ME 354 Mechanics of Materials Laboratory Overview and

ME 354 Mechanics of Materials Laboratory

Overview and Summary

Prerequisites - ENGR 170 "Introduction to Materials Science"- ENGR 220 "Mechanics of Materials"

General DescriptionMechanics of Materials Laboratory (5 Credits)Study of the properties, behavior and performance of engineering materials including stress-strain relations, strength, deformation, fracture, creep, and cyclic fatigue.

Introduction to experimental techniques common to structural engineering, interpretation of experimental data, comparison of measurements to numerical/analytical predictions, and formal, engineering report writing.

Three 1-hr lectures/week + One 1-hr recitation/week+ One 3-hr laboratory/week

Laboratory exercises1) Measurement, Significant Figures, And Statistics

2) Strains, Deflections And Beam Bending

3) Curved Beams

4) Mechanical Properties and Performance of MaterialsTensionHardnessTorsionCharpy V-Notch Impact

5) Stress Concentrations

6) Fracture

7) Fatigue (Time-Dependent Failure)

8) Creep (Time-Dependent Deformation)

9) Structures

10) Compression and Buckling

ME 354, MECHANICS OF MATERIALS LABORATORY

MEASUREMENT, SIGNIFICANT FIGURES, AND STATISTICS

PURPOSE

The purpose of this exercise is familiarize you with measurements, significant figures,and statistics.

ASPECTS OF MECHANICS OF MATERIALS

Measurements, inherent error, discrepancy, magnitude and units, direct measurements,Indirect measurements, accuracy, precision

Absolute error, relative error and percentage error

err X X absolute

errX X

X=

errX

relative

%errX X

X= 100 err percentage

where X is the true value and X is the measured value

a t m

rt m

m

a

m

t m

mr

t m

= −

= −

= −

( )

( )

( )100

Significant figures and statistics

X

X

n sample mean

X X

n -1 sample standard deviatio

ii 1

n

ii 1

n 2

=

=−( )

=

=

∑

∑

( )

ˆ ( )σs


STRAINS, DEFLECTIONS AND BEAM BENDING LABORATORY

PURPOSE

The purpose of this exercise is a) to familiarize the user with strain gages and associatedinstrumentation, b) to measure deflections and compare these to predicted deflections,and c) to verify certain aspects of stress-strain relations and simple beam theory.


1) Strain transformationsεθ = εxcos2θ + εy sin2 θ + γxycosθsinθ

applied to strain gages45°

45°

60°

60°

a) Rectangular Rosette b) Delta Rosette c) Biaxiial Strain Gage

Rectangular Deltaεx = ε0° εx = ε0°

εy = ε90° εy = 13

2(ε60° + ε120° ) − ε0°[ ]

γxy = 2ε45° − (ε0° + ε90° ) γxy = 2 33

ε60° − ε120°[ ]

2) Constitutive relationsStress Strain

σ{ } = C[ ] ε{ }

σx = E(1+ ν )

εx + νE(1+ ν )(1− 2ν )

(εx + εy + εz )

σy = E(1+ ν )

εy + νE(1+ ν )(1− 2ν )

(εx + εy + εz )

σz = E(1+ ν )

εz + νE(1+ ν )(1− 2ν )

(εx + εy + εz )

τxy = Gγxy

τ yz = Gγyz

τxz = Gγxz

ε{ } = S[ ] σ{ }

εx = 1E

σx − ν(σy + σz )[ ]εy = 1

Eσy − ν(σx + σz )[ ]

εz = 1E

σz − ν(σx + σy )[ ]γxy = 1

Gτ xy

γyz = 1G

τ yz

γxz = 1G

τ xz

3) Beams in bending (Analytical and Numerical)

σ σ= -MyI

and =McImax

50.8 mm

30°

30°(1)

(2)(3)

(4) (5)

(6) Beam LongitudinalAxis

RectangularRosette

Delta Rosette

FIGURE 2 - Top View of Specimen Geometry Showing Orientations of 3-Element Strain Gage Rosettes. Note: Strain Gage Channel Numbers are Shown in Parentheses.

FIGURE 3 - Strain Gage Locations and Cross Sectional Dimensions of the Beam. Note: Strain Gage Channel Numbers are Shown in Parentheses.

3.175 mm12.700 mm

22.225 mm

RectangularRosette

38.100 mm 3.048 mm

25.400 mm

3.048 mm

b) Cross Section of Beama) Side View of Uniaxial Strain Gage Locations.

(9)

(8)

(7)

685.8 mm457.2, 406,4, or 355.6 mm

228.6 mm177.8 mmToggle Bolt (Load)

Uniaxial S.G. (3)

RectangularRosette

Delta Rosette

Load CellRing

FIGURE 1 - Overall view of Test Specimen Geometry and Setup

A

ASection A-A

Reaction

Actual Experimental Setup Finite Element Analysis


STRESSES IN STRAIGHT AND CURVED BEAMS

PURPOSE

The purpose of this exercise is to the study the limitations of conventional beam bendingrelations applied to curved beams and to use photo elasticity to determine the actualstresses in a curved beam for comparison to analytical and numerical solutions.


1) Nonlinear strain distribution and therefore nonlinear stress distribution in curvedbeams in bending

ε x = − Roεo

r

y

ht

σ x = − Roσo

r

y

ht

= − My

(y + ρ)Ay

2) Photoelasticity, birefringence and the stress optic law

(σ1 − σ2 ) = fNt

Ø 0.255 TYP

Photo elastic t est specimen

Dimensions in mm

19. 05

38.105 REF

22.225

47.63

85.730

9. 525

No te: Nominal t hickness = 6.35 mm

R 38.10

R 19.0576.2

3) Finite element analysis consistent with analytical and empirical results

Sigma_Y

-4.760

-2.270

0.2270

2.7200

5.2100

7.7000

10.200

-7 250

12.700

Lin STRESS Lc=1

X

Y

Z

σ y = +12.7 MPa σ y =- 7.3 MPa


MECHANICAL PROPERTIES AND PERFORMANCE OF MATERIALS:TENSILE TESTING

PURPOSE

The purpose of this exercise is to obtain a number of experimental results important forthe characterization of the mechanical behavior of materials. The tensile test is afundamental mechanical test for material properties which are used in engineeringdesign, analysis of structures, and materials development.

MECHANICAL PROPERTIES AND PERFORMANCE OF MATERIALS:HARDNESS TESTING

PURPOSE

The purpose of this exercise is to obtain a number of experimental results important forthe characterization of the mechanical behavior of materials. The hardness test is amechanical test for material properties which are used in engineering design, analysis ofstructures, and materials development.

MECHANICAL PROPERTIES AND PERFORMANCE OF MATERIALS:TORSION TESTING

PURPOSE

The purpose of this exercise is to obtain a number of experimental results important forthe characterization of materials. In particular, the results from the torsion test will becompared to the results of the engineering tensile test for a particular alloy using theeffective stress-effective strain concept.

MECHANICAL PROPERTIES AND PERFORMANCE OF MATERIALS:CHARPY V-NOTCH IMPACT

PURPOSE The purpose of this exercise is to obtain a number of experimental results important forthe characterization of the mechanical behavior of materials. The Charpy V-notch impactis a mechanical test for determining qualitative results for material properties andperformance which are useful in engineering design, analysis of structures, andmaterials development.


1) Uniform, uniaxial stress state used to extract material properties in elastic and plasticranges (tension)

P

Ao

Lo

σ =P/Ao

ε=(Li-Lo)/Lo1

τ

σσ1

σ = σ = 02 3Mohr's Circle for UniaxialTension

2) Localized deformation evaluation resistance to penetration and allows empiricalcorrelation to tensile strengths (hardness).

Brinell Rockwell

P=3000 kg or 500 kgD=10 mm

dt

Steel or tungstencarbideball

Side view

Top view

0 1 2

d

h1

h2

P minor = 10 kg

P major = 60, 100 or 150 kg

Brale (diamond cone)or ball indentor

Brale (diamond cone)or ball indentor

BHN HBPDt

P

D D D d= = =

− −( )[ ]π π

22 2

Rockwell Scale Indentor Pmajor

(X ) (kg)

B 1/16" ball 100

C Brale (diamond) 150

M 100 for C scale

M 130 for B scale

=

= = −−( )

==

HRX R Mh h

X2 1

0 002.

3) Non uniform plastic stress state can be analyzed using results from the uniform stressstate of the tension test and effective stress-strain relations for plasticity (torsion)

τ =TR/Jγ =R /Lθ

2R

θφ=γ

L

T

τ

σσ1

σ =−τ2 =τ

4) Environmental and geometrical effects such as temperature, notches, and loading ratecan cause materials to behave much differently than they may in tension tests (impact).

h1

h2

mass, m

IZOD CHARPYV-NOTCH

IMPACT ENERGY=mg(h1-h2)

TEMPERATUREIMP

AC

T E

NE

RG

Y

Brittle

Ductile

Ductile/BrittleTransition


STRESS CONCENTRATIONSPURPOSE

The purpose of this exercise is to the study the effects of geometric discontinuities on thestress states in structures and to use photo elasticity to determine the stressconcentration factor in a simple structure.


1) Stress concentration factors due to stress raisers in components

ktlocal

remote

= σσ

2) Photoelasticity, birefringence and the stress optic law

(σ1 − σ2 ) = fNt

38.10

279.40254.00

101.60

9.525

Note: Nominal thickness =6.35 mm Photo elastic test specimens

9.525

R 12.70

Dimesions in mm

Von Mises

1.940000

3.880000

5.820000

7.760000

9.690000

11.60000

13.60000

0 006930

15.50000

Lin STRESS Lc=1

X

Y

Z


FRACTUREPURPOSE

The purpose of this exercise is to study the effects of cracks in decreasing the load-carrying ability of structures and to determine the plane strain critical stress intensityfactor, KIc, for single-edge notched specimens.


1) Stress intensity factor

K F

Fractures if K KI

I Ic

=

=

( )α σ πa

2) Fracture toughness and the reduction of the load-carrying capability of structuresbecause of cracks

σα π

= K cI

F a( )

σALLOWABLESTRESS,

CRACK LENGTH, a

Low KHigh K Ic

Ic

σALLOWABLESTRESS,

CRACK LENGTH, a

σοat = transition crack length

between yield and fractu

W a

B

2h

PP


COMPRESSION AND BUCKLING

PURPOSE

The purpose of this exercise is to study the effects of end conditions, column length, andmaterial properties on compressive behaviour and buckling in columns.


1) Geometric instability of buckling in compression

σ σ π= =cre

EL k

2

2( / )2) Material yielding in compression

σ σ= o

0

100

200

300

400

0 100 200

Le/k

Allo

wa

ble

S

tre

ss

(MP

a)

YieldBuckle

Pinned/Pinned Fixed/Free Fixed/Fixed Pinned/FixedL Le = L Le = 2 L L/2e = L 0.7 Le =


FATIGUEPURPOSE

The purpose of this exercise is to determine the effect of cyclic loads on the long-termbehaviour of structures and to determine the fatigue lives (Nf) as functions of uniaxialtensile stress for an aluminum alloy. Axial fatigue tests are used to obtain the fatiguestrength of materials where the strains are predominately elastic both upon initial loadingand throughout the test.


1) S-N curves and the effects of stress ratio, surface finish and maximum stress on time-dependent failure due to fluctuating loads.

2) Variability in fatigue data

0

5 0

100

150

200

250

300

350

400

1E-01 1E+01 1E+03 1E+05 1E+07 1E+09Nf

Max

imum

Str

ess

(M

Pa

)

R = - 1 R=0.1

Bounds on Fatigue Data


CREEP

PURPOSE

The purpose of this exercise is study the effect of loading on the time-dependentdeformation (i.e., creep) and to characterize the room temperature creep behaviour of asoft alloy under various loads. Specifically, short-term creep tests will be used to identify

constants in the «εmin = Aσn relation. Predictions using these constants will be comparedto results measured from long-term creep tests of this same alloy.


1) Strain-curves and the effect of applied stress on the time-dependent deformation at ahomologous temperature of about 0.5 for this low-temperature alloy

0

0.005

0.01

0.015

0.02

0.025

0 500 1000 1500 2000 2500TIME (s)

ST

RA

IN

(m/m

)

m=1.8 kgm=2.7 kgm=3.2 kgm=3.6 kg

2) Predicting long-term creep behaviour with test results that have creep times much less

than the required creep times using the relation «εmin = Aσn

- 8

- 7

- 6

- 5

- 4

- 3

- 2

- 1

0

0 0.2 0 .4 0 .6 0 .8 1log σ

log ε

min

Long-term tests

Short-term tests

ME 354, MECHANICS OF MATERIALS LABORATORYSTRUCTURES

PURPOSE

The purpose of this exercise is to the study the effects of various assumptions inanalyzing the stresses and loads in an engineering structure.


1) Analyzing a complex engineering structure (bicycle frame) using an assumed simpletruss analysis

Top Tube

Sea

t T

ubeDown Tube

For

k

Hea

d T

ube

Seat Stay

Chain Stay

570 mm

620 mm

45 mm

455 mm

455 mm

435 mm

495 mm

Bottom BracketP

Front AxleRear Axle32 mm

SG A (top)SG B (top)

SG C (top)

SG D (side)

SG G (bottom) SG H

(side)

SG E (side)

SG I (side)

SG F (back)

1080 mm

SG J (bottom)

2) Experimentally-measured strains together with stresses calculated from constitutiverelations used to evaluate actual stress distributions and loads for comparison to thosedetermined in the truss analysis.

3) Application of a simple interpretation of a virtual work technique to analyze a complexstructure

∆ = ∑ N N LEAU L

4) Use of a finite element analysis (FEA) model to allow analysis of complex structureswithout the need to oversimplify

X

Y

Z

X

Y

Z

Documents

ME 354 Mechanics of Materials Laboratory Overview and