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Class #1.2Class #1.2Civil Engineering Materials – CIVE 2110Civil Engineering Materials – CIVE 2110

Strength of MaterialsStrength of Materials

Mechanical Properties Mechanical Properties of Materialsof Materials

Fall 2010Fall 2010Dr. GuptaDr. GuptaDr. PickettDr. Pickett

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Stress and Strain Stress and Strain

Mechanical Properties of Materials Mechanical Properties of Materials - used to develop relationships between- used to develop relationships between Stress and Strain.Stress and Strain.- determined Experimentally.- determined Experimentally.

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Experimental DeterminationExperimental Determinationof Mechanical Properties of of Mechanical Properties of MaterialsMaterials

Stress and Strain experiments:Stress and Strain experiments:A standard specimen of the Material:A standard specimen of the Material:- Pulled in Uniaxial Tension.- Pulled in Uniaxial Tension.- Diameter = 0.5 in.- Diameter = 0.5 in.- Gauge Length = 2.0 in. - Gauge Length = 2.0 in. Or Or - a strain gauge is placed- a strain gauge is placed parallel to long axis of specimenparallel to long axis of specimen

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Stress-Strain DiagramsStress-Strain DiagramsConventionalConventional Stress-Strain Diagram : Stress-Strain Diagram :

Normal StressNormal Stress calculated using calculated usingORIGINAL cross sectional area, AORIGINAL cross sectional area, AOO..

Called Called Engineering Stress.Engineering Stress.Even though cross section decreases, necks down.Even though cross section decreases, necks down.

Assumes stress is CONSTANT over cross section.Assumes stress is CONSTANT over cross section.

Normal StrainNormal Strain calculated using calculated using ORIGINAL length, LORIGINAL length, LOO..

Assumes strain is CONSTANT over cross sectionAssumes strain is CONSTANT over cross section

O

O

LLL

OALoadP

P

P

P

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Stress-Strain DiagramsStress-Strain DiagramsTRUETRUE Stress-Strain Diagram : Stress-Strain Diagram :

Normal StressNormal Stress calculated using calculated usingACTUAL cross sectional area, AACTUAL cross sectional area, Aii, at any instant., at any instant.

Called Called TRUE Stress.TRUE Stress.Because cross section decreases, necks down.Because cross section decreases, necks down.

Normal StrainNormal Strain calculated using calculated using ACTUAL length, LACTUAL length, Li i , at any instant., at any instant.

i

i

LLL

iALoadP

P

P

P

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Stress-Strain Diagram – Ductile MaterialStress-Strain Diagram – Ductile MaterialConventional (Engineering)Conventional (Engineering) and and TRUETRUEStress-Strain diagramsStress-Strain diagramsDiffer only in regions of Differ only in regions of LARGELARGE strain . strain .

i

i

LLL

iALoadP

OALoadP

O

O

LLL

σ (K

SI)

ε (μ in./in.)

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Stress-Strain Diagram – Ductile MaterialStress-Strain Diagram – Ductile MaterialEngineering Engineering Stress-Strain diagrams:Stress-Strain diagrams:- - Label the axes.Label the axes.- - REGIONS:REGIONS: - Elastic – region where specimen will return to- Elastic – region where specimen will return to original size & shape original size & shape after loading & unloadingafter loading & unloading - Plastic - region where specimen will NOT return - Plastic - region where specimen will NOT return to original size &to original size & shape after shape after loading & unloadingloading & unloading - Yielding – region where- Yielding – region where specimen will continuespecimen will continue to elongate with littleto elongate with little or no increase in loador no increase in load σ

(KS

I)

ε (μ in./in.)

88

Stress-Strain Diagram – Ductile MaterialStress-Strain Diagram – Ductile MaterialEngineering Engineering Stress-Strain diagrams:Stress-Strain diagrams:- - REGIONS:REGIONS: - Strain Hardening – region where specimen will elongate - Strain Hardening – region where specimen will elongate only with increasing load, and theonly with increasing load, and the cross sectional area will decreasecross sectional area will decrease uniformly over entire specimen gauge length uniformly over entire specimen gauge length - Necking - region where - Necking - region where specimen cross sectionalspecimen cross sectional area will decrease in aarea will decrease in a localized spot, and localized spot, and load carrying capacityload carrying capacity will decrease, will decrease, uncontrollably uncontrollably

σ (K

SI)

ε (μ in./in.)

99

Stress-Strain Diagram – Ductile MaterialStress-Strain Diagram – Ductile MaterialEngineering Engineering Stress-Strain diagrams:Stress-Strain diagrams:- - POINTS:POINTS: - Proportional Limit, - Proportional Limit, σσPLPL – highest stress at which – highest stress at which

Stress and StrainStress and Strain are linearly proportional, via E are linearly proportional, via E - Modulus of Elasticity (Young’s Modulus) – - Modulus of Elasticity (Young’s Modulus) – the slope of the the slope of the Stress-Strain curve in theStress-Strain curve in the Linear-Elastic region, Linear-Elastic region, slope up to slope up to σσPLPL

(E(ESTEELSTEEL=29x10=29x1066 psi) psi) σ (K

SI)

ε (μ in./in.)

EE=

Δσ/Δ

ε

1010

Stress-Strain Diagram – Ductile MaterialStress-Strain Diagram – Ductile MaterialEngineering Engineering Stress-Strain diagrams:Stress-Strain diagrams:- - POINTS:POINTS: - Modulus of Elasticity (Young’s Modulus) - Modulus of Elasticity (Young’s Modulus) the slope of the the slope of the Stress-Strain curve in theStress-Strain curve in the Linear-Elastic region,Linear-Elastic region, slope up to the slope up to the Proportional LimitProportional Limit (E(ESTEELSTEEL=29x10=29x1066 psi) psi)

- Hooke’s Law- Hooke’s Law σ (K

SI)

ε (μ in./in.)

E

E=Δσ

/Δε

E

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Stress-Strain Diagram – Ductile MaterialStress-Strain Diagram – Ductile MaterialEngineering Engineering Stress-Strain diagrams:Stress-Strain diagrams:- - POINTS:POINTS: - Modulus of Elasticity - Modulus of Elasticity (Young’s Modulus) = (Young’s Modulus) = the slope of the the slope of the Stress-Strain curve in theStress-Strain curve in the Linear-Elastic regionLinear-Elastic region - alloy content affects- alloy content affects - Proportional Limit- Proportional Limit - not Modulus of Elasticity- not Modulus of Elasticity (E(ESTEELSTEEL=29x10=29x1066 psi) psi)

σ (K

SI)

ε (μ in./in.)

StiffnessE

E=Δσ

/Δε

E=Δσ

/Δε

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Stress-Strain Diagram – Ductile MaterialStress-Strain Diagram – Ductile MaterialEngineering Engineering Stress-Strain diagrams:Stress-Strain diagrams:- - POINTS:POINTS: - Elastic Limit, - Elastic Limit, σσELEL – highest stress at which the specimen – highest stress at which the specimen will return to original size and shapewill return to original size and shape after loading and unloading after loading and unloading (Steel, very close to Proportional Limit) (Steel, very close to Proportional Limit) - Yield Point, - Yield Point, σσYY – stress – stress at which specimenat which specimen will have permanentwill have permanent (plastic) deformation(plastic) deformation after loading & unloading, after loading & unloading, specimen will elongatespecimen will elongate with LITTLE or NO load with LITTLE or NO load increaseincrease (Steel, very close to (Steel, very close to Proportional Limit)Proportional Limit)

σ (K

SI)

ε (μ in./in.)

E=Δσ

/Δε

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Stress-Strain Diagram – Ductile MaterialStress-Strain Diagram – Ductile Material

Engineering Engineering Stress-Strain diagrams:Stress-Strain diagrams:- - POINTS:POINTS: - Ultimate Stress, - Ultimate Stress, σσUU – highest stress on the specimen – highest stress on the specimen

- Fracture Stress, - Fracture Stress, σσFF – stress at which specimen breaks – stress at which specimen breaks

σ (K

SI)

ε (μ in./in.)

E=Δσ

/Δε

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Stress-Strain Diagram – Ductile MaterialStress-Strain Diagram – Ductile MaterialEngineering Engineering Stress-Strain diagrams:Stress-Strain diagrams:

- - DUCTILE vs. BRITTLE Behavior:DUCTILE vs. BRITTLE Behavior: - Ductile behavior – specimen exhibits significant - Ductile behavior – specimen exhibits significant permanent (plastic) deformationpermanent (plastic) deformation before failure,before failure, (good, gives warning before failure) (good, gives warning before failure) - Brittle behavior – - Brittle behavior – specimen exhibitsspecimen exhibits little or no little or no permanent (plastic)permanent (plastic) deformation deformation before failure,before failure, (bad, no warning before(bad, no warning before failure) failure)

σ (K

SI)

ε (μ in./in.)

E=Δσ

/Δε

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Engineering Stress-Strain Diagram – DuctileEngineering Stress-Strain Diagram – DuctileSome Ductile materials have Some Ductile materials have distinct Yield Point anddistinct Yield Point and large yielding region:large yielding region: - alloys such as; - alloys such as; steel, brass, steel, brass, - elements such as; - elements such as; Molybdenum, ZincMolybdenum, Zinc

Some Ductile materials have Some Ductile materials have no distinct Yield Point: no distinct Yield Point: - Such as; Aluminum- Such as; Aluminum- need to define a YIELD STRENGTH- need to define a YIELD STRENGTH - Stress at 0.2% strain - Stress at 0.2% strain - draw a line from - draw a line from εε = 2000x10 = 2000x10-6-6 = 0.002 = 0.2% = 0.002 = 0.2% - parallel to the linear elastic portion of curve- parallel to the linear elastic portion of curve- - σσYS YS = = stress at intersection of parallel line with curvestress at intersection of parallel line with curve

σ (K

SI)

ε (μ in./in.)

E=Δσ

/Δε

ε (μ in./in.)

1616

Engineering Stress-Strain Diagram – BrittleEngineering Stress-Strain Diagram – BrittleBrittle materials exhibit Brittle materials exhibit LITTLE or NO permanent deformationLITTLE or NO permanent deformationBEFORE failure (BAD).BEFORE failure (BAD).

Examples:Examples:- Gray Cast Iron- Gray Cast Iron- Concrete- Concrete - Low Tensile Strength- Low Tensile Strength - High Compression Strength - High Compression Strength - No failure warning (Tension or Compression)- No failure warning (Tension or Compression) - BAD- BAD - need to get people- need to get people - off bridge- off bridge - out of building- out of building

σ (K

SI) ε (μ in./in.)

E=Δσ

/Δε

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Brittle - ConcreteBrittle - ConcreteConcreteConcrete - Low Tensile Strength- Low Tensile Strength - High Compression Strength - High Compression Strength - No failure warning (Tension or Compression)- No failure warning (Tension or Compression) - BAD- BAD - need to get people- need to get people - off bridge- off bridge - out of building- out of building

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Brittle - ConcreteBrittle - ConcreteConcreteConcrete - Low Tensile Strength- Low Tensile Strength - High Compression Strength - High Compression Strength - No failure warning (Tension or Compression)- No failure warning (Tension or Compression) - BAD- BAD - need to get people- need to get people - off bridge- off bridge - out of building- out of building

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DuctilityDuctilityDUCTILITY can be measured by:DUCTILITY can be measured by:

- Percent Elongation:- Percent Elongation:

- Percent Reduction of Area:- Percent Reduction of Area:

σ (K

SI) ε (μ in./in.)

E=Δσ

/Δε

original

originalfracture

original

originalfracture

AAA

Areaofduction

LLL

Elongation

__Re%

%

2020

Strain Hardening & HysteresisStrain Hardening & HysteresisIf a specimen is loaded past If a specimen is loaded past σσELEL,,

Then unloaded,Then unloaded,Some deformation will be removed,Some deformation will be removed,Some deformation will remain.Some deformation will remain.

The slope of the UNLOADING curve will be The slope of the UNLOADING curve will be the SAME SLOPE, E, as the LOADING curve.the SAME SLOPE, E, as the LOADING curve.

If specimen is loaded again,If specimen is loaded again,The slope of the re-load curve will be E.The slope of the re-load curve will be E.A higher A higher σσYY will be reached because of will be reached because of

STRAIN HARDENING.STRAIN HARDENING.

But there will be a smaller plastic region But there will be a smaller plastic region Remaining, Remaining, So DUCTILITY will be less.So DUCTILITY will be less.

σ (K

SI) ε (μ in./in.)

E=Δσ

/Δε

Original Yield PointYield Point after

Strain Hardening

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Strain Hardening & HysteresisStrain Hardening & HysteresisSome energy will be Some energy will be LOST or USED or DISSIPATED LOST or USED or DISSIPATED In the LOAD and UNLOAD processes.In the LOAD and UNLOAD processes.

Energy dissipated is a function of the Energy dissipated is a function of the area inside the LOAD-UNLOAD curves,area inside the LOAD-UNLOAD curves,called HYSTERESIS loops.called HYSTERESIS loops.

Mechanical hysteresis devices are used toMechanical hysteresis devices are used toReduce EARTHQUAKE forces on structures.Reduce EARTHQUAKE forces on structures.

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Hysteresis DevicesHysteresis Devices

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Hysteresis DevicesHysteresis Devices

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StressStress-Strain-Strain

Diagrams Diagrams

Example: Example: Problem 3-4Problem 3-4Hibbeler 7Hibbeler 7thth edition, edition, pg. 102pg. 102

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StressStress-Strain-Strain

Diagrams Diagrams

Example:Example:

Problem 3-4Problem 3-4Hibbeler 7Hibbeler 7thth edition, edition, pg. 102pg. 102