloo Maple   (<a href=www.waterloomaple.com) provides a range of userwritten programs for a few specialized engineering problems. For instance, under the Fluid section, a program describes equations and mechanics for skydiving. Users can download the equations and run them in Maple software.">

The Web page for Maple software from Waterloo Maple (www.waterloomaple.com) provides a range of userwritten programs for a few specialized engineering problems. For instance, under the Fluid section, a program describes equations and mechanics for skydiving. Users can download the equations and run them in Maple software.


Experts at stress analysis say FEA users should think a bit like trial lawyers — never ask a question through simulation without a good idea of the answer. The estimate then serves as a sanity check against answers that could be way off the mark. And the estimate should not depend on finite elements.

One way to generate accurate estimates is to punch out a few equations on a calculator while referring to a handbook for a particular discipline. For instance, Roark's Formulas for Stress & Strain, by Warren Young, McGraw-Hill, N.Y., is a classic. There are others.

A software version of a stress handbook provides another route. Mathsoft Inc., Cambridge, Mass., for example, has Roark's equations in its electronic-handbook format. It lets users cut and paste useful equations into worksheets. The software also stores the worksheets so they can be recalled, modified, and rerun for new analyses.

There are other tactics for conducting these sanity checks. For example, "Think of analysis as made of several concentric targets," says Steve Kensinger, CEO of Savage Enterprises Inc., Minneapolis. "Each smaller ring represents a more sophisticated level of analysis. The simplest tools are in the outer ring and complex FEA is at the center. Handbooks in software, such as TK Solver and MathCAD, and physical testing lets users start zeroing in on the best answer. They draw you closer to the answer at the center and allow more informed investigations in the next level of complexity."

Look for equations and physical tests that help set the boundaries for the answer, suggests Kensinger. "Such tools identify key variables and the most sensitive ones. These variables become the focus of more sophisticated analysis."

A caveat here is to take care in comparing variables between different modeling tools. For instance, deflection is the primary unknown in FEA. Stress is secondary. "So start comparing deflections," says Kensinger. When comparing stresses between tools, be sure to include all components and calculate them at the same point in their respective models.

"Of course, formulas are not conclusive," he adds. "You make a lot of assumptions using them, so you need to look at their potential effects. For example, suppose you apply a standard beam formula to a simple cantilever made of a channel section. Standard beam formulae give one result. FEA provides a very different answer. Standard beam theory assumes a channel's shear center coincides with its mass center. But in channel sections, they do not coincide." Properly constructed FE models account for this deviation. The equation probably won't. Although, at least one formula in Roark allows an offset shear center.

"Ansys FEA software, for instance, has a beam element with an offset shear center for modeling channels," says Kensinger. "But if your FEA program doesn't have such an element, you'd have to model the channel geometry using plate or solid elements to get the correct answer."

An FEA model can also be used to determine at what condition a structure deviates from the linear behavior defined by most simple equations. In later analyses, the equation can be used until nonlinearity affects the performance of similar structures. This reduces the time needed to run FEA. "And while you're doing this, you should be learning," he adds. "By the time you get a final model, you should have identified known and unknown variables, a max and min range for variables, whether they are linear or not, and what type of nonlinearities exist." Each ring of the target clarifies the objective of the analysis and leads closer to the bull's eye.

Another perspective comes from an analyst at a Midwest aerospace manufacturer. For example, to find the frequency of a plate with irregular or complex geometry, he suggests first get a feel for the answer by finding the frequency of a plate with regular geometry — one that's rectangular or square with dimensions similar to the irregular plate. The value can be checked with a handbook to get a ballpark figure.

FFT analysis is another tool for frequency analysis. It's essentially an oscilloscope working with a computer. In a nutshell, the user hits a sample part with a hammer and the machine finds the part's natural frequencies. Analysts are usually interested in the fundamental mode. So if the material properties of a prototype are completely unknown, the analyzer can sense the stiffness characteristics, then give the frequency along with clues for the material's other properties.

For instance, experts suggest working backward with an appropriate equation. You insert values for stiffness, modulus of elasticity, or density, and solve for the unknown. Stiffness is a function of modulus of elasticity and moment of inertia. When the geometry is known, the missing variable is the modulus of elasticity.

This is a simple yet powerful method, say experts, because as more plastic comes into use, vendor-provided information has such a wide range, it is not as useful as it should be. Finding properties becomes the responsibility of the analyst.

Analysts also suggest keeping summaries of past analyses to check for similar information to provide approximate answers.

For thermal analysis, classical thermal equations from handbooks become one place to start. MathCAD software, for example, can incorporate equations for free convection and radiation in the geometry. Experience can tweak the equation when the box or enclosure is not a convenient rectangular configuration. Heat is not necessarily uniformly distributed, even though one assumes a uniform distribution for determining preliminary values. That means a temperature from an equation is the low end of the answer range. FEA should give slightly higher answers. And expect isolated hot spots. Then define a range for the final temperature. Based on experience and a few assumptions, you can always check what you did with MathCAD or other software. You should find it is within a few degrees, say experts.

"Successful and cost effective analysis depends on several elements", says Kensinger. "And not all are related to the analysis tool. First of all, management should clearly define design objectives and product requirements to engineering. Without them, an analysis program cannot run efficiently."

Another necessity is access to all technology behind the design requirements — reference books through sophisticated analysis tools. Everyone on the design team must understand the physics of the product as well as the physics behind the tools. Finally, the design team must have access to personnel trained using each tool. These people can be in-house or at a consultancy.

"Analysis aids are information tools," says Kensinger. "They don't give answers, they provide information. They characterize physical concepts, not the real world. It's up to the engineer to decide what the answer is."