Normal Stress and Strain

8/2/2019 Normal Stress and Strain

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TENSION, COMPRESSION, AND

SHEAR

Sridhar Condoor

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What is Mechanics of Materials?

Deals with the behavior of solid bodies

subjected to different types of loads

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Skyscrapers of Fazlur Khan

Khan used a taper in theJohn Hancock building toreduce the wind loadsand 18-story tall exterior

steel diagonalsthefamous X-bracestocarry this load.

The buildingsconfiguration is a tubestiffened by the floors.

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Khan designed theWillis Tower using thebundled tubeapproach.

Nine separate steel-framed square

buildings, each 75 feetsquare, are interlockedto form a massivestructure.

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Benefits:

Reduced the material requirements. The net

saving in steel alone is about $10 million.

The tubes terminated at different levels creating

more rentable windowed space (space with

windows as opposed to internal space).

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Design Task

Need Float coins using aluminum foil.

The goal is to float as much weight as possiblewithout wetting the coins.

The size of the foil is 5 X 5.

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Insights

Interrelated subjects

Strength

Stiffness

One or multiple boats

Sheet size translates

into cost

Tipping

Loading

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Prismatic Bar

A straight structural member of constant

cross-section.

Cross-section is taken perpendicular to the

length of the beam.

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Tension - Compression

Tension - The tensile

force or tension

stretches the bar and

increases its length.

Compression - A

compressive force or

compression decreases

the length of the bar.

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Is this a valid tensile loading?

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Examples

Tension

The cable supporting the

elevator

Belt in the belt/pulley

system

Tensile members of a truss

Compression

Legs of a stool

Walking stick

Nail being hammered

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Uniaxial Load

Tension

Not fail-safe

Does not buckle

Compression

Fail-safe

Buckling

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Normal Stress

Intensity of force

where F is the force

andA is the cross-

sectional area.

The stress is independent of structural material and

the length of the prismatic bar.3:24 AM


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Units - Stress

Force/Area

N/m^2 or Pascal

lb/in^2 or psi

7000 Pascal = 1 psi

Sign convention

Tensile stress positive

Compressive stress

negative

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Stress - Strength

The magnitude of the stress alone does not

provide any insight into whether a structure

safe or not.

The stress must be compared to what the

material can safely withstand the strength of

the material.

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Normal Strain

Elongation per unit

length

Dimensionless quantity

Sign convention

Tensile strain positive

Compressive strain

negative

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PROBLEM

A circular aluminum tube

of length L 400 mm is

loaded in compression by

forces P. The outside and

inside diameters are 60

mm and 50 mm,

respectively. A strain gage

is placed on the outside of

the bar to measure normal

strains in the longitudinal

direction.

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PROBLEM

(a) If the measured strain

is 550x10-6, what is the

shortening of the bar?

(b) If the compressive

stress in the bar is

intended to be 40 MPa,

what should be the load

P?

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SOLUTION

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PROBLEM

The cross section of aconcrete pier that isloaded uniformly incompression.

(a) Determine the averagecompressive stress inthe concrete if the loadis equal to 2500 k.

(b) Determine thecoordinates x and y of

the point where theresultant load must actin order to produceuniform normal stress.

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SOLUTION

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SOLUTION

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PROBLEM

A loading crane consistingof a steel girder ABCsupported by a cable BDis subjected to a load P.

The cable has an effectivecross-sectional area A =481 mm2.

The dimensions of thecrane are H = 1.6 m, L1 =3.0 m, and L2 = 1.5 m.

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PROBLEM

(a) If the load P = 32 kN,

what is the average

tensile stress in the

cable?

(b) If the cable stretches

by 5.1 mm, what is the

average strain?

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SOLUTION

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SOLUTION

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SOLUTION

T=102 kN

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STRESS-STRAIN DIAGRAMS

TENSION TEST

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ENGINEERING STRESS-STRAIN DIAGRAM

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ENGINEERING STRESS-STRAIN

DIAGRAM

Nominal stress

Nominal strain

0

PA

0L

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PROPORTIONALITY & ELASTIC LIMIT

Proportional limit: the highest stress at which

stress is directly proportional to strain.

Elastic limit: the greatest stress the material

can withstand without any measurable

permanent strain after unloading.

Elastic limit > Proportional limit

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YIELD STRENGTH

A. Stress at whichconsiderableelongation occurswithout change in

the stress level

B. Offset yield strength -the stress required to

produce a smallspecific amount ofdeformation.

Strain

Stress

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ULTIMATE TENSILE STRENGTH

Ultimate tensile strength or tensile strength Su

is the maximum load Pmax divided by the

original cross-sectional area Ao of the

specimen.

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Engineering Stress-Strain Curve for

Brittle Materials

Perfectly linear

Do not have a yield

point.

Do not strain-harden -

the ultimate strengthand breaking strength

are the same.

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True Stress-Strain or Flow Curve

A true indication of

deformation

characteristics

Based on the true

instantaneous area of

the specimen.

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Specific Stiffness Ratios

E (psi) Density

(lb/in3)

Specific

stiffness

Steel 30 E6 0.283 .106 E 9

Aluminum 10.5 E6 0.1 .105 E 9

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Hookes Law

Strain is directly proportional to stress

E

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Poissons Ratio ()

The negative ratio of

the transverse strain

(perpendicular to the

applied load), to theaxial strain (in the

direction of the applied

load)

Poissons ratio 0 0.5

lateral

longitudinal

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Poissons Ratio ()

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Problem

A steel bar of length 2.5

m with a square cross

section 100 mm on

each side is subjectedto an axial tensile force

of 1300 kN.

Assume that E 200 GPa

and v 0.3.

Determine the increase in volumeof the bar.

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E 200 GPa and v 0.3.

Solution

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Solution

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SHEAR STRESS

Stresses acting parallel

to the surface

PA

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Single vs Double Shear

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BEARING STRESS

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BEARING STRESS

Applied load over

bearing area

Bearing area is theprojected area

In this case, projectedarea is equal to ??????

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PROBLEM

Three steel plates, each 16 mmthick, are joined by two 20-mm diameter rivets.

A. If the load P 50 kN, what isthe largest bearing stressacting on the rivets?

B. If the ultimate shear stressfor the rivets is 180 MPa,what force Pult is required tocause the rivets to fail inshear?

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SOLUTION

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SOLUTION

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Ductility vs Brittleness

DOES A CD FAIL IN A DUCTILE OR BRITTLE

MANNER?

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DUCTILITY VS BRITTLENESS

Ductility - Is a qualitative measure of a

material that indicates the ability of the metal

to flow plastically before failure.

Normal Stress and Strain

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