Problem 4. Breaking spaghetti

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PROBLEM NO. 4
BREAKING SPAGHETTI
Find the conditions under which dry spaghetti
falling on a hard floor does not break.
OVERVIEW

mechanical properties



impact
buckling



tube, camera, debris
results




fracture points
experimental setup


Euler’s critical buckling load
modes
simulation


Young’s modulus
weakest fracture force, various sizes
surface, number of spaghetti, angle dependence
comparison
conclusion
SPAGHETTI PROPERITES




lenght = 25.5cm
mass, density – five sizes
mass (g)
Young’s modulus (E)

stress/strain ratio

material characteristic
measured from beam deflection


w+ w0, w0 initial deflection (spaghetti mass)
F load applied at end
l
w0
w
F
0.474
0.627
0.81
0.980
1.177
density (kg/m3)
1515.474
1489.091
1486.303
1429.764
1399.133
SPAGHETII PROPERTIES

Young’s modulus E – beam deflection

r 
4
area moment of inertia - circular cross-section I 
2
l
w0+w
F
x

applied load F


m spaghetti mass , l lenght, E Young’s modulus
deflection for  = 0,    ℎ

bending moment
2
M  M
F

d w
dx
2
EI
3 
=
 3
BEAM DEFLECTION – YOUNG’S MODULUS
 determined from the coefficient
different applied loads F
 deflection measurement ymax

3
 = 0 + 
3
sp a g h e tti °2
•

1,76E+10 N/m
2
2
1,04E+10 N/m
2
9,13E+09 N/m
2
°5
•
0 ,0 2 2
1,31E+10 N/m
°4
•

2
°3
•

1,81E+10 N/m
°2
•

0 ,0 2 4
°1
b e a m d e fle c tio n / m

0 ,0 2 0
0 ,0 1 8
0 ,0 1 6
0 ,0 1 4
0 ,0 1 2
0 ,0 1 0
0 ,0 0 4
0 ,0 0 6
0 ,0 0 8
a p p lie d lo a d / N
0 ,0 1 0
0 ,0 1 2
IMPACT
elastic
 spaghetti fall accellerated (g)

impact with the surface
 both surface and spaghetti

acting like springs that obey Hooke's law
 force is proportional to the amount of deformations
velocity
height

time
time
IMPACT

momentum  is the surface force impulse 



force is small at first
enlarges to a maximum when spaghetti reverses directions
drops down as it jumps-off
• approximated constant F
• interested in maximum
• varies for different surfaces
• ∆ = ∆
• ∆  ∆   
fo rc e
•   − 
• causes spaghetti do deform
• break
Fg
tim e
F-reaction force
BUCKLING

displacement of structure transverse to load
F
δ
F


buckling model (spring)
elastic force moment
 Mel =  = δl


k-spring constant
load moment
 M = δ

M < Mel stable equilibrium - beam returns to the initial position

M = Mel indifferent equilibrium – remains at δ:  = l


initial buckling occurs
M > Mel unstable equilibrium – plastic deformations
F
β
l
BUCKLING

at M = Mel buckling occurs
critical condition
 depends on the beam support type


beam support
lower end simple (can rotate and slide)
 upper end free
 = 
f

x
B
w
A – deflection at point A
B – deflection at point B (f)
A
BUCKLING


buckling moment, equation of the beam elastic line
 = −  −  =
2

2
2

2

= 
x l
2 
− 2 

=
B
w
A
−



 = 
f
 harmonic oscillator equation, α2 =

to simplify calculations


 =   −   + , integrated equation of the beam elastic line

boundary conditions at point A,  = 0


 0 =  0 −  0 +  = 0  = −

 ′ 0 =  0 +  0 =  0

cos() = 0 critical states ,  = 2 − 1
minimal critical force n=1

 =
π2
42
π
,
2
= 1,2,3, …
BUCKLING

buckling modes

if the force  =
2 − 1
2
,  ∈  related to  = 2 − 1 2

spaghetti forms a sinusoidal line
depending on the relation – different buckling modes

greatest deflection – highest stress point


π2
42
critical buckling force
π2
  =
n=1
n=2
42
 °1 0.33 N, °2 0.58 N, °3 0.72 N, °4 0.91 N, °5 1.20 N

even the smallest impact forces exceed these values!


buckling deformation occurs
since surface reaction force is not related to 

IRREGULAR BUCKING MODE

greatest probability fracture points - simulation
n=3
FRACTURE POINT
irregular buckling modes
 debris lenght measured




most probable values and simulation compared
simulation
AutoCAD, Autodesk simulation multiphysics
 measured material properties and spaghetti dimension
 force acting conditions


whole surface, directioned through spaghetti
~gradual mesh

highest stress point
• center
FRACTURE POINT


highest stress points
 most probable fracture point
mashing conditions
 free ends
 force acting on the whole cross-section
EXPERIMENTAL SETUP
directed through a
long vertical pipe
 obtaining ~equal
impact velocities
 recording the process
camera

120 fps
 impact time and
velocity evaluation


debris measured
fracture point
 probability of fracture

PARAMETERS
weakest fracture force
 spaghetti size


Young’s modulus, area inertia moment, mass
surface hardness
 impact angle


buckling and bending
surface roughness
 number of spaghetti


interactions during the fall
SPAGHETTI SIZE DEPENDENCE
YOUNG’S MODULUS
π2
 2  - relation to critical buckling force
4

=

 =  ∆ - evaluated from video

/~2 = (2 − 1)2
∆

buckling mode ~ = 1.21
2 ,4
°5
w e a k e s t fra c tu re fo rc e [N ]
2 ,2
repeated measurements
marked spaghetti
image sequence observed
2 ,0
1 ,8
1 ,6
°3
1 ,4
°4
1 ,2
1 ,0
0 ,8
°1
°2
0 ,6
0 ,4
0 ,0 0 2
0 ,0 0 4
0 ,0 0 6
0 ,0 0 8
2
Y o u n g 's m o d u lu s *a re a in e rtia m o m e n tu m E I [N m ]
SPAGHETTI SIZE DEPENDENCE
YOUNG’S MODULUS

on a narrow force scale

smaller debris lenght is proportional to impact force


mode slightly changes
simulation and measured values agreement
a v e ra g e d e b ris le n g h t [c m ]

°1 spaghetti – 3 initial heights
metal surface – steel

debris lenght zero at

4 ,0
sim u la tio n re g re ssio n
m e a su re d va lu e s
3 ,5
3 ,0
2 ,5


2 ,0

experimenal value
1 ,5
0 ,7 0
0 ,7 5
0 ,8 0
fo rc e [N ]
0 ,8 5
0 ,9 0
 = 0.56 N
estimated from the simulation

 = 0.59 ± 0.03
SURFACE DEPENDENCE

HB – Brinell hardness
steel 120HB
 (oak) wood 3.8HB
 rubber not comparable
 rough/smooth stone
35HB


DIFFERENT
SURFACE
impact duration
 velocity after impact


losses due to surface
deformation
SURFACE DEPENDENCE HARDNES

necessary force remains the same
=

∆
0 + 1
=
∆
∆
0 velocity before impact ~shared, 1 velocity after impact varies!

linear fit coefficient =



 = 1.09 ± 0.05,    = 1.10 ± 0.02 spaghetti °2
IMPACT ANGLE DEPENDENCE

tube remains vertical



surface changes angle, smooth stone surface
surface reaction force is vertical to the surface Fs
 buckling 1 =  and bending 2 =  component
as the impact angle  
 bending force becomes more significant ()


strucutures are more sensitive to bending displacements
   


friction force is not great enough to keep the spaghetti steady
it slides of the surface – no fracture
Fs
α
F1
α
α
F2
α
IMPACT ANGLE DEPENDENCE

complex buckling/bending relation
 as the angle increases, bending gains significance over buckling


strucutures break more easily under bending loads
angle ~30° friction force is not great enough to keep the spaghetti
steady

slides – no fracture
•
•
•
tube height 3.25 m
spaghetti °2
at angles exceeding 80° no fracture
Fs
F1
α
F2
α
SURFACE DEPENDENCE ROUGHNESS



spaghetti °4, same stone two sides – rough, smooth
rough stone surface changes the spaghetti impact angle (surface imperfections)
 greater angle results in more bending deformation – longer debris
debris lenght zero for smooth surface (regression linear coefficient)

 = 1.46 ± 0.02 expected value (smooth)  = 1.48 ± 0.01
a v e ra g e d e b ris le n g h t [c m ]
3 ,5
sm o o th sto n e su rfa ce
ro u g h sto n e su rfa ce
3 ,0
2 ,5
2 ,0
1 ,5
1 ,0
0 ,5
0 ,0
1 ,5
1 ,6
1 ,7
1 ,8
1 ,9
2 ,0
fo rc e [N ] (e v a lu a te d fo r s m o o th s u rfa c e )
2 ,1
SINGLE ROD / BULK
DEBRIS LENGHT COMPARISON

too many movements for the force to be evaluated on camera
force and debris lenght are proportional
 on the same height≈  


spaghetti interact in a bulk
 change direction, hit the surface under a small angle


greater angle results in more bending deformation – longer debris
collide with each other
α
smooth stone surface
CONCLUSION


theoretical explanation buckling
conducted experiment


lowest fracture impact forces at vertical fall






= 0.59 ± 0.03
= 1.10 ± 0.02
= 1.29 ± 0.02 
= 1.48 ± 0.01
= 2.26 ± 0.03 
predicted using simulation and measured – agreement
surface hardness dependence

same minimum fracture forces


°1 
°2 
°3 
°4 
°5 
debris lenght at a force →   ℎ


conditions under which spaghetti does not break
different impact duration and velocity change - confiration
impact angle dependence


surface roughness dependence
number of spaghetti falling

changes the bending/buckling influence on dispacements
REFERENCES

V.Šimić, Otpornost materijala 1, Školska knjiga, 1995.
V.Šimić, Otpornost materijala 2, Školska knjiga, 1995.
Halliday, Resnick, Walker, Fundamentals of physics, 2003.

B. Audoly, S. Neukirch, http://www.lmm.jussieu.fr/spaghetti/


THANK YOU!
IMPACT
typical stress strain curve for brittle materials
 Hook’s diagram

yield strenght
material becomes permanently deformed
stress
fracture point

∆
=

0
∆

0
  =

fracture modes

proportional limit
Hook’s law velid
strain
  =  

for a long thin object
AREA MOMENT OF INERTIA

property of a cross section

geometrically: the strain in the beam
maximum at the top
 decrease linearly to zero at the medial axis
 continues to decrease linearly to the bottom



energy stored in a cross-sectional slice of the bent beam
• proportional to the sum of the square of the distance to the medial
axis
circle

symmetrical (same on every axis)
ρ
•  = 2

=

2 =
dρ
dA
4



3
2 =
0
2
r
BEAM DEFLECTION METHOD
beams with complex loads, boundary deflections
2
d w
EI
 equation of the elastic line for a beam M  M F 
2

2
2

load intensity and bending moment relation

consider probe beam with load intensity of  = 


dx
= −
same shaped stress diagram as the bending moment of our beam
   
2
2
= − = − =
2
− 2 

2 = 2 
  =  +  +  −    ,  = 0,  = 0


=


BEAM DEFLECTION METHOD
l
applied load-small weight
w0
w
F

C




 = 
2
2
3
= =
3
3

2

   

    

3
=
=
 −  
 3
SIMULATION REGRESSION
a v e ra g e d e b ris le n g h t [c m ]
4 ,0
3 ,5
sim u la tio n re g re ssio n
sim u la tio n
m e a su re d va lu e s
3 ,0
2 ,5
2 ,0
1 ,5
0 ,7 0
0 ,7 5
0 ,8 0
fo rc e [N ]
0 ,8 5
0 ,9 0
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