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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