FAILURE THEORIES AND MATERIAL STRENGTHOn the base of engineering mechanics, this chapter will develop more specific

understanding in failure theories and material strength for machinery design. The failure usually reflect one of the most important perspectives ([pE5spektiv]) in assessing the safety of mechanical system and components.

3.1 THEORIES OF FATIGUE The maximum-normal-stress theory (the first strength theory)The maximum-normal-strain theory (the second strength theory)The maximum-shear-stress theory (the third strength theory) The distortion-energy theory (the fourth strength theory)

3.2 BULK STRENGTHS OF THE MACHINE COMPONENTWS

3.2.1 Load and Stress static load static stress

Load stress varying load varying (fluctuating) stress

Load: Loads are the external action between two bodies. In terms of the behaviour, load can be presented in a fashion of force, bending moment and torque.

Static load: The magnitudes or directions of the load remain unchanged or change a little and slowly within a given time.

Varying load: The magnitudes or direction of the load is continuously time-varying.Stress: Stresses are the external resistances or forces, which are set up in the

material when a load acts on a component.Static stress: static stresses are the stresses whose magnitudes and directions remain

unchanged or change a little and slowly within a given time.

Varying (fluctuating) stress: A varying stress varies its magnitude and directions all the time. Static or varying load both can cause varying stress.

o t

σ

σ=constant

Fig.3.1 Stress-time relationships for some typical stress

(a) Static stress;(b) and (c) Non-symmetrical (or nonsinusoidal) fluctuating stresses; (d) Sinusoidal fluctuating stress; (e) Repeated alternating (pulsant) stress; (f) Completely reversed (symmetrical) stress Stress

Working stress: Using the formula in mechanics of materials calculate the stress on section plane of element

Calculated stress: Followed the theory of strength, calculate the stress that is equal to simple tension

Ultimate stress: Some utmost of mechanical characters of material Strength utmost: off set limit, limit of fatigue,Allowable stress: Calculate the allowable maximum of stress

Safety factor :

Calculated value of safety factor

The reason of introduce the safety factor:① The inexactness in calculation of stress;

② The discrepancy of mechanical model and actual state；

③ The nonhomogeneity of mechanical characters ；

④  The importance of occasion that the elements apply in.

3.2.2 Stress-Strength Design Method Stress-strength method of factor-of-safety method of design is a method as old as

engineering design itself. That is why it is often called the classical method of design. It mainly includes two parts.

The maximum stress condition:

The safety factor condition:

3.3 STRENGTH DESIGN UNDER STATIC VARYING STRESSESStatic strength: Both the design for ductile materials and brittle materials are not so

difficult under static stresses. We just need to pay attention to the fatigue theories.

3.3.1 The Strength Calculation of Machine Elements Under the Static Stresses

Where ——calculated stress

——ultimate stress (strength utmost, off set limit)

S——safety factor [σ]——allowable stress

3.3.2 The Strength Calculation of Machine Elements Under the Static Stresses

3.3.2.1 Cycle Performance and Fatigue Limit 1.Cycle Performance

The parameters used to describing the stress condition：

σmin=minimum stress σmax=maximum stressσa =alternating stress (or stress amplitude)σm=mean stressσr=stress rangeσs=steady ,or static stress

From fig.3.1, the mean stress and alternating stress can be calculated by

Cycle Performance : -1, completely reversed stress

= 0, repeated alternating stress

1, static stress

2. σ-N fatigue curve (material fatigue curve Ⅰ)The σ-N fatigue curve is shown as follow. It taken the limit of fatigue of material

by the parameter, based on the experiment we can get the curve from.

Repeated alternating stress

r =0

Sinusoidal fluctuating

stress

Completely reversed

stress

r =-1 σmax

σm

T

σmaxσmin

σa

σa

σm σmax

σmin σa

σa

o t

σ

o t

σ

σa

σa

σmin

r =+1

o

σ

t

σ max

A

N=1/4

BC

D

NND104103 N0

σ max

A

N=1/4

BC

D

NND104103 N0

A

N=1/4

BC

D

NND104103 N0N=1/4

BC

D

NND104103 N0

In the origin, the number of stress cycles is N=1/4. It means the material will be

snapped when it is loaded to the maximum, so the value is the strength limit .

σ ma

x

A

N=1/4

BC

D

NND104103 N0

σ ma

x

A

N=1/4

BC

D

NND104103 N0

σ ma

x

A

N=1/4

BC

D

NND104103 N0

A

N=1/4

BC

D

NND104103 N0N=1/4

BC

D

NND104103 N0

σt

AB：N< , has little changed, so it can be approximated to static stress

strength.

BC：As the N augments, declines, at the point C, ,so this kind of

destroy is called low-cycle fatigue or strain fatigue.

CD: As the N augments, declines rapidly, this phase is called high cycle

fatigue or finite lifetime. Most fatigue of machine element happen in this phase.

Describe as follow:

Explanation: machinery design mainly discuss the high cycle fatigue. Sometimes it is necessary to do the strength check of static stress for the element which have prodigious peak value but little action.

Since point D: Curve goes to be horizontal, it means as the N augments no longer declines, it is called infinite life.

The equation is:

Because of the value of is large, when doing the fatigue experiment we always

ordain a cycle index (cycle radix). We use and its fatigue limit

approximatively stand for ,

σmax

Nσr

N0≈107

C

DσrN

N

σB A

N=1/4 104

CB

103

interval CD: the relationship between N and fatigue limit ：

Among the equation above, the value of is definited by material testing.As the testing result shown in interval CD, after relevant numbers of varying

stresses the check bar will happen endurance failure. But since the point D, if the maximum stress less than the stress of the point D, In that way no matter how many times of the circles, the material will never be destroyed.

High circle fatigue:CD-----fatigue for finite life After D----fatigue for infinite life

life factor

Eg: In the strength calculation of gear and worm, use the following equation to calculate the allowable stress.

3. Constant Life Fatigue Curve (Material Fatigue Curve Ⅱ)Material fatigue curve can also be shown by using the relationship among limit

stress amplitudes in the given N, it is called constant life fatigue curve.

σ a

σ m

σ Sσ S

σ -1σ -1

Two reduced method in practical application

σ a

σ m

σ Sσ S

σ -1σ -1

σ a

σ m

σ Sσ S

σ -1σ -1

45˚45˚

3.3.2.2.Stress Limit Curve( Curve)

Allowable fatigue design diagram:1.The stress limit of material

The explanation of stress limit：

(1)Every point on the curve stands for limiting fatigue stress of material under the different cycle performance.

(2)Point A’ correspond the cycle of r=-1,

Point D’ correspond the cycle of r=0,

Line OC correspond the static stress r=1, ,choose the coordinate values of

point C be equal to the off set limit of material.

(3) A straight line is drawn from point C with an angle of 45°to the mean-stress axis and stops at point G. Every point on line CG’ stands for the varying static condition

that .

(4)The area inside the broken line is yielding and fatigue safety zone.

(5)For simplifying the calculation, use the equation but not the curve.

2.The stress limit of element ：

Because of the geometrical shape, size dimension,Surface texture and intensifying factor, the fatigue value of material is above and beyond the Fatigue value of element. fatigue value of material fatigue limit fatigue value of element influence coefficient KKσ——bending fatigue limit influential factorsKτ——shear fatigue limit influential factors

eg：

σ-1e——symmetric cycle bending fatigue limit of elementσ-1 ——symmetric cycle bending fatigue limit of material

Amending the stress limit of material diagram A’G’:The coordinate value changed:

A’(0,σ-1)→A（0， ）or（0， ）

D’( , )→D( , )

The equation of AD：

=

Where is a constant with regarding to the material and can be determined by

fatigue tests of following equation:

----material characteristic

---- material bending characteristic

--- material shearing characteristic

Where is a constant with regarding to the material and can be determined by

fatigue tests of following equation:

The equation of line CG( considered by static stress )

In the condition of shearing stress , the similar form linear equation can get

3.3.2.3 The Strength Calculation of Machine Elements Under the Static Stresses

The characteristic of static stress: holds the value on minimum life section

Or holds the value on minimum life section. Pure static varying stress: one of

the normal or tangential stress Doubleaction static varying stress: both normal and tangential stress.

The design method: theoretical designUse for calculation: safety factor checking

Machine element’s fatigue value of pure static varying stress when calculating the endurance bending strength of machine element, first calculate the maximum (minimum) stress on minimum life section, second calculate the, and then definite the operating point M on the limit stress curve of element . Which point is the value of the limit stress depends on the varying discipline of the element working stress.1.In the case of r=C (a constant),the cycle performance of the varying stress remains the same .It is the stress condition of the majority of the rotating shaft.

N1‘N1‘

M1’

σ a

D

G

σ m

A

M

C

N

M1’M1’

σ a

D

G

σ m

A

M

C

N

σ a

D

G

σ m

A

M

C

N

The limit stress of the point M

The calculation of safety factor and stress condition:

Static stress condition:

2. In the case of , the mean stress of the varying stress remains the same.

It is the stress condition of the vibrating loaded spring.The limit stress of point M:

The calculation of safety factor and stress condition:

Static stress condition:

3.In the case of theminimum stress of the varying stress remains the same.

σ a

JG

σ m

A

C

M

N

σ a

JG

σ m

A

C

M

N

σ minM

M3’

σ minM

M3’

σ minM

M3’M3’

N3‘

σ minN

N3‘N3‘

σ minN

It is the stress condition of the bolt joining under the axial varying load.

The limit stress of point M:

The calculation of safety factor and stress condition:

illustration：when the law of how the stress varies is dubious,we always choose the

calculation method of r=C.

Also , for the shear stress, the only thing we have to do is using the τ instead of σ in the equation above.

Machine element’s fatigue value of pure unstable varying stressEg: The stress condition of automobile leaf-spring is affected by the load, vehicle

speed, pavement, tyre and road sense, etc. Fig3-9 regular unstable varying stress diagram Fig3-10 unstable varying stress on the σ-N curveAdopt: fatigue damage accumulation hypothesis

Regular unstable varying stress diagram

Unstable varying stress on the curve

If every circle of stresses make the equicohesive destruct on the material, the

material damage ratio of stress is in every circle, so in circles the material

damage ratio of is and in circles the material damage ratio of is

.

If the stress is lower than , we can believe that the stress is not damaging.

When the damage ratio is up to 100%,the material happens to endurance failure. So we obtain:

As the experiment shown：

1)When the sequence of stress operation is increased and then decreased, we obtain:

2) When the sequence of stress operation is decreased and then increased, we obtain:

In general case:

In limit case:

If under these stress the material isn’t destroyed,we can obtain:

Let the calculation stress of unstable stress to be:

So , the strength condition is :

The fatigue value of doubleaction static varying stress

The element which is under the alternate stress of bending and torsion is belong to this kind of fatigue value.        

Eg: when the shaft is under the action of bending and torsion, the calculation value of safety factor and strength condition is :                        

      

Among this:

——only under the normal stress , the calculation value of safety factor

——only under the shearing stress , the calculation value of safety factor

3.3.3 the Measure for Increasing the Fatigue Value of Machine ElementIn the design phase the design measure should be taken to increasing the fatigue

value of machine element :

①. Lower the stress concentration as much as possible；

②. Choose the material of high fatigue value; ③.Choose the heat treatment method and reinforced technics to increasing the

fatigue value of machine element;④. Increase the surface texture of machine element; ⑤.Decrease or eliminate the original crack size of the element surface.

For the case of constant ratio r, When the working points are on fatigue safety zone:

Where is a constant with regarding to the material and can be determined by

fatigue tests of following equation:

When the working points are on plastic safety zone:

Illumination :Two other cases (σm =constant and σmin =constant) are not given in detail here for

they are not very common.

MAIN CONTENTS OF THE CHAPTERKnowing the significance and use of the fatigue curve and limit stress diagram, can

draw the simplified limit stress diagram by knowing some basic mechanical

characters ( )and the geometrical property.

Grasp the calculation method of strength under the pure varying stress, and know the concept of equivalent stress.

Know the significance and application mode of fatigue damage accumulation hypothesis (Miner law).

Grasp the checking method of doubleaction varying stress.Can use the line graph and numerical tables in the appendix.