Faculty of Engineering University of Wollongong

APPLIED FINITE ELEMENT ANALYSIS FOR CIVIL ENGINEERS

CIVL491/980

Course Notes for Part 1

A/Prof Alex Remennikov

Autumn Semester 2014

LECTURE 1

The objective of this Lecture is to introduce you to basic concepts in finite element formulation. The main topics in this session will include the following:

Solution of engineering problems Numerical methods A brief history of the Finite Element Method (FEM) Basic steps in the Finite Element Method Verification of results Understanding the problem

Learning objectives: At the end of this session, you should be:

Aware of the aims and content of the course Aware of the course resources and their location Aware of the assessment procedure Familiar with basic steps required to perform finite element analysis Familiar with the role of finite element analysis in the engineering design

cycle Aware that a good understanding of the fundamental concepts of the finite

element method will benefit you by enabling to use STRAND7 more effectively.

1.1 INTRODUCTION The finite element method is a numerical procedure that can be used to obtain solutions to a large class of engineering problems involving stress analysis, heat transfer, and fluid flow. This course is designed to help you gain a clear understanding of the fundamental concepts of finite element modelling. Having a clear understanding of the basic concepts will enable you to use general-purpose finite element software, such as STRAND7 and SpaceGass, more effectively. Practical use of these computer packages is an integral part of this course. The relevant basic theory behind each fundamental concept will be discussed first. The discussion is followed by examples that are solved manually and then the results are verified against your findings from finite element analysis (FEA). It is our hope that this course will serve as a starting point for future design engineers who are beginning to get involved in finite element modelling and need to know the underlying concepts of FEA. 1.2 ENGINNERING PROBLEMS In general, engineering problems are mathematical models of physical situations. Mathematical models are represented by differential equations with a set of

corresponding boundary and initial conditions. The differential equations are derived by applying the fundamental laws and principles of nature (conservation of mass, conservation of momentum) to an analysed system. These governing equations represent balance of mass, force, or energy. The analytical solutions are composed of two parts: (1) a homogenous solution and (2) a particular solution. In any given engineering problem, there are two sets of parameters that influence the way in which a system behaves. First, there are parameters that provide information regarding the natural behaviour of a given system. These parameters include physical properties such as elasticity, thermal conductivity, and viscosity. Table 1.1 below gives the physical properties that are required to characterise the natural characteristics of various engineering systems. Table 1.1

Problem Type Physical Properties That Characterise a System

Modulus of elasticity, E

Modulus of elasticity, E

Modulus of elasticity, E; Moment of inertia, I

Thermal conductivity, K

On the other hand, there are parameters that produce disturbances in a system. Examples of these parameters include external forces, moments, temperature difference across a medium, and pressure difference in fluid mechanics. These types of parameters are shown in Table 1.2. Table 1.2

Problem Type Parameters causing disturbances

in engineering systems

Solid Mechanics External forces and moments; support excitation; support settlement

Heat Transfer Temperature difference; heat input

Fluid Flow Pressure difference; rate of flow

Electrical Network Voltage difference

The system characteristics as shown in Table 1.1 dictate the natural behaviour of a system, and they always appear in the homogenous part of the solution of a governing differential equation. In contrast, the parameters that cause the disturbances appear in the particular solution. It is important to understand the role of these parameters in finite element modelling in terms of their respective appearances in stiffness and load (or forcing) matrices. The system characteristics will always show up in the stiffness matrix or resistance matrix, whereas the disturbance parameters will always appear in the load matrix. 1.3 NUMERICAL METHODS There are many practical engineering problems for which we cannot obtain exact solutions. This inability to obtain an exact solution may be attributed to either the complex nature of governing differential equations or the difficulties that arise from dealing with the boundary and initial conditions. To deal with such problems, we resort to numerical approximations. In contrast to analytical solutions, which show the exact behaviour of a system at any point within the system, numerical solutions approximate exact solutions only at discrete points, called nodes. At this point, it is recommended to dig out and look at the Lecture Notes for CIVL392 Engineering Computing 2. The first step of any numerical procedure is discretisation. This process divides the medium of interest into a number of small subregions and nodes. There are two common classes of numerical methods:

1. finite difference methods, and 2. finite element methods

With finite difference methods, the differential equation is written for each node, and the derivatives are replaced by finite difference equations. This approach

results in a set of simultaneous linear equations. Although finite difference methods are easy to understand and employ in simple problems, they become difficult to apply to problems with complex geometries or complex boundary conditions. In contrast, the finite element method uses integral formulations rather that finite difference equations to create a system of algebraic equations. Moreover, an approximate continuous function is assumed to represent the solution for each element. The complete solution is then generated by connecting or assembling the individual solutions, allowing for continuity at the interelemental boundaries. Classification of structural analysis techniques in relation to the methods employed is demonstrated in Figure 1.1:

Figure 1.1 Classification of structural analysis techniques Discrete element methods (matrix method) assume the structure consists of a finite number of elements joined at nodes. The behaviour of a single element is analysed using the basic methods of structural analysis or energy methods, allowing the contributions of all the elements to be assembled together to give a set of simultaneous equations, which can be solved using matrix algebra. This method can readily deal with complex structures and is ideally suited for computer programming. Matrix methods may be further classified for discrete (or skeletal) structures as:

The flexibility (force) method that solves for the internal forces in terms of unknown displacements.

Structural Analysis

Analytical

Methods

Numerical

Methods

Differential Equations

Methods

Energy

Methods

Discrete Element

Methods

Finite

Difference

Method

Numerical

Integration

Method

Stiffness

Method

Flexibility

Method

The stiffness (displacement) method that solves for the displacements in terms of unknown forces.

The finite element method may be thought of as being numerical integration technique, which is simply an extension of the stiffness matrix method to continuous structures, which was developed in the late 1940s and early 1950s, at the same time as the emergence of the digital computer. 1.4 A BRIEF HISTORY OF THE FINITE ELEMENT METHOD Finite Element Method was first developed in 1943 by R.Courant, who utilised the Ritz method of numerical analysis and minimisation of variational calculus to obtain approximate solutions to vibration problems. The next significant step in the utilisation of finite element methods was taken by Boeing in the 1950s when Boeing, followed by others, used triangular stress elements to model airplane wings. Yet, it was not until 1960 that R.Clough made the term “finite element” popular. During the 1960s, investigators began to apply the finite element method to other areas of engineering, such as heat transfer and seepage flow problems. Zienkewicz wrote the first book entirely devoted to the finite element method in 1967. By early 1970s, FEA was limited to expensive mainframe computers generally owned by the aeronautics, automotive, defence, and nuclear industries. Since the rapid decline in the cost of computers and the phenomenal increase in computing power, FEA has been developed to an incredible precision. Present day supercomputers are now able to produce accurate results for all kinds of engineering problems. 1.5 WHAT IS FINITE ELEMENT ANALYSIS? FEA constitutes a part of the overall engineering design cycle. Figure 1.2 shows how finite element analysis fits in the design cycle, with computer aided design and with manufacturing and testing. FEA includes a computer model of a material or design that is stressed and analysed for specific results. It is used for new product design as well as for refinement of existing products. 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 performed on a relatively slow 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. Depending on the nature of a problem at hand, FEA can be linear or non-linear. 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 capable of testing a material all the way to fracture. FEA has become a solution to the task of predicting failure due to unknown stresses by showing critical areas in a material and allowing designers to see all of the theoretical stresses within the object. 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.

Figure 1.2 The Engineering Design Cycle

1.6 BASIC STEPS IN THE FINITE ELEMENT METHOD The basic steps involved in any finite element analysis consist of the following: Pre-processing Phase

1. Create and discretise the solution domain into finite elements; that is, subdivide the problem into nodes and elements.

2. Assume a shape function to represent the physical behaviour of an element; that is, an approximate continuous function is assumed to represent the solution of an element.

3. Develop equations for an element. 4. Assemble the elements to represent the entire problem. Construct the global

stiffness matrix. 5. Apply boundary conditions, initial conditions, and loading.

Solution Phase

6. Solve a set of linear or non-linear algebraic simultaneous equations to obtain nodal results, such as nodal displacements or nodal temperatures in a heat transfer problem.

Post-processing Phase

C O N C E P T

D E S I G N

A N A L Y S I S

T E S T I N G

M A N U F A C T U R I N G

CAD

CAE

CAM

7. Obtain other important information. At this point, the analyst may be interested in values of principal stresses, axial forces and bending moments, damage distribution, etc.

In a large software package the analysis portion is accompanied by the pre-processor and post-processor portions of the software. There also exist stand-alone pre- and post-processors (e.g., FEMAP, PATRAN, TrueGrid, HyperMesh) that can communicate with other large programs (e.g., NASTRAN, ANSYS, ABAQUS). Specific procedures of “pre” and “post” are different in different programs. Learning to use them is often a matter of trial, assisted by tutorials, manuals, and on-line documentation. Fluency with pre- and post-processors is helpful to the user but is unrelated to the accuracy of FE results produced. 1.7 VERIFICATION OF RESULTS In recent years, the use of finite element analysis as a design tool has grown rapidly. Easy-to-use finite element analysis packages such as STRAND7 have become a common tool in hands of design engineers. Unfortunately, many engineers without the proper training or solid understanding of the underlying concepts have been using finite element analysis. Engineers who use finite element analysis must understand the limitations of the finite element procedures. There are various sources of error that can contribute to incorrect results. They include:

1. Wrong input data, such as physical properties and dimensions This mistake can be corrected by simply listing and verifying physical properties and coordinates of nodes before processing any further with the analyses.

2. Selecting inappropriate types of elements Understanding the underlying theory will benefit you the most in this respect. You need to fully grasp the limitations of a given type of element and understand to which type of problems it applies.

3. Poor element shape and size after meshing This area is very important part of any finite element analysis. Inappropriate element shape and size will influence the accuracy of your results. It is important that the user understands the principles of developing a suitable finite element mesh and the mesh refinement techniques.

4. Applying wrong boundary conditions and loads This step is usually the most difficult aspect of FE modelling. It involves taking an actual problem and estimating the loading and the appropriate boundary conditions for a finite element model. This step requires good judgement and some experience.

You must always find ways to check your results. While experimental testing of your model may be the best way to do it, it may be expensive or time consuming. Indeed experimenting with mesh density and distribution, element type etc. may be an important part of the verification process. You should always start by applying

equilibrium conditions to different portions of a model to ensure that the physical laws are not violated. For example, for static models, the sum of the forces acting on a free body diagram of your model must be zero. This concept will allow you to check for the accuracy of computed internal forces and reactions. You may consider defining and mapping stresses along an arbitrary cross section and integrating this information. The resultant internal forces computed in this way must balance against external forces. One of the lectures in this course will be devoted entirely to verifying the results of your finite element models. 1.7 IMPORTANCE OF UNDERSTANDING THE PROBLEM You can save a lot of time and money if you first spend a little time with a piece of paper and a pencil trying to understand the problem you are planning to analyse. Before starting finite element modelling using a computer, try to develop a feel for the problem. A good engineer would try to intuitively predict structural behaviour and ask such questions as: Is the problem linear or non-linear? Is the material under axial loading? Is the body under bending moments or twisting moments or their combination? Do you need to worry about buckling? Can we approximate the behaviour of the material with a two-dimensional model? If you choose to employ FEA, “back-of-the-envelope” calculations will greatly enhance your understanding of the problem, in turn helping you to develop a good, reasonable finite element model.

Figure 1.3 Example of a good finite element mesh