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EGA324 – Stress Concentrations
创建日期：2024-05-20 15:54:26
所属分类：物理Physical
标签： 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…
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COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 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