ME362 Stress Analysis
ME 362: Stress Analysis
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
Instructor:
Prof. Sridhar Krishnaswamy
323 Catalysis Building
Tel: 1-4006
TA:
Yunyoung Kim
331 Catalysis Building
Tel: 1-4006
email: s-krishnaswamy@northwestern.edu
email: yykim@northwestern.edu
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
COURSE POLICY:
• Homework assignments: assigned every week, due start of class Wednesday of
the following week. Common discussion encouraged, but individual write-up.
• Computer Lab: work in groups of three; individual reports
• Project: Group project (teams of three); common report
• Exams: One 50-minute midterm (May 1, 2006) and one two-hour final (9-11am
Jun 6, 2006). Open handwritten notes; printed matter other than class handouts
and homework sets not allowed..
GRADING:
Homework
Midterm exam
Computer Lab
Project
Final exam
Option I
15%
15%
10%
20%
40%
Option II*
15%
0%
10%
20%
55%
*Option II is only available to those who have submitted all the homework
assignments and taken the midterm examination.
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
What is Stress Analysis all about?
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
What is Stress Analysis all about?
www.jokesandpictures.com/ funpages/fplosers.htm
This is not Psychology 362!
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
What is Stress Analysis all about?
Deals with deformation / motion and internal forces in engineering
structures under the action of applied loads.
Ansys
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
Why Stress Analysis?
Ensure that deformation is within acceptable limits and that
internal forces are below what the structure can withstand
for the expected loads on the structure.
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
Why Stress Analysis?
Ensure that deformation is within acceptable limits and that
internal forces are below what the structure can withstand
for the expected loads on the structure.
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
Big Dig Ceiling Collapse - 2006
http://www.boston.com/news/traffic/bigdig/articles/2006/07/28/bolt_system_graphic/
poor construction?
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
Paris Airport Terminal Roof Collapse - 2004
Causes of collapse:
innovative architectural design
But poor structural design?
http://www.istructe.org/thestructurale
ngineer/HC/getfile.asp?id=1117.
Northwestern University
“The French Transport Ministry commission found
that the structure had a low initial reserve of
strength and a combination of factors led to the
major collapse including:
The high ‘flexibility’ in the structure under dead load
and external actions (temperature, etc) was increased
by cracking, which may have been the result of
insufficient or misplaced reinforcement.
• A lack of robustness and redundancy to transfer
loads away from a local failure.
• High local punching stresses.
This innovative structure has a form, detailing and
sensitivity to construction misfits and tolerances and
to creep and thermal effects which are well outside
the range envisaged by normal code requirements and
analysis. The commission recommends that there
should be a requirement for all unusual and complex
structures to be subject to more rigorous modelling
and analysis with independent checking.”
Ref: J.G. Woods IstructE.org
Sridhar Krishnaswamy
ME362 Stress Analysis
Montreal Overpass Collapse - 2006
http://www.foxnews.com/story/0,2933,216926,00.html
poor construction?
Northwestern University
Misplaced rebar caused overpass collapse:
report
Updated Thu. Oct. 12 2006 10:52 PM ET
CTV.ca News Staff
The devastating overpass collapse in
Laval, Que. that left five people dead
happened because steel reinforcing rods
were installed incorrectly in the structure,
according to reports.
Sridhar Krishnaswamy
ME362 Stress Analysis
Aloha airlines flight 243 - 1988
A major portion of the upper crown skin and
structure of section 43 separated in flight
causing an explosive decompression of the
cabin. The damaged area extended from
slightly aft of the main cabin entrance door,
rear ward about 18 feet to the area just
forward of the wings and from the left side
of the cabin at the floor level to the right
side window level.
From the NTSB Accident Report
http://www.aloha.net/~icarus/index.htm
Environmental degradation
Æ multi-site damage
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
Liberty ships:
Liberty Ships were lightly armed cargo vessels built
in the US for transporting desperately needed
supplies across the U-boat infested Atlantic to a
beleaguered Europe in WWII. Some 2700 vessels
were built from 1942 until the end of the war. Such
huge numbers were possible only through
prefabricated all-welded construction to a standard
design, together with a massive investment of capital,
materials and workers. Towards the end of the
programme one vessel was completed in less than
five days.
It later became clear that the failures could be attributed to:
- the all- welded construction which eliminated crack- arresting plate boundaries which
are present in riveted joints
- the presence of crack- like flaws in welded joints performed by inexperienced operators
pressed into service by the exigencies of the programme
- the use of materials whose low resistance to crack advance ( toughness ) was further
reduced by low temperatures.
http://www.mech.uwa.edu.au/DANotes/fracture/maritime/maritime.html
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
INTELLIGENT STRUCTURAL HEALTH MANAGEMENT
design correctly / construct properly / monitor structural degradation
Probability of
detection
Measured
state of structure
Structural Health
Monitoring System
PROGNOSTICS
Current
state of structure
Damage growth
characteristics
Failure Model
DIAGNOSTICS
Structural Model
Probabilistic prognosis of damage evolution
(damage vs time or cycles)
low
Failure probability
within preset interval
high
Inspection and Repairs
at maintenance facility
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
Stress Analysis
Determine deformation / motion and internal forces in engineering
structures under the action of applied loads.
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
A brief review – Hooke’s Experiment
Consider uniaxial stretching of a uniform cylindrical rod.
fracture
F
proportional limit
L
Initial length L
Cross-sectional area A
Northwestern University
L
δ
F
δ
Hooke’s law:
extension is in proportion to force
up to a point : F = K δ
Sridhar Krishnaswamy
ME362 Stress Analysis
Eine kleine problem with Hooke’s Law
F
L
δ
F
F
geometry 1
geometry 2
same material
material 1
material 2
same geometry
δ
δ
L
δ
F=Kδ
Κ(geometry,material)
F
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
Stress and Strain make their appearance
In order to separate the effect of geometry and material, we scale out the
geometric effect by defining:
Stress: σ = F/A = (force) / (cross-sectional area) [N.m-2]
Strain: ε = δ/L = (stretch) / (original length)
[dimensionless]
σ
L
m aterial 1 (in d ep en d en t o f A , l)
m aterial 2 (in d ep en d en t o f A, l)
δ
F
ε
Hooke’s law – better version:
stress is in proportion to strain
up to a point : σ = E ε
where E is Young’s modulus of the material
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
Deeper Meaning of Stress
F
F
F
F
Stress is a “ continuum” measure of the intensity of
internal forces that arise when atoms are displaced from
their equilibrium position due to applied loads.
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
Deeper Meaning of Strain
A
A’
Δx
B
L
B’
δ
F
change in length A' B '− AB
Strain: ε =
=
original length
AB
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
The concept of displacement
x
A
Δx
u(x)
B
A’
L
u(x+Δx)
B’
δ
F
Strain-displacement relation
A' B'− AB u ( x + Δx) − u ( x)
ε
=
=
Strain:
AB
Δx
Northwestern University
du
ε=
dx
Sridhar Krishnaswamy
ME362 Stress Analysis
A consequence of Newton’s laws
σ (x + Δx)A − σ (x)A + ρgA Δx = 0
σ(x) A
x
a
b
c
d
b
a
L
⎧ σ (x + Δx) − σ (x) ⎫
⎨
⎬ + ρg = 0
⎩
⎭
Δx
W=ρg AΔx
Δx
c
d
dσ
+ ρ g = 0 Equilibrium
equation
dx
σ(x+Δx) A
P
P
σ (x) = + ρ g(L − x)
A
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
Stress Analysis
Course Content
Characterization
-internal forces
-geometry of deformation
-material response
Solution methods
-“exact” analytical
- energy methods
- finite element method
Analysis of structures
Northwestern University
Cast of characters: {σ, ε, u}
Newton’s law:
dσ
+ ρg = 0
dx
Geometry of
deformation:
du
ε=
dx
Material
response:
σ=Eε
Sridhar Krishnaswamy
ME362 Stress Analysis
Finite Element Analysis
- Ansys -
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
Team Projects
Airplane Wing
Northwestern University
St. Louis Arch
Suspension Bridge
Chicago Porch
Collapse
Sridhar Krishnaswamy
ME362 Stress Analysis
and now for something completely different
Monty Python
and yet the same…
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
SATELLITE
λ
a
ω
EARTH
GEOSYNCHRONOUS
ORBIT
mg
Northwestern University
re2
λ2
(ωλ )
2
= m
λ
⎧ gre ⎫
⇒ λ = ⎨ 2⎬
⎩ω ⎭
2
1/ 3
= 42x10 6 m
Sridhar Krishnaswamy
ME362 Stress Analysis
SATELLITE
λ
a
ω
SKY-HOOK
EARTH
GEOSYNCHRONOUS
ORBIT
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
SATELLITE
λ
a
ω
SPACE ELEVATOR
EARTH
GEOSYNCHRONOUS
ORBIT
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
SATELLITE
λ
a
ω
ORBITAL TOWER
EARTH
GEOSYNCHRONOUS
ORBIT
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
SATELLITE
λ
a
ω
ORBITAL TOWER
EARTH
GEOSYNCHRONOUS
ORBIT
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
one enormous 1D stress analysis problem
SATELLITE
λ
b
d
a
c
a
r
ω
TOWER
EARTH
GEOSYNCHRONOUS
ORBIT
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
Can we really build an orbital tower?
References:
• Isaacs, A.C. Vine, H. Bradner and G.E. Bachus, (1966), `Satellite Elongation into a True
Sky-Hook,' Science, vol.151, 11th February, 1966.
• Arthur C. Clarke, (1978), `Fountains of Paradise,' (fiction), Ballantine Books, New York.
• Check out the web (http://liftoff.msfc.nasa.gov/academy/tether/spacetowers.html) for
variations on the above theme
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
one enormous 1D stress analysis problem
SATELLITE
λ
b
d
a
c
a
r
ω
TOWER
EARTH
GEOSYNCHRONOUS
ORBIT
d
weight = [mass][acceleration due to
gravity at location of the element]
b
1
2
W
Fc
a
c
2
A
r
+
A
⎧
⎫
⎡ 1
2 ⎤
W = ρ
Δr ⎨g e2 ⎬ e n
⎣ 2 ⎦
⎩ r ⎭
Δr
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
one enormous 1D stress analysis problem
SATELLITE
λ
b
d
a
c
a
r
ω
TOWER
EARTH
GEOSYNCHRONOUS
ORBIT
d
b
1
2
W
Fc
a
centrifugal force = [mass][(angular
velocity)2.radial distance from center of
rotation]
⎡ ⎧ A1 + A2 ⎫
⎤ 2
Δr
FC = − ⎢ ρ ⎨
ω r ] en
⎬
[
⎥
⎣ ⎩ 2 ⎭
⎦
c
Δr
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
one enormous 1D stress analysis problem
SATELLITE
λ
b
d
a
c
a
r
ω
TOWER
EARTH
GEOSYNCHRONOUS
ORBIT
d
internal forces = [maximum allowable
stress].[cross-sectional area]
b
1
2
W
Fc
a
c
F1 = + σ max A1 en
F2 = − σ max A2 e n
Δr
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
one enormous 1D stress analysis problem
W + F1 + F2 + FC = 0
⎡ ⎧ A1 + A2 ⎫
⎤
⎧ re2 ⎫
⎡ ⎧ A1 + A2 ⎫
⎤
Δr ⎨g 2 ⎬ + σ max A1 − σ max A2 − ⎢ ρ ⎨
Δr ⎥ [ω 2 r ] = 0
ρ ⎨
⎬
⎬
⎢⎣ ⎩ 2 ⎭
⎥⎦
⎣ ⎩ 2 ⎭
⎦
⎩ r ⎭
d
σ max
b
1
⎡ A2 − A1 ⎤ = ρ ⎧ A1 + A2 ⎫
⎬
⎨
⎩ 2 ⎭
⎣ Δr ⎦
⎡ re2
2 ⎤
g
− ω r
⎢⎣ r 2
⎥⎦
2
W
Fc
a
c
ρ A
dA
=
σ max
dr
⎡ re
2 ⎤
g
− ω r
⎢⎣ r2
⎥⎦
2
Δr
⎡ ρ ⎧ 2 ⎛ 1 1⎞
ω 2 2 2 ⎫⎤
A(r ) = A(λ ) exp −
−
λ − r )⎬
+
⎨ gr
(
⎢⎣ σ max ⎩ e ⎝ λ r ⎠
2
⎭⎦⎥
Northwestern University
Sridhar Krishnaswamy
ME362 Stress Analysis
Can we really build an orbital tower?
Parameters:
Radius of earth, re = 6.4x106m
Acceleration due to gravity on earth's surface, g = 9.81ms-2
Angular velocity of earth/tower/satellite = ω = 2π/(24*60*60) rad/sec.
Radius of geosynchronous orbit, λ = 42x106m.
Density of orbital tower material = ρ
Maximum allowable tensile stress in orbital tower material = σmax.
⎡
A(re )
ρ
7 ⎤
= exp ⎢−
4.86x10 )⎥
(
A( λ )
⎣ σ max
⎦
What materials can we use to build such a tower?
Northwestern University
Sridhar Krishnaswamy