Mechanics in Energy Resources Engineering - Chapter 2. Axially...

Week 4 22 March

Mechanics in Energy Resources Engineering- Chapter 2 Axially Loaded Members (2)

Ki B k Mi PhDKi-Bok Min PhD

Assistant ProfessorE R E i iEnergy Resources EngineeringSeoul National University

Preview

bull Introduction

bull Changes in Lengths of Axially Loaded Members (축하중을받는g g y (부재의길이변화)

bull Changes in Lengths Under Nonuniform Conditions (균일봉길이변화)g g ( )bull Statically Indeterminate Structures (부정정구조물)

Th l Eff t Mi fit d P t i (열효과 어긋남및사전변형)bull Thermal Effects Misfits and Prestrains (열효과 어긋남및사전변형)bull Stresses on Inclined Sections (경사면에서의응력)bull Strain Energy (변형율에너지)bull Stress Concentrations (응력집중)Stress Concentrations (응력집중)bull Impact Loading (충격하중)

Thermal Effects misfits and prestrains

bull Other sources of stresses and strains other than lsquoexternal loadsrsquo

ndash Thermal effects arises from temperature changendash Misfits results from imperfections in constructionndash Misfits results from imperfections in constructionndash Prestrains produced by initial deformation

Thermal Effects

bull Changes in temperature produce expansion or contraction thermal strains

bull Thermal strain εT T

ndash α coefficient of thermal expansion (1K or 1degC) eg granitite ~1 0 times10-5 degC

T T

10 times10 Cndash Heated Expansion (+) Cooled contraction (-)

bull Displacement by thermal expansion T T L T L T T

Thermal Effects

bull No restraints free expansion or contractionndash Thermal strain is NOT followed by thermal stressThermal strain is NOT followed by thermal stressndash Generally statically determinate structures do not produce thermal

stressstress

bull With supports that prevent free expansion and contraction Thermal stress generatedThermal stress generated

ndash How much thermal stress

Thermal Effects

bull Two bars in the left were under uniform temperature increase of ΔT

Fixed support

RA1

Fixed support

RA2

ndash E Elastic Modulus α Coefficient of Thermal Expansion

A

RA1

ARA2

ndash If E1=E2 and α1gt α2 which bar will generate the higher thermal stress1 2

ndash If α1= α2 and E1gtE2 which bar will generate the higher thermal stressg g

ndash Will RA and RB the sameB

R

B

RFixed support

RB1

Fixed support

RB2

Thermal EffectsCalc lation of Thermal stress (E ample 2 7)Calculation of Thermal stress (Example 2-7)

ndash Equilibrium Eq0F 0ver B AF R R

ndash Compatibility Eq

0AB T R

ndash Displacement Relations

T T L AR

R LEA

ndash Compat Eq Displ RelEA

0AT R

R LT LEA

ndash ReactionsEA

A BR R EA T

ndash Thermal Stress in the bar A BT

R R E TA A

Thermal EffectsC l l ti f Th l tCalculation of Thermal stress

bull Thermal Stress in the bar

A BR R E T

ndash Stress independent of the length (L) amp cross-sectional area (A) T E T

A A

ndash Assumptions ΔT uniform homogeneous linearly elastic material

ndash Lateral strain

Stresses on inclined sections

bull Stresses on inclined sections a more complete picture

ndash Finding the stresses on section pq

ndash Resultant of stresses still PN Normal Force

ndash Normal Force (N) and Shear Force (V)

V Shear ForcecosN P sinV P

ndash Normal Stress (σ) and shear stress (τ)

N V

1A

1A

AA area of cross-sectionA1 area of inclined section 1 cos

AA


bull Based on the sign convention (note minus shear stress)2cosN P sin cosV P

1

cosA A

1

sin cosA A

2 1cos 1 cos 2 1sin cos sin 2 2cos 1 cos 2

2 sin cos sin 2

2

2 1 1 2 i i 2x

ndash Above equation are independent of material (property and elastichellip)

2cos 1 cos 22x x sin cos sin 2

2x

x

ndash Maximum stresseshellipwhy is this importantWhen θ = -45deg

max x max 2x

When θ = 0


bull Element A ndash maximum normal stress maximum normal stress ndash no shear Maximum normal stress Maximum shear stressmax x

bull Element Bndash The stresses at θ = 135deg -45deg

and -135deg can be obtained from previous equations

ndash Maximum shear stressesndash One-half the maximum normal

max 2x

One half the maximum normal stress


bull Same equations can be used for uniaxial compressionbull What will happen if material is much weaker in shear than in bull What will happen if material is much weaker in shear than in

compression (or tension)Sh t f ilndash Shear stress may cause failure

Strain Energy (변형율에너지)

bull Static loadndash Load applied slowly without dynamic Load applied slowly without dynamic

effects due to motion

bull P moves through distance δ and bull P moves through distance δ and does a certain amount of work

Strain Energy

bull The work (W) done by the incremental loading 1 10

W Pd

ndash The work done by the load is equal to the

area below the load-displacement curve

0

bull Strain Energy (U)E b b d b th b d i th ndash Energy absorbed by the bar during the loading process internal work

St i k d b th l d ndash Strain energy = work done by the load (when no E subtracted in the form of heat)

1 10U W Pd

Unit Nm = J

Strain Energy

bull Elastic and Inelastic Strain Energy

Strain EnergyLi l El ti B h iLinearly Elastic Behavior

bull Strain Energy for linear elastic barPU W

2

U WPLEA

2 2P L EA

Positive for both (+) amp ( ) P

2 2P L EAUEA L

ndash Positive for both (+) amp (-) Pndash With unchanged load (P) Luarr Uuarrndash However Euarr or Auarr - - gt Udarr

2 2

EAk 2 2

2 2P kUk

L

Strain EnergyN if BNonuniform Bars

bull Total Strain Energy U of a bar consisting of several segments

2n n N L

St i i t li f ti f th l d

2

1 1 2

n ni i

ii i i i

N LU UE A

ndash Strain energy is not a linear function of the loads even when the material is linearly elastic

2 2( )P P L P LL1 E1 A1

1 2 1 2 2

1 1 2 2

( )2 2

P P L P LUE A E A

L2 E2 A2

2 2

0

( )2 ( )

L N x dxU

EA x

Strain EnergyE l 2 12Example 2-12

bull Compare the amounts of strain energy stored in the bars assuming linearly elastic behavior

Strain EnergyE ample 2 14Example 2-14

bull Determine the vertical displacement δB of joint B


bull Determine the strain energy of a prismatic bar suspended from its upper end

bull Two cases( ) i ht f th b it lfndash (a) weight of the bar itself

ndash (b) weight + a load P at the end

Strain EnergyDi l t d b Si l L dDisplacement caused by a Single Load

bull Strain energy stored in the structure

2PU W

2UP

ndash The displacement of a structure can be determined directly from the strain energy

ndash ConditionsStructure behave in a Linearly elastic manner

Only one load may act

Strain EnergySt i E D itStrain-Energy Density

bull Strain-Energy Density (u)ndash Strain energy per unit volume of materialStrain energy per unit volume of material

2 2

2 22 2U P EuAL EA L

ndash Strain-energy density in terms of stress and strain2 2

2 2Eu

E

2 2

2 2Eu

E

ndash Geometrical interpretationσ

ε

Stress Concentration

bull Stress concentrationndash Occurs when uniform stress are disturbed Occurs when uniform stress are disturbed

by abrupt change in geometry (eg hole)

bull Saint Venantrsquos Principlebull Saint Venant s Principlendash Peak stress directly under concentrated

load P gtgt Pbtload P gtgt Pbtndash Maximum stress diminish rapidly as we

move away from the point of load move away from the point of load applicationAt di t f th d f th b ndash At a distance from the end of the bar equal to the width b of the bar stress distribution is nearly uniform

Stress concentrationStress concentration ca sed b the holeStress concentration caused by the hole

bull Stress concentration factorndash K =3 for this case

max

nom

K

K 3 for this case

Stress concentration

With b gtgt d Typical underground operation

Mechanics in Energy Resources EngDifference from Ci ilMechanical EngDifference from CivilMechanical Eng

bull Mechanics of addition (덧셈의역학) ndash CivilArchitectural Engineering Machineries

ndash Ex) building with bricks (load on the column)

bull Mechanics of removal (뺄셈의역학) Underground bull Mechanics of removal (뺄셈의역학) ndash Underground structure undergroundsurface Mines

Deflected steel beamndash Ex) Drilling a borehole Excavation of rock

Hendersen Mine Colorado USA (Molybdenum)

Civil structural problem Mechanics of ldquoAdditionrdquo

Mechanics in Energy Resources Eng Mechanics of ldquoRemovalrdquo

Side view

1 2

Mechanics of Addition gMonitoring points

1

2

Before drillingexcavation

hellip

2

hellip

4

3

Start of drillingexcavation

stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain

Further advance of drillingexcavation

Impact Loading

bull Static loads applied slowly and remain constant with time

bull Dynamic loads applied and removed suddenly or vary with timevary with time

bull Impact of an object falling onto the lower end of a i ti b M i l ti prismatic bar Maximum elongationndash Response is very complicatedndash Approximate analysis by using the concept of strain

energy

Impact Loading

bull Maximum elongationndash Potential energy lost by the falling mass (M) = Potential energy lost by the falling mass (M)

maximum strain energy acquired by the bar2max( ) ( ) EAM h W h max

max max( ) ( )2

Mg h W hL

122

max 2WL WL WLhEA EA EA

ndash Elongation due to the weight of the collar under static

EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading

bull Maximum stressmaxE max L

122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122

max2 21 1st st st st

st

hE hEL L

bull Impact factor

max

st

Impact Loadingdd l li d l disuddenly applied loading

bull A load is applied suddenly with no initial velocity ndash What is the difference from the static loadingWhat is the difference from the static loadingndash h 0 from

12

max21 1st

st

h

st

max 2 st

ndash Impact factor is 2

Impact LoadingE l 2 16Example 2-16

bull Calculate the maximum elongation of the bar due to the impact Impact factor

Summary

bull Introduction



Th l Eff t Mi fit d P t i (열효과 어긋남및사전변형)bull Thermal Effects Misfits and Prestrains (열효과 어긋남및사전변형)bull Stresses on Inclined Sections (경사면에서의응력)bull Strain Energy (변형율에너지)bull Impact Loading (충격하중)Impact Loading (충격하중)bull Stress Concentrations (응력집중)

Chapter 3 Torsion

bull Introductionbull Torsional Deformations of a circular barbull Circular bars of linearly elastic materials

Nonuniform torsionbull Nonuniform torsionbull Stresses and Strains in Pure Shearbull Relationship Between Moduli of Elasticity E and Gbull Transmission of Power by Circular ShaftsTransmission of Power by Circular Shaftsbull Statically Indeterminate Torsional Membersbull Strain Energy in Torsion and Pure Shear

Preview

bull Introduction



Th l Eff t Mi fit d P t i (열효과 어긋남및사전변형)bull Thermal Effects Misfits and Prestrains (열효과 어긋남및사전변형)bull Stresses on Inclined Sections (경사면에서의응력)bull Strain Energy (변형율에너지)bull Stress Concentrations (응력집중)Stress Concentrations (응력집중)bull Impact Loading (충격하중)




Thermal Effects




T T



Thermal Effects


stressstress



Thermal Effects


Fixed support

RA1

Fixed support

RA2


A

RA1

ARA2




R

B

RFixed support

RB1

Fixed support

RB2




0AB T R


T T L AR

R LEA


0AT R

R LT LEA

ndash ReactionsEA

A BR R EA T


R R E TA A



A BR R E T


A A










N V

1A

1A


AA



1

cosA A

1

sin cosA A


2 sin cos sin 2

2

2 1 1 2 i i 2x



2x

x


max x max 2x

When θ = 0






max 2x









Strain Energy


W Pd



0



1 10U W Pd

Unit Nm = J

Strain Energy




2

U WPLEA

2 2P L EA


2 2P L EAUEA L


2 2

EAk 2 2

2 2P kUk

L



2n n N L


2

1 1 2

n ni i

ii i i i

N LU UE A


2 2( )P P L P LL1 E1 A1

1 2 1 2 2

1 1 2 2

( )2 2

P P L P LUE A E A

L2 E2 A2

2 2

0

( )2 ( )

L N x dxU

EA x











2PU W

2UP






2 2

2 22 2U P EuAL EA L


2 2Eu

E

2 2

2 2Eu

E


ε









max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion






Thermal Effects




T T



Thermal Effects


stressstress



Thermal Effects


Fixed support

RA1

Fixed support

RA2


A

RA1

ARA2




R

B

RFixed support

RB1

Fixed support

RB2




0AB T R


T T L AR

R LEA


0AT R

R LT LEA

ndash ReactionsEA

A BR R EA T


R R E TA A



A BR R E T


A A










N V

1A

1A


AA



1

cosA A

1

sin cosA A


2 sin cos sin 2

2

2 1 1 2 i i 2x



2x

x


max x max 2x

When θ = 0






max 2x









Strain Energy


W Pd



0



1 10U W Pd

Unit Nm = J

Strain Energy




2

U WPLEA

2 2P L EA


2 2P L EAUEA L


2 2

EAk 2 2

2 2P kUk

L



2n n N L


2

1 1 2

n ni i

ii i i i

N LU UE A


2 2( )P P L P LL1 E1 A1

1 2 1 2 2

1 1 2 2

( )2 2

P P L P LUE A E A

L2 E2 A2

2 2

0

( )2 ( )

L N x dxU

EA x











2PU W

2UP






2 2

2 22 2U P EuAL EA L


2 2Eu

E

2 2

2 2Eu

E


ε









max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion



Thermal Effects


stressstress



Thermal Effects


Fixed support

RA1

Fixed support

RA2


A

RA1

ARA2




R

B

RFixed support

RB1

Fixed support

RB2




0AB T R


T T L AR

R LEA


0AT R

R LT LEA

ndash ReactionsEA

A BR R EA T


R R E TA A



A BR R E T


A A










N V

1A

1A


AA



1

cosA A

1

sin cosA A


2 sin cos sin 2

2

2 1 1 2 i i 2x



2x

x


max x max 2x

When θ = 0






max 2x









Strain Energy


W Pd



0



1 10U W Pd

Unit Nm = J

Strain Energy




2

U WPLEA

2 2P L EA


2 2P L EAUEA L


2 2

EAk 2 2

2 2P kUk

L



2n n N L


2

1 1 2

n ni i

ii i i i

N LU UE A


2 2( )P P L P LL1 E1 A1

1 2 1 2 2

1 1 2 2

( )2 2

P P L P LUE A E A

L2 E2 A2

2 2

0

( )2 ( )

L N x dxU

EA x











2PU W

2UP






2 2

2 22 2U P EuAL EA L


2 2Eu

E

2 2

2 2Eu

E


ε









max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion



Thermal Effects


Fixed support

RA1

Fixed support

RA2


A

RA1

ARA2




R

B

RFixed support

RB1

Fixed support

RB2




0AB T R


T T L AR

R LEA


0AT R

R LT LEA

ndash ReactionsEA

A BR R EA T


R R E TA A



A BR R E T


A A










N V

1A

1A


AA



1

cosA A

1

sin cosA A


2 sin cos sin 2

2

2 1 1 2 i i 2x



2x

x


max x max 2x

When θ = 0






max 2x









Strain Energy


W Pd



0



1 10U W Pd

Unit Nm = J

Strain Energy




2

U WPLEA

2 2P L EA


2 2P L EAUEA L


2 2

EAk 2 2

2 2P kUk

L



2n n N L


2

1 1 2

n ni i

ii i i i

N LU UE A


2 2( )P P L P LL1 E1 A1

1 2 1 2 2

1 1 2 2

( )2 2

P P L P LUE A E A

L2 E2 A2

2 2

0

( )2 ( )

L N x dxU

EA x











2PU W

2UP






2 2

2 22 2U P EuAL EA L


2 2Eu

E

2 2

2 2Eu

E


ε









max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion






0AB T R


T T L AR

R LEA


0AT R

R LT LEA

ndash ReactionsEA

A BR R EA T


R R E TA A



A BR R E T


A A










N V

1A

1A


AA



1

cosA A

1

sin cosA A


2 sin cos sin 2

2

2 1 1 2 i i 2x



2x

x


max x max 2x

When θ = 0






max 2x









Strain Energy


W Pd



0



1 10U W Pd

Unit Nm = J

Strain Energy




2

U WPLEA

2 2P L EA


2 2P L EAUEA L


2 2

EAk 2 2

2 2P kUk

L



2n n N L


2

1 1 2

n ni i

ii i i i

N LU UE A


2 2( )P P L P LL1 E1 A1

1 2 1 2 2

1 1 2 2

( )2 2

P P L P LUE A E A

L2 E2 A2

2 2

0

( )2 ( )

L N x dxU

EA x











2PU W

2UP






2 2

2 22 2U P EuAL EA L


2 2Eu

E

2 2

2 2Eu

E


ε









max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion





A BR R E T


A A










N V

1A

1A


AA



1

cosA A

1

sin cosA A


2 sin cos sin 2

2

2 1 1 2 i i 2x



2x

x


max x max 2x

When θ = 0






max 2x









Strain Energy


W Pd



0



1 10U W Pd

Unit Nm = J

Strain Energy




2

U WPLEA

2 2P L EA


2 2P L EAUEA L


2 2

EAk 2 2

2 2P kUk

L



2n n N L


2

1 1 2

n ni i

ii i i i

N LU UE A


2 2( )P P L P LL1 E1 A1

1 2 1 2 2

1 1 2 2

( )2 2

P P L P LUE A E A

L2 E2 A2

2 2

0

( )2 ( )

L N x dxU

EA x











2PU W

2UP






2 2

2 22 2U P EuAL EA L


2 2Eu

E

2 2

2 2Eu

E


ε









max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion










N V

1A

1A


AA



1

cosA A

1

sin cosA A


2 sin cos sin 2

2

2 1 1 2 i i 2x



2x

x


max x max 2x

When θ = 0






max 2x









Strain Energy


W Pd



0



1 10U W Pd

Unit Nm = J

Strain Energy




2

U WPLEA

2 2P L EA


2 2P L EAUEA L


2 2

EAk 2 2

2 2P kUk

L



2n n N L


2

1 1 2

n ni i

ii i i i

N LU UE A


2 2( )P P L P LL1 E1 A1

1 2 1 2 2

1 1 2 2

( )2 2

P P L P LUE A E A

L2 E2 A2

2 2

0

( )2 ( )

L N x dxU

EA x











2PU W

2UP






2 2

2 22 2U P EuAL EA L


2 2Eu

E

2 2

2 2Eu

E


ε









max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion





1

cosA A

1

sin cosA A


2 sin cos sin 2

2

2 1 1 2 i i 2x



2x

x


max x max 2x

When θ = 0






max 2x









Strain Energy


W Pd



0



1 10U W Pd

Unit Nm = J

Strain Energy




2

U WPLEA

2 2P L EA


2 2P L EAUEA L


2 2

EAk 2 2

2 2P kUk

L



2n n N L


2

1 1 2

n ni i

ii i i i

N LU UE A


2 2( )P P L P LL1 E1 A1

1 2 1 2 2

1 1 2 2

( )2 2

P P L P LUE A E A

L2 E2 A2

2 2

0

( )2 ( )

L N x dxU

EA x











2PU W

2UP






2 2

2 22 2U P EuAL EA L


2 2Eu

E

2 2

2 2Eu

E


ε









max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion








max 2x









Strain Energy


W Pd



0



1 10U W Pd

Unit Nm = J

Strain Energy




2

U WPLEA

2 2P L EA


2 2P L EAUEA L


2 2

EAk 2 2

2 2P kUk

L



2n n N L


2

1 1 2

n ni i

ii i i i

N LU UE A


2 2( )P P L P LL1 E1 A1

1 2 1 2 2

1 1 2 2

( )2 2

P P L P LUE A E A

L2 E2 A2

2 2

0

( )2 ( )

L N x dxU

EA x











2PU W

2UP






2 2

2 22 2U P EuAL EA L


2 2Eu

E

2 2

2 2Eu

E


ε









max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion










Strain Energy


W Pd



0



1 10U W Pd

Unit Nm = J

Strain Energy




2

U WPLEA

2 2P L EA


2 2P L EAUEA L


2 2

EAk 2 2

2 2P kUk

L



2n n N L


2

1 1 2

n ni i

ii i i i

N LU UE A


2 2( )P P L P LL1 E1 A1

1 2 1 2 2

1 1 2 2

( )2 2

P P L P LUE A E A

L2 E2 A2

2 2

0

( )2 ( )

L N x dxU

EA x











2PU W

2UP






2 2

2 22 2U P EuAL EA L


2 2Eu

E

2 2

2 2Eu

E


ε









max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion



Strain Energy




2

U WPLEA

2 2P L EA


2 2P L EAUEA L


2 2

EAk 2 2

2 2P kUk

L



2n n N L


2

1 1 2

n ni i

ii i i i

N LU UE A


2 2( )P P L P LL1 E1 A1

1 2 1 2 2

1 1 2 2

( )2 2

P P L P LUE A E A

L2 E2 A2

2 2

0

( )2 ( )

L N x dxU

EA x











2PU W

2UP






2 2

2 22 2U P EuAL EA L


2 2Eu

E

2 2

2 2Eu

E


ε









max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion





2

U WPLEA

2 2P L EA


2 2P L EAUEA L


2 2

EAk 2 2

2 2P kUk

L



2n n N L


2

1 1 2

n ni i

ii i i i

N LU UE A


2 2( )P P L P LL1 E1 A1

1 2 1 2 2

1 1 2 2

( )2 2

P P L P LUE A E A

L2 E2 A2

2 2

0

( )2 ( )

L N x dxU

EA x











2PU W

2UP






2 2

2 22 2U P EuAL EA L


2 2Eu

E

2 2

2 2Eu

E


ε









max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion





2n n N L


2

1 1 2

n ni i

ii i i i

N LU UE A


2 2( )P P L P LL1 E1 A1

1 2 1 2 2

1 1 2 2

( )2 2

P P L P LUE A E A

L2 E2 A2

2 2

0

( )2 ( )

L N x dxU

EA x











2PU W

2UP






2 2

2 22 2U P EuAL EA L


2 2Eu

E

2 2

2 2Eu

E


ε









max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion













2PU W

2UP






2 2

2 22 2U P EuAL EA L


2 2Eu

E

2 2

2 2Eu

E


ε









max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion





2 2

2 22 2U P EuAL EA L


2 2Eu

E

2 2

2 2Eu

E


ε









max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion











max

nom

K

K 3 for this case











Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion













Side view

1 2


1

2


hellip

2

hellip

4

3


stre

ss

1

2

drillingexcavation

t i

3

4 1

4

strain


Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion



Impact Loading




energy

Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion



Impact Loading



max max( ) ( )2

Mg h W hL

122



EA EA EA

loading conditionst

WLEA

12

max21 1st

st

h

Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion



Impact Loading


122

max2W W WhE

A A AL

A A AL

stst

EW MgA A L

A A L

12122


st

hE hEL L

bull Impact factor

max

st



12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion





12

max21 1st

st

h

st

max 2 st




Summary

bull Introduction




Chapter 3 Torsion





Summary

bull Introduction




Chapter 3 Torsion



Chapter 3 Torsion



Mechanics in Energy Resources Engineering - Chapter 2. Axially...

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Transcript of Mechanics in Energy Resources Engineering - Chapter 2. Axially...