Coss section deendence on ski ole sti ness Johan Bystöm and Leonid Kuzmin Abstact Ski equiment oduce SWIX has ecently esented a new ai of ski oles, called SWIX Tiac, which di es fom conventional (ound) ski oles by having a tiangula coss section. SWIX claims that the main objective fo this design is that it has sueio sti ness to weight atio comaed to common ski oles. We ove in this ae that this claim in geneal is not tue. Moe seci c, we show that fo thin walled coss sections, a hollow cicula coss section has u to 36% bette sti ness to weight atio than a coesonding tiangula coss section. 1 Motivation fo this ae The insiation to this ae was some statements we found in a PDF document (see [5]) on the web age of ski equiment oduce SWIX (htt://www.swixsot.com). Moe seci c, this document contained some facts about thei ecently designed ski oles SWIX Tiac, which in contast to conventional (ound) ski oles have a tiangula coss section. Many of the to coss county skies in the wold today use SWIX Tiac ski oles, see Figue 1 below. One of SWIX main objectives fo this choice of design is found in the following statement: "Weight, sti ness and stength ae dictated by the amount of mateial used and the cicumfeence of the shaft. A tiangle encicled by a cicle has a smalle cicumfeence than the cicle. By using a tiangle geomety less mateial is needed and weight is saved. The distance fom the cente of the shaft, to the oute edge of the shaft is the dimension that gives a ole shaft its sti ness. A tiangle has the same distance fom the cente to a cone as the cicle has although only in thee diections. By oienting the cones in otimum diections we can achieve the same geomety sti ness (if not incease) with a educed amount of mateial and thus deceased oveall weight of the shaft." The conclusions in this statement does not coincide with what ou engineeing intuition tells us, thus the goal with this ae is to ove that the statement above in geneal is not tue. 1
Figue 1: Nowegian coss county skie Andes Gløesen in Oteää, Estonia, 010. Intoduction A long and slende constuction element (called a stut) is likely to fail in (Eule) buckling duing high comessive loading. Eule buckling is caused by elastic instability of the stut, whee the actual comessive stess is lowe than the ultimate comessive stess the mateial can handle. The maximum (comessive) axial load a long, slende stut can cay without buckling is called the citical load, F c. This load causes the stut to be in a state of unstable equilibium, whee the slightest lateal load will make the stut to fail by buckling. The fomula fo the citical load of a staight homogeneous column was found in 1757 by Swiss mathematician Leonhad Eule and states that F c = EI K l ; whee l is the length of the column, E is Young s modulus of the mateial, I is the aea moment of inetia (in the diection whee it has the lowest value) and K is a constant deending on the tye of suot at the ends. Assuming that the length is xed and that we have aleady chosen one mateial ( xed E), the only method we have to incease the citical load is to choose a suitable coss section with lage aea moment of inetia I; given by ZZ I = y dx dy; S
whee S is the coss section. Assume now that we want to comae two di eent coss sections with the same aea A (and hence the same total mass) ZZ A = aea(s) = dx dy: In ode to detemine the e ectiveness of a given coss section, we theefoe intoduce a so called shae facto, which is a dimensionless aamete which comaes the aea moment of inetia I of ou coss section with that of some given e-detemined simle coss section I 0 ; unde the assumtion of equal coss section aea A = A 0. To this end, let us take a solid cicula coss section with adius as ou simle coss section S 0 : Then the coss section aea A 0 and aea moment of inetia I 0 ae A 0 = ; S I 0 = = A 0 : We now de ne the shae facto fo a given coss section S to be the atio = I I 0 unde the assumtion of equal coss section aea A = A 0, which imlies that = I A 0 = I A : Thus, if a given coss section has a shae facto of fo instance = ; it means that it can withstand times highe comessive foce without buckling than a solid cicula coss section with the same coss section aea. Let us st comae the shae facto of a solid tiangula coss section. Then we will comute aoximate shae factos fo thin-walled coss sections (unde the assumtion that the thicknesses ae much lowe than the oveall dimension) and afte that comae with a numeical examle. Finally, we will comute the exact exessions fo the coss sections and fom that conclude that a hollow tiangula coss section is infeio to a hollow cicula coss section when one wants to maximize the citical foce without inceasing the mass, when the thickness is elatively small comaed to the oveall dimension. Fo eades inteested in theoy of buckling and beam bending, we efe to any of the classic books witten by Stehen Timoshenko, fo instance [3] o [] o the classic book by Jones []. 3
3 Stut buckling It is easonable to assume the ski ole is comessed by loads F that ae alied with a small eccenticity e measued fom the axis of the stut. This gives ise to a centic load F and a moment coule M 0 = F e: This moment causes the stut to de ect w (z) at the onset of loading. Theefoe, the bending moment M in the stut at height z ove the lowe end is M (z) = M 0 + F ( w) = F (e w (z)) : This bending moment will cuve the ski oles with a cuvatue that is ootional to the alied moment M; with the ootional coe cent being the eciocal of the oduct of Young s modulus E and the aea moment of inetia I (EI is called the bending sti ness), i.e. = d w dz = M F (e w (z)) = : EI EI That is, the di eential equation of the de ection cuve is whee This equation has the geneal solution w 00 + w = e; = F EI : w (z) = C sin z + D cos z + e; in which A and B ae constants of integation. It is natual to assume that the ends of the ski ole act as in joints, giving the bounday conditions Hence w (0) = D + e = 0 w (l) = C sin l + D cos l + e = 0: C = e (1 cos l) sin l D = e; = e tan l ; giving w (z) = e 1 tan l sin z cos z :
This gives the coesonding bending moment M (z) = F (e w (z)) = F e tan l sin z + cos z : To nd the maximum de ection ; we take the deivative of this exession and nd the value of z fo which it is zeo, that is w 0 (z) = e sin z tan l cos z = 0; which imlies that sin z l tan z = = tan cos z : Thus the maximum de ection oduced by the eccentic load occus at the midoint of the column z = l= and is l = w = e tan l! l l 1 sin + cos 1 = e 1 : cos l The coesonding maximum moment along the stut is then l M max = M = F e : cos l We see that the maximum de ection becomes lage when! c = l fom below. The so called citical load F c is then the limit F c = EI c = EI l : Let us now look fo the maximum stess in the stut. It will occu at the coss section whee the de ection and bending moments have thei lagest values, that is, at the midoint z = l=. Acting at this coss section ae the comessive foce F and the bending moment M max ; giving the stesses F=A and M max y max =I; esectively, whee A is the coss section aea and y max the distance fom the centoidal axis to exteme oint on the concave side of the stut. Thus the total stess is! = F A + M maxy max = F I A 1 + Ay max e : I cos l 5
Shae facto of a solid tiangula coss section Fist, let us comae a solid equilateal tiangula coss section to a solid cicula coss section. Let L be the side of the tiangle. Then the tiangula coss section, called S T ; has coss section aea and aea moment of inetia (see fo instance [1]) A T = aea(s T ) = I T = L 3 3 ; 3L ; giving a shae facto of T = I T A T = L 3 3 3L = 3 3 1:1: Hence a solid tiangula coss section is 1% sti e than a solid cicula coss section with the same coss section aea. Remak 1 The aea moment of inetia fo cicula and equilateal tiangula coss sections ae equal in all diections. This follows since I xx = I yy and I xy = 0; whee the last identity follows fom integation of an odd function ove a symmetic egion. Assume now that we otate the coodinate system the angle : Let us call the otated x-axis and the otated y-axis : Then the aea moments of inetia in the new coodinate system follow fom the exessions I = 1 (I xx + I yy ) + 1 (I xx I yy ) cos I xy sin = I xx ; I = 1 (I xx + I yy ) 1 (I xx I yy ) cos + I xy sin = I xx ; I = 1 (I xx I yy ) sin + I xy cos = 0: In othe wods, o les having cicula o equilateal tiangula coss sections ae equally sti in all diections. The same conclusion is also tue fo hollow cicula o equilateal tiangula coss sections. 6
5 Comaison of two di eent thin-walled hollow coss sections Let us now comae the e ectiveness of two di eent hollow coss sections. Let the st coss section SR be a hollow cicula coss section with mean adius Ro + Ri = and thickness t = Ro Ri ; whee Ro and Ri denote oute and inne adii, esectively. Let the second coss section ST be a hollow equilateal tiangula coss section with mean side L= Lo + L i and same thickness L o Li L; 3 whee Lo and Li denote oute and inne sides, esectively. Note that we assume that the thickness t is same fo both coss sections, see Figue 1 below. t= Figue : Two hollow coss sections. Then the coss sections have aeas AR = aea(sr ) = Ro AT = aea(st ) = 3Lo " t Ri = + 3Li 3 = L + 3t 7 t # L = t; 3t = 3Lt;
and aea moments of inetia " I R = R o R i = + t I T = L o 3 3 L i 3 3 = 1 3 3 L + 3t L # t = t ( + t ) 3 t; 3t = Lt (L + 3t ) L3 t : This gives aoximate shae factos fo the thin-walled hollow cicula and tiangula coss sections of R = I R A R T = I T A T = t ; = L 9t : Since we want to know which coss section is most e cient, we assume that we have the same coss section aea A R = A T ; giving L = 3 : Hence, exessed in tems of only and t, the aoximate shae factos fo equal coss section aeas ae Since R = t ; T = 7 t : R = 7 T 1:3678; we can conclude that the thin walled cicula coss section is aoximately 36 37% sti e than the tiangula coss section with equal coss section aea, egadless of the actual dimensions. Note that we have made a small aoximation when we discaded the tem containing t ; since if fo instance t=l < 0:1; we get that L < L + 3t < 1:03L ; imlying that the aoximation L + 3t L is valid and analogously fo the facto + t : 8
6 Induced stesses We saw in the intoduction that if we comess the ole with a given foce F; the induced stess is! = F A + M maxy max = F I A 1 + Ay max e ; I cos l whee F = EI Let us st conside the cosine facto. Since cosine is locally deceasing when its agument is inceasing fom zeo, we have that I 1 > I, 1 <, 1 cos 1l < 1 cos l Theefoe, a lage value of I will make the contibution fom the cosine facto to the stess smalle. Let us st assume that the contibution fom the cosine facto is indeendent of I: Again, we assume that we have the same coss section A R = A T : Then the induced stess deends on the quotient Z = y max I ; which is Z R = = 1 I R t and Z T = 3 L 3 = I T 3L t ; since two thids of the tiangula height is the maximal distance fom the centoidal axis to the oute suface fo the tiangula coss section. Hence, using we get that Z R Z T = L = 3 ; 3L = 3 3 0:60: If we now also include the fact that the aea moment of inetia fo a thin walled cicula coss section is lage than fo the coesonding tiangula coss section, the tem Ay max I 9 e cos l :
in the exession fo the induced stess in the cicula coss section is less than 60% of the cosonding value fo a thin walled tiangula section. All in all, the stess induced in a comessed ski ole is signi cantly lowe in a ole with cicula coss section than in a ski ole with tiangula coss section. 7 Comaison with some numeical values Let us comae the aoximate esults above with some exact numeical values. As an examle, we let the oute adius be R 0 = 8:0 mm and the inne adius be R i = 7:0 mm. This gives a mean adius of = 7:5 mm and thickness t = 1:0 mm. Then the mean tiangle side is L = 3 = 15:71mm giving oute and inne sides of L o = L + 3t = 17: mm, L i = L 3t = 13:98 mm. Then the cicula and tiangula coss section aeas ae and the aea moments of inetia A R = A T = 7:1 mm I R = R o R i = 1331: mm ; I T = esectively. Hence the cicula coss section is 1 3 3 L o L i = 980:7 mm ; 1331: 980:7 1:36 times sti e than the tiangula coss section. Remak In ode to detemine how lage eo one makes when doing the aoximation in the evious section, we can comae the aoximate values fo the thin walled coss sections with the exact values calculated in the examle above. The aoximate values ae which is about 1% o the coect values. I R = 3 t = 135: mm ; I T = L3 t = 968:95 mm ; 10
8 Exact values fo the shae factos We saw that while a solid equilateal tiangula coss section was bette than a solid cicula coss section, the oosite was tue fo thin walled hollow coss sections. It/R o I/ R o R i / R o Figue 3: I R and I T (nomalized by Ro) lotted against the atio R i =R o : Thus we ealize that thee must be some oint in between these extemes, whee a hollow cicula coss section is equally e ective as a hollow tiangula coss section, see Figue below whee we have lotted I R and I T (nomalized by Ro) against the atio R i =R o : Let us look fo this oint. We had that I R = t ( + t ) ; I T = Lt (L + 3t ) ; when L = 3 fo equal coss section aea. This gives the atio I R = ( + t ) I T L (L + 3t ) = 3 ( + t ) = 7 1 + 1 + 3t 1 + 7 3 11 t t :
Let us theefoe check the behavio of the function Fist, we see that f (x) = 7 1 + 1 x 1 + 7 x : f (0) = 7 1:36 which is the limit in the case when the thickness goes to zeo. Next, let us ty to solve the equation f (x) = 1; x > 0; which gives the oint whee a hollow cicula and equilateal tiangula coss sections ae equally e ective, see Figue 3 below. I/ It R i / R o Figue : I R =I T lotted against R i =R o : Intesection with 1 gives the oint whee hollow cicula and tiangula coss sections ae equally e ective. In this oint, we must have 7 1 + 1 x = 1 + 7 x ; that is, x = t = 1 8 7 1:037:
Hence a hollow cicula coss section is moe e cient than a hollow equilateal tiangula coss section when the thickness t is less than 1:037 times the mean adius and oosite when the thickness is lage than 1:037 times the mean adius. Since R o = + t ; R i = we have that R i = t R o + t = 7 3 81 6 1 0:317 in this case. Summing u, a hollow cicula coss section is moe e cient than a hollow equilateal tiangula coss section when the atio between the inne adius and the oute adius is lage than 0:317 and oosite when the atio is less than 0:317. 9 Conclusion We have shown that a cicula hollow coss section is sti e than a tiangula hollow coss section with the same coss section aea when inne adius is lage than about one thid of the oute adius. It theefoe does not seems to be coect fom the oint of otimizing bending sti ness to choose a tiangula coss section ove a cicula coss section fo thin walled stuctues. Howeve, thee might be othe hysical easons fo such a choice, not coveed hee o discussed in the document ublished by SWIX [5]. t ; Refeences [1] M. F. Ashby. Mateials selection in Mechanical Design. Elsevie/Buttewoth- Heinemann, 3d ed., 005. [] R. Jones. Buckling of Bas, Plates, and Shells, Bull Ridge Publishing, 006. [3] S. Timoshenko and J. M. Gee, Theoy of Elastic Stability, ed., McGaw-Hill, 1961. [] S. Timoshenko and D. H. Young. Theoy of Stuctues. McGaw-Hill Book Comany, nd Ed., 1965. [5] htt://www.swixtiac.com/ess_kit/swix_tiac_booklet.df 13