ONE DIMENSIONAL ANALYSIS OF A STEP BAR

CONTENTS

INTRODUCTION..................................................................................................................................... 1

FEM [FINITE ELEMENT METHOD]........................................................................................................... 2

FEA [FINITE ELEMENT ANALYSIS]............................................................................................................ 6

INTRODUCTION TO ANSYS..................................................................................................................... 9

STEP BAR............................................................................................................................................. 12

BIBLIOGRAPHY.................................................................................................................................... 17

YADAVRAO TASGAONKAR INSTITUTE OF Page ENGINEERING AND TECHNOLOGY (MECH.)

ONE DIMENSIONAL ANALYSIS OF A STEP BAR

INTRODUCTION

This report will explain you about need to have a proper analysis of a new design in order to get the knowledge about different methods of analysis.

While it is difficult to quote a date of the invention of the finite element method, the method originated from the need to solve complex elasticity and structural analysis problems in civil and aeronautical engineering. Its development can be traced back to the work by A. Hrennikoff and R. Courant. In China, in the later 1950s and early 1960s, based on the computations of dam constructions, K. Feng proposed a systematic numerical method for solving partial differential equations. The method was called the finite difference method based on variation principle, which was another independent invention of finite element method. Although the approaches used by these pioneers are different, they share one essential characteristic: mesh discretization of a continuous domain into a set of discrete sub-domains, usually called elements.

Hrennikoff's work discretizes the domain by using a lattice analogy, while Courant's approach divides the domain into finite triangular sub regions to solve second order elliptic partial differential equations (PDEs) that arise from the problem of torsion of a cylinder. Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by Rayleigh Ritz, and Galerkin.

The finite element method obtained its real impetus in the 1960s and 1970s by the developments of J. H. Argyris with co-workers at the University of Stuttgart, R. W. Clough with co-workers at UC Berkeley, O. C. Zienkiewicz with co-workers Ernest Hinton, Bruce Irons and others at the University of Swansea, Philippe G. Ciarlet at the University of Paris 6 and Richard Gallagher[5] with co-workers at Cornell University. Further impetus was provided in these years by available open source finite element software programs. NASA sponsored the original version of NASTRAN, and UC Berkeley made the finite element program SAP IV widely available. A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by Strang and Fix. The method has since been generalized for the numerical modeling of physical systems in a wide variety of engineering disciplines, e.g., electromagnetism, heat transfer, and fluid dynamics

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ONE DIMENSIONAL ANALYSIS OF A STEP BAR

FEM [FINITE ELEMENT METHOD]

In mathematics, the finite element method (FEM) is a numerical technique for finding approximate solutions to boundary value problems for partial differential equations. It uses subdivision of a whole problem domain into simpler parts, called finite elements, and variational methods from the calculus of variations to solve the problem by minimizing an associated error function. Analogous to the idea that connecting many tiny straight lines can approximate a larger circle, FEM encompasses methods for connecting many simple element equations over many small sub domains, named finite elements, to approximate a more complex equation over a larger domain

Basic concept

The subdivision of a whole domain into simpler parts has several advantages: Accurate representation of complex geometry Inclusion of dissimilar material properties Easy representation of the total solution Capture of local effects.

A typical work out of the method involves (1) dividing the domain of the problem into a collection of sub domains, with each sub domain represented by a set of element equations to the original problem, followed by (2) systematically recombining all sets of element equations into a global system of equations for the final calculation. The global system of equations has known solution techniques, and can be calculated from the initial values of the original problem to obtain a numerical answer.

In the first step above, the element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are often partial differential equations (PDE). To explain the approximation in this process, FEM is commonly introduced as a special case of Galerkin method. The process, in mathematical language, is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with a set of algebraic equations for steady state problems, a set of ordinary differential equations for transient problems.

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ONE DIMENSIONAL ANALYSIS OF A STEP BAR

These equation sets are the element equations. They are linear if the underlying PDE is linear, and vice versa. Algebraic equation sets that arise in the steady state problems are solved using numerical linear algebra methods, while ordinary differential equation sets that arise in the transient problems are solved by numerical integration using standard techniques such as Euler's method or the Runge-Kutta method.

In step (2) above, a global system of equations is generated from the element equations through a transformation of coordinates from the sub domains' local nodes to the domain's global nodes. This spatial transformation includes appropriate orientation adjustments as applied in relation to the reference coordinate system. The process is often carried out by FEM software using coordinate data generated from the sub domains.

FEM is best understood from its practical application, known as finite element analysis (FEA). FEA as applied in engineering is a computational tool for performing engineering analysis. It includes the use of mesh generation techniques for dividing a complex problem into small elements, as well as the use of software program coded with FEM algorithm. In applying FEA, the complex problem is usually a physical system with the underlying physics such as the Euler-Bernoulli beam equation, the heat equation, or the Navier-Stokes equations expressed in either PDE or integral equations, while the divided small elements of the complex problem represent different areas in the physical system.

FEA is a good choice for analyzing problems over complicated domains (like cars and oil pipelines), when the domain changes (as during a solid state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. For instance, in a frontal crash simulation it is possible to increase prediction accuracy in "important" areas like the front of the car and reduce it in its rear (thus reducing cost of the simulation). Another example would be in numerical weather prediction, where it is more important to have accurate predictions over developing highly nonlinear phenomena (such as tropical cyclones in the atmosphere, or eddies in the ocean) rather than relatively calm areas.

The structure of finite element methods

Finite element methods are numerical methods for approximating the solutions of mathematical problems that are usually formulated so as to precisely state an idea of some aspect of physical reality. A finite element method is characterized by a variational formulation, a discretization strategy, one or more solution algorithms and post-processing procedures.Examples of variational formulation are the Galerkin method, the discontinuous Galerkin method, mixed methods, etc.

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ONE DIMENSIONAL ANALYSIS OF A STEP BAR

A discretization strategy is understood to mean a clearly defined set of procedures that cover (a) the creation of finite element meshes, (b) the definition of basis function on reference elements (also called shape functions) and (c) the mapping of reference elements onto the elements of the mesh. Examples of discretization strategies are the h-version, p-version, hp-version, x-FEM, isogeometric analysis, etc. Each discretization strategy has certain advantages and disadvantages. A reasonable criterion in selecting a discretization strategy is to realize nearly optimal performance for the broadest set of mathematical models in a particular model class.

There are various numerical solution algorithms that can be classified into two broad categories; direct and iterative solvers. These algorithms are designed to exploit the sparsity of matrices that depend on the choices of variational formulation and discretization strategy.

Postprocessing procedures are designed for the extraction of the data of interest from a finite element solution. In order to meet the requirements of solution verification, postprocessors need to provide for a posteriori error estimation in terms of the quantities of interest. When the errors of approximation are larger than what is considered acceptable then the discretization has to be changed either by an automated adaptive process or by action of the analyst. There are some very efficient postprocessors that provide for the realization of superconvergence.

General form of the finite element method

In general, the finite element method is characterized by the following process. One chooses a grid for . In the preceding treatment, the grid consisted of triangles,

but one can also use squares or curvilinear polygons. Then, one chooses basis functions. In our discussion, we used piecewise linear basis

functions, but it is also common to use piecewise polynomial basis functions.

A separate consideration is the smoothness of the basis functions. For second order elliptic boundary value problems, piecewise polynomial basis function that are merely continuous suffice (i.e., the derivatives are discontinuous.) For higher order partial differential equations, one must use smoother basis functions. For instance, for a

fourth order problem such as , one may use piecewise quadratic basis functions that are  .

Another consideration is the relation of the finite-dimensional space   to its infinite-

dimensional counterpart, in the examples above . Aconforming element method is one in which the space   is a subspace of the element space for the continuous problem. The example above is such a method. If this condition is not satisfied, we obtain a nonconforming element method, an example of which is the space of piecewise linear functions over the mesh which are continuous at each edge midpoint. Since these

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ONE DIMENSIONAL ANALYSIS OF A STEP BAR

functions are in general discontinuous along the edges, this finite-dimensional space is

not a subspace of the original .

Typically, one has an algorithm for taking a given mesh and subdividing it. If the main method for increasing precision is to subdivide the mesh, one has an h-method (h is customarily the diameter of the largest element in the mesh.) In this manner, if one shows that the error with a grid   is bounded above by  , for some   and  , then one has an order p method. Under certain hypotheses (for instance, if the domain is convex), a piecewise polynomial of order   method will have an error of order  .

If instead of making h smaller, one increase the degree of the polynomials used in the basis function, one has a p-method. If one combines these two refinement types, one obtains an hp-method (hp-FEM). In the hp-FEM, the polynomial degrees can vary from element to element. High order methods with large uniform p are called spectral finite element methods (SFEM). These are not to be confused with spectral methods.For vector partial differential equations, the basis functions may take values in .

VARIOUS TYPES OF FEM

AEM Generalized finite element method Mixed Finite Element method. Hp-FEM Hpk-FEM XFEM S-FEM Fiber Beam method Spectral Element method Meshfree methods Finite Element Limit Analysis Discontinous Galerkin Methods Streched Grid Method

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ONE DIMENSIONAL ANALYSIS OF A STEP BAR

FEA [FINITE ELEMENT ANALYSIS]

What is Finite Element Analysis?

FEA consists of a computer model of a material or design that is stressed and analyzed for specific results. It is used in new product design, and existing product refinement. A company is able to verify 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.There are generally two types of analysis that are used in industry: 2-D modeling, and 3-D modeling. While 2-D modeling conserves simplicity and allows the analysis to be run on a relatively normal computer, it tends to yield less accurate results. 3-D modeling, however, produces more accurate results while sacrificing the ability to run on all but the fastest computers effectively. Within each of these modeling schemes, the programmer can insert numerous algorithms (functions) which may make the system behave linearly or non-linearly. Linear systems are far less complex and generally do not take into account plastic deformation. Non-linear systems do account for plastic deformation, and many also are capable of testing a material all the way to fracture. 

How Does Finite Element Analysis Work?

FEA uses a complex system of points called nodes which make a grid called a mesh (Figure 2). This mesh is programmed to contain the material and structural properties which define how the structure will react to certain loading conditions. Nodes are assigned at a certain density throughout the material depending on the anticipated stress levels of a particular area. Regions which will receive large amounts of stress usually have a higher node density than those which experience little or no stress. Points of interest may consist of: fracture point of previously tested material, fillets, corners, complex detail, and high stress areas. The mesh acts like a spider web in that from each node, there extends a mesh element to each of the adjacent nodes. This

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ONE DIMENSIONAL ANALYSIS OF A STEP BAR

web of vectors is what carries the material properties to the object, creating many elements. (Theory)

A wide range of objective functions (variables within the system) are available for minimization or maximization:

Mass, volume, temperature Strain energy, stress strain Force, displacement, velocity, acceleration Synthetic (User defined)

There are multiple loading conditions which may be applied to a system. Next to Figure 3, some examples are shown:

Point, pressure (Figure 3), thermal, gravity, and centrifugal static loads

Thermal loads from solution of heat transfer analysis

Enforced displacements Heat flux and convection Point, pressure and gravity

dynamic loads

Each FEA program may come with an element library, or one is constructed over time. Some sample elements are:

Rod elements Beam elements Plate/Shell/Composite elements Shear panel Solid elements Spring elements Mass elements Rigid elements Viscous damping elements

Many FEA programs also are equipped with the capability to use multiple materials within the structure such as:

Isotropic, identical throughout Orthotropic, identical at 90 degrees General anisotropic, different throughout

Types of Engineering Analysis

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ONE DIMENSIONAL ANALYSIS OF A STEP BAR

Structural analysis consists of linear and non-linear models. Linear models use simple parameters and assume that the material is not plastically deformed. Non-linear models consist of stressing the material past its elastic capabilities. The stresses in the material then vary with the amount of deformation as in Figure 4.Vibrational analysis is used to test a material against random vibrations, shock, and impact. Each of these incidences may act on the natural vibrational frequency of the material which, in turn, may cause resonance and subsequent failure.

Fatigue analysis helps designers to predict the life of a material or structure by showing the effects of cyclic loading on the specimen. Such analysis can show the areas where crack propagation is most likely to occur. Failure due to fatigue may also show the damage tolerance of the material (Figure 5).

Heat Transfer analysis models the conductivity or thermal fluid dynamics of the material or structure (Figure 1). This may consist of a steady-state or transient transfer. Steady-state transfer refers to constant thermoproperties in the material that yield linear heat diffusion. 

Results of Finite Element AnalysisFEA has become a solution to the task of predicting failure due to unknown stresses by showing problem areas in a material and

allowing designers to see all of the theoretical stresses within. This method of product design and testing is far superior to the manufacturing costs which would accrue if each sample was actually built and tested. 

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ONE DIMENSIONAL ANALYSIS OF A STEP BAR

INTRODUCTION TO ANSYS

The FEM Process

The finite element process is generally divided into three distinct phases1. PREPROCESSING - Build the FEM model.2. SOLVING - Solve the equations.3. POSTPROCESSING - Display and evaluate the results.

The ANSYS Interface

Figure shows the ANSYS interface with Utility Menu across the top followed by theANSYS Command Input area, Main Menu, Graphics display, and ANSYS Toolbar.

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ONE DIMENSIONAL ANALYSIS OF A STEP BAR

Many general purpose software are readily available for analysis of mechanical, civil and aircraft structures based on FEM. Even though actual commands may vary from one software to another, the general features can be broadly classified under the following three categories.

(i) Pre-processor PhaseIn this phase, data is input by the user regarding

(a) Idealised I-D, 2-D or 3-D geometric model consisting of:• Element type (discrete structure with truss, 2-D beam, 3-D beam or pipe elements;continuum with 2-D plane stress, plane strain, thin shell, 3-D solid or thick shellelements)• Appropriate nodal coordinates• Element attributes and element connectivity.

In some large components, it is also possible to create a large 2-D or 3-D model using key points and Boolean operations on areas or volumes. These areas or volumes can be meshed by the software into many elements of equal or different sizes depending on user's choice as per the expected stress distribution. The software makes sure that the generated elements satisfy aspect ratio norms.

(b) Properties of materials such as

Modulus of elasticity, Poisson's ratio, mass density, coefficient of linear thermalexpansion etc. for structural analysis thermal conductivity, specific heat etc. for a thermalanalysis; with options of:- Isotropic or orthotropic- Constant or temperature-dependent material data.Some softwares have standard material properties in their database. User need to specify thematerial type only. Care must be taken to see that the units in the material databasecorrespond to the units of the other data input by the user

(c) Section properties like:• Area for a truss element• Moment of inertia and section depth in a beam element• Thickness in a 2-d plate etc.

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ONE DIMENSIONAL ANALYSIS OF A STEP BAR

Some software include the facility of choosing a standard section shape such as C, I, L, H, from its database with their dimensions specified by the user and then calculate values like area, moment of inertia etc. Material and section properties may be identified with material numbers and section numbers so that each element can be associated with a particular material number and a particular section number (called element attributes). In this way, material data and section data, common to many elements, need not be input repeatedly, saving considerable time and effort in data preparation.

(d) Load particulars such as:• Distributed loads due to self weight, wind load• Concentrated or point loads• Steady-state or transient temperature distribution over the entire model, for astructural analysis• Free-stream temperature, constant temperature on some part of boundary for a thermalanalysis etc.It is also possible to analyse the same structure for different sets of loads (defined in somesoftware as load steps or load cases)

(e) Boundary conditions or restraints for translation or rotation OOF at various nodes(including restraints on rigid body motion), indication of symmetry for a structural analysis or insulated wall for a thermal analysis etc

(ii) Solution Phase

In this phase, the program uses the data file generated by the pre-processor stage and carries out desired analysis.

Different options usually available are:(a) Static structural analysis, which calculates nodal displacements(b) Dynamic structural analysis, which calculates natural frequencies and mode shapes ortime history response (corresponding to load vs time data) or response to earthquake(corresponding to frequency vs amplitude data)(c) Thermal analysis, which calculates nodal temperatures due to thermal conduction in asolid body with specified temperature and/ or convection boundary conditions.

(iii) Post-processor phaseThe output of solution phase is a large set of nodal displacement or temperature values. The post processor phase reads these values as well as geometry data of pre-processor phase and presents in a more easily readable form such as iso-stress or iso-temperature contours, plots of deformed shape etc. Some software also has the facility of presenting output for a specific combination of different loads (or load steps). Many general purpose software, such as ANSYS, ADINA, NASTRAN, PAFEC, NISA, PAFEC, STRUDL, etc., are commercially available in the market. The specific format and sequence of input data may vary between them. Also, modelling options as well as loads and boundary conditions that each of the software can handle may also vary. But, data to be input generally remains the same.

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ONE DIMENSIONAL ANALYSIS OF A STEP BAR

STEP BAR

AIM: Determine the Nodal Displacement, Reaction, Stress and Strain of the stepped bar.

PROCEDURE: - Input Data in ANSYS

Preferences - StructuralPreprocessor –Element type - Add - Structural link – 2D spar 1Real constants - Add - Set No. 1; Area 150 Set No. 2; Area 100Material props –Material Models- Structural- Linear- Elastic - Isotropic – Material No 1; EX 2e5; PRXY 0.3 Material No 2; EX 0.7e5; PRXY 0.3

Modeling- Create - Nodes – In Active CS - (0,0),(50,0),(100,0)

Elements -Elem attributes - Real const. Set no. 1, Matl No. 1 Auto Numbered - Thru Nodes - (1, 2)

Elem attributes - Real const. Set no.2, Matl No.2 Auto Numbered - Thru Nodes - (2, 3)

{Note:- Use the mouse to pick node #1 (i.e. click on it). It will now be marked bya small yellow box. Now move the mouse toward node #2. A line will now showon the screen joining these two points. Left click and a permanent line will appear.Connect the remaining nodes using the same method.OR :- Enter node numbers to be joined as in figure shown.}

Tool Bar – PlotCtrls – Style – Size and Shape – Display of Element {ON}– Plot - Elements

Loads – Define Lodes- Apply – Structural - Displacement - on Node - 1 All DOF=0 Structural - Force/Moment - on Node - 2 => -10000

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ONE DIMENSIONAL ANALYSIS OF A STEP BAR

(Apply) Structural - Force/Moment - on Node - 3 => 5000 (OK)Solution - Analysis type - New Analysis – Static Solve - Current LS - Solution is done - Close

General Postproc –Element Table- Define Table- Add- Label – Force - Item, comp Result data item- By Sequence num SMISC, 1 (Apply) -Label – Stress - Item, comp Result data item- By Sequence num LS, 1 (Apply)

{Note: When entering the final data point, click on 'OK' to indicate that you are finished entering nodes. If you first press 'Apply' and then 'OK' for the final node, you will have defined it twice! If you did press 'Apply' for the final point, simply press 'Cancel' to close this dialog box. Units- Note the units of measure (ie mm) were not specified. It is the responsibility of the user to ensure that a consistent set of units are used for the problem; thus making any conversions where necessary. Correcting Mistakes- When defining nodes, lines, areas, volumes, elements, constraints and loads you are bound to make mistakes. Fortunately these are easily corrected so that you don't need to begin from scratch every time an error is made! Every 'Create' menu for generating these various entities also have a corresponding 'Delete' menu for fixing things up.}

-Label – Strain - Item, comp Result data item- By Sequence num LEPEL, 1 (OK)Plot results – Contour Plot- Line Element Results – FORCE (OK) - Line Element Results – STRESS (OK) - Line Element Results – STRAIN (OK)

Plot results – Contour Plot- Nodal Solu- DOF solution - Translation UXList results - Nodal solution - DOF solution - Translation UX Node UX 1,2,3 _____

- Reaction solution - Structural force FX Node FX 1 _____

List results – Element Table Data- Items to be listed- FORCE + STRESS + STRAIN

YADAVRAO TASGAONKAR INSTITUTE OF Page ENGINEERING AND TECHNOLOGY (MECH.)

ONE DIMENSIONAL ANALYSIS OF A STEP BAR

Results Obtained: Displacements: u2 = -0.83333×10-2 mm, u3 = 0.27381×10-1 mmReaction Forces: R1 = 5000 NELEMENT FORCE STRESS STRAIN1 -5000 -33.333 -0.16667×10-32 5000 50 0.71429×10-3

Figure 1:- FEM model of Stepped Bar

Figure 3:- Contour Plot of Displacement

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ONE DIMENSIONAL ANALYSIS OF A STEP BAR

Figure 2:- Deformed + undeformed shape

Table 1:- Nodal Solution

YADAVRAO TASGAONKAR INSTITUTE OF Page ENGINEERING AND TECHNOLOGY (MECH.)

ONE DIMENSIONAL ANALYSIS OF A STEP BAR

Table 2:- Reaction Solution

Table 3:- Element Table Data

YADAVRAO TASGAONKAR INSTITUTE OF Page ENGINEERING AND TECHNOLOGY (MECH.)

ONE DIMENSIONAL ANALYSIS OF A STEP BAR

BIBLIOGRAPHY

‘FINITE ELEMENT METHODhttp://en.wikipedia.org/wiki/Finite_element_method

‘INTRODUCTION TO FINITE ELEMENT METHOD’http://www.sv.vt.edu/classes/MSE2094_NoteBook/97ClassProj/num/widas/history.html

YADAVRAO TASGAONKAR INSTITUTE OF Page ENGINEERING AND TECHNOLOGY (MECH.)