lecture_slides_chapter_1.pdf

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Practical Stress Analysis withnd

Dr. Br an J Mac Donald

© Bryan Mac Donald/Glasnevin Publishing 2007 ‐2011


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Part 1: Introduction to FEA Pages 1 to 2 of book

Why do We Need Finite Element Analysis?

Consider the simple beam problem shown

Are you able to determine the maximum stress and

predict where in the beam it occurs?y

WL

I

3max

max6

bh

hWL

What if you are asked to say whether the beam hadfailed due to the applied load?

ow wou you o s

© Bryan Mac Donald/Glasnevin Publishing 2007-2011


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What happens if the problem gets more complex?

1) There is a taper in the beam which our theorydoesn’t take account of.

2) There are a number of holes in the beam whichour theory doesn’t take account of.

3) There is a more complex loading system presentwhich our theory doesn’t allow for.



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So how do we overcome these problems?

“ ”into our calculations.

The FOS results in a safe design,but:

-

2) Too much material/weight

oo muc was e



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The FOS method has been used for centuries to produce many innovative designs.

experimental tests build prototypes and test them!

building and testing prototypes (knownas a “build-and-break cycle)

t ta es a ong t me to get an answer towhether the design works or not.

’for the behaviour of the product untilnear the end of the process

ere mus e a e er way

There is – it’s called Finite Element Anal sis!



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What is FEA?

s mp e e n on:The finite element method (FEM) is a mathematical technique that is used to

obtain approximate answers to complex problems that cannot be solved usingbasic theories

An even simpler definition:The fundamental concept of the FEM is that it splits up a complex problem into

“ ”together the answers to all the simple problems to give an approximate solutionto the complex problem.

A more scientific definition:The FEM divides the domain of interest into a finite number of simple sub-

domains and uses variational concepts to construct an approximation of theso u on over e co ec on o su - oma ns.



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What is FEA? A Plane Stress Problem:

’

0

yx

xyx

0yx

yyx

xxE

01

01

xv

yu

yv

xu

xyyx ;;

zzxy

2100 /)(

Linear Elastic Stress-Strain Relationshipfor Plane Strain

Strain-displacement relationships forPlane Strain

Great! But how do we solve it?

Well, we can’t! ………………….so we use FEA to get an approximate answer!



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What is FEA? A Plane Stress Problem:

Divide into finite elementsand apply loads andboundary conditions

Solve FE problem to obtaindisplacement of each “node”

Further process thedisplacement results

in order to obtainstrain and stress



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How Does FEA Work?

•Divides structure into small pieces called elements.

•These elements are simple shapes, so simple theory can work with them.

•The collection of elements is called a finite element “mesh”•As the number of elements in the mesh increases their size decreases and so theapproximate solution (should) become more accurate

•Analogy with Integration / Simpson’s rule:



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How Does FEA Work?

All Finite Element Analyses follow the same procedure:

1. Step 1: Evaluate the problem and make assumptions

2. Step 2: Describe how the finite elements will behave.

4. Step 4: Form element equations5. Step 5: Assemble each element equation into a global problem equation6. Step 6: Specify loads and boundary conditions7. Step 7: Solve the global problem8. Step 8: Evaluate the results.

Let’s look at an example to illustrate the above steps.- We will cover the theory in this example later in the book.- For now don’t worry too much, just try and understand the concepts



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A tapered steel beam of rectangular cross section isused to support a horizontal load of F = 10,000 N.Use FEA to determine the deflection of the freeend of the beam and the max stress in the beam.

Step1: Evaluate the problem and make assumptions

•This is a relatively simple problem that has an analytical solution and probably doesn’t require FEA to obtain an approximate solution, however, for the purposes of example let’s assume that an anal tical solution does not exist.

•We are told that the beam is made from steel so let’s assume that the load isn’t great enough

to cause permanent plastic deformation which means we can assume a linear elastic material mo e governe y oo es aw w usua ma er a proper es or s ee .

Assumption 1: Linear elastic material model with E = 210 x 109 Pa



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We know that the beam is embedded in a concrete

edge of the beam cannot move.

Assumption 2: The left hand edge of the beam is

• Applied horizontal load on the right hand edge of the beam.• e eam s own we g t w , owever, cause t to e ect s g t y ownwar s.• Let’s assume self ‐weight to be negligible ‐ beam can only deflect in the horizontal direction.• All loads and reactions in horizontal direction = a one ‐dimensional problem

Assumption 3: A one ‐dimensional analysis is sufficient to solve this problem

We are not told much about the manner that the load is applied to the right hand edge of theeam. e s assume a e oa s app e very s ow y. s means a ynam c v ra oneffects can be ignored and that loads are applied very slowly.

Assum tion 4: A linear static anal sis is sufficient to solve this roblem



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• We are also not told about the environment in which the beam is placed.• As a first approximation let’s assume that environmental factors are not significant and henceonly a structural analysis is required.

Assumption 5: A structural analysis is sufficient to solve this problem

ep : esc e ow e n e e emen s w e ave

• Assumption 3 above means that we can use a one ‐dimensional element for this analysis. • The simplest element type is a 1D linear truss element.

• This element

effectively

behaves

like

a spring

that

can

only

deform

in

one

direction.

• 2 nodes – Ni and N jue o no e moves a s ance i an no e moves y j

• So the elements displacement is a function of the nodal displacement

(e)

• Where Si and S j are “shape functions” which relate displacement of the element as a whole to the specific displacement of the nodes that make up the element.

i i j j



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• We now define a local coordinate system in the element (natural coordinate system)• given by • where is measured from the centre of the element• such that N i has a coordinate = ‐1 and N j has a coordinate = 1.

• ,

Si = ½ (1 ‐ ) and S j = ½ (1 + ) Eqn 1.07

• Equation 1.06 is more commonly used in its matrix form:

.

• We can also get an expression relating element strain (general) to nodal displacement (specific)



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• From assumption 4: material obeys Hooke’s law

• Putting equation 1.10 into equation 1.11 gives us:

Step 3: Build the Finite Element Model

Split the tapered beam into 3 regularly shaped elements:• based on average cross section at mid point of element



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• The finalised model shows three elastic springs (i.e. truss elements) with equivalent stiffnessto the three uniform cross section elements shown in the previous figure:

Step 4: Form Element Equations

• We will discuss this step in some detail in chapters 3 and 4. For now let’s just accept that thegoverning equation for this problem is:

• It can be shown (see chapter 4) that for this element type:

• Evaluating this equation (i.e. plugging in [B] and integrating) gives us:



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• Calculating the cross sectional area for each element and plugging in values of E andelement length, l, gives us an equation for each element:

• This allows us to form the element equations using equation 1.13

• Combining these three equations into a matrix equation:



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Step 6: Specify Loads and Boundary Conditions

Step 7: Solve the Global Problem

• This example is relatively easy to solve – use Gaussian elimination or similar• Solving gives:

U1 = 0,U2 = 2.199 x 10

‐5m,= ‐53 . ,

U4 = 11.062 x 10‐5m

• So, the global nodal displacement vector is:Deflection of free end of the

beam is 11.062 x 10 ‐5 m



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Step 8: Evaluate the Results

• Let’s find the strain and stress in the elements:

• Using the strain ‐displacement equations (equation 1.09)

•equation 1.11 (i.e. Hooke’s Law)

Max stress in the beam is .



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Summary of Chapter 1:

After com letin cha ter 1, ou should:

1. Be able to explain why finite element analysis is an important tool for engineering analysis

and design

2. Be able to explain what finite element analysis is and how it works.

3. Be able to carr out a sim le one ‐dimensional finite element anal sis of an en ineerinproblem on a piece of paper, by following the methodology of example 1.1.

4. Understand the various steps required for any finite element analysis


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