chapter 2 design for strength

8/12/2019 Chapter 2 Design for Strength

http://slidepdf.com/reader/full/chapter-2-design-for-strength 1/66

MECHANICAL ENGINEERING DESIGN 1

MEC 531

Part B:

Design for Strength

By:

NURZAKI IKHSAN



Chapter Outline

1. Static Strength

2. Failure theories

3. Stress Concentration

4. Fatigue Strength.5. Introduction to Fracture Mechanics

Chapter 2:Design for Strength



In engineering practices, there are many cases in whichmachine members are subjected to combined stresses due

to simultaneous action of either tensile or compressive

stresses combined with shear stresses. E.g. propeller shaft,

crankshaft.

Understanding the basic principal stresses are important to

determine the yield strength.

Static strength



Static strength

Free Body Diagram (FBD)

Simplifying a body by isolating each element with its physical

attributes and showing all forces that are acting on it to be in

an equilibrium state.



Static strength



Static strength

Four types of internal loading

Normal force, N.

This force act perpendicular to the area

Shear Force, V.

This force lies in the plane of the area (parallel)

Torsional Moment or Torque, T.

This torque is developed when the external loads tend to twist one

segment of the body with respect to the other

Bending Moment, M.

This moment is developed when the external loads tend to bend

the body. Normal force, N. This force act perpendicular to the area.



Static strength

Normal Stress

The intensity of the force acting normal to ΔA

It called as tensile stress if the normal force ‘pulls’ and called as

compressive stress when it “pushes”.

A

F z A

z

0lim



Static strength

Shear Stress:

• Stress that acts parallel to the surface of a material creating shear.• The intensity of the force acting tangent to ΔA

2

r

V

A

V

ShearStress

Single Double

2

)2/(

r

F

A

V



Static strength

Normal Strain

Deformation of a body by changes in length of line segments and the

changes in the angles between.

s

s s

avg

'

Shear Strain

The change in angle between two line segments that were originally

perpendicular

'2

nt



Static strength

Torque:

Moment that tends to a twist a member about its longitudinal axis.

Cross-sections for hollow and solid circular

shafts remain plain and undistorted because a

circular shaft is axisymmetric.

Cross-sections of noncircular (non-

axisymmetric) shafts are distorted when

subjected to torsion.



Static strength

and max J

T

J

Tc

4

2c J

44

2 io cc J

Solid Shaft

The polar moment of inertia J can bedetermined using an area element in the

form a differential ring, thus:

Tubular shaft

The polar moment of inertia J can be

determined by substrating J for a shaft

radius ci from that determined for a shaft ofradius c0, thus;



Static strength

The motor delivers a torque of 50 N.m to the shaft AB. This torque is

transmitted to shaft CD using the gears at E and F. Determine the equilibrium

torque T’ on shaft CD and the maximum shear stress in each shaft. The

bearings B, C, and D allow free rotation of the shafts.

Exercise Review 2



Static strength

Linear ElasticMaterial

Behavior

E

G

y

xv

E= Modulus of Elasticity / Young’s modulus

G= Shear Modulus of Elasticity / Modulus of Rigidity

Poisson’s Ratio



Static strength

The maximum normal stress due to bending,

S

M

I

Mcm

Consider a rectangular beam cross section,

Ahbhh

bh

c

I S

6

13

6

1

3

12

1

2

Bending Moment Rotational forces within the beam that cause bending. At any point

within a beam, the Bending Moment is the sum of: each external force

multiplied by the distance that is perpendicular to the direction of the

force.



Static strength

Maximum shearing stress occurs for ave x

xy

y x

s

xy

y x R

22tan

2

2

2

max

Principal stresses occur on the principal planes of stress with zero shearing

stresses.

2

2

minmax,22

xy

y x y x

o90 byseparated

anglestwodefines : Note

oo 45 byfromoffsetand90 by

separatedanglestwodefines: Note

p

y x

xy p

22tan



Static strength

The engine crane is used to support the engine,which has a weight of 6kN

.Draw the shear and moment diagrams of the boom ABCwhen it is in the

horizontal position shown.

Exercise Review 4



Stress Concentration

• All the shape or holes on parts and components have potentialto contribute to failure or cracks.

• Avoiding cross-section, holes, notches, shoulders, etc. is quite

impossible in machine members.

• Examples of machine members leading to stress concentration:




• Any discontinuity increases the stress in the surrounding area

of the discontinuity which acts as the ‘Stress Riser’. • The regions in which they occur are called area of stress

concentration.

• Ratio of maximum and nominal stress is known as stress

concentration factor or kt (normal stress) and kts (shear

stress).

• The factors relates the maximum stress at the discontinuityover the nominal stress (free from the stress riser).

• The possibility of crack initiated is higher especially when

stress concentration factor is greater than critical stress

concentration.




• For an elliptical hole in an infinite plate,

subjected to a uniform tensile stress 1, stress

distribution around the discontinuity is

disturbed and at points remote from the

discontinuity the effect is insignificant.

• It is shown as:

• If a=b the hole reduces to a circular one and therefore σ3 =

3σ1 which gives kt =3 (circular hole).

• If, however ‘b’ is large compared to ‘a’ then the stress at the

edge of transverse crack is very large and consequently k is

also very large.

• If ‘b’ is small compared to ‘a’ then the stress at the edge of a

longitudinal crack does not rise and kt =1.




A number of methods are available to reduce stress concentration in

machine parts:

1. Provide a fillet radius so that the cross-section may change gradually.

2. Sometimes an elliptical fillet is also used.

3. If a notch is unavoidable it is better to provide a number of small

notches rather than a long one. This reduces the stress concentration to

a large extent.

4. If a projection is unavoidable from design considerations it is preferable

to provide a narrow notch than a wide notch.

5. Stress relieving groove are sometimes provided.



Failure theories

Failure can mean a part has separated into two or more pieces; has

become permanently distorted, thus ruining its geometry; has had its

reliability downgraded; or has had its function compromised, whatever

the reason.



Failure theories

• Why we need Failure Theories?

• To design structural components and calculate margin of safety.

• To guide in materials development.

• To determine weak and strong directions.



Ductile failure

Imagine that the matrix of circlesshown below represent anisotropic material.

A ductile material deforms significantly

before fracturing. (extensive plastic

deformation and energy absorption

(toughness) before fracture

Ductility is measured by % elongation at

the fracture point.

Materials with 5% or more elongation are

considered ductile.



Ductile failure

The material continues to stretch linearlyuntil the yield stress of the material is

reached.

At this point the material begins tobehave differently. Planes of

maximum shear exist in the material

at 45°, and the material begins to

slide along these planes.

As the material is being loaded itstretches linearly. As the material is

being pulled further apart, its

resistance becomes greater.



Ductile failure

The sliding between relative

planes of material allow the

specimen to deform noticeably

without any increase in stress. We

call this a yield of the material.



Brittle failure

The brittle material also behaves in a

linear fashion as it is being loaded.

Brittle material yields very little before

fracturing (Little plastic deformation and

low energy absorption before failure)

The yield strength is approximately the

same as the ultimate strength in tension.

The ultimate strength in compression is

much larger than the ultimate strength in

tension.



Brittle failure

When the normal stress in the specimen

reaches the ultimate stress, σult , the

material fails suddenly by fracture. This

tensile failure occurs without warning,

and is initiated by stress concentrationsdue to irregularities in the material at the

microscopic level.

The material continues to stretch as

more and more load is applied.



Failure theories

• There is no universal theory of failure for the general case of material

properties and stress state. Instead, over the years several hypotheses havebeen formulated and tested, leading to today’s accepted practices mostdesigners do.

• The generally accepted theories are:

• Ductile materials (yield criteria)

• Maximum shear stress (MSS) a.k.a Tresca Theory

• Distortion energy (DE) a.k.a Von Misses

• Ductile Coulomb-Mohr (DCM)

• Brittle materials (fracture criteria)

• Maximum normal stress (MNS)

• Brittle Coulomb-Mohr (BCM)

• Modified Mohr (MM)



Failure theories

Assuming a plane stress problem with σ A ≥ σB , there are three cases to consider

Case 1: σ A ≥ σB ≥ 0. For this case, σ1 = σ A and σ3 = 0.

Case 2: σ A

≥ 0 ≥ σB . Here, σ1 = σ

A and σ3 = σ

B

Case 3: 0 ≥ σ A ≥ σ

B . For this case, σ1 = 0 and σ3 = σ

B,

Ductile Materials -Maximum-Shear-Stress Theory

The maximum-shear-stress theory predicts that

yielding begins whenever the maximum shear stress

in any element equals or exceeds the maximum shear

stress in a tension test specimen of the same material

when that specimen begins to yield.

The MSS theory is also referred to as the Tresca or

Guest theory.



Failure theories

Ductile Materials-Distortion-Energy Theory

• The distortion-energy theory predicts that yielding occurs when the

distortion strain energy per unit volume reaches or exceeds the distortion

strain energy per unit volume for yield in simple tension or compression of

the same material.

•For unit volume subjected to any three-dimensional stress state designated

by the stresses σ1, σ2, and σ3, effective stress is usually called the von Mises

stress , σ′ as

• Using xyz components of three-dimensional stress, the von Mises stress canbe written as

and for plane stress,



Failure theories

Consider a case of pure shear τ xy ,where for plane stress

σ x = σy = 0. For yield

Thus, the shear yield strength predicted by th distortionenergy theory is

Ductile Materials-Distortion-Energy Theory



Failure theories

Coulomb-Mohr Theory for Ductile Materials

Not all materials have compressive strengths equal to their

corresponding tensile values.

The idea of Mohr is based on three “simple” tests: tension,

compression, and shear, to yielding if the material can yield, or to

rupture.

The practical difficulties lies in the form of the failure envelope.A variation of Mohr’s theory, called the Coulomb-Mohr theory or

the internal-friction theory , assumes that the boundary is straight.

For plane stress, when the two nonzero principal stresses are σ A ≥

σB , we have a situation similar to the three cases given for the MSS

theory

Case 1: σ A ≥ σB ≥ 0. For

this case, σ1 = σ A and σ3

= 0. Equation (5 –22)

reduces to a failure

condition of

Case 2: σ A ≥ 0 ≥ σB . Here,σ1 = σ A and σ3 = σB , andEq. (5 –22) becomes

Case 3: 0 ≥ σ A ≥ σB . Forthis case, σ1 = 0 and σ3 =σB , and Eq. (5 –22) gives



Failure theories

Failure of Ductile Materials Summary

• Either the maximum-shear-stress theory or

the distortion-energy theory is acceptable

for design and analysis of materials that

would fail in a ductile manner.

•For design purposes the maximum-shear-

stress theory is easy, quick to use, and

conservative.

• If the problem is to learn why a part failed,

then the distortion-energy theory may be

the best to use.

• For ductile materials with unequal yield

strengths, Syt in tension and Syc in

compression, the Mohr theory is the best

available.



Failure theories

Brittle Materials-Maximum-Normal-Stress Theory

• The maximum-normal-stress (MNS) theory

states that failure occurs whenever one of

the three principal stresses equals or

exceeds the strength.

• For a general stress state in the ordered form

σ1 ≥ σ2 ≥ σ3. This theory then predicts that

failure occurs whenever

where Sut and Suc are the ultimate tensile

and compressive strengths, respectively,

given as positive quantities.



Failure theories

Failure of Brittle Materials Summary

• Brittle materials have true strain at

fracture is 0.05 or less.

• In the first quadrant the data appear on

both sides and along the failure curvesof maximum-normal-stress, Coulomb-

Mohr, and modified Mohr. All failure

curves are the same, and data fit well.

• In the fourth quadrant the modified

Mohr theory represents the data best.

• In the third quadrant the points A, B, C ,

and D are too few to make any

suggestion concerning a fracture locus.



Fatigue Strength

• Machine members are found to fail under the action of fluctuating

stresses. The actual maximum stresses were well below the ultimatestrength of the material, and even below the yield strength.

• Properties of materials and the material behavior can be observed using

the stress-strain diagrams or S-N curve

• The most distinguishing characteristic of the fatigue failure is that thestresses have been repeated a very large number of times.

• Fatigue failure gives no warning. It is sudden and total, and hence

dangerous.

• Example of fatigue failure: shaft of electric motor, which rotate at 1725

rev/min., that means it have stresses in tension and compression 1725

each min.

• A fatigue failure has an appearance similar to a brittle fracture as the

fracture surface are flat and perpendicular to the stress axis with the

absence of necking



• Static conditions : loads are applied gradually, to give sufficient time for

the strain to fully develop.

• Variable conditions : stresses vary with time or fluctuate between

different levels, also called repeated, alternating, or fluctuating stresses.

• When machine members are found to have failed under fluctuatingstresses, the actual maximum stresses were well below the ultimate

strength of the material, even below yielding strength.

• Since these failures are due to stresses repeating for a large number of

times, they are called fatigue failures.

• When machine parts fails statically, the usually develop a very large

deflection, thus visible warning can be observed in advance; a fatigue

failure gives no warning!

Fatigue Strength

h



Fatigue Strength

Fatigue failures may contribute in 2 areas of failure:

1. Due to progressive development of crack.

2. Due to sudden fracture.

Cracks are initiated at the discontinuity, for example:

– Change in cross-section.

– A key way. – A hole.

Fatigue failure is quite different from a static brittle fracture as it

arise from three stages of development.

1. Crack initiation.2. Crack propagation.

3. Final catastrophic failure.



F i S h



Fatigue Strength

Stage 3: Final catastrophic failure

• Remaining area (ligament) cannot sustain

loading anymore.• Unstable (significant) crack propagation and

rapid failure.

• Fracture when the remaining material cannot

support the loads.

initiation

propagation

fracture

F i S h



Fatigue Strength

F ti St th



Fatigue Strength

F ti Lif M th d i F ti F il A l i



Fatigue Life Methods in Fatigue Failure Analysis

Let be the number of cycles to fatigue for a specified level of loading

-For , generally classified as low-cycle fatigue

-For , generally classified as high-cycle fatigue

Three major fatigue life methods used in design and analysis are

1.stress-life method : is based on stress only, least accurate especially for

low-cycle fatigue; however, it is the most traditional and easiest to

implement for a wide range of applications.

2.strain-life method : involves more detailed analysis, especially good for

low-cycle fatigue; however, idealizations in the methods make it less

practical when uncertainties are present.

3.linear-elastic fracture mechanics method : assumes a crack is already

present. Practical with computer codes in predicting in crack growth with

respect to stress intensity factor





Stress-Life Method : R. R. Moore

• The most widely used fatigue-testing device is the R. R. Moore high-

speed rotating-beam machine.

• Specimens in R.R. Moore machines are subjected to pure bending

by means of added weights.

• Other fatigue-testing machines are available for applying fluctuating

or reversed axial stresses, torsional stresses, or combined stresses

to the test specimens.





S-N Curve

• In R. R. Moore machine tests, a constant

bending load is applied, and the number

of revolutions of the beam required for

failure is recorded.

• Tests at various bending stress levels are

conducted.

• These results are plotted as an S-N

diagram.

• Log plot is generally used to emphasize

the bend in the S-N curve.

• Ordinate of S-N curve is fatigue strength,

, at a specific number of cycles


S-N diagram from the results of completely reversed axial

fatigue test. Material : UNS G41300 steel.




• In the case of steels, a knee occurs in the

graph, and beyond this knee failure will

not occur, no matter how great the

number of cycles - this knee is called the

endurance limit , denoted as

• Non-ferrous metals and alloys do not have

an endurance limit, since their S-N curve

never become horizontal.

• For materials with no endurance limit, the

fatigue strength is normally reported at

• is the simple tension test


Characteristics of S-N Curves for Metals





• The best approach yet advanced to explain the nature

of fatigue failure. However, it needs to compound

several idealizations, and so uncertainties will exist in

the results.

• A fatigue failure begins at a local stress raisers. Whenthe stress at these discontinuity exceeds the elastic

limit, plastic strain occurs.

• Bairstow using experiments to verify that elastic limits

of iron and steel can be changed by the cyclic

variation of stress.

• A stress-strain plot of controlled cyclic loads could

show the strength variation due to stress repetitions.

The Strain-Life Method





• The total-strain amplitude is the sum of elastic and

plastic strain

• is the fatigue strength coefficient, the true stress corresponding to fracture in one

reversal.

• is the fatigue strength exponent as the slope of the elastic-strain line.

• is the fatigue ductility coefficient, the true strain corresponding to fracture in one

reversal.

• is the fatigue strength exponent as the slope of the elastic-strain line.

The Strain-Life Method

Manson-Coffin Relationship






Fatigue cracking consists three stages

✓Stage I : crack initiation,

invisible to the observer.

✓Stage II : crack propagation,most of a crack’s life

✓Stage III : final fracture due torapid acceleration of crackgrowth.

Linear-Elastic Fracture Mechanics Method





Linear-Elastic Fracture Mechanics Method

Paris Law for Crack Growth

Assuming a crack is discovered early in stage II, the crack growth can be

approximated by the Paris equation as

is the variation in stress intensity factor due to fluctuating stresses.

crack length

number of cycles material constants

Endurance Limit for Steels



Endurance Limit for Steels

For steels, the endurance limit relatesdirectly to the minimum tensile strength

as observed in experimental

measurements.

From the observations, the endurance of

steels can be estimated as

with the prime mark on the endurancelimit referring to the rotating-beam

specimen.

Fatigue Strength : Basics



Fatigue Strength : Basics

• Low-cycle fatigue considers the range from N=1 to about 1000

cycles.

• In this region, the fatigue strength is only slightly smaller than

the tensile strength .

• High-cycle fatigue domain extends from 103 to the endurance limit

life (106 to 107 cycles).

• Experience has shown that high-cycle fatigue data are rectified by

a logarithmic transform to both stress and cycles-to-failure.

Fatigue Strength : General



For actual mechanical applications, the fatigue strength calculatedabove is extended to a more general form as

: cycle to failure

Fatigue Strength : General

Fatigue Strength at Different N



Fatigue Strength at Different N

• Define the fatigue strength at a specified number of cycles as

• By combining the elastic strain relations, we can get

• Define f as the fraction of tensile strength. The value of f at 103 cycles is then

• To find b, substitute the endurance strength and the corresponding cycles and

solving for b as

• For example, for steels when

Endurance Limit Modifying Factors



Endurance Limit Modifying Factors

The endurance limit of the rotating-beam specimen might differ from the

actual application due to the following differences from laboratory tests.

Material : composition, basis of failure, variability

Manufacturing : method, heat treatment, fretting corrosion, surface

condition, stress concentration

Environment : corrosion, temperature, stress state, relaxation times.

Design : size, shape, life, stress state, stress concentration, speed,

fretting, galling

Modifying factors of surface condition, size, loading, temperature, and

miscellaneous items are proposed by Marin to quantify these differences.

Marin Modification Factors on Endurance Limit




where

= surface condition modification factor

= size modification factor

= load modification factor

= temperature modification factor

= reliability factor= miscellaneous-effects modification factor

= rotary-beam test specimen endurance limit





Surface Factor :

It depends on the finishing quality of the actual part surface and on the tensile

strength of the part material. It can be calculated as

Loading Factor :

The axial and torsional loadings results in different endurance limit than that of

a standard rotating-bending test. The load factor applies to other loadingconditions as





Size Factor :

the size factor has been evaluated using 133 set of data points in the literature.For axial loading, . For bending and torsion can be expressed as

Temperature Factor :

If only tensile-strength data are available, polynomial fitting to the

data could provide the temperature factor at various temperature

values.

If the rotating-beam endurance limit is known at room temperature,

we have





Reliability Factor :

Most endurance strength data are reported as mean values.

To account for the scatter of measurement data, the reliability modification

factor is written as

iscellaneous effect Factor :

The miscellaneous factor intends to account for the reduction in endurance

limit due to all other effects, such as residual stresses, different material

treatments, directional characteristics of operations, and corrosion.

Fracture Mechanics : Introduction



• The linear elastic fracture mechanics (LEFM) assume that cracks can growduring service.

• The use of elastic stress-concentration factors provides an indication of

the average load required on a part for the onset of plastic deformation,

or yielding.

• For the infinite plate loaded by an applied uniaxial stress σ, the maximumstress occurs at (±a, 0) and is given by

such that the crack growth occurs when the energy release rate fromapplied loading is greater than the rate of energy for crack growth.

Fracture Mechanics : Introduction

Fracture Mechanics : Stress Intensity Factor



• Three distinct modes of crack propagation exist

– Mode I : the opening crack propagation mode( the most common and important mode)

– Mode II : the sliding mode

– Mode III : the tearing mode

• Consider a mode I crack of length 2a in the infinite plate, the stress field on a dx dy

element in the vicinity of the crack tip is given by

where K I is the stress intensity factor with a mode I crack defined as

• For various load and geometric configurations,

where β is the stress intensity modification factor

Fracture Mechanics : Stress Intensity Factor

Fracture Mechanics : Fracture Toughness



• When the magnitude of the mode I stress

intensity factor reaches a critical value, the

critical stress intensity factor K I c crack

propagation initiates.

• The critical stress intensity factor K I c is also

called the fracture toughness of the

material.

• Fracture toughness K I c for engineering metals lies in the range 20 ≤ K I c ≤

200 MPa · √m; for engineering polymers and ceramics, 1 ≤ K I c ≤ 5 MPa · √m.

For a 4340 steel, where the yield strength due to heat treatment ranges

from 800 to 1600 MPa, K I c decreases from 190 to 40 MPa · √m.

• The strength-to-stress ratio K I c /K I can be used as a factor of safety as

Fracture Mechanics : Fracture Toughness

chapter 2 design for strength

Documents