me 354 mechanics of materials laboratory overview and
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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
ME 354, MECHANICS OF MATERIALS LABORATORY
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.
ASPECTS OF MECHANICS OF MATERIALS
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
ME 354, MECHANICS OF MATERIALS LABORATORY
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.
ASPECTS OF MECHANICS OF MATERIALS
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
ME 354, MECHANICS OF MATERIALS LABORATORY
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.
ASPECTS OF MECHANICS OF MATERIALS
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
ME 354, MECHANICS OF MATERIALS LABORATORY
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.
ASPECTS OF MECHANICS OF MATERIALS
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
ME 354, MECHANICS OF MATERIALS LABORATORY
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.
ASPECTS OF MECHANICS OF MATERIALS
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
ME 354, MECHANICS OF MATERIALS LABORATORY
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.
ASPECTS OF MECHANICS OF MATERIALS
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 =
ME 354, MECHANICS OF MATERIALS LABORATORY
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.
ASPECTS OF MECHANICS OF MATERIALS
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
ME 354, MECHANICS OF MATERIALS LABORATORY
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.
ASPECTS OF MECHANICS OF MATERIALS
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.
ASPECTS OF MECHANICS OF MATERIALS
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.