Professor of Mechanics
St. Louis, Mo.
Professor of Aerospace Engineering
University of Texas at Austin
Senior Research Engineer
Engineering Software Research & Development Inc.
St. Louis, Mo.
Simple math models found in conventional handbooks such as Roarks formulas for stress and strain have done a good job accumulating, preserving, and distributing expert knowledge. They help examine whether or not a particular design is feasible, rank alternative designs with respect to some criterion, and estimate natural frequencies or fatigue life. What’s more, they’re readily available, need no special training in numerical simulation methods, and reveal answers in minutes.
But handbook models also have serious disadvantages. For instance, they handle only those problems which can be characterized with a few parameters. The solution sets are not easily enlarged. Widely used handbooks are typically weak in problems that involve orthotropic materials, laminated plates, filament-wound composites, or modal analyses. They can’t handle nonlinear analyses, and the solutions from conventional handbooks are not consistently reliable.
The good news, however, is that modern computers and software can remove these drawbacks yet retain the advantages of conventional handbooks. These systems can incorporate handbook-type problems in parametric form that work with a finite-element analysis program.
A lot of mechanical and structural work is routine or so-called variant design. Its goal is to modify an existing part to satisfy some new criterion or requirement. Modifications usually involve changing dimensions, material properties, or both, and examining several load cases.
Parametric finite-element models can efficiently treat repetitive design problems. These are models created once and used many times for each type of problem. Assigning variables within a specified range resizes the model while preserving the FE-mesh integrity. This arrangement frees users from the burden of having to design meshes or to check their adequacy. Even automated error indicators can be provided. One quick benefit of conducting analysis this way is that completing several parametric analysis on the same part allows developing a family of design curves.
An efficient handbook framework for an FEA program should provide capabilities such as:
• A parametric definition of the topology, material properties, loads, and a means for enforcing logical constraints among parameters.
• Associativity between the topological data and the FE mesh. This means that when a parameter changes the mesh automatically updates.
• Hierarchic modeling capabilities which allow examining whether the data of interest are sensitive to the modeling decisions. It should be possible to test, for example, whether a particular plate or shell model can approximate the data of interest with sufficient accuracy, or whether material or geometric nonlinearities are important.
• Flexible postprocessing lets users access graphical or tabular engineering information obtained by the solution. It should be convenient to examine the effects of various load combinations in postprocessing operations.
An FEA-based handbook should provide reliable information in much the same way conventional handbooks do, even when used by people without expertise in FEA technology. The user interface has to provide convenient access to available solutions and choice of the free parameters with logical checks on their values. The user interface must provide problem-dependent solution strategies. There must be a seamless transition from a linear model to a nonlinear one when certain conditions occur. For example, a nonlinear solution should be computed when some region exceeds the yield strain in a linear solution. The type and format of output must be designed to suit the problem and provide an indication of its accuracy.
A FEW EXAMPLES
Many types of structural connections can be parameterized and entered into the library of an FEA-based handbook. Numerical simulation of structural connections is challenging because of complicated interactions between fasteners and joined components. The formulation must account for physical factors such as partial contact between the fasteners and plate. In this case, the fastener’s shank is loaded in compression only. Formulations must also account for the shear stiffness of fasteners, initial clearance or interference between the fastener and joined plates, and material and geometric nonlinearities.
The model shows a “rigid core” for the fastener, surrounded by a Winklertype spring with a stiffness coefficient k which can be selected to represent the fastener’s spring rate. The rigid core has two degrees of freedom corresponding to displacement vector components δx and δy. The interaction between the fastener shank and the fastener hole is modeled by the relationship
where Tn is the normal traction acting on the fastener hole; un is the normal displacement along the perimeter (positive inward); θ is a polar coordinate of a system the origin of which is located in the center of the fastener hole, and Δr represents the interference between the fastener and hole. A negative Δr means the fastener fits loosely. When Δr = 0 the fastener is said to be “neatly fitted.” Any number of fastener cores can be connected by assigning the same variable name to the corresponding displacement vector components δx and δy.
Tn = 2 k(un + δx cosθ + δy sinθ + Δr)
In subsequent iterations the spring rate k comes from the following equation.
Because it’s possible that the material yields in the neighborhood of the fastener holes, the model accounts for yielding in the connected plates by means of the deformation theory of plasticity. Convergence occurs in less than 10 iterations when the material does not yield. When it does, the number of iterations depends on the extent of yielding.
where Ef = fastener’s modulus of elasticity, vf = Poisson’s ratio for the fastener material, and rf = the radius of the fastener hole. All fasteners are assumed neatly fitted (Δr = 0).
The spliced plates are 6061-T6 aluminum alloy with E = 10.1 3 106 psi and Poisson’s ratio n = 0.365. The yield stress is 3.53 103 psi and the yield law is of the Ramberg-Osgood type. The fasteners are steel with Ef = 29.0 × 106 psi, vf = 0.295. One symmetry plane reduces the problem size. The other symmetry plane is not used so the problem is more easily visualized.
The estimated relative error of the linear solution at p = 8 was 0.05% in the energy norm. Nonlinear solutions are performed at p = 8. Equivalent stress contours corresponding to the nonlinear solution (p = 8) appear in How the joint behaves.
In another example, fastener holes and attachment lugs are often cold-worked to increase durability and fatigue resistance. In these problems the magnitude and distribution of residual stresses are sought. Because of the problem’s complexity, a classical treatment is limited to holes in infinite plates. In engineering practice, however, cold-worked holes close enough to an external boundary or other holes should be included in the problem. They require better numerical treatment. Such problems lend themselves to standardization in FEA-based handbook libraries.
A typical cold-working problem shows the distribution of the equivalent (von Mises) stress in an attachment lug where the mandrel is forced into the hole. The mandrel material is assumed to be linearly elastic. The lug is made of a material described by the Ramberg-Osgood stress-strain law with E = 16.03 106 psi, n = 0.31, S70E = 1.373 105 psi; n = 33, and σy = 1.313 105 psi. The problem specifies a diametric interference level of 0.03 in. And material properties of the mandrel are E = 30 × 106 psi and n = 0.30. The lug is thick enough to consider only plane strain conditions. The analysis is based on the deformation theory of plasticity and the solution uses the p-version finite-element numerical-computation method.
Residual stresses are computed by superposition, which means that tractions imposed by the mandrel are computed and applied in the opposite direction to simulate stress-free boundary conditions after removing the mandrel. The combined stress fields equal the residual stress distribution.
With such reliable theory and fast computers available, handbook limitations no longer exist. What’s a little harder to come by are qualified people to plug new models into the FEA-based handbooks.
The professional qualifications required for creating entries into such handbook libraries include expertise in the particular engineering field of the intended end user, theoretical aspects of mechanics or other relevant disciplines, and applied numerical simulation technology. Admittedly, such expertise is relatively rare. But once located, the FEA-based handbook framework is well suited for best use of that important resource.