Machine design - sintesi

Sommario

BEAM THEORY............................................................................................................................ 5

CONSTRAINTS.......................................................................................................................... 5

SLENDER STRUCTURES........................................................................................................... 5

STATIC DETERMINACY.............................................................................................................. 5

EULERO-BERNOULLI BEAM THEORY......................................................................................... 5

ASSUMPTIONS................................................................................................................... 5

NORMAL STRESS.................................................................................................................. 6

ENERGY................................................................................................................................ 6

BENDING MOMENT............................................................................................................... 7

AXIAL + BENDING................................................................................................................ 7

SHEAR STRESS..................................................................................................................... 7

TORQUE............................................................................................................................... 8

MECHANICAL PROPERTIES OF MATERIALS.................................................................................. 9

MATERIAL CHARACTERISTICS.................................................................................................. 9

TENSILE TEST..................................................................................................................... 10

DUCTILE vs BRITTLE behaviour.......................................................................................... 13

ASHBY DIAGRAM............................................................................................................. 13

NOTCH EFFECT......................................................................................................................... 14

THEORETICAL STRESS CONCENTRATION FACTOR.................................................................14

TYPICAL NOTCHES IN ENGINEERING STRUCTURES.............................................................16

CIRCULAR HOLE.............................................................................................................. 16

ELLIPTICAL HOLE............................................................................................................. 16

OTHER TYPES OF NOTCHES: VARIATION IN SECTION.......................................................16

EXPERIMENTAL STRESS CONCENTRATION FACTOR................................................................17

BRITTLE MATERIALS........................................................................................................... 17

DUCTILE MATERIALS........................................................................................................... 17

RESIDUAL STRESSES (ON DUCTILE MATERIAL)................................................................18

STRESS STATE.......................................................................................................................... 19

EQUILIBRIUM OF BODIES....................................................................................................... 19

RIGID BODY........................................................................................................................ 19

DEFORMABLE BODY........................................................................................................... 19

Calculation of stress state along a generic direction.............................................................20

Physical meaning of this kind of approach:.....................................................................21

PRINCIPAL STRESSES.......................................................................................................... 21

STRESS INVARIANTS........................................................................................................... 22

HYDROSTATIC & DEVIATOR TENSOR..................................................................................22

OCTAHEDRAL STRESSES AND PLANE................................................................................. 23

MOHR’S CIRCLES................................................................................................................ 23

PLANE STRESS................................................................................................................ 24

PLAIN STRAIN.................................................................................................................. 24

ASSESSMENT OF STRUCTURES................................................................................................. 25

STATIC STRENGTH ASSESSMENT........................................................................................... 25

STATIC FAILURE.................................................................................................................. 25

SAFETY COEFFICIENT (η)....................................................................................................... 28

JOINTS: BOLTED, WELDED (AND GLUED)..................................................................................29

BOLTED JOINTS...................................................................................................................... 29

ASSESSMENT OF SHEAR BOLTED JOINTS............................................................................30

STANDARDS.................................................................................................................... 30

Bolt classification......................................................................................................... 30

FAILURE MODES OF BOLTED JOINTS................................................................................31

Particular cases: torsion and bending................................................................................32

Shear + torsion............................................................................................................... 32

Shear + bending............................................................................................................. 32

ASSESSMENT OF FRICTIONAL BOLTED JOINTS....................................................................33

WELDED JOINTS..................................................................................................................... 34

WELDING TECHNIQUES...................................................................................................... 34

TYPES OF WELDING............................................................................................................ 35

Fillet welds...................................................................................................................... 35

Butt joints....................................................................................................................... 35

DEFECTS IN WELDING........................................................................................................ 36

NON-DESTRUCTIVE TESTS (NDT):................................................................................... 36

STANDARDS....................................................................................................................... 36

Assessment for BUTT WELDS.......................................................................................... 37

Assessment for FILLET WELDS........................................................................................ 37

GEARS...................................................................................................................................... 40

INVOLUTE........................................................................................................................... 40

KINEMATIC AND FORCES.................................................................................................... 41

FORCES........................................................................................................................... 41

Typical FAILURE of modern gears....................................................................................... 42

FATIGUE OF METALS................................................................................................................. 43

Fatigue crack NUCLEATION CLASSES................................................................................. 43

Fatigue PROPAGATION........................................................................................................ 43

LOAD PARAMETERS for fatigue.............................................................................................. 44

WHOLER CURVES............................................................................................................... 44

Fatigue TEST MACHINE....................................................................................................... 45

STAIR-CASE METHOD...................................................................................................... 46

Typical RELATIONS between ULTIMATE TENSILE STRESS and FATIGUE LIMIT...................46

From SPECIMEN to COMPONENT............................................................................................ 46

DIMENSIONAL effect........................................................................................................... 46

SURFACE effect.................................................................................................................. 47

NOTCH effect...................................................................................................................... 47

FATIGUE LIMIT OF COMPONENTS..................................................................................... 48

FATIGUE ASSESSMENT considering MEAN STRESS EFFECT....................................................48

HAIGH DIAGRAM for NORMAL LOAD (axial & bending).......................................................48

Possible SIMPLIFICATIONS of HAIGH DIAGRAM................................................................49

GOODMAN’S SIMPLIFICATION of the HAIGH DIAGRAM in the compressive region...........49

HAIGH DIAGRAM for SHEAR STRESS (applied in case of TORSION)....................................50

PALMGREEN-MINER APPROACH for FINITE-LIFE at different STRESS...................................50

MULTI-AXIAL FATIGUE................................................................................................................ 50

GOUGH-POLLARD criterion.................................................................................................... 51

CONSTANT TORSION and VARYING NORMAL loads.............................................................51

SINES CRITERION.................................................................................................................. 51

COURSE SUMMARY................................................................................................................... 53

STRUCTURAL DESIGN: design for avoiding failure (mainly static or fatigue)

STRUCTURE: object that reacts to and transmits a force -> we need to know how this load

enter and propagate in the structure. To find it out we need to know:

1) Geometry

2) Loads

3) Boundaries

Loads and boundaries are very similar (to work, a boundary applies a force)

Local load has a limit, below which it is safe, above will fail.

N.B.1): components do not always fail due to local stresses, can also fail by a global

phenomenon, generally instability, but we are not going to study them.

N.B.2): almost every object is requested also to be a structure (even if it is not its main aim)

Sometimes the requirement is not only not to fail, but for example a proper stiffness (rigidità);

in some cases, a sort of controlled failure is designed in order to absorb some energy (usually

in crashes).

A lot of times this is not a straightforward analysis, but an iterative one.

First thing to do is to reduce the complex system the several smaller problems (to make it

simple) and then sum together the results through models that increase the complexity little

by little.

DESIGN TASKS: the final design will be a balance between different requirements and needs:

1) Green and sustainable design

2) Functional design

3) Cost reduction

4) High performances

5) Structural design

BEAM THEORY

The solid mechanics theory of beams, more commonly referred to simply as “beam theory,”

plays an important role in structural analysis because it provides the designer with a simple

tool to analyse numerous structures. Although more sophisticated tools, such as the finite

element method, are now widely available for the stress analysis of complex structures, beam

models are often used at a pre-design stage because they provide valuable insight into the

behaviour of structures. Such calculations are also quite useful when trying to validate purely

computational solutions.

CONSTRAINTS

ROLLER blocks 1 DOF

 HINGE or PIN blocks 2 DOF

 SLIDER blocks 2 DOF

 ENCASTRE or CLAMP blocks 3 DOF



SLENDER STRUCTURES

Slender member = a member in which one dimension is much higher than the other 2.

Assumptions:

(i) The segment representing the body is called axis of the member itself

(ii) The axis of the member is kept straight

(iii) The cross section can vary along the axis

Slender structures are usually classified according to their geometrical and structural

features, i.e. the kind of loads they are able to transfer:

1) Axle (rotating member, transferring transverse load)

2) Shaft (rotating member, transferring torque)

3) Bar (fixed members, transferring axial load)

4) Beam (fixed members, transferring transverse load)

STATIC DETERMINACY

Let us consider a structure with m of degrees of constraints (d.o.c.) given by the

external supports, and l degrees of freedom (d.o.f.). m is also the number of

unknown reactions. Then if m = l the structure is statically determinate and the

reactions can be calculated using the simple equilibrium equations.

Therefore, we can calculate the stress state in every point.

EULERO-BERNOULLI BEAM THEORY

ASSUMPTIONS

1) Long beam with constant properties (material homogeneous and isotropic)

2) The bending moment and physical properties are all constant along the

beam’s span. Hence, the deformation of the beam must be identical at all points

along its axis.

3) The cross-section is infinitely rigid in its own plane, i.e. no deformations occur in

the plane of the cross-section

4) The cross-section of a beam remains plane after deformation.

5) The cross-section remains normal to the deformed axis of the beam.

6) The cross-section of the beam is assumed to be symmetric (the bending takes

place in that plane of symmetry).

7) The cross-section is uniform (no holes).

8) The axial strain is uniform

9) The deformations undergone by the beam are considered small, therefore it is

possible to refer to the un-deformed configuration of the beam (Lagrangian

description)

The member is thought to be composed by a number of longitudinal fibers, that are

considered one independent of each other.

Eulero-Bernoulli theory allow to predict the solution only for UNLOADED (from external forces,

internal stresses are acceptable) cross sections; therefore, only far from the application of

constraints and forces.

NORMAL STRESS

Due to equilibrium in x direction the resultants of normal stress

that are uniform over the surface A, is equal to N.

NA

∫ σdA=N → σ=

A

The axial strain is uniform too (over the cross section of the

beam).

δ

=

ε L =elongation =strain

with δ , ε , L=initial length

The relation between stress and strain is the constitutive law and depends on the material

type. For a linear elastic material stress and strain are related by Hooke’s law:

NL

=Eε =

σ → δ AE

An axial deformation (elongation) provokes also a transversal deformation (contraction): if the

material of the beam is also isotropic (mechanical behaviour does not depend on the

orientation), the ratio between transversal and axial(longitudinal) strain (for simple axial load)

is called Poisson’s coefficient:

−ε tranversal

=

ν ε longitudinal

Transversal strain is a remarkable example of the presence of a strain without

stress in that direction

Typical values for Poisson’s coefficient are in the range: 0.25-0.35

Normal stresses provoke a change in volume:

=abc =abc(1+ )(1−ν )(1−ν )

V →V ε ε ε

i f 1 1 1 2 3

∧ε

Neglecting the higher order of infinitesimal ε 1 1

( )

=abc −2

V 1+ ε ν ε

f 1 1 ∆V

( )

=abc −2 =ε −2

∆ V ε ν ε → ν ε

So we obtain: 1 1 1 1

V

v

Therefore, the variation of volume is positive for < 0,5.

ENERGY

A deformed body accumulates energy:

δ f

∫

=Pdδ

dW → L= Pdδ

0

BENDING MOMENT

In bent sections the fibres on one side (upper or lower) are stretched, while the ones on the

other side are compressed; somewhere in the middle there will be a NEUTRAL SURFACE/AXIS,

a portion where fibres do not change in length.

For small θ, I can calculate the deformation for every fibre:

y

ds=( ) =dx−

ρ− y dϑ dx=ρdϑ dx

ρ

→

−y

∆ L=ds−dx= dx

ρ

−

∆ L y

= = =−Γy

ε x dx ρ =Eε =−EΓy

σ

1) x x 1 =¿

Γ=

ρ

dx= initial length of the fibre, ds= final length of the fibre, = curvature radius, ρ

curvature, y= distance from neutral axes

But usually we do not know the curvature, we have to RELATE STRESS TO THE BENDING

MOMENT: −¿ EΓydA=0

Through equilibrium: ∫ ∫ ¿

σ dA=

x

A A

(minus is just because y-axes is directe