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EGA324 – Stress Concentrations
Creation date：2024-05-20 15:54:26
Belonging category：物理Physical
Tag EGA324
Stress Concentrations

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EGA324 – Stress Concentrations
Overview
❑ Stress Concentrations & Raisers
❑ Industrial Examples
❑ Infinite Circular Plate
❑ Stress Concentration Factors (SCF)
❑ Neuber’s rule to calculate SCF from stress-
strain curves
❑ Tensile Test Experiment
❑ Assignment
EGA324 – Stress Concentrations
Stress Concentrations & Raisers
❑ What is a Stress Concentration?
▪ A stress concentration (often called a stress
raiser) is a location in an object where stress in
concentrated
▪ An object is strongest when load is evenly
distributed over its areas so any reduction in area
results in a localised increase in stress
▪ Stress concentrations can arise from notches,
grooves, holes, threads, or any other abrupt
changes in geometry or in poor designs of sharp
corners and even from inspections/stamping
▪ These discontinuities are known as stress raisers
EGA324 – Stress Concentrations
Stress Concentrations
=


– stress concentration
– maximum stress
– reference (nominal) stress
?
EGA324 – Stress Concentrations
Examples of Stress
Concentrations
❑ Can anyone name an example?
❑ Pressure Vessels
❑ Poor weld profile
❑Alexander Kielland drilling rig disaster
❑ Bolt/Rivet holes
❑de Havilland DH.106 Comet (Commercial Airliner)
EGA324 – Stress Concentrations
Alexander Kielland drilling rig
disaster
EGA324 – Stress Concentrations
de Havilland Comet Accident
EGA324 – Stress Concentrations
Infinite Plate with Small Hole
❑ Kirsch Equations (1898)
=
∞
2
1 −
2
2
+
∞
2
1 −
2
2
1 −
32
2
cos 2
=
∞
2
1 +
2
2
−
∞
2
1 +
34
4
cos 2
= −
∞
2
1 −
2
2
1 +
32
2
cos 2
r – radial co-ordinate; a – hole radius
– loading direction; – stress
EGA324 – Stress Concentrations
Infinite Plate with Small Hole
❑ at the hole: r = a
= 0; = 0
= ∞ 1 − 2 cos 2 ;
Maximum stress at = ± 90° ( = 3∞)
=


=
EGA324 – Stress Concentrations
Notch Sensitivity Index
❑ Notch sensitivity (q) of material is a measure of
how sensitive a material is to notches or geometric
continuities
=
(ℎ )
(ℎ)
= 1 + − 1 1 ≤ ≤
=
− 1
− 1
0 ≤ ≤ 1
q is normally between 0-0.1 for ductile materials, 0.5
for cast irons and 0.5-1.0 for brittle materials
EGA324 – Stress Concentrations
SCF (SCC) Tables
W. C. Young and R. G. Budynas, Roark’s Formulas for Stress and Strain, 7th ed. New York: McGraw-Hill, 2002
EGA324 – Stress Concentrations
SCF (SCC) look-up graphs for stress-
raisers
EGA324 – Stress Concentrations
Neuber’s Rule
❑ Neuber’s rule states that the product of nominal
(elastic) stresses and strains is proportional to the
product of the local (elastic plastic) stresses and
strains
=
2 =
& – the nominal stress based on
specimen geometry
– theoretical stress concentration factor
(tables/look-up)
& – stress/strain value beyond yield point
of the material
EGA324 – Stress Concentrations
Neuber’s Rule
❑Typically know &
❑Need to determine the
effective stress concentration

EGA324 – Stress Concentrations
Example 1 – SCE for single
hole
❑ or determined by tables/equations:
= 3 − 3.140
20
100
+ 3.667
20
100
2
− 1.527
20
100
3
= 2.51
❑ found using net cross sectional area
=

−
=
64
100 − 20 8
∙
103
10−6
= 100 MPa
❑ Based on elastic behaviour peak stress
at edge of hole would be
= = 2.51 ∙ 100 = 251 MPa
EGA324 – Stress Concentrations
Example 1 – SCE for single
hole
❑ Nominal strain is read from the stress-strain
curve at = 100 MPa (A: = 5 ∙ 10
−4)
❑ Applying Neuber’s Rule:
=
2 = 2.51
2 100 5 × 10−4 = 0.315
❑ Intersection of = 0.315 (Point B)
yields = 243 MPa & = 13 ∙ 10−4. Effective
/ experimental stress concentration is:
or =


=
243
100
= .
❑ Results show 251 MPa exceeds the yield point of the material,
so actual peak stress at hole edge found using Neuber’s Rule
EGA324 – Stress Concentrations
Example 2 – SCE for two notches
❑ or determined by look up graph
,


=
5
60
= 0.08 ሶ3 → = 2.45
❑ found using net cross sectional area
=

∙
=
110
60∙12.5
∙
103
10−6
= 147 MPa
❑ Based on elastic behaviour peak stress at edge of notch would be
= = 2.45 ∙ 147 = 360 MPa
d=60mm; ρ=5mm;
t=12.5mm; P=110kN
EGA324 – Stress Concentrations
Example 2 – SCE for two
notches
❑ Results show 360 MPa exceeds the yield point of the material, so
actual peak stress at hole edge found using Neuber’s Rule
❑ A: = 147 MPa; = 7.2 ∙ 10
−4 –
can also be done using = when
E is known
❑ Applying Neuber’s Rule:
=
2 = 2.45
2 147 7.2 × 10−4 = 0.635
EGA324 – Stress Concentrations
Example 2 – SCE for two
notches
❑ B: Intersection of =
0.635 gives = 275 MPa & =
23 ∙ 10−4.
❑ The effective/experimental stress
concentration is:
or =


=
275
147
= .
= 0.635
EGA324 – Stress Concentrations
The Experiment…please watch in your own time
https://canvas.swansea.ac.uk/courses/36971/modules/items/1968427
EGA324 – Stress Concentrations
Experimental Aim
❑ The aim of this experimental investigation is to assess the
effects of introducing a stress concentration feature, such as
a hole or notch, to a component or structure
❑ A number of mild steel tensile specimens are provided with
and without stress raisers and the load/displacement
(stress/strain) characteristics of these specimens under
tensile loading need to be compared and contrasted
❑ An extensometer will be demonstrated on the plain
samples for more accurate strain measurements that would
typically be best practice
EGA324 – Stress Concentrations
Specimens
Plain
Sample
2.5 mm
single
notch
5 mm
Hole
2 mm
Hole
2x2.5 mm
double
notch
❑A total of 10 mild steel tensile test specimens will be provided
with five different geometries:
❑Key dimensions: thickness – 1mm, length – 180 mm, gauge length –
70mm, width of gauge in plain specimen – 15mm
EGA324 – Stress Concentrations
Specimens – Manufacture Route
2.5 mm
single
notch
5 mm
Hole
2 mm
Hole
2x2.5 mm
double
notch
❑ Tensile specimens have been manufactured through water
jet cutting
EGA324 – Stress Concentrations
Equipment
❑ The tests are performed
using a Hounsfield 25kN
electrically operated
tensile test machine
EGA324 – Stress Concentrations
Experimental Procedure
❑ Specimens are located between the jaws (wedge grips on the
Hounsfield machine)
❑ Setting the instrumentation to zero, tensile test is started with the
cross-head of the machine being driven at a displacement rate of
3 mm/min
❑ Force is measured using a load cell within the Hounsfield. Time,
load and displacement are recorded continuously during the test
❑ Once the specimen fractures, yield load and ultimate tensile load
should be noted to initially gauge any differences
❑ Start with the plain specimens and repeat for all – additional
details given in video & in the lab
EGA324 – Stress Concentrations
Table of Results
Yield Load [N] Ultimate Tensile Load [N]
Specimen 1 Specimen 2 Specimen 1 Specimen 2
Plain
2 mm central hole
5 mm central hole
Single-sided notch
Double-sided notch
❑ Initial table of results gives an insight to the effect of
the different stress concentration features
❑ Would expect to see a more complete results table in your
final reports, that may not include loads – with stress a better
comparator
EGA324 – Stress Concentrations
Analysing Tensile Data
❑ Data comes from the machine as
▪ time (s), displacement (µm), force (N)
❑ Needs to be converted to engineering stress, and
engineering strain, via:
=

0
& =
∆
0
Where is the applied force, 0 is the cross-sectional
area of the test piece, 0 is the gauge length and ∆ is
the extension
EGA324 – Stress Concentrations
Analysing Tensile Data
❑ This needs to be converted to true stress, and true
strain, which is approximated by:
= ( + 1) ∙ & = ln(1 + )
❑ Use Neuber’s rule as described above and in the
examples together with the calculated stress-strain to
compare the experimental SCFs with the empirical SCFs
and how do they compare?
❑ Nominal stress () values for the Neubers rule analysis for
the experimental data are to be taken from the elastic region
of the curve – using a consistent approach
EGA324 – Stress Concentrations
Typical Experimental Results
0
50
100
150
200
250
300
350
0 0.1 0.2 0.3 0.4 0.5 0.6
St
re
ss
(
M
P
a)
Strain
Eng Stress/Strain True Stress/Strain
0
50
100
150
200
250
300
350
0 0.1 0.2 0.3 0.4 0.5 0.6
Use Scatter with Smooth Lines, way too many data points to include every point!
EGA324 – Stress Concentrations
Fracture/Failure Surfaces
Available on Canvas:
❑What do they look like?
❑ What do the angles at which the surface
orientated tell you?
❑ Ductile or Brittle type fracture?
EGA324 – Stress Concentrations
Some Open Questions
❑ How does the plain sample properties compare to that of mild steel seen in
the literature?
❑How does the size of the hole affect the tensile characteristics of the
components and why ?
❑ How does the number of notches affect the tensile characteristics and why ?
❑ How do the tensile characteristics for a 5 mm hole compare with those for
two R2.5 mm notches and why ?
❑ Rank the five stress raisers in terms of their impact on the components’
behavior under load.
❑ Compare and contrast the degree and form of the deformation (strain) at
failure for the different stress raisers and how do they compare with the plain
specimen ?
❑ How do the shapes of the fracture surfaces compare and why might they be
different ?
❑ Comment on the repeatability of the tests and how could accuracy be
improved ?
❑ In addition – Computational Modelling aspects to be considered…
EGA324 – Stress Concentrations
Examples
Format:
❑ Review experimental protocol & raw data sets
❑ Individual tensile test data analysis
❑ SCFs – empirical & experimental (Neuber’s)
❑ Going further…
EGA324 – Stress Concentrations
Experimental Procedure
❑ Important points:
• Linear measurement – callipers/micrometer
• How would this influence your stress/strain results?
• Displacement rate – strain rate effects on materials
• Data acquisition rate
• Consistent jaw separation (returning jaws to same
starting position between tests)
• Testing standards:
• e.g. BS, ISO
EGA324 – Stress Concentrations
Experimental Practice
❑ Nominal Dimensions
– we are using these…
Case category
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COMSM0093 COMP281 AFM 333 IBUS1001 CP2501 Math 105A VE320 TEC101 SOCY2166 COM240 EFIM20033 ENGI4467 EFIM20034 EC 979 BE432 7EJ502 CSSE6400 COMP90025 Econ 312 ENG2031 ECO0006 EXERSCI 207 CAN209 ECO 137W FT312 ECO 2199 MATH32052 BDW4213 APS 360 SAS 6 CHME0017 PO92Q Economics 01AV MATH39512 MATH 5486 COMP8620 ISE102 ECON235 ECON 206 FI 345 PHAY0020 MATH0099 MATH0085 EFB106 PHY 134 MAE115 circuits Chemistry CSE12 MATH-UA 233 PHAS0038 COMP2140 CONMGNT 7049 COMP151101 ECON 485 ACX5903 ENGN6536 ARCH1162 Law PHAS0042 CCF1C03 MATH 351 Phys 139 AG942 MGT001371 APH106 ECON1051 MA4AM MA3Z7 NUMBER THEORY REST0006 SESS0026 STAT0023 TABL 2741 CMP4008Y EFIM10015 Acct 560 PSYC735 GEOG 5 ACCT3104 CSEC5616 PHIL1012 ITLS6111 ECON8022 5CCM226A compX123 MA4XJ R001 EWB LLP120 COMP643 ENG 103 ECON 23950 ACCT 207 FINC 616 Bio99 Math 137A Psychology BCPM0054 MATH20212 EC3450 CENV6141 PSY 205 CE335 CS365 Calculus ACSD INFT6800 MCIT INFO6030 PHYS 111LV H63EEJ ECON2151 ENGI 2181 MTRX1701 PLIT08009 CS240 Math 270 CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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