applications of FEA in mechanical engineering

ABSTRACT

In the recent years of Manufacturing everybody is trying to reduce manufacturing lead time, trying to know at what load the product will fail ?, for how many time it will be in working condition ? Answers to all these questions is not impossible but it is difficult. For getting answers there is one and only one technique FEA.

FEA means Finite Element Analysis. Here in this method it consists of preparation of model of product before its actual manufacturing . This model is analyzed by applying loads, results are compared with the customers need and finally conclusion about the behavior of the product is made.

Applications of  FEA are not restricted to any branch or any side, but it can be used every where there is problem. But here in this seminar its applications in the Mechanical Engineering are discussed such as Stress Analysis, Fluid Flow problems etc.

Thus FEA is one and only one technique, which can be used for finding the behavior of the product before its actual manufacturing.

 1. INTRODUCTION

Finite element analysis was first developed for use in the aerospace and nuclear industries where the safety of structures is critical. Today, the growth in usage of the method is directly attributable to the rapid advances in computer technology in recent years. As a result, commercial finite element packages exist that are capable of solving the most sophisticated problems, not just in structural analysis, but for a wide range of phenomena such as steady state and dynamic temperature distributions, fluid flow and manufacturing processes such as injection molding and metal forming.

                                     .
FEA consists of a computer model of a material or design that is loaded and analyzed for specific results. It is used in new product design, and existing product refinement. A company is able to verify that a proposed design will be able to perform to the client's specifications prior to manufacturing or construction. Modifying an existing product or structure is utilized to qualify the product or structure for a new service condition. In case of structural failure, FEA may be used to help determine the design modifications to meet the new condition.

                                                                                          .
Mathematically, the structure to be analyzed is subdivided into a mesh of finite sized elements of simple shape. Within each element, the variation of displacement is assumed to be determined by simple polynomial shape functions and nodal displacements. Equations for the strains and stresses are developed in terms of the unknown nodal displacements. From this, the equations of equilibrium are assembled in a matrix form which can be easily be programmed and solved on a computer. After applying the appropriate boundary conditions, the nodal displacements are found by solving the matrix stiffness equation. Once the nodal displacements are known, element stresses and strains can be calculated.

          2.  HISTORY OF FEM & FEA

Finite Element Analysis (FEA) was first developed in 1943 by R. Courant, who utilized the Ritz method of numerical analysis and minimization of variational calculus to obtain approximate solutions to vibration systems. Shortly thereafter, a paper published in 1956 by Turner, Clough, Martin, & Topp established a broader definition of numerical analysis. This paper centered on the "stiffness and deflection of complex structures"..

                                                                                                                             
By the early 70's, FEA was limited to expensive mainframe computers generally owned by the aeronautics, automotive, defense, and nuclear industries, and the scope of analyses were considerably limited. Finite Element technology was further enhanced during the 70's by such people as Zeinkiewicz & Cheung, when they applied the technology to general problems described by Laplace & Poisson's equations. Mathematicians were developing better solution algorithms, the Galerkin, Ritz & Rayleigh-Ritz methods emerged as the optimum solutions for certain categories of general type problems. Later, considerable research was carried out into the modeling & solution of non-linear problems, Hinton & Crisfield being major contributors.

3. THE DIFFERENCE BETWEEN FEM & FEA

The terms 'finite element method' & 'finite element analysis' seem to be used interchangeably in most documentation, so the question arises is there a difference between FEM & FEA?? The answer is yes, there is a difference.

                                                                                                                                
The finite element method is a mathematical method for solving ordinary & elliptic partial differential equations via a piecewise polynomial interpolation scheme. Put simply, FEM evaluates a differential equation curve by using a number of polynomial curves to follow the shape of the underlying & more complex differential equation curve. Each polynomial in the solution can be represented by a number of points and so FEM evaluates the solution at the points only. A linear polynomial requires 2 points, while a quadratic requires 3. The points are known as node points or nodes. There are essentially three mathematical ways that FEM can evaluate the values at the nodes, there is the non-variational method (Ritz), the residual method (Galerkin) & the variational method (Rayleigh-Ritz).

FEA is an implementation of FEM to solve a certain type of problem. For example if we were intending to solve a 2D stress problem. For the FEM mathematical solution, we would probably use the minimum potential energy principle, which is a variational solution. As part of this, we need to generate a suitable element for our analysis. We may choose a plane stress, plane strain or an ax symmetric type formulation, with linear or higher order polynomials. Using a piecewise polynomial solution to solve the underlying differential equation is FEM, while applying the specifics of element formulation is FEA, e.g. a plane strain triangular quadratic element.

 4.  TYPICAL ANALYSIS

In the real world, no analysis is typical, as there are usually facets that cause it to differ from others. There is however a main procedure that most FE investigations take. This procedure is detailed below:

4.1  PLANNING THE ANALYSIS

This is arguably the most important part of any analysis, as it helps ensure the success of the simulation. Oddly enough, it is usually the one analysts leave out. The purpose of an FE analysis is to model the behavior of a structure under a system of loads. In order to do so, all influencing factors must be considered & determined whether their effects are considerable or negligible on the final result. The degree of accuracy to which any system can be modeled is very much dependant on the level of planning that has been carried out. Answers to many questions need to be found. 'Planning an analysis' is dealt with in detail in the 'improving results' section of this site. 

                                                                          

·           PRE-PROCESSOR
The preprocessor stage in general FE packages involves the following:

·           Specifying the title that is the name of the problem. This is optional but very useful, especially if a number of design iterations are to be completed on the same base model.

·           Setting the type of analysis to be used, e.g. structural, fluid, thermal or electromagnetic, etc. (sometimes this can only be done by selecting a particular element type).

·           Creating the model. The model is drawn in 1D, 2D or 3D space in the appropriate units (M, mm, in, etc..). The model may be created in the pre-processor, or it can be imported from another CAD drafting package via a neutral file format (IGES, STEP, ACIS, Para solid, DXF, etc.). If a model is drawn in mm for example and the material properties are defined in SI units, then the results will be out of scale by factors of 10⁶. The same units should be applied in all directions, otherwise results will be difficult to interpret, or in extreme cases the results will not show up mistakes made during the loading and restraining of the model.

·           Defining the element type, this may be 1D, 2D or 3D, and specific to the analysis type being carried out (you need thermal elements to do thermal analyses).

·           Applying a Mesh. Mesh generation is the process of dividing the analysis continuum into a number of discrete parts or finite elements. The finer the mesh, the better the result, but the longer the analysis time. Therefore, a compromise between accuracy & solution speed is usually made. The mesh may be created manually, such as the one on the right, or generated automatically like the one below. In the manually created mesh, you will notice that the elements are smaller at the joint. This is known as mesh refinement, and it enables the stresses to be captured at the geometric discontinuity (the junction).

·           Manual meshing is a long & tedious process for models with any degree of geometric complication, but with useful tools emerging in pre-processors, the task is becoming easier. Automatic mesh generators are very useful & popular. The mesh is created automatically by a mesh engine, the only requirement is to define the mesh density along the model's edges. Automatic meshing has limitations as regards mesh quality & solution accuracy. Automatic brick element (hex) meshers are limited in function, but are steadily improving. Any mesh is usually applied to the model by simply selecting the mesh command on the preprocessor list of the gui.

·           Assigning properties. Material properties (Young’s modulus, Poisson’s ratio, the density, & if applicable, coefficients of expansion, friction, thermal conductivity, damping effect, specific heat etc.) will have to be defined. In addition element properties may need to be set. If 2D elements are being used, the thickness property is required. 1D beam elements require area, I_xx, I_yy, I_xy, J, & a direction cosine property, which defines the direction of the beam axis in 3D space. Shell elements, which are 2½D in nature (2D elements in 3D space), require orientation & neutral surface offset parameters to be defined. Special elements (mass, contact, spring, gap, coupling, damper etc.) require properties (specific to the element type) to be defined for their use.

·           Apply Loads. Some type of load is usually applied to the analysis model. The loading may be in the form of a point load, a pressure or a displacement in a stress (displacement) analysis, a temperature or a heat flux in a thermal analysis & a fluid pressure or velocity in a fluid analysis. The loads may be applied to a point, an edge, a surface or a even a complete body. The loads should be in the same units as the model geometry & material properties specified. In the cases of modal (vibration) & buckling analyses, a load does not have to be specified for the analysis to run.

·           Applying Boundary Conditions. If you apply a load to the model, then in order to stop it accelerating infinitely through the computer's virtual ether (mathematically known as a zero pivot), at least one constraint or boundary condition must be applied. Structural boundary conditions are usually in the form of zero displacements, thermal BCs are usually specified temperatures, fluid BCs are usually specified pressures. A boundary condition may be specified to act in all directions (x, y, z), or in certain directions only. They can be placed on nodes, key points, areas or on lines. BC's on lines can be in the form of symmetric or anti-symmetric type boundary conditions, one allowing in plane rotations and out of plane translations, the other allowing in plane translations and out of plane rotations for a given line. The applications of correct boundary conditions are a critical to the accurate solution of the design problem. At least one BC has to be applied to every model, even modal & buckling analyses with no loads applied. See the 'Advanced BCs' section for explanations on more advanced boundary condition types.

4.2 SOLUTION
Thankfully, this part is fully automatic. The FE solver can be logically divided into three main parts, the pre-solver, the mathematical-engine & the post-solver. The pre-solver reads in the model created by the pre-processor and formulates the mathematical representation of the model. All parameters defined in the pre-processing stage are used to do this, so if you left something out, chances are the pre-solver will complain & cancel the call to the mathematical-engine. If the model is correct the solver proceeds to form the element-stiffness matrix for the problem & calls the mathematical-engine, which calculates the result (displacement, temperatures, pressures, etc.). The results are returned to the solver & the post-solver is used to calculate strains, stresses, heat fluxes, velocities, etc.) for each node within the component or continuum. All these results are sent to a results file, which may be read by the post-processor.

                                              

4.3 POST-PROCESSOR
Here the results of the analysis are read & interpreted. They can be presented in the form of a table, a contour plot, deformed shape of the component or the mode shapes and natural frequencies if frequency analysis is involved. Other results are available for fluids, thermal and electrical analysis types. Most post-processors provide an animation service, which produces an animation & brings your model to life.

                                                                                         .
Contour plots are usually the most effective way of viewing results for structural type problems. Slices can be made through 3D models to facilite the viewing of internal stress patterns.

                                                                                                                  .
All post-processors now include the calculation of stress & strains in any of the x, y or z directions, or indeed in a direction at an angle to the coordinate axes. The principal stresses and strains may also be plotted, or if required the yield stresses and strains according to the main theories of failure (von mises, St. Venant, Tresca etc.). Other information such as the strain energy, plastic strain and creep strain may be obtained for certain types of analyses.

   5. APPLICATIONS  OF  FEA

The finite element method is a very important tool for those involved in engineering design, it is now used routinely to solve problems in the following areas:

• Structural strength design
• Structural interactions with fluid flows
• Analysis of Shock (underwater & in materials)
• Acoustics
• Thermal analysis
• Vibrations
• Crash simulations
• Fluid flows
• Electrical analyses
• Mass diffusion
• Buckling problems
• Dynamic analyses
• Electromagnetic evaluations
• Metal forming
• Coupled analyses

6. STRESS & DISPLACEMENT ANALYSIS

6.1 INTRODUCTION

The most common application of FEA is the solution of stress related design problems. As a result, all commercial packages have an extensive range of stress analysis capabilities.

6.2 WHAT IS STRESS??

Stress can be described as a measurement of intensity of force. As all engineers know, if this intensity increases beyond a limit known as yield, the component's material will undergo a permanent change in shape or may even be subjected a to dramatic failure.

From a formal point of view, three conditions have to be met in any stress analysis, equilibrium of forces (or stresses), compatibility of displacements and satisfaction of the state of stress at continuum boundaries. These conditions, which are usually described mathematically in good undergraduate strength of material texts, are also applicable to non-linear analyses.

6.3 PRPCESS:-

It all starts off with the formulation of the components 'stiffness' matrix. This square matrix is formed from details of the material properties, the model geometry & any assumptions of the stress-strain field (plane stress or strain).

• Once the stiffness matrix is created, it may be used with the knowledge of the forces to evaluate the displacements of the structure (hence the term displacement analysis).
• On evaluation of the displacements, they are differentiated to give six strain distributions, 3 mutually perpendicular direct strains & 3 corresponding shear strains.
• Finally six stress distributions are determined via the stress/strain relationships of the material.
• Commercial packages usually go one further & calculate a range of more usable stress fields from the six stress components such as the principal stresses & a host of failure prediction stresses as described by the most common yield criteria (Von Mises/Maxwell/Heckney, Guest/Tresca, Heubner/Thornton, etc.). The displacements can be used in conjunction with the element stiff nesses to determine the reaction forces & the forces internal to each element (otherwise known as the stress resultants).
• A point to note is that at least one of the displacements must be known before the rest can be determined (before the system of equations can be solved). These known displacements are referred to as boundary conditions and are oftentimes a zero value. Without these boundary conditions, we would get the familiar singularity or zero-pivot error message from the solver, indicating that no unique solution was obtainable.

6.4 AN ALTERNATIVE SOLUTION

An alternative solution may be obtained via the force matrix method (otherwise known as the flexibility method). In the previous description, the displacements were the unknown, and solution is said to be obtained via the stiffness method. In the force method, the forces are the nodal unknowns, while the displacements are known. The solution is obtained for the unknown forces via the flexibility matrix & the known displacements. The stiffness method is more powerful & applicable than the flexibility approach.

6.5 NON-LINEAR ANALYSES

In order to explain non-linearity in stress analyses, lets examine the nature of linear solutions. Many assumptions are made in linear analyses, the two primary ones being the stress/strain relationship & the deformation behavior. The stress is assumed to be directly proportional to strain and the structure deformations are proportional to the loads. The second assumption is oftentimes mistaken to derive from the first, a fishing rod is an example of a non-linear structure made of linear material. A stress analysis problem is linear only if all conditions of proportionality hold. If any one of them is violated, then we have a Non-Linear problem.

Most real life structures, especially plastics, are non-linear, perhaps both in structure and in material. Most plastic materials have a non-linear stress strain relationship. The non-linearity arising from the nature of material is called 'Material Non-linearity'. Furthermore, thin walled plastic structures exhibit a non-linear load-deflection relationship, which could arise even if the material were linear (fishing rod). This kind is called geometric non-linearity.

All non-linearties are solved by applying the load slowly (dividing it into a number of small loads increments). The model is assumed to behave linearly for each load increment, and the change in model shape is calculated at each increment. Stresses are updated from increment to increment, until the full-applied load is reached

In a nonlinear analysis, initial conditions at the start of each increment is the state of the model at the end of the previous one. This dependency provides a convenient method for following complex loading histories, such as a manufacturing process. At each increment, the solver iterates for equilibrium using a numerical technique such as the Newton-Raphson method. Due to the iterative nature of the calculations, non-linear FEA is computationally expensive, but reflects the real life conditions more accurately than linear analyses. The big challenge is to provide a convergent solution at minimum cost (the minimum number of increments).

See the 'non-linear' section of solution types for more details of such analyses & how solutions                        

are achieved. For details on applying loads & boundary conditions, see the 'improving results'             and 'faster analysis' sections on the menu.

 7. ANALYSIS OF TEMPERATURE & HEAT FLOW

7.1 INTRODUCTION

Thermal analysis is used to determine the temperature distribution, heat accumulation or dissipation, and other related thermal quantities in an object. The nodal degrees of freedom (primary unknown data) are the temperatures. The primary heat transfer mechanisms are conduction, convection and radiation. In addition, less dominant phenomena such as change of phase (melting or freezing) & internal heat generation can occur.

 The heat transfer coefficient is dependent on many factors such as fluid pressure, velocity, density, specific heat (ratio of specific heats if the fluid is compressible), viscosity & conductivity. It is also dependent on surface properties such as roughness & geometry. Due to the extreme non-linear nature of convection type phenomena, solutions are usually based on empirical relations such as log laws.

In order to implement convective heat transfer in FEA, boundary conditions for specific cases have been developed. Examples of which are Vertical Plate in horizontal flows, flow over isothermal inclined flat plates, flow through horizontal cylinders, flow over an inclined surface, vertical enclosed space flow, flow in horizontal tubes & ducts, generic convection as a function of temperature difference or grashof & prandtl numbers, flow along a rotating disk, etc.. Each boundary condition may have automatic implementation for each of the three flow types, laminar, Transition & turbulent.

                                                                                                                                                                                                                                        

Information defining all parameters of each BC type must be input, this can usually be carried out manually or via tables of data. An important point to note is that very few packages have the capability to apply convective BC's to the level described here. MSc/Thermal and SC03, the Rolls-Royce proprietary code, both have extensive capabilities for applying convective type BC's. At least two other FEA software vendors are currently considering implementing such capability into their codes.

7.2 NON-LINEAR & TRANSIENT ANALYSIS

If temperatures are much higher or lower than the average temperature in certain locations of the model, there is a good chance that the heat transfer coefficients & conductivity will themselves become temperature dependent. The problem becomes non-linear, as the heat transfer rate is not directly proportional to temperature. The approach to a solution is similar to that of a non-linear displacement analysis, the load is divided into a number of smaller ones that are applied incrementally. The solution becomes an iterative procedure rather than one of matrix factorization alone.

If the thermal load is impulsive in nature (time dependent), then a solution through time is required. This is carried out by dividing the overall time range into a number of smaller time steps & applying time integration techniques to handle the evaluation of the solution from one time step to the next. As with structural analyses, there are three main types of time integration techniques, Implicit, explicit & central difference (Crank-Nicholson). Implicit analyses are stable but computationally expensive, explicit integration is fast but unstable, and Crank-Nicholson is a mix of the two, but is also unstable.

7.3 THERMAL STRESS ANALYSIS

Often an object will fail because of stresses induced by uneven heating, rapid temperature change or differences in thermal properties. A coupled analysis, which models both thermal and stress variations, can be effective in predicting the overall structural response. It facilitates effective prediction of incidents where thermal expansion is an important consideration, such as in reciprocating & gas turbine engine design 

                                                                                  .
The usual procedure is to carry out a thermal analysis, which evaluates the temperature distribution. These temperatures can then be used to prime the displacement analysis, and hence thermal deflections, strains & stresses can be evaluated

                                                                 .
It is also possible to have fully coupled analyses where the temperatures & displacements are a function of each other. This is most evident in analyses that involve fluid flows, such as in a gas turbine or rocket. The heat transfer rates are dependent on mass flow rates, but mass flow rates are a function of valve & seal clearances (labrynth seals in gas turbines, nozzles in rockets). Therefore, we end up with a scenario where clearances are a function of temperature and temperature is a function of clearances. This type of problem can only be solved via a non-linear and fully coupled solution.

7.4 ITERATIVE ANALYSES & SUBMODELLING

In analysis of large & complex systems (such as reciprocating & turbine engines), it is usual to carry out isolated design work on specific components of the overall system. For example, we may want to refine the design of a piston, and would like a realistic temperature distribution, but don't want to incorporate the rest of the engine into the analysis model. In order to do so effectively, information is required about the state of temperature at the sub-system extremities.

                                                                           

One way of providing these temperatures is to carry out a coarse analysis on the overall model and use the temperatures at relevant points to 'prime' the sub-system model (otherwise known as a sub model). 

                                                                                                                                        

FEA Packages with such facilities have the capability of writing out the temperature-time values from the coarse model to a file. This file can be subsequently used in the sub model, and so an accurate representation of the thermal environment can be provided while studying the finer details contained within the sub-system model. This approach is extremely rich as regards saving on analysis times. Despite the high level of idealization being implemented, it provides very accurate & realistic modeling of the physical conditions being investigated.

 8. BUCKLING ANALYSES: SUDDEN COLLAPSE

8.1 INTRODUCTION

Buckling is a critical state of stress and deformation, at which a slight disturbance causes a gross additional deformation, or perhaps a total structural failure of the part. Structural behavior of the part near or beyond 'buckling' is not evident from the normal arguments of static. Buckling failures do not depend on the strength of the material, but are a function of the component dimensions & modulus of elasticity. Therefore, materials with a high strength will buckle just as quickly as low strength ones.

                                                                                                       .
If a structure has one or more dimensions that are small relative to the others (slender or thin-walled), and is subject to compressive loads, then a buckling analysis may be necessary.

From an FE analysis point of view, a buckling analysis is used to find the lowest multiplication factor for the load that will make a structure buckle. The result of such an analysis is a number of buckling load factors (BLF). The first BLF (the lowest factor) is always the one of interest. If it is less than unity, then buckling will occur due to the load being applied to the structure. The analysis is also used to find the shape of the buckled structure.

8.2 EVALUATING LINEAR INSTABILITIES

From a formal point of view, buckling is an Eigen value problem that is a function of the material & geometric stiffness matrices. Consequently, there will be a number of buckling modes and corresponding mode shapes.

                                                                                                              .
As with a frequency analysis, eigen value extraction may be carried out using a number of available methods, the best choice depends on the form of the equations being solved. The main methods are the power, subspace, LR, QR, Givens, Householder & Lanczos methods.

An important note is that the eigen value method does not take into account of any initial imperfections in the structure and so the results rarely correspond with practical tests. Eigen value solutions usually over estimate the buckling load and give no information about the post-buckling state of the structure. Sudden buckling simply does not occur in the real world.

                                                                                                                      
So how should we know if a linear buckling analysis is sufficient?? Carry out both a linear static analysis and a linear (eigen value) buckling analysis. If the max stress is significantly less than yield, and the buckling load factor is greater than 1.0, then buckling will probably not occur. If however the BLF is less than 1.0, then the buckling analysis will be linear provided that the max stress is far below yield. In all other cases, a non-linear buckling analysis should be carried out. If the component is critical to the safe operation of a system, full displacement analyses should be carried out.

8.3 NON-LINEAR BUCKLING

A more practical approach is to carry out a large displacement analysis, where buckling can be detected by the change of displacement in the model. A large displacement problem is non-linear in nature. Geometric non-linearity arises when deformations are large enough to significantly alter the way load is applied, or load is resisted by the structure. 

                                                  .
The approach to a non-linear buckling solution is achieved by applying the load slowly (dividing it into a number of small loads increments). The model is assumed to behave linearly for each load increment, and the change in model shape is calculated at each increment. Stresses are updated from increment to increment, until the full-applied load is reached. The solution becomes an iterative procedure rather than one of matrix factorization alone, and consequently is computationally expensive.

An interesting variation arises in the case of automotive applications. In the case of front-end collision, the hood is expected to crumple (buckle) in order to absorb the energy of collision, as well as to save the passenger compartment. In such cases, we are not designing against, but for buckling.

8.4 AVOIDING INSTABILITIES

Any structure is most efficient when subjected to evenly distributed tensile or compressive stress, such as occurring in cables, strings etc. Evidently, such modes of loading makes the best use of the material, and its strength. On the other hand bending (flexing) is the least efficient way of loading a structure. A high flexural stiffness of the structure means high resistance to buckling. This is true even if the load is entirely in plane, since when buckling is imminent; the only stiffness that counts is flexural.

                                                                                                 .
Eccentricity of loading promotes buckling. Eccentricity means that the resultant load does not pass through the centroid of the load bearing cross section. It is safe to assume that in 100% of practical applications, loads are eccentric.

When buckling occurs, symmetry of the part does not apply. There is no symmetry of the buckled shape, although both the part, and the loading may be symmetric. Correspondingly, when carrying out an FE buckling investigations, it is advisable to implement a full 3D analysis of the structure under inspection.

                                                                                                                               .
The non-linear stress strain behavior of the material reduces the stiffness at higher stress (load) levels, and hence elastic formulas from the handbooks tend to be highly unconservative.

If a component is structurally slender, and is made of plastic, then the component faces buckling from three directions; from the low material stiffness, the large deflections producing eccentricity during deformation, and from the non-linearity of the material itself.  

                                     
                                        
By and large it is true that buckling usually occurs when compressive stress is present. But what is not evident that compressive stress can prevail in un-expected places. Shallow domes under internal pressure can develop local compressive stress regions, and make it vulnerable to instabilities.

8.5 BIFURCATION & SNAP THROUGH BUCKLING

In many systems a smooth change in a control parameter (the load) can lead to an abrupt change in the behavior of the system. A simple example is the buckling of a rod. If a straight rod is compressed by a small load, it shrinks to some extent, but remains straight. For larger loads, however, it starts to buckle. Mathematically, the solution corresponding to a straight rod still exists, but it is unstable for the large load applied and very small transverse perturbations make the rod buckle. The transition from the unbuckled to the buckled state occurs via a bifurcation, that is, at the onset of the instability a new solution corresponding to the buckled rod comes into existence. In bifurcation buckling, there are two equilibrium solutions at the bifurcation point, the ordinary static strength of materials solution and the instable (buckling) solution.

                                                                     
Snap through buckling occurs when a structure is subject to an increasing load that at some point causes the structure to undergo a gross deformation. Subsequent to this deformation, the structure regains sufficient stability to carry load, usually in a configuration that changes the structural load from being initially compressive to tensile. An example of this is a shallow dome in compression. If the load becomes too great, it buckles and snaps through so that the load is supported in tension.

 9. FE ANALYSIS OF FLUID FLOW PROBLEMS

9.1 INTRODUCTION

Fluid flow problems arise in almost all industrial sectors: food processing, water treatment, marine engineering, automotive, aerodynamics, and gas turbine design. FEA facilitates the prediction of fluid flow, heat & mass transfer, and chemical reactions (explosions) and related phenomena.

By solving the fundamental equations governing fluid flow processes, FE analyses provide information on important flow characteristics such as pressure loss, flow distribution, and mixing rates. This results in better designs, lower risk, and faster time to the marketplace for product or processes. Models can be developed for physical phenomena such as turbulence, multiphase flow, chemical reactions, and radiative heat transfer.

9.2 SOLUTION APPROACH

The foundation of fluid dynamics is based on the Navier-Stokes equations, the set of partial differential equations that describe fluid flow. In FEA, this equation is rewritten as algebraic equations that relate the velocity, temperature, pressure, and other variables, such as species concentrations. The resulting equations are then solved numerically, yielding a complete picture of the flow.

The equations are solved iteratively using the method of weighted residuals. The main method of solution is achieved via the Galerkin method, but others exist. One such variant is the Petrov-Galerkin method, which is used to solve instances of viscous, high Reynolds number flows.

                                                                                                    
During the solution, asymmetric solution matrices may exist. Therefore, it is often necessary to use many relatively small & higher order elements in order to obtain convergence. Due to this problem, it is not uncommon to implement reduced integration type elements in an analysis.

                                                                                            .

For transient problems, a special time integration technique known as the semi-explicit scheme is used in large analyses, as it is more economical than other methods available.

9.3 COUPLED ANALYSES

Due to the complex nature of the physical processes being modeled, it is not unusual to conduct coupled analyses as part of a design program. Fluid-structural, fluid-thermal & fluid-acoustic analyses are not uncommon.

                                                                                                             .

Fluid-structural interaction is important in the design of offshore structures & their pipelines (also known as risers). Strong tidal currents & wave bombardment have a critical influence on the life of such components. The structural design of dams usually requires a model which involves hydrostatic loading & seepage flows. Fluid-thermal analyses are important in many applications such as viscous flows through restrictors & valves (both process & medical applications). Fluid-acoustic evaluation is required in order to ensure that noise emissions from systems such as gas turbines are within tolerable limits.

9.4 METAL FORMING TECHNIQUES

Solidification modeling of complex castings can significantly reduce casting porosity and improve casting yields. Large strain analysis of forging operations can provide residual strains/stresses generated in products and provide estimates of product "spring back". Despite the difficulty one may have in visualizing a metal behaving as a fluid, FEA uses slow non-Newtonian flow techniques for simulating metal forming processes. This type of analysis is divided into two primary sections, steady state problems such as extrusion & rolling processes, and transient problems such as forging & stamping.

                                                                                         
Both steady state & transient solutions are usually highly non-linear in nature. Therefore they are computationally expensive to implement.

9.5 CFD: AN ALTERNATIVE TECHNIQUE

Despite the fact that most fluid-flow type problems can be implemented successfully using FEA, it is not the paramount technology. Due to the nature of the fluid formulations for solution via FEM, long solution times & poor convergence can be experienced. As a result, a more convenient solution is obtained by using a method known as CFD. This method is based around finite- difference & finite volume solution techniques.

                                                                            .
The solution-adaptive grid capability of CFD is particularly useful for accurately predicting flow fields in regions with large gradients, such as free shear layers and boundary layers. In comparison to solutions on structured or block-structured grids, this feature significantly reduces the time required to generate a "good" grid and arrive at a suitable solution.

10. ESTIMATING PRODUCT LIFE USING FEA: CREEP, FATIGUE & FRACTURE

10.1 INTRODUCTION

Structures and products may fail in-service prior to its expected life. A service environment containing dynamic or vibration characteristics has the potential for causing premature failure. Stress-life fatigue procedures may be applied to the analysis of structures or products to determine cycles to failure. Strain-life fatigue procedures may be applied to the analysis of structures or products to determine cycles to crack initiation. Linear elastic fracture mechanics (LEFM) may be applied to the analysis of structures or products to determine the life expectancy from crack initiation to final fracture. 

                                                                                               
Components may also fail prematurely if they are subject to significant loads over long periods of time. Creep is the slow change in dimensions of a material from prolonged exposure to stress.

                                                                                                                

The interaction of creep & stress rupture with cyclic stressing and the fatigue process is not yet understood, but is of great importance in many high performance-engineering applications.

10.2 CREEP

Creep is effectively a pre-failure of the material. Creep may occur under the following conditions:

• Under a fixed level of continuous force, deformation increases with the passage of time and the deformation is not completely recovered even if the force is removed. This is known as creep deformation.
• When a fixed amount of deformation is maintained for a long time, the resistance to load decreases with the passage of time. This is known as stress relaxation.
• If the loading time is further extended, rupture occurs. This is known as creep rupture.
  
  Creep strains are of significant importance are not usually encountered until the operating temperatures reach a range of approximately 35% to 70% of the melting point on a scale of absolute temperature (Kelvin).
  
  From a formal point of view, the phenomenon of creep is manifested by a time-dependent deformation under a constant strain. The material develops creep strains, which increase with duration of loading. The constitutive law of creep is usually defines the rate of creep as a function of stresses and total creep strains. The result is usually a system of first order differential equations with non-linear coefficients. This system is usually solved using an iterative procedure such as Newton-Raphson.                   
• Finite Element creep analyses can be carried out with time or strain hardening laws. Indeed, some packages facilitate the creation of special creep laws as defined by the user. FE packages also have the capability of automatically switching from explicit to implicit time stepping when the explicit time step is restricted by numerical stability considerations, thus providing for efficient solution of long-time creep problems.

10.3 FATIGUE

Cyclic or repeated loading can cause failure at lower stresses than static loading. This aspect is central to fatigue performance. Fatigue can be described as a progressive failure phenomenon that proceeds by the initiation and propagation of cracks to an unstable size.
                                         
High frequencies with low amplitudes are characteristic of noise and vibration studies while the low frequencies with moderate amplitudes represent classical fatigue. Finally, low frequency with high amplitude is typical of impact fatigue.

                                                                               
Test specimens are tested in a chosen mode -- tensile or flexural -- for thousands or millions of cycles. The yield stress for a given number of cycles is termed the fatigue strength. The fatigue life of a part is the number of cycles to failure at a given fluctuating load.

S-N data can be used reliably for design only if the test conditions for generating S-N data match the service conditions for the component.

                                                                                 
The most critical choice for tests is between load controlled or displacement controlled cyclic loading. Other test variables are temperature, mean stress, amplitude of fluctuation and frequency. Elevated temperatures hasten failure.

                                                                                                
FEA can predict fatigue stresses. However, stresses do not allow life predictions unless the fatigue characteristics of the material are known. Unfortunately, there are significant problems in determining fatigue characteristics. Fatigue can be affected by the frequency of vibration, so that conventional (low frequency) handbook data may not predict the fatigue at ultrasonic frequencies. Even where it might be reliable, low frequency data is usually too limited to provide life predictions at ultrasonic frequencies. For example, low frequency tests are often stopped at 500 million cycles, which represent only seven hours of continuous ultrasonics at 20 kHz.

                                                                                    

Further, fatigue is affected by the raw stock type (rod, bar, or plate), the raw stock size, and the direction of vibration relative to the material's grain. For nominally equivalent material, fatigue may also vary from heat-to-heat (especially for titanium) or among different manufacturers.

                                                                                                      
Fatigue is also affected by machining, which can leave residual tensile stresses that shorten fatigue life. These stresses are difficult to predict, since they depend on such factors as material removal rate, the type of machining coolant, the tool sharpness, etc. FEA cannot predict these surface stresses.

Thus, unless the material's fatigue properties and the effects of machining are well known, the stresses predicted by FEA probably cannot be used to predict fatigue life. However, the FEA stress data can be used to redesign fatigues that have known failure problems.

Although this is a limitation of FEA, it is also a limitation of any other method of fatigue analysis. The fatigue life cannot be predicted from stress unless the material's fatigue characteristics are known.

10.4 FRACTURE

Fracture occurs when new cracks appear or existing cracks become extended. This phenomenon is very complex in crystalline solids. Fractures are generally categorized as being brittle or ductile.

                                                                                                          
Brittle fracture manifests itself as a very rapid propagation after little or no plastic deformation. The speed at which cracks propagate rises rapidly to a terminal velocity, which is usually around one third the speed of sound in the material. In polycrystalline materials the crack front proceeds along cleavage planes within each crystal, giving the fracture surface a granular appearance. If brittle fracture proceeds along grain boundaries it is referred to as intergranular fracture.

                                                                           
Ductile rupture is fracture that takes place after extensive plastic deformation. It proceeds by slow propagation resulting from the formation & coalescence of voids. The fracture surface is dull & fibrous in appearance. Three distinct stages occur in polycrystalline materials. The component begins to neck-down locally & small discrete cavities appear in the neck region. Next, the cavities coalesce into a crack in the center of the cross-section. Finally, the crack spreads to the surface along shear planes oriented at 45° to the tensile axis.                                                                                                                                                                                                                                                                                                                                                                                                                                           

                                                          .
                                                                      
Most FE solutions are loosely based around contour integral approaches, as they include the influence of significant plastic deformation accompanying the initiation and stable growth of cracks. 

                                                                                                                
                                                                                                               
The J-Integral uses a path independent contour integral that encloses the crack tip. Mathematically it is a line integral in 2D where an arbitrary chosen path for the line integral is required. Failure by initiation & growth of a crack is associated with a critical value of J. Once it is found, the critical stress intensity factor can be determined. In order to ensure that the path used in the FE model is independent, multiple contours around the crack tip are usually extracted automatically.

                                                                   
It is usual to use 8 nodded quadrilaterals in such an analyses, with one edge collapsed to zero length at the crack tip. The midside nodes of the remaining non-zero length sides are usually placed at quarter the distance to the edge not in contact with the crack tip. The reason being that it produces a singularity at the crack tip & hence a solution is guaranteed.