11

Class #2.1Class #2.1Civil Engineering Materials – CIVE 2110Civil Engineering Materials – CIVE 2110

Strength of MaterialsStrength of Materials

Mechanical Properties Mechanical Properties

of Ductile Materialsof Ductile Materials

Fall 2010Fall 2010

Dr. GuptaDr. Gupta

Dr. PickettDr. Pickett

22

Strain Energy Strain Energy

External energy is required to deform specimen.External energy is required to deform specimen.Internal energy is stored in specimenInternal energy is stored in specimen during deformation process.during deformation process.Work = Energy = Force x DisplacementWork = Energy = Force x Displacement Force starts from ZERO, Force starts from ZERO, Strain Energy = Strain Energy = ΔΔU = U = “work done” “work done” ΔΔU =U = due to average force over load process. due to average force over load process.

VE

VolVE

Vol

AreaStressFU

ntDisplacemeForceUEnergyStrain

ZZZZ

ZYXZZZZYXZ

ZZZZ

2

2

1

2

1

2

1

2

12

1

2

12

1

33

Strain Energy Strain Energy

Strain Energy = Strain Energy = ΔΔU = U = “work done” “work done”

ΔΔU =U = due to average force over load process. due to average force over load process.

EV

UuDensityEnergyStrain

EV

UuVolumeunitperEnergyStrain

VE

VolVE

Vol

AreaStressFU

ntDisplacemeForceUEnergyStrain

Z

Z

ZZZZ

ZYXZZZZYXZ

ZZZZ

2

2

2

2

1

2

1

2

1

2

1

2

1

2

12

1

2

12

1

44

Strain Energy – Modulus of ResilienceStrain Energy – Modulus of Resilience

Modulus of Resilience:Modulus of Resilience:

- A measurement of a material’s ability to - A measurement of a material’s ability to

absorb energyabsorb energy

WITHOUT PERMANENT deformation.WITHOUT PERMANENT deformation.

- Area under ELASTIC portion of - Area under ELASTIC portion of

stress-strain diagram.stress-strain diagram.

EusilienceofModulus PL

PLPLr

2

2

1

2

1Re

55

Strain Energy – Modulus of ToughnessStrain Energy – Modulus of Toughness

Modulus of Toughness:Modulus of Toughness:

- A measurement of a material’s ability to - A measurement of a material’s ability to

absorb energy BEFORE FAILURE.absorb energy BEFORE FAILURE.

- Area under ENTIRE stress-strain diagram.- Area under ENTIRE stress-strain diagram.

- Materials with high toughness- Materials with high toughness

will give warning before failure (GOOD)will give warning before failure (GOOD)

AreaPlasticE

uToughnessofModulus PLt _2

1 2

66

Modulus of Toughness - ConcreteModulus of Toughness - ConcreteConcrete - need warning before failureConcrete - need warning before failure

- put in steel reinforcement bars- put in steel reinforcement bars

High Toughness Low ToughnessHigh Toughness Low Toughness

77

Poisson’s RatioPoisson’s RatioPoisson’s Ratio:Poisson’s Ratio:- For material that is:- For material that is:

- Homogenous- Homogenous- Isotropic- Isotropic- in linear elastic range- in linear elastic range

the material VOLUME must remain CONSTANTthe material VOLUME must remain CONSTANT - deformations in the Longitudinal direction- deformations in the Longitudinal direction must be compensated for by deformationsmust be compensated for by deformations in the TWO directions PERPENDICULAR to thein the TWO directions PERPENDICULAR to the Longitudinal direction.Longitudinal direction.

L

rRatiosPoissonalLongitudin

Lateral

''

88

Poisson’s RatioPoisson’s Ratio

Poisson’s Ratio:Poisson’s Ratio:

- Typical values:- Typical values:

steel = 0.27 - 0.32steel = 0.27 - 0.32

aluminum = 0.35aluminum = 0.35

cast iron = 0.28cast iron = 0.28

copper alloys = 0.34 - 0.35copper alloys = 0.34 - 0.35

concrete = 0.15concrete = 0.15

wood = 0.29 – 0.31wood = 0.29 – 0.31

L

rRatiosPoissonalLongitudin

Lateral

''

99

Shear Stress-Strain DiagramsShear Stress-Strain DiagramsAssumptions:Assumptions:- Material is: - Material is:

- loaded in pure shear- loaded in pure shear- homogeneous- homogeneous- isotropic- isotropic- ductile- ductile- in linear elastic range, has- in linear elastic range, has - proportional limit, - proportional limit, - Shear Modulus of Elasticity, G- Shear Modulus of Elasticity, G - Modulus of Rigidity, G- Modulus of Rigidity, G

12'

EGGLawsHooke

PL

PLG

1010

Shear Stress-Strain DiagramsShear Stress-Strain Diagrams

GGSteelSteel = 11x10 = 11x1066 psi psi

GGAluminumAluminum = 4x10 = 4x1066 psi psi

GGCopperCopper = 5.5x10 = 5.5x1066 psi psi

12'

EGGLawsHooke

PL

PLG

1111

CreepCreepCreep = time dependent Creep = time dependent permanent deformationpermanent deformation

Due to – Load Due to – Load - Temperature- Temperature

Creep Strength = highest initial stress that theCreep Strength = highest initial stress that the material can be subjected tomaterial can be subjected to

in order to avoid a specifiedin order to avoid a specified creep strain creep strain over a specified timeover a specified time

1212

FatigueFatigueFatigue = BRITTLE fracture at Fatigue = BRITTLE fracture at

stress < material’s Yield Stressstress < material’s Yield Stress

due to repeated load cyclesdue to repeated load cycles

Cause: Localized stress > average stressCause: Localized stress > average stress

Fatigue Limit = stress below which Fatigue Limit = stress below which

NO failure occurs for s NO failure occurs for s

specified number of cyclesspecified number of cycles

Endurance Limit = Fatigue LimitEndurance Limit = Fatigue Limit

1313

FatigueFatigueFatigue Limit = stress below which Fatigue Limit = stress below which

NO failure occurs for s NO failure occurs for s

specified number of cyclesspecified number of cycles

Endurance Limit = Fatigue LimitEndurance Limit = Fatigue Limit

1414

Shear StrainShear StrainExample:Example:

Problem 3-30Problem 3-30

Hibbeler 7Hibbeler 7thth edition, edition,

pg. 116pg. 116

1515

Shear StrainShear StrainExample:Example:

Problem 3-32Problem 3-32

Hibbeler 7Hibbeler 7thth edition, edition,

pg. 117pg. 117