Six Sigma and Beyond: Design for Six Sigma, Volume VI

This technique is not thought of as being a reliability improvement method, yet it can contribute significantly to its enhancement. Finite Element Analysis (FEA) is a technique of modeling a complex structure into a collection of structural elements that are interconnected at a given number of nodes. The model is subjected to known loads, whereby the displacement of the structure can be determined through a set of mathematical equations that account for the element interactions. The reader is encouraged to read Buchanan (1994) and Cook (1995) for a more complete and easy understanding of the theoretical aspects of FEA.

In commercial use, FEA is a computer-based procedure for analyzing a complex structure by dividing it into a number of smaller, interconnected pieces (the "finite elements"), each with easily definable load and deflection characteristics.

TYPES OF FINITE ELEMENTS

The library of finite elements available in general purpose codes can be subdivided into the following categories:

1. Point elements: An example of a point element is a lumped mass element or an element specifically created to represent a particular constraint or loading present at that point.

2. Line elements: Truss links, rods, beams, pipes, cables, rigid links, springs, and gaps are examples of line elements. This type of element is usually characterized by two grid points or nodes at each end.

3. Surface elements: Membranes, plates, shells and certain types of fluid and thermal elements fall into this category. The surface elements can be triangular or quadrilateral, and thin or thick; accordingly they are characterized by a connectivity of three or more grid points or nodes.

4. Solid elements: Examples of solid elements include wedges, prisms, cubes, parallelepipeds and three-dimensional fluid and thermal elements. Elements in this category are usually defined using six or more grid points or nodes.

5. Special purpose elements: Combinations of springs, gaps, dampers, electrical conductors, acoustic, fluid, magnetic, mass, superelement, crack tips, radiation links, etc., are included in this category.

For example, commonly used elements in the automotive industry (body engineering) are:

• Beams

• Rigid links

• Thin plates ” triangular and quadrilateral

• Solid elements

• Springs

• Gaps (contact or interface elements)

TYPES OF ANALYSES

There are many combinations of analyses one may perform with FEA as the driving tool. However, the two predominant types are nonlinear and dynamic. Using these types one may focus on specific analysis of ” for example nonlinearities types such as:

Geometric

• Stress less than yield strength

• Euler (elastic) buckling

• Examples: quarter panel under jacking and towing; hood following front crash

Material

• Stress greater than yield strength or material is nonlinear elastic

• Plastic flow

• Examples: seat belt pull; door intrusion beam bending

Combination of geometric and material

• Stress is greater than yield strength and buckling takes place

• Crippling

• Examples: rails during crash; roof crush

The reader should also recognize that combinations of these types exist as well, for example linear/static ” the easiest and most economical. Most of the FEA applications involve this kind of analysis. Examples include joint stiffness and door sag. Nonlinear/static is less frequently used. Examples include door intrusion beam, roof crush, and seat belt pull. Linear/dynamic is rarely used. Examples include windshield wipers or latch mechanism. Nonlinear/dynamic is the most complex and most expensive. Examples include knee bolster crash, front crash, and rear crash. Let us look at these combinations a little more closely:

• Linear static analysis: This is the simplest form of analytical application and is used most frequently for a wide range of structures. The desired results are usually the stress contours , deformed geometry, strain energy distribution, unknown reaction forces, and design optimization. Typical examples are door sag simulation, margin/fit problems, joint stiffness evaluation, high stress location search for all components , spot weld forces, and thermal stresses.

• Euler buckling analysis: This analysis is also relatively simple to perform and is used to calculate critical buckling loads. Caution should be exercised when performing this analysis because it produces analytical results that are not conservative. In other words, the critical buckling load thus calculated is usually higher than the actual load that would be determined through testing. A typical application is hood buckling.

• Normal modes analysis: This is an extremely useful technique for determining the natural frequencies (eigenvalues) of components and also the corresponding eigenvectors which represent the modes of deformation. Strictly speaking, this category does not fall under dynamic analysis since the problem is not time dependent. Typical examples include instrument panels, total vehicle or component NVH evaluation, door inner panel flutter, and steering column shake.

• Nonlinear static analysis: In general, all nonlinear analysis requires advanced methodology and is not recommended for use by inexperienced analysts. Usually, a graduate degree or several graduate level courses in the theory of elasticity, plasticity, vibrations, and solid and fluid mechanics are required to understand nonlinear behavior. Nonlinear FEA tends to be as much an art as it is a science, and familiarity with the subject structure is essential. Typical examples are seat back distortion, door beam bending rigidity studies, underbody components such as front and rear rails and wheel housings, bumper design, and crush analysis of several components.

• Nonlinear dynamic analysis: This FEA category is the most advanced. It involves very complex ideas and techniques and has become practicable only due to the availability of super-high-speed computers. This class of analysis involves all the complexities of nonlinear static analysis as well as additional problems involved with iterative time step selection and contact simulation at impact. Typical applications are related to crash evaluation and energy management.

PROCEDURES INVOLVED IN FEA

The procedures involved in FEA include:

1. Problem definition: Specification of concerns and expected results

2. Planning of analysis: Making decisions regarding the applicability of FEA, which code to use, and the size and the type of model to be constructed

3. Digitizing: The translation of a drawing into line data that is available to the modeler

4. Modeling: Creating the desired finite element model as planned (Many sophisticated tools are available such as the PDGS-FAST system, PAT RAN, and so on.)

5. Input of data: Creating, editing and storing a formatted data file that includes a description of the model geometry, material properties, constraints, applied loading, and desired output

6. Execution: Processing the input data in either the batch or the interactive mode through the finite element code residing on the computer system and receiving the output in the form of a printout and/or post-processor data

7. Interpretation of output: A study of the output to check the validity of the input parameters as well as the solution of the structural problem

8. Feasibility considerations: Utilizing the output to make intelligent technical decisions about the acceptability of the structural design and the scope for design enhancement

9. Parametric studies: Redesign using parametric variation (The easiest changes to study are those involving different gages, materials, constraints, and loading. Geometric changes require repetition of steps 3 through 8; the same is true about remodeling of the existing geometry.)

10. Design optimization: An iterative process involving the repetition of steps 3 through 9 to optimize the design from considerations of weight, cost, manufacturing feasibility, and durability

STEPS IN ANALYSIS PROCEDURE

The steps in the analysis procedure are:

1. Establish objective.

2. What type of analysis? What program?

   Statics

   • Mechanical Loads

     • Forces

     • Displacements

     • Pressure

     • Temperatures

   • Heat Transfer

     • Conduction

     • Convection

     • 1-D radiation

   Dynamics

   • Mode frequency

   • Mechanical load

     • Transient (direct or reduced) linear

     • Sinusoidal

   • Shock spectra

   • Heat transfer direct transient

   Special features

   • Nonlinear

     • Buckling

     • Large displacement

     • Elasticity

     • Creep

     • Friction, gaps

   • Substructuring

3. What is minimum portion of system or structure required?

   • Known forces or displacements at a point

     • Allows for separation

       • Structural symmetry

       • Isolation through test data

       • Cyclic symmetry

4. What are loading and boundary conditions?

   • Loading known

   • Loading can be calculated from simplistic analysis

   • Loading to be determined from test data

   • Support of excluded part of system established on modeled portion

   • Test data taken to establish stiffness of partial constraints

5. Determine model grid.

   • Choose element types.

   • Establish grid size to satisfy cost versus accuracy criterion.

6. Develop bulk data.

   • Establish coordinate systems.

   • Number node or order elements to minimize cost.

   • Develop node coordinates and element connectivity description.

   • Code load and B.C. description.

   • Check geometry description by plotting.

OVERVIEW OF FINITE ELEMENT ANALYSIS ” SOLUTION PROCEDURE

The process of FEA may be summarized with a flow chart of linear static structural analysis in seven steps. The steps are:

1. Represent continuous structure as a collection of discrete elements connected by node points.

2. Formulate element stiffness matrices from element properties, geometry, and material.

3. Assemble all element stiffness matrices into global stiffness matrix.

4. Apply boundary conditions to constrain model (i.e., remove certain degrees of freedom).

5. Apply loads to model (forces, moments, pressure, etc.).

6. Solve matrix equation {F} = [K]{u} for displacements.

7. Calculate element forces and stresses from displacement results.

INPUT TO THE FINITE ELEMENT MODEL

Once the user is satisfied with the model subdivision, the following classes of input data must be prepared to provide a detailed description of the finite element model to typical FEA software such as MSC/NASTRAN (1998):

• Geometry: This refers to the locations of grid points and the orientations of coordinate systems that will be used to record components of displacements and forces at grid points.

• Element connectivities: This refers to identification numbers of the grid points to which each element is connected.

• Element properties: Examples of element properties are the thickness of a surface element and the cross-sectional area of a line element. Each element type has a specific list of properties.

• Material properties: Examples of material properties are Young's modulus , density, and thermal expansion coefficient. There are several material types available in MSC/NASTRAN. Each has a specific list of properties.

• Constraints: Constraints are used to specify boundary conditions, symmetry conditions, and a variety of other useful relationships. Constraints are essential because an unconstrained structure is capable of free-body motion, which will cause the analysis to fail.

• Loads and enforced displacements: Loads may be applied at grid points or within elements.

OUTPUTS FROM THE FINITE ELEMENT ANALYSIS

Once the data describing the finite element model have been assembled and submitted to the computer, they will be processed by a software package such as MSC/NASTRAN to produce information requested by the user. The classes of output data are:

1. Components of displacements at grid points

2. Element data recovery: stresses, strains, strain energy, and internal forces and moments

3. Grid point data recovery: applied loads, forces of constraint, and forces due to elements

It is the responsibility of the user to verify the accuracy of the finite element analysis results. Some suggested checks to perform are:

• Generate plots to visually verify the geometry.

• Verify overall model response for loadings applied.

• Check input loads with reaction forces.

• Perform hand checks of results whenever possible.

• Review and check results.

• Plot deformation and stress contour.

• Check equilibrium and reaction forces.

• Check concentration region for fineness of grid (compare calculated stress distribution with assumed element distribution).

• Check peak deflection and/or stress for ballpark accuracy.

ANALYSIS OF REDESIGNS OF REFINED MODEL

At this stage, generally a correlation is attempted even though it is very difficult and presents many potential problems. These problems are about 60% associated with analysis and 40% associated with the actual testing. Remember that correlations at this stage commonly (over 50 projects) may run from 5 to 30%.

Obviously, the focus should be on testing and test-related correlation with real world usage. Items of concern should be:

Loads:

• Isolation of single component of assembly

• Hard to put assumed load in controlled lab test (linear loads causing moments)

Strain gages:

• Gage locations and orientation

• Single leg gages versus rosettes

• Improper gage lead hookup

Non-linearities:

• Plasticity

• Pin joint clearance

• Bolted joints

In a typical analysis, the related correlation issues/problems/concerns examples are:

• Mesh size (for localized stress concentration, isolate concentration region and refine mesh)

• Element type

• Load distribution and B.C. isolation

• Input error/bad data

• Weld details

Common problems that may be encountered in the FEA are:

• Part not to size

• Misunderstanding or interpretation of results

Therefore, to make sure that the FEA is worth the effort, the following steps are recommended:

1. Initially, take simple, well-isolated components, with simple well-defined loads.

2. Do not expect miracles .

3. Use a joint test/analysis program. It can improve the capabilities of each step and serves as a check on techniques.

4. Work together. This is the key. The test results supplement weakness of analysis and vice versa.

SUMMARY ” FINITE ELEMENT TECHNIQUE: A DESIGN TOOL

• Proven tool ” approximate but very accurate if applied properly.

• Fine enough grid to match true strain field.

• Need to know loads accurately.

• Are supports rigid? What spring stiffness?

• Do not let FEA become just a research tool searching for an absolute answer. Use in all stages of design cycle as relative comparison tool in conjunction with test.

• FEA if nothing else forces someone to examine in detail a component design.

• A check on geometry itself.

• The experimenter must think in detail about loads and interaction with rest of system