1

Introduction to Finite Element Analysis Andrei Lozzi 2017

References: V Adams & A Askenazi, Building Better products with FEA, On World Press

R D Cook et al, Concepts & Applications of FEA, Wiley

SolidWork 2012 Simulation training manual

R G Budynas, Ch8, Advanced Strength & Applied Stress Analysis, McGraw Hill

1 Introduction.

There are many, many different FEA systems, some dealing with acoustic fields, others with heat flow,

and those that are relevant to us here that deal with stress and strain. Because of their technical advantages

and consequential commercial value these systems have been evolving continuously and rapidly, resulting

in a huge variety on the market today. Their uses have been largely in the hands of specialists, whom

speak in strange languages and become isolated above the rest of us, a little like monks. These specialists

also become suspiciously committed to just their system, they speak badly of all other systems, and of the

uninitiated (that is the rest of us). Of late the user interface of some FEA packages have become quite

comprehensible and relatively easy to apply, to the point that they can be used by the more aware and

respectful of engineers. This change has displeased the monks.

This is an introduction to one such user friendly system – Simulation, a version of which operates within

the SolidWorks environment. What is said here is intended to be basic, generally applicable and

indicative of the capabilities built into modern systems.

This overview of the mathematical operations that take place within an FEA package is based upon that

published by Budynas. In an application of FEA, a solid object which may be of any shape is discretized

into a large number of blocks or elements, with all the elements usually of the same basic shape. These

blocks contact each other and transmit loads only at their nodes, which are usually the vertices of the

elements. Internally all these blocks usually have the same idealized stress/strain distribution. It suits

Budynas’ approach to begin by considering how points within a loaded component are displaced in space

(by ui ) and ends with the forces ( fi ) applied to the boundaries of the component that causes those

displacements. Once we arrive at that overall relationship ( ui fi ) we can invert the functions, and then

calculate what is more commonly required of us: arrive at the stress from the applied forces ( fi ui ).

There are many standard (FEA) blocks or elements in use, of different shapes, with different number

of nodes and with different rules for their internal distribution of stress. Each may have their own strong

points and none are of course quite perfect. If you are sufficiently driven it is possible for you to

eventually define new elements to better approximate new or existing materials. The trend has been to use

more elaborate elements that give more precise and realistic answers, which are more automatic in their

application, but which are increasingly more computationally demanding. Here I propose to use an FEA

package as much as possible like a black box, knowing relatively little of what goes on inside, but

making adjustments to the analyses depending upon the consistency and realism of the inputs and of the

results. I expect that in the near future this is how FEA will be used for the majority of its applications.

2 One Dimension FEA

We can exemplify FEA procedures by considering a one dimensional model in which the steps are

deceptively obvious. Consider the bar below of variable height but uniform thickness being approximated

by string of rectangular blocks of various heights but the same thickness, subjected to tension F. The

strain and stress within each element will be uniform, but will differ between elements.

2

Fig 1 One dimension FEA model. The upper

curved flat bar is approximated by a sequence

rectangular elements of uniform thickness

but different height: hi, sharing force

F = fi but different displacements ui at the nodes.

iu is the displacement of each element’s node, with respect to a common external coordinate system,

F the applied load and is transmitted undiminished to each node fi = F, and il & i are the length and

elongation of each element.

B represents the operation that arrives at the strain i from the displacements ui (or elongations i )

for each element:

i

i

i

i Bll

uu

21 i, i, & li are all elements of vectors Eq 1

[D] is the operation that using the material properties (Young’s modulus) allows us to derive the stress

from the strain:

iii DE Eq 2

A is the operation that finally gives us the force on an element from the stress (here stress multiplied by

the area)

FAAreaf iiii Eq 3

If we apply these operations in sequence we get:

iii kBDAf Eq 4

The product of these operations in Eq4 obviously give us the stiffness array [ k]. Note that if we invert [ k]

we can calculate elongations from the nodal forces: ii fk1

, Then with [B] in Eq 1 we can calculate

the distribution of strain, with [D] Eq 2 we arrive at the stresses, and with the areas we have the forces.

All this may seem like a waste of thinking time, but it describes the operations that can be carried out to

give very useful results in 2 or 3D when everything is not so obvious. Thus if we can define for the type

of element that we wish to use, the geometric operations required to extract internal strain from the

displacements of the nodes: [B], relate stress to strain in a general way: [D], and describe the stress

distribution within each element from the forces at their nodes: [A], we can by inverting the product

of these operations calculate the distortion within a component from just the forces applied to it! The

internal forces balance out, just like they did in the linear model above, so only the externally applied

forces and reactions are required. Applications of these principles to 2D elements can be found in any

introductory text on FEA but they take many pages and clarify little more than the above few lines do.

The big attractions of the method is that firstly it can be applied to complex components for which

existing stress analysis methods provide only rough approximations to their stress levels, requiring the use

large factors of safety and or prototype testing. Secondly, the method can be applied by engineers that are

3

a little challenged by the theory of solids. As computers and software continue to develop we can expect

more reliable answers ever more quickly using FEA. These are all good reasons.

3 Higher dimensions FEA

In 2 or 3 dimensions things become elaborate but in principle no different to the above. We not only have

displacements being a function of several variables ie δi ( ui, vi, ... ), we also have normal and shear

elongation εi (εx, εy, γxy, … ), normal and shear stress and related modulus of elasticity. All the operators

become matrix operators, but as before:

ii B relates strain to displacements

ii D relates stress to strain by means of Young’s modulus and Poisson’s ratio.

ii Af and, for each different type of ‘Finite’ element we need a matrix operation that

relates stress within the elements

to forces at its nodes.

Again as before:

ii BDAf

or:

ii kf

Fig 2 A bracket and its FEA model

The matrices [B] and [D] use material properties to arrive at stress as is done in Mohr’s analysis. Matrix

[A] is determined by the element’s definition. As mentioned before the number of standard elements in

general use is large, more than 50. The simplest and possibly the earliest is a planar triangle for which the

stress is assumed constant along the triangle’s boundaries. For such elements each corner or node has 2

degrees of freedom, giving a total of 6 for the whole element. A small component divided in 1000

element will have its matrix [A] of about 6000 rows by 6000 columns. Very few components are currently

discretised in just 1000 elements, 20 000 is more typical. More realistic and elaborate elements are 3D

bricks with 8 to 20 nodes, each with possibly 6 degrees of freedom, resulting in gigantic matrices. All this

makes modern FEA very CPU and RAM hungry and getting more so by the day. Given that CPUs and

RAM are becoming exponentially more capable and cost effective, this is a problem that is happily

being met by a solution.

4 Elements

Shown below in Fig 3 is range of FEA elements ranging from 2D plates to 3D bricks. Not included are

simpler trusses and beams elements, which are appropriate to structural frames. The earliest triangular

elements could not cope well with bending loads, since these elements only described uniform stress

along their boundaries. Since bending moments generate non uniform stress across the section of a beam,

triangular elements could give very inaccurate results from bending loads. These outdated elements were

necessary when computing power was gauged in kiloflops not in megaflops (tetraflops was just science

fiction), are held as examples of what can be wrong with FEA. Simple triangular elements are not put to

any serious use any more.

4

Fig 3 Three planar and three solid elements

5 H and P Elements

Modern elements are sometimes categorized simply as being of H or P type. H elements are

predominantly linear but allow for up to second order variation in stress within each element. Type P

begins with second order and allows up to maybe the 9th order. Many if not most elements presume

straight sides, but some modern ones can cope with extraordinary distortions, the likes that could be

applied to Salvadore Dali figures. High orders consumes disproportionate amount of CPU time and give

false confidence and little return for the extra computing time. It is generally said that with H elements

more precise solutions (not necessarily more accurate) can be obtained by using more elements, varying

the element size, increasing their density in the regions where the stress varies rapidly, and finally to use

quadratic elements in the high stress regions. This procedure which is efficient of computer time has

required more human intervention.

With most P systems it is typical to reduce element size together with increasing their order in the region

of rapid stress change. Some proprietary P elements systems discretise the components in very few

elements, 10s not 10,000s. The number of elements remains unchanged while very flexible stress

distributions are applied within the elements. Currently sophisticated P elements seem to be the most

effective. Convergence is the term used to described the approach to a calculated asymptotic level

achieved by either continual refinement of the H element sizes or by increasing the order of P

elements. This asymptotic condition indicates that there is reduced benefits in ever longer computation

times.

Fig 4. Convergence with H elements Fig 5 Convergence with P elements

5

6 Classes of FEA systems

A Most FEA systems have been developed to deal with contact between components in an

assembly, and stress up to the elastic limit of the material, Fig 6 a). – (eg Simulatio)

B Others deal with non linear and plastic strain, Fig 6 b)

C Hysterisis for rubbery items, Fig 6 c).

D Large deformation for buckling and folding, contact between the deformed surfaces.

E Dynamic loads from accelerations, damping loads varying with element velocities, impacts

between components and explosions, Fig 7.

Fig 6. a), elastic stress/strain relation b), elastic-plastic c), non-linear, rubbery

The equations governing A, B, & C may be of the form: ii kf . These are called implicit systems

The equations governing systems E may be of the form: iiii amvckf These are called

explicit and take into account damping (velocity dependent) and inertia forces (acceleration dependent)

forces.

Fig 7 Example of

an explicit system

type E, applied to

a crash simulation.

6

7 Simulation and deformations.

Simulation belongs to the possibly simplest of the above FEA systems, that is type A. Examples of A and

B systems do not redeveloped the CAD model during or after loading. Probably an approximate

exaggerated deformed model is usually displayed on the screen, but it cannot be used to identify specific

locations to determine the stress level at points of interest. The software has to be set up to display stress

levels in an un-deformed display before the operator can pin point particular locations where the stress

levels are required.

To carry out analysis with system C to E (and possibly some B) the dimensions of the CAD model

has to be progressively reconstructed, requiring greater computer time.

8 Simulation and element types

Currently Simulation can deal most effectively with solid bulky objects, like machine components,

using solid elements. Traditional users of FEA systems use shell or plate elements wherever possible

because of their computational efficiency. Shell beam and strut elements have of late been added to

Simulation repertoire but SolidWorks does not automatically provide you with a surface or plate

version of its model. Simulation can now deal with articulated members attached to a frame, as for

example with suspension components hinged to a frame.

9 An area of difficulties – stress concentration

FEA appears to be the answer’s to an engineer’s prayer’s, it can be used to analyse messy components for

which classical analysis can only provide uncertain approximations, but FEA has 2 problem areas. The

first and the lesser problem deals with stress concentration. In areas where a component’s surface changes

shape rapidly, the stress distribution will be typically uneven and in concave corners ‘stress

concentration’ will result. With modern sophisticated elements and fast capable computers with huge

RAM, stress concentration is a fading problem. In Fig 2 there is likely to be above average stress in the

region around the hole. Modern systems can automatically generate higher element density (H elements)

or of higher order (P elements) in the region of profile changes and in the presence of stress gradients.

Most FEA vendor’s method of dealing with uncertainties associated with stress concentration is to

provide test cases for which there are analytic and experimental evidence. Here the distribution of

stress for an analytically solved set-piece is compared with the stresses determined by the vendor’s

software. One such set-piece is the stress distribution around a central hole in a rectangular plate loaded in

tension (provided by Simulation). Usually several elements and densities are applied in these

demonstrations to engender confidence in the vendor’s package. The accuracy of stress prediction can be

surprisingly high, but do not be lulled in a false sense of security. Try one of these demonstrations

without looking at their carefully developed procedure, to see just how off the mark your answers can be.

10 A bigger area of difficulties – boundary conditions

The second and often a greater source of difficulties are generally referred to as the boundary

conditions. Note again in Fig 2 the bracket seems to be attached to a perfectly rigid wall and the pressure

is applied unevenly at exactly 3 nodes. Neither of these conditions is faithful to a real bracket, hence there

certainly will be areas of inaccuracies in the vicinity of the load and the reaction. The stresses in these

areas will converge to ‘precise’ levels, but will be inaccurate by some unknown amount.

7

Δy = 0

Δx = 0

11 The Poisson effect

a) b)

Fig 8 a), b) & b) Rectangular plate under tension,

showing Poisson’s effect due to an over constrained

left boundary. At right in b) there is uniform tension,

at left the nodes are fixed, ie Δx = Δy = 0.

At c) a section of the plate is loaded, but the boundary c)

conditions are set appropriately for a simple tension.

Fig 8 shows a particular effect of restraints on a component that is intended to be placed in simple tension.

In a) the horizontal and vertical displacement of the nodes at the left boundary of a plate component are

fixed. Then tension is applied on the right edge of the component. The Poisson’s effect, which is the

shrinkage in the vertical direction of the elements, due to the elongation in the horizontal direction, takes

place as expected in b), but only on the right half of the plate. This clearly results in distortion of some

elements on the left, which would be inappropriate for simple tension. A means of dealing with this sort

of difficulties is to identify the area of a component where accurate results are required, and to ensure that

any poor modeling takes place well away from that area. It is often unnecessary and impractical to

model all areas of a component or assembly equally faithfully, if this is the case for your study then

place the simplified poor modeling well away from the critical areas.

9 Prediction of component failure

FEA simply applied does not provide us with a reliable means of predicting the location and load at which

failure will occur. Next to maximum stresses, residual stresses from the manufacturing processes,

dimensional faults, manufacturing errors and material imperfections seem to dominate where and for

what load components fail. On the positive side, strain gauge readings do verify the results of (good)

FEA calculations, giving confidence in the methods, whereas quality control can help with the others.

10 The future (shortly at a PC near you)

20 years ago mainframe computers had up to Mbs of RAM, now PCs can have several Gbs.

20 years ago it took 10 hours to do a 10 000 element FEA problem, now PC does it in seconds.

It is a little difficult to predict the future but is seems reasonably safe to expect that in a few years PCs

will handle FEA elements down to crystalline size, with loads and restraints detailed to match, and it will

happen almost automatically. We just have to get ready for the future.

8