Appunti completi Chassis and body design

Sommario

FEA – INTRODUCTION ..................................................................................................................... 2

Solutions steps of FEA .................................................................................................................. 3

DISCRETISATIONS .................................................................................................................... 4

STIFFNESS MATRIX ................................................................................................................... 4

Shape function ............................................................................................................................. 6

ANALYSIS METHOD ........................................................................................................................ 8

Deformation method .................................................................................................................... 8

FRAMES ........................................................................................................................................ 12

Rotating node frames ................................................................................................................. 14

Translating node frame ............................................................................................................... 21

THREADED JOINT .......................................................................................................................... 27

Stainless steel screw .................................................................................................................. 28

Threaded joint: basics ................................................................................................................ 31

Screw stress calculation. ...................................................................................................... 36

Design calculation...................................................................................................................... 46

Design Calculation ..................................................................................................................... 49

Friction coefficient ..................................................................................................................... 50

ADHESIVE JOINT ........................................................................................................................... 52

Structural adhesives................................................................................................................... 63

Structural adhesives: enlarged view of Volkersen’s model ........................................................... 64

Goland and Reissner’s theory ..................................................................................................... 66

Cylindrical joint: some practical applications of anaerobic adhesives ....................................... 66

Cryogenic fit ........................................................................................................................... 70

Loctite calculation method ......................................................................................................... 72

EXPERIMENTAL STRESS ANALYSIS ............................................................................................... 74

STRAIN GAGE ............................................................................................................................. 75

How strain gage works? .......................................................................................................... 76

STRAIN GAGE CIRCUITS ............................................................................................................... 82

The Wheatstone bridge ............................................................................................................... 82

Limitations of the Wheatstone bridge ...................................................................................... 84

SOME CIRCUIT ARRANGMENT ................................................................................................ 84

LOAD CELLS ................................................................................................................................. 89

Strain rosettes ............................................................................................................................ 90

Basic formula ......................................................................................................................... 90

1

FEA – INTRODUCTION

What’s the aim of FEA? – FINITE ELEMENT METHOD.

Fea became popular during the ’50, and its main aim was to find approximate solution for

high degrees of static indeterminacy: structure problem characterized by a high number

of redundant constraints.

It was originally intended just for structure problem but then it’s spread to a wider range of

applications including hydraulic problem, thermal problem… nowadays int covers all kind of

physics engineering application.

What is FEA/FEM all about?

Let’s take this beam into account and let’s assume that we have a load distribution q – function

of x – x coordinate stars from A to B and the beam length it’s L. we assume that we know both

the cross section of the beam and they are constant and the material of the beam is known

with the Young’s modulus and Poisson ratio.

The problem is to find out which the vertical displacements are for this given boundary

condition. As we know, this solution can be found in this case because the problem it’s easy.

4

()

= [1]

4

In the case of this beam, we can both face the problem by leveraging 2 different methods:

1. Direct integration of the function u(x)=g[f(x)]

2. Assuming an approximated function which is called which depends on k

�()

parameters (polynomial function e.g) – and this function is defined to respect some

boundary conditions and these conditions can be determined by minimizing the total

energy of the body and enforcing some condition that we’ve called k.

FEA method uses second approach, and it defines an a priory shape of solution but not on the

whole model it defines a priory shape on just a small portion of the continuum which is called

2

fine elements: we subdivide the domain into small simple elements over which a priory

solution is known and it’s imposed.

All this portions of the whole body are connected to each other just at specified position which

are named nodes.

The number of nodes and elements are important parameters in the FEA.

Why don’t we solve problems characterized by many degrees of static indeterminacy by direct

integration? It’s impossible.

We must leverage some approximated solution algorithms.

Solutions steps of FEA

1. Discretization: of continuum body by a grid – which is called mesh. Meshing defines

the position of nodes and then the elements which both to the elements or even one the

elements, in any case, the dimension of the elements is defined thus length surface

volume. Then we must establish a match with nodes and elements. Then we set a

condition for which any element interact to other just at nodes. Therefore, there isn’t

contact between elements, other than at nodes. Lastly, we need to define the stiffness

matrix of each elements: we need to assemble the matrix of all elements to obtain the

stiffness matrix of the all system.

2. Simple displacement functions are assumed: inside the element volume.

3. Compatibility of nodal displacement is enforced.

4. Nodal displacements are retrieved through a ‘displacement method’ like algorithm:

we are simply assuming that stiffness of problem is known, and it’s known by the

stiffness matrix. It’s known also the number of boundary condition (some nodes cannot

move through some direction or along all direction) and then we know the forces of the

system. What we do is find minimum energy condition of the system to retrieve the

displacement function which should satisfy boundary condition.

Due to the fact that we have divided our continuum body into a finite number of elements we

are replacing the initial differential equation [1] with a sort of number of algebraic equation

which are many in number but very easy to solve. There will be an equation for any DOF of

problems. = ° ∙ °

5. Assemble the global system of equation: equilibrium, compatibility and boundary

conditions are enforced.

6. Solve the global system 3

DISCRETISATIONS

That’s about dividing a continuum into a finite number of discrete domains. From the Figure we

can see that by switching from the original domain to a some of triangle, we are losing some

surfaces, and these lost surfaces are smaller as the number of triangles becomes bigger.

By simple relations we can retrieve the formula below:

� ∙ cos � �� ∙ �2 ∙ ∙ sin � �� − 2

= ∙

2 2

= = % �

% = ∙

2

The greater the number of discretisation domains the smaller the error.

STIFFNESS MATRIX

It is the matrix which relates the force components on the i-th nodes for given displacement j-

th nodes. If we move the j-th nodes, should we measure the displacement, our system will

undergo under some strain, and this implies that we should experience a force.

Example:

Fx force applied to node A which makes node B move by Uv=1 mm along y – axis.

The stiffness matrix of the system will characterize this behaviour and if the beam, is for

example, perfectly rigid, the vertical displacement will be 0.

The stiffness matrix of a finite element can be written by one of the following procedures:

1. DIRECT FORMULATION: we will use the known formulas for basics case of solid

mechanics which are valid for a restricted class of problems. 4

2. VARIATIONAL FORMULATION: we will use a formulation which relies on the

minimization of the total energy of the element.

We will see this method to a very simple element: TRUSS ELEMENT. The truss element is an

element which react to normal load (no bending, no torsion, no shear)

1. DIRECT FORMULATION:

Truss element is between two nodes A – B and is loaded by forces . The nodes undergo

,

displacement along the horizontal direction.

Problems in solid mechanics are solved by enforcing 3 conditions:

1. Equilibrium

2. Compatibility of displacement

3. Constitutive law

+ = 0 → = − = → = ∙ ( − )

= + ∆ → ∆ = ∙ +

=

= ∙

= → ℎℎ ℎ .

1 −1

{} []

� � = � � ∙ � � = ∙ {}

−1 1

5

2. VARIATIONAL FORMULATION

For this element the displacement function is linear between nodes A and B, thus the

longitudinal strain can be calculated:

= −

= − =0

1

= �( ∙ )

2

= ∙ + ∙

∆ −

= =

2 2 2

( )

− − 2 +

1 ∆

= �( ∙ ) = � � � = ∙

2

2 2

1 −1

1. �

� = � ∙ � �

−1 1

1 0

2. � = � � ∙ � �

0 1

1 −1

� � = ∙� � ∙ � �

−1 1

The variational method is valid although is quite cumbersome compared to the direct method

for simple cases:

- In the case of solving of variational method we’ve chosen an ‘a priori’ shape of the

element displacement

Shape function

How to define the displacement field inside the element? What we use are shape function.

Shape functions are functions of nodal displacement and they allow for weighing the

displacement on the element based on the proximity to the one node or the other: in other

words, in the case of a truss element, we can see that a couple of shape function can be

defined to describe the overall linear trend of displacement in the element. 6

How can we do that?

We choose 2 functions:

1. which attains a value of one when x =0, and

= −

it takes 0 when this function is one when we are

= ;

at node A and becomes 0 when we move at node B.

2. behaves the opposite. Becomes 0 when we are

=

at A and becomes maximum (=1) when we reach the B

node.

What we do then is making use of these functions by superpositions of effect meaning the

overall displacement at the generic point inside the element u(x) is given by:

�1 �

() = ∙ + ∙ = ∙ − ∙

+

( − )

() = + ∙

At A: the displacement we experience our is the same of

(1

= 0 − 0) + ∙ 0: ()

node A

At B:

= �1 − � + ∙ =

In between it’s a superposition of both contributions. We approach node A to displacement of

our point inside the element is governed by the behaviour of node A and this is the opposite

when we move to node B.

By 2 nodes we can define linear shape function.

From the figure we see a simple beam, we calculate strain based on derivation of

displacement: ( − )

()

= =

If the displacement function is linear, the only type of strain function we can get is a constant

function. Therefore, since strain and stress are related by a constant, the stresses are constant

on each element, meaning that neighbouring element have different stress levels and we

cannot describe stress increases (blue line).

To describe stress concentration, we can operate in 2 ways. 7

At the hole edge we have stress increase and there is a sharp decrease in stresses as we move

out to worst the plate edge. How can we solve this problem?

If we do a zoom, the step increases in stresses can be caught by increasing the number of

elements: we refine the mesh locally, by using simple elements.

Or we can choose larger elements characterized by high order function:

So we have less elements with higher order functions.

ANALYSIS METHOD

Manca force method

Deformation method

Fixed support and let’s take a number of trusses which are link to a lower part of the structure

which is a block (perfectly rigid).

Element 1 →

1 1

Element 2 →

2 2

Element 3 →

3 3 8

In the general case, these 3 elements can be different and to apply a downwards pointing force

on the block and the length of the trusses is the same

(=L). Using the force method,