# The Finite Element Analysis Engineering Essay

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Spot welds are widely used in automobile industry and due to their application in automobiles industries, lighter materials are used day by day in the automobile; there are more than about 4000 spot welds on a car body component which are used to join the important parts of the body to each other and to the chassis as well. So it is very important to know that the method of joining is the ideal one and it can withstand all the vibrations, impacts or jerks it might come across during its application. Hence for this purpose software is developed which is called NASTRAN in which it is possible to investigate the flaws and the capacity of the material to resist vibrations. Results are also supposed to be obtained by actual experimental basis and both the two obtained from the software and the experiments are to be compared.

CONTENTS

NOTATIONS

INTRODUCTION

## 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 modelling, and 3-D modelling. While 2-D modelling conserves simplicity and allows the analysis to be run on a relatively normal computer, it tends to yield less accurate results. 3-D modelling, however, produces more accurate results while sacrificing the ability to run on all but the fastest computers effectively. Within each of these modelling 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.

(http://www.sv.vt.edu/classes/MSE2094_NoteBook/97ClassProj/num/widas/history.html)

fea-software-finite-element-analysis-395516.jpg

SPOT WELDING

Â Spot welding is a process for joining steel sheets. The two parts are held between electrodes and the heat generated at the interface between the sheets causes local welding when pressure is applied.

(http://metals.about.com/library/bldef-Spot-Welding.htm)

welding-cills.jpg

(http://www.rs-ford.co.uk/series-one-restoration-project/restoration-project-part-2/)

PROJECT AIMS

In general a car body is expected to have thousands of spot welds. As the major contributor to a car's dynamic characteristics, spot weld joints must be properly designed, modelled and made to deliver good dynamic performance and resistance to premature failure.

Structural analysis software package MSC PATRAN/NASTRAN will be used to develop a finite element model and carry out Normal Modes analysis in order to obtain natural frequencies and modes of test structures.

The results of natural frequencies and modes computed from NASTRAN will be used to compare with those obtained from experimental work for validation purposes

To understand theory of vibration and finite element method.

To minimize the error by comparison or experimental and FE modal data

To gain knowledge and skills in FE models with the help of PhD students.

Detailed experimental study on car body spot welding.

## Background information

## Background study on vibration analysis

Vibration analysis is the study of the movement of an object about an equilibrium point. The need to study vibration in system comes about since vibration affects how these systems operate and may cause them to fail. Different methods of vibration analysis have been developed; their use depends on the type of system being analysed. Vibration analysis is usually done by deriving mathematical models and finding solution to governing equations and analysing these results. Vibration have been extensively researched and studied by experts for a long time.

## Previous research related to finite element modelling and updating

The finite element method originated from the need for solving complexÂ elasticityÂ andÂ structural analysisÂ problems inÂ civilÂ andÂ aeronautical engineering. Its development can be traced back to the work ofÂ Alexander HrennikoffÂ (1941) andÂ Richard CourantÂ (1942). While the approaches used by these pioneers are different, they share one essential characteristic:Â mesh discretization of a continuous domain into a set of discrete sub-domains, usually called elements. Starting in 1947,Â Olgierd ZienkiewiczÂ fromÂ Imperial CollegeÂ gathered those methods together into what would be called the Finite Element Method, building the pioneering mathematical formalism of the method.

Hrennikoff's work discretizes the domain by using aÂ latticeÂ analogy, while Courant's approach divides the domain into finite triangular sub regions to solveÂ second orderÂ ellipticÂ partial differential equationsÂ (PDEs) that arise from the problem ofÂ torsionÂ of a cylinder. Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by Rayleigh,Â Ritz, andÂ Galerkin.

Development of the finite element method began in earnest in the middle to late 1950s forÂ airframeÂ andÂ structural analysis and gathered momentum at theÂ University of Stuttgartthrough the work ofÂ John ArgyrisÂ and atÂ BerkeleyÂ through the work ofÂ Ray W. CloughÂ in the 1960s for use inÂ civil engineering. By late 1950s, the key concepts ofÂ stiffness matrixÂ and element assembly existed essentially in the form used today. NASA issued a request for proposals for the development of the finite elementÂ softwareÂ NASTRANÂ in 1965. The method was again provided with a rigorous mathematical foundation in 1973 with the publication ofÂ StrangÂ andÂ Fix'sÂ An Analysis of The Finite Element Method and has since been generalized into a branch of applied mathematics for numerical modelling of physical systems in a wide variety ofÂ engineeringÂ disciplines, e.g.,Â electromagnetism, thanks toÂ Peter P. Silvester andÂ fluid dynamics.

## Research related to resistance spot welding (RSW)

Elihu Thompson originated resistance spot welding.Â After accidentally fusing copper wires during an experiment years earlier, Elihu set out to develop electric resistance welding in 1885. His numerous patents were dated from then until 1900. Eventually, a merger of his company and Thomas Edison's formed the General Electric Company (GE).

Elihu described the basic principle of resistance spot welding as follows:

"All that was required was a transformer with a primary to be connected to the lighting circuit and a secondary of a few turns of massive copper cable. The ends of this cable were fitted with strong clamps that grasped the pieces of metal to be welded and forced them tightly together. The heavy current flowing through the joint created such a high heat that the metal was melted and run together."

## Resistance spot welding in vehicle manufacturing:

Earliest use in automotive industry dates to 1930's, achieving higher strength and productivity in manufacturing and repair.

(http://www.carolinacollisionequipment.com/automotive-resistance-spot-welding-history)

## Theory

WELDING

The process of joining together two pieces of metal so that bonding accompanied by appreciable inter atomic penetration takes place at their original boundary surfaces. The boundaries more or less disappear at the weld, and integrating crystals develop across them. Welding is carried out by the use of heat or pressure or both and with or without added metal. There are many types of welding including Metal Arc, Atomic Hydrogen, Submerged Arc, Resistance Butt, Flash, Spot, Stitch, Stud and Projection.

(http://metals.about.com/library/bldef-Welding.htm)

## Spot-welding

One of the oldest electric welding processes, resistance welding is widely used in the industry even today. The weld is made by a combination of heat, pressure, and time. As the name resistance welding implies, it is the resistance of the material to be welded to current flow that causes a localized heating in the part. The pressure exerted by the tongs and electric flow through the electrode tips, holds the parts to be welded in intimate connection prior to the welding process and also during and after the welding time period. The required amount of time current flows in the joint is determined by thickness of the material and form, the extent to which the current flows, and also the cross-sectional part of the welding tip contact surfaces.

(http://www.millerwelds.com/pdf/Resistance.pdf)

Spot welding parameters

Spot welding parameters include:

Electrode force

Diameter of the electrode contact surface

Squeeze time

Weld time

Hold time

Weld current

The determination of suitable welding parameters for spot welding is a very complex matter. A minute change of one parameter will affect all the other parameters.

As the instrument is being used, after some time the contact surface of the electrodes keeps increasing. This makes it difficult to plan the welding parameter table, which shows the most favourable welding parameters for different conditions. However, this table shows mark values for the welding parameters.Â

Sheet thickness, tÂ

[mm]Â Â Â

Electrode force, FÂ

[kN]Â Â Â

Weld current, IÂ

[A]Â Â Â

Weld timeÂ

[cycles]Â Â Â

Hold timeÂ

[cycles]Â Â Â

Electrode diameter, dÂ

[mm]

0.63 + 0.63Â Â Â

2.00Â Â Â

8 500Â Â Â

6Â Â Â

1Â Â Â

6

0.71 + 0.71Â Â Â

2.12Â Â Â

8 750Â Â Â

7Â Â Â

1Â Â Â

6

0.80 + 0.80Â Â Â

2.24Â Â Â

9 000Â Â Â

8Â Â Â

2Â Â Â

6

0.90 + 0.90Â Â Â

2.36Â Â Â

9 250Â Â Â

9Â Â Â

2Â Â Â

6

1.00 + 1.00Â Â Â

2.50Â Â Â

9 500Â Â Â

10Â Â Â

2Â Â Â

6

1.12 + 1.12Â Â Â

2.80Â Â Â

9 750Â Â Â

11Â Â Â

2Â Â Â

6

1.25 + 1.25Â Â Â

3.15Â Â Â

10 000Â Â Â

13Â Â Â

3Â Â Â

6Â Â Â 7

1.40 + 1.40Â Â Â

3.55Â Â Â

10 300Â Â Â

14Â Â Â

3Â Â Â

6Â Â Â 7

1.50 + 1.50Â Â Â

3.65Â Â Â

10 450Â Â Â

15Â Â Â

3Â Â Â

6Â Â Â 7

1.60 + 1.60Â Â Â

4.00Â Â Â

10 600Â Â Â

16Â Â Â

3Â Â Â

6Â Â Â 7

1.80 + 1.80Â Â Â

4.50Â Â Â

10 900Â Â Â

18Â Â Â

3Â Â Â

6Â Â Â 7

2.00 + 2.00Â Â Â

5.00Â Â Â

11 200Â Â Â

3x7+2Â Â Â

4Â Â Â

Â Â Â Â Â 7Â Â Â 8

2.24 + 2.24Â Â Â

5.30Â Â Â

11 500Â Â Â

3x8+2Â Â Â

4Â Â Â

Â Â Â Â Â 7Â Â Â 8

2.50 + 2.50Â Â Â

5.60Â Â Â

11 800Â Â Â

3x9+3Â Â Â

5Â Â Â

Â Â Â Â Â Â Â Â Â Â Â 8

2.80 + 2.80Â Â Â

6.00Â Â Â

12 200Â Â Â

4x8+2Â Â Â

6Â Â Â

Â Â Â Â Â Â Â Â Â Â Â 8

3.00 + 3.00Â Â Â

6.15Â Â Â

12 350Â Â Â

4x9+2Â Â Â

6Â Â Â

Â Â Â Â Â Â Â Â Â Â Â 8

3.15 + 3.15Â Â Â

6.30Â Â Â

12 500Â Â Â

4x9+2Â Â Â

6Â Â Â

Â Â Â Â Â Â Â Â Â Â Â 8

## Electrode force

The idea of the electrode force is to clutch the metal sheets to be joined together. This requires a large electrode force or else the weld quality will not be good enough. However, the force must not be too large as it might lead to other problem. When the electrode force goes up the heat energy will go down. This means that the higher electrode force requires a higher weld current. When weld current becomes too high spatter will arise between electrodes and sheets. This will cause the electrodes to get caught with the sheet.Â

An enough target value for the electrode force is 90 N per mm2. As that the size of the contactÂ surface will increase during welding. To keep the same conditions during the entire welding process, the electrode force has to be steadily increased. As it is rather difficult to change the electrode force in the same rate as the electrodes wear off, generally a mean value is selected.Â Â Â

## Diameter of the electrode contact surface

One general criterion of resistance spot-welding is that the weld shall have a nugget diameter of 5*t1/2, "t" being the thickness of the steel sheet. Thus, a spot weld made in two sheets, each 1 mm in thickness, would generate a nugget 5 mm in diameter according to the 5*tÂ½-rule. Diameter of the electrode contact surface should be slightly larger than the nugget diameter. For example, spot welding two sheets of 1 mm thickness would require an electrode with a contact diameter of 6 mm. In practice, an electrode with a contact diameter of 6 mm is standard for sheet thickness of 0.5 to 1.25 mm. This contact diameter of 6 mm conforms to the ISO standard for new electrodes.Â Â Â Â

Squeeze time

Squeeze Time is the time interval between the period when the electrode force is applied initially on the material and the first application of current. Squeeze time is necessary to delay the weld current until the electrode force has attained the desired level.

## Weld time

Weld time is mainly the time of application of the welding current to the metal sheets. The weld time is measured and adjustedÂ in cycles of line voltage as are all timing functions. One cycle is 1/50 of a second in a 50 Hz power system. (When the weld time is taken from American literature, theÂ number of cycles has to be reduced due to the higher frequency (60Hz) that is used in the USA.)Â

As the weld time is, more or less, related to what is required for the weld spot, it is hard to provide a precise value of the ideal weld time. For instance:Â

Weld time should be as short as possible.Â

The weld current should be able to provide the best weld feature as possible.Â

The weld parameters should be chosen to give as little wearing of the electrodes as possible. (Often this means a short weld time.)Â

The weld time shall cause the nugget diameter to be big when welding thick sheets.

The weld time might have to be adjusted to fit the welding equipment in case it does not fulfil the requirements for the weld current and the electrode force. (This means that a longer weld time may be needed.)Â Â

The weld time shall cause the indentation due to the electrode to be as small as possible. (This is achieved by using a short weld time.)Â Â Â

The weld time shall be accustomed to welding with programmed tip-dressing, where the size of the electrode contact surface can be kept at a continual value. (This means a shorter welding time.)Â

When welding sheets with a thickness greater than 2 mm it might be appropriate to divide the weld time into a number of impulses to avoid the heat energy to increase. This method will give good-looking spot welds but the strength of the weld might be poor.Â Â

By multiplying the thickness of the sheet by ten, a good target value for the weld time can be reached. When welding two sheets with the thickness 1 mm each, an appropriate weld time is 10 periods (50Hz).

## Hold time (cooling-time)

Hold time is the time, after the welding, when in order to chill the weld the electrodes will still be applied.Â Considered from a welding technical point of view, the hold time is the most interesting welding parameter.Â Hold time is necessary to allow the weld nugget to solidify before releasing the welded parts, but it must not be to long as this may causeÂ the heat in the weld spot to spread to the electrode and heat it. The electrode will then get more exposed to wear. Further, if the hold time is too long and the carbon content of the material is high (more than 0.1%), there is a risk the weld will become brittle. If welding is done on galvanized carbon steel it is preferable to have a longer hold time.Â

## Weld current

The current present in the welding circuit at the time of making of a weld is the weld current. The amount of weld current is controlled by two things; first, the setting of the transformer tap switch determines the maximum amount of weld current available; second the percent of current control determines the percent of the available current to be used for making the weld. Low percent current settings are not normally recommended as this may impair the quality of the weld. Adjust the tap switch so that proper welding current can be obtained with the percent current set between seventy and ninety percent.Â Â

The weld current should be kept as low as possible. When determining the current to be used, the current is gradually increased until weld spatter occurs between the metal sheets. This indicates that the correct weld current has been reached.

## UPDATING OF A MODELS

Model updating is used to minimize the 'distance' between FEA and reference test data. It is viable approach to improve the correlation between FE models and experimental data by minimising the differences between results from the two approaches. In this project updating is cast as a structural optimisation problem and optimisation algorithm (SOL 200) of FE code NASTRAN is used to perform updating

The equation of motion for undamped free vibration of a structure can b expressed ass,

Where M and K are the nxn mass and stiffness matrices of the structure, and u is the nx1 modal displacement vector (with n being the number of DOFs of the whole structure).ï€ ï¬ï€½ï€ ï·ï€² is the eigenvalue and ï·ï€ is the natural frequency of the structure.

When the vector ï±ï€ of structural parameters (such as the Young's modulus) are changed, the vector ï¬ï€ of m eigenvalues will normally change as well. Approximations used in NASTRAN are based on simple first order Taylor Series expansion and the general form of this expansion for ï¬ï€ is

## ï¬i+1 = ï¬i + Si (Î´Î¸)

In equation (2), Si is an mxn sensitivity matrix at ith iteration, which denotes the rates of change of the structural eigenvalues (ï¬i) with respect to changes in parameters (ï¤ï±ï€ ), which can be expressed as

It should be noted that any modifications made to the system parameters could affect the modal properties of the system. Therefore, the parameters and modal properties involved in the updating process must be selected properly. Selection of the right parameters and modal properties for the updating procedure are briefly explained in the next sections.

An objective function based on residuals between the experimental modal data (e.g., natural frequencies, mode shapes, etc.) and their predictions is set for minimisation in the updating procedure.

The procedure continues until convergence is accomplished when the difference between values of the objective function (J) from consecutive iterations is sufficiently small. In this work, the objective function is constructed based on eigenvalue residuals, given by where is the jth experimental eigenvalue and is the jth eigenvalue predicted by the FE model. It

is important to note that equation (4) only holds if the measured and its predicted counterpart are

paired correctly, and therefore it is vital to ensure that the experimental and numerical data relate to

the same mode. In addition, it is generally preferable to use a larger number of experimental modal

properties in the updating process. Obviously, this would be more difficult but the updated model

should be more predictive than using only a few modal data.

THEORY

## MSC NASTRAN

MSC NastranÂ is the abbreviation of MacNeal-Schwendler Corp set up by two developers that worked on a NASA contracted to create a general purpose FEA for portability of government programs

NASTRAN contains over one million lines of code. NASTRAN is compatible with a large variety of computers andÂ operating systemsÂ ranging from smallÂ workstationsÂ to the largestÂ supercomputers.

NASTRAN was designed from the beginning to consist of several modules. A module is a collection of subroutines designed to perform a specific task-processing model geometry, assembling matrices, applying constraints, solving matrix problems, calculating output quantities, conversing with the database, printing the solution, and so on. The modules are controlled by an internal language called theÂ Direct Matrix Abstraction Program (DMAP).

Each type of analysis available is called a solution sequence.

Some of the most common solution sequence codes are:

101 - Linear Static

103 - Modal

105 - Buckling

106 - Non-Linear Static

107 - Direct Complex Eigen value

108 - Direct Frequency Response

109 - Direct Transient Response

110 - Modal Complex Eigen value

111 - Modal Frequency Response

112 - Modal Transient Response

129 - Nonlinear Transient

144 - Static Aeroelastic Analysis

145 - Flutter / Aeroservoelastic analysis

146 - Dynamic Aeroelastic Analysis

153 - Non-Linear static coupled with heat transfer

159 - Nonlinear Transient coupled with Heat transfer

187 - DDAM

200 - Design Optimization and Sensitivity analysis

400 - Non-Linear Static and Dynamic (implicit) (MSC.NASTRAN native, supersedes 106, 129, 153 and 159 - part of MD.NASTRAN)

600 - Non-Linear Static and Dynamic (implicit) (front end to MSC.Marc - part of MD.NASTRAN)

601 - Implicit Non-Linear (Adina for NX Nastran)

700 - Explicit Non-Linear (LS Dyna plus MSC.Dytran - part of MD.NASTRAN)

701 - Explicit Non-Linear (Adina NX Nastran)

When building finite element models, a lot of simplifying assumptions and estimates has to be made. Idealisation, discretisation and parameter evaluation are all possible error sources. If these models have to be used in lifetime estimations, optimisation processes or system synthesis computations, they have to be a valid representation of reality. MSC/NASTRAN sol200 is used in the process of validation and verification of dynamic finite element models

NON DISTRUCTIVE TESTING

Non-destructive testingÂ (NDT) is a wide set of investigation techniques used in science and industry to assess the properties of a material, component or system without causing damage. Because NDT does not permanently alter the article being inspected, it is a highly-valuable technique that can save both money and time in product evaluation, troubleshooting, and research. Common NDT methods includeÂ ultrasonic,Â magnetic-particle,Â liquid penetrant,Â radiographic, remote visual inspection (RVI),Â eddy-current testingÂ andÂ low coherence interferometry.Â NDT is a commonly-used tool inÂ forensic engineering,Â mechanical engineering,Â electrical engineering,Â civil engineering,Â systems engineering,Â aeronautical engineering,Â medicine, andÂ art.

PROJECT MANAGEMENT

At the start of the project, the project purpose and objectives were not quite clear. The project objectives were built on the principle of what the project title was. Up to the point of the proposal report submission, different background topics related to the project were researched, and it seemed that the project had two main elements to it, which were finite element analysis and modal updating. The project was allotted during the start of the year so there was not much research done during the summer vacation and so coming in to the start of the year there was not much knowledge and understanding on the finite element modelling. After the submission of the proposal stage report, the purpose of the project became clear and the project was divided in to two parts, developing a finite element model and carrying out Normal Modes analysis in order to obtain natural frequencies and modes of test structures. Initially, meetings were arranged with the supervisor and the necessary arrangements were made for the requirements of the project.

By the end of the proposal stage, the topic FE modelling and experimental study of car body components was clear and research on technology and the working of the program MSC PARTRAN/NASTRAN was done. Various theories were to be cleared in order to get the clear picture of the project. Various topics like non destructive testing, forward and inverse analysis, finite element analysis and vibration theory were discussed in detail with the project supervisor in the weekly meetings. During the later weeks of the 1st semester various article and papers were read on the previous research work and the progress made by others so far. A Gantt chart was planned along with the proposal report and all the basic steps were included in it for the successful progress of the project. As time passed by many changes were made in the Gantt chart.

With regards to comparing the initial Gantt chart plan was upheld, the vibration theory was continually enhanced throughout the length of the project, and so did the enhancing of the understanding of the finite element modelling. The research on finite element analysis and model updating did take place in the 1st semester, although during the second semester more extensive research was done. Initially there was some trouble with trying to understand engineering concepts and the purpose of model updating that took place, and also to identify what if of importance and relevant to the project, and what is not.