Origin of Unusually High Rigidity in Selected Helical Coil Structures

David Tom´anek1, ∗ and Arthur G. Every2 1Physics and Astronomy Department, Michigan State University, East Lansing, Michigan 48824, USA 2School of Physics, University of the Witwatersrand, Private Bag 3, 2050 Johannesburg, South Africa (Dated: February 18, 2018) Using continuum elasticity theory, we describe the elastic behavior of helical coils with an asymmetric double-helix structure and identify conditions, under which they become very rigid. Theo- retical insight gained for macro-structures including a stretched telephone cord and an unsupported helical staircase is universal and of interest for the elastic behavior of helical structures on the micro- and nanometer scale.

PACS numbers: 63.22.-m, 62.20.de, 62.25.Jk

I. INTRODUCTION (a) (b) (c)

Helical coil structures, ranging from a stretched telephone cord in Fig. 1(a) and an unsupported spiral staircase in Fig. 1(b) on the macro-scale to DNA and proteins L = on the micro-scale abound in Nature. Since their elastic inner [H2+(4πR)2]½ behavior is governed by the same laws of Physics inde- stringer pendent of scale, insight obtained on the macro-scale will H axis H benefit the understanding of helical micro- and nanos- R tructures. An intriguing example of unusual high rigidity i on the macro-scale, which has remained unexplained to outer date, is the unsupported all-wooden spiral staircase in the stringer Loretto Chapel1 in Santa Fe, New Mexico, constructed Ro around 1878 and shown in Fig. 1(b). In the following we explore the elastic behavior of this structure using continuum elasticity theory in order to identify the reason φ for its rigidity2–4. Since continuum elasticity theory applies from nanometer-sized fullerenes and nanotubes5–7 4πR Figure 1 to the macro-scale, we expect our approach to be useful to explore the rigidity of helical structures on the micro- FIG. 1. (a) Photograph of a coiled telephone cord with the same topology as an unsupported spiral staircase. (b) Re- and nanometer scale. touched photograph of the unsupported spiral staircase at the The use of continuum elasticity theory rather than the Loretto Chapel as constructed. (c) Trace of the helical inner case-specific finite-element method3 in this case is mo- or outer stringers of the staircase on the surface of a cylinder, tivated by our objective to identify the universal origin which can be unwrapped into a rectangle. of the high rigidity of the Loretto spiral staircase and related helical coils with an asymmetric double-helix structure. Theoretical insight gained for macro-structures in- can be equivalently described as a helicoid, or a “filled- cluding a stretched telephone cord and an unsupported in” helix, with finite nonzero inner and outer radii. helical staircase is universal and of interest for the elastic There is an extensive literature on the properties of he- behavior of helical structures on the micro- and nanome- 2 ter scale. lical springs, which dates back to Love’s treatise , as well as the more recent Refs. [3,4] and literature cited therein. arXiv:1706.03288v1 [cond-mat.soft] 10 Jun 2017 Yet the compound helical structure of the Loretto staircase and related helical coils appears to have escaped II. ELASTIC BEHAVIOR OF A HELICAL COIL attention in publications so far. The rigidity of the con- necting steps provides the spiral staircase with a remark- From a Physics viewpoint, the spiral staircase of able degree of stiffness, as we show below. We propose Fig. 1(b) is a compression coil or a helical spring with a that this property is shared by similar helical structures rather high pitch. It can be characterized as an asymmet- independent of their scale. ric double-helix structure consisting of an inner stringer Any coil of radius R and total height H, such as the coil of radius Ri and an outer stringer coil of radius Ro, inner and the outer stringer of the staircase, lies within and spans two turns in total. The two stringer coils are a cylindrical wall of the same height, which can be un- connected by rigid steps of width Ro − Ri. The staircase wrapped onto a triangle, as shown in Fig. 1(c). Let us 2

2 2 2 first consider the case of incompressible inner and outer Ignoring δLi , δH and δRi terms in the limit of small stringers of height H that are separated by the constant deformations, we obtain distance Ro − Ri, which defines the step width. 2 2 2 2 According to Fig. 1(c), the equilibrium length Li of Li + 2LiδLi = H + 2HδH + 16π (Ri + 2RiδRi) . (9) the inner stringer coil with two turns is related to its Subtracting Eq. (1) from Eq. (9), we obtain equilibrium radius Ri and its equilibrium height H by 2 2 2 2 2 2LiδLi = 2HδH + 16π 2RiδRi . (10) Li = H + (4π) Ri . (1) The equivalent relation applies, of course, to the outer With the assumption δRi = δRo = δR, we can rewrite Eq. (10) and its counterpart for the outer stringer as stringer of radius Ro. We first consider an incompressible stringer of constant 2 LiδLi = HδH + 16π RiδR , length Li, which is stretched axially by δH, causing the 2 LoδLo = HδH + 16π RoδR . (11) radius Ri to change by δRi. Then,

2 2 2 2 Combining all terms containing δR on one side, we can Li = (H + δH) + 16π (Ri + δRi) . (2) eliminate δR by dividing the two equations. This leads Subtracting Eq. (1) from Eq. (2) and ignoring δH2 and to δR2 terms in the limit of small deformations, we obtain L δL − HδH R i i i = i (12) 2 LoδLo − HδH Ro 2HδH + 16π (2RiδRi) = 0 (3) and, by rearranging terms, to and consequently Li H Lo H H δL − δH = δL − δH = κ , (13) R i R R o R δRi = − 2 δH . (4) i i o o 16π Ri where κ is a variable to be determined by minimizing the Considering the outer helical stringer to behave inde- strain energy U of the deformed stringers. U is given by pendently for the moment, we expect the counterpart 2 2 of Eq. (4) 1 δLi 1 δLo U = C Li + C Lo H 2 Li 2 Lo 2 2 δRo = − 2 δH (5) 1 δL δL 16π Ro = C i + o , (14) 2 Li Lo to describe the outer stringer. The step width should then change by where C is the force constant describing the elastic response of the stringers to stretching. For a macroscopic HδH 1 1 stringer with the Young’s modulus E and the cross- δ(Ro − Ri) = δRo − δRi = − 2 − . (6) 16π Ro Ri sectional area A, C = EA. From Eq. (13) we get

The only way to keep the step width constant, corre- HδH + κRi δLi = (15) sponding to δ(Ro − Ri) = 0, is to have either zero step Li width Ri = Ro, reducing the double-helix to a single- helix, or to suppress any change in height with δH = 0. for the inner stringer. Similarly, we get Even though each individual stringer coil can change HδH + κRo its height H while keeping its length L constant, the δLo = (16) assumed rigid connection between the inner and outer Lo stringers makes the staircase completely rigid. for the outer stringer and can now rewrite the strain en- Next, we relax the constraint that each stringer should ergy as maintain its length when the staircase changes its height H. Nevertheless, we will still maintain the assumption 1 (HδH + κR )2 (HδH + κR )2 U = C i + o . (17) of a ﬁxed step width 3 3 2 Li Lo

Ro − Ri = const. (7) The optimum value of κ is obtained from requiring ∂U/∂κ = 0. This leads to that translates to δRi = δRo = δR. The compressible inner stringer helix will still be characterized by its equi- (HδH + κR )R (HδH + κR )R i i + o o = 0 , (18) librium length Li and equilibrium radius Ri. Its defor- 3 3 L Lo mation caused by changes of its axial height by δH is i then described by which can be rewritten as

2 2 2 2 3 3 (Li + δLi) = (H + δH) + (4π) (Ri + δRi) . (8) (HδH + κRi)RiLo + (HδH + κRo)RoLi = 0 . (19) 3

We can regroup the terms to get For the sake of a fair comparison to a straight staircase with a slope given by tan(ϕ), as seen in Fig. 1(c), we do 3 3 2 3 2 3 HδH RiLo + RoLi + κ Ri Lo + RoLi = 0 , (20) not use the pitch, but rather the local slope tan(ϕi) = H/(4πRi) of the inner stringer to characterize how steep which yields expressions for the optimum values of κ, δLi the staircase is. The Loretto staircase is rather steep near and δLo. We get the inner stringer with tan(ϕi) = 1.9, corresponding to ◦ 3 3 ϕi≈61 . RiLo + RoLi The stringers of the Loretto staircase have a rectan- κ = −HδH 2 3 2 3 , (21) Ri Lo + RoLi gular cross-section of 6.4 cm×19.0 cm, and so we have for the cross-section area A = 121 cm2. Considering the elastic modulus E≈1010 N/m2 for wood along the HδH R L3 + R L3 i o o i grains, we obtain C = E·A = 1.2×108 J/m. Thus, the δLi = 1 − Ri 2 3 2 3 Li Ri Lo + RoLi spring constant of the double-helix structure describing 2 6 Li Ro the staircase could be as high as k = 4.8×10 N/m. = HδH 2 3 2 3 (Ro − Ri) , (22) Ri Lo + RoLi Now consider the staircase suspended at the top and free to deform in the axial direction. The largest defor- and mation will occur when a load is applied on the lowest 3 3 step. A person of 100 kg in that location would apply HδH RiLo + RoLi δLo = 1 − Ro net force F = 981 N to the staircase, causing an axial L R2L3 + R2L3 o i o o i elongation of δH = F/k = 0.2 mm, which is very small. 2 LoRi In reality, the staircase is anchored both at the top = HδH 2 3 2 3 (Ri − Ro) . (23) Ri Lo + RoLi and the bottom, and its total height is constrained. The weight of a person climbing up the stairs is supported To interpret this result, let us first consider the inner and by the fraction x of the staircase below, which is under outer stringers to be independent first and only then con- compression, and the fraction (1 − x) of the staircase sider the effect of a constant step width separating them. above, which is under tension. The local axial deflection In response to δH > 0, the inner stringer prefers to re- δh along the staircase is then given by duce its radius significantly, but this reduction is limited by the constant-step-width constraint. Thus, the length F of the inner stringer is increased and it is in tension. For δh(x) = x(1 − x) . (26) k this to occur, the stairs must have been pulling the inner stringer outwards, and so the steps are subject to tensile The largest deflection occurs in the mid-point of the stair- stress. In response to increasing its height, also the outer case, with x(1 − x) = 1/4. The local vertical deflection stringer prefers to reduce its radius. But the constant- caused by a person of 100 kg standing at this point should step-width constraint reduces its radius even more, so be only δh≈0.05 mm. As expected intuitively, there is no that the outer stringer ends up in compression. To ac- deflection for a person standing either at the top or at complish this, the steps must be pulling it inward and the bottom. again should be subjected to tensile stress. In response to δH < 0, the strains in the inner and the outer stringers will change sign and the steps will be under compressive III. BENDING DEFORMATION OF A HELICAL stress. COIL The total strain energy amounts to

1 H2δH2(R − R )2 1 Structurally, the telephone cord in Fig. 1(a), the un- U = C o i = kδH2 , (24) 2 R2L3 + R2L3 2 supported helical staircase in Fig. 1(b), and a rubber hose i o o i share one important property: all elastic material is on where k is the spring constant of the entire double-helix the surface of a hollow cylinder, forming a tube. In a fur- structure, given by ther degree of simplification, we may ignore the interior structure of this elastic tube and describe its stretching, H2(R − R )2 twisting or bending deformations using continuum elas- k = C o i . (25) 2 3 2 3 ticity theory7. So far, we have considered stretching as Ri Lo + RoLi the dominant response to tensile stress. When a compres- We note that in in a single-stringer case, characterized sive load F is applied to the helical coil, there will always by Ro − Ri = 0, the axial spring constant k would vanish be a reduction in the height H proportional to F/H due in our model. to compression. But there will only be bending, which For the initially mentioned spiral staircase in the is synonymous with buckling, if FH2 exceeds a critical 8,9 Loretto chapel, Ri = 0.26 m, Ro = 1.00 m, and value . Our task will be to identify this critical value. H = 6.10 m. From Eq. (1), we get Li = 6.92 m and This finding agrees with published continuum elastic- Lo = 13.97 m. ity results for long-wavelength acoustic phonon modes in 4 tubular structures7, which suggest a fundamentally dif- The critical condition for bending to occur is that this 2 ferent dispersion relation ωZA∝k for bending modes, in work should exceed the total bending energy, stark contrast to ωLA,T A∝k for stretching and torsion. Since the vibration frequency is proportional to the de- F δH > Ub,tot . (34) formation energy, it makes sense that bending is preferred to compression at small values of k corresponding to long and translates to wavelengths and large H values, and vice versa for short FH2 > 2π2D . (35) wavelengths and small H values. t As expanded upon further in the Appendix, we con- We can see from the parameters of the Loretto staircase sider an elastic tube of radius R and height H that could that it is very stable against buckling. From the above be either compressed or bent by the displacement am- 2 3 3 2 2 equations, we obtain 2π Dt = 2π c11R = π R Hk/(1− plitude A. We will consider the tube material to be de- 2 2 2 2 α ), which simplifies to 2π Dt = π R HEA/[H(1 − scribed by the 2D elastic constant c11 and the Poisson 2 2 2 2 α )] = π R EA/(1 − α ). Since R≈Ro = 1 m, α = 0.24, ratio α. Then according to the equation Eq. (A4) in the 8 2 9 and EA = 1.2×10 J/m, we get 2π Dt = 1.25×10 Jm. Appendix, we obtain for the total axial compression en- Assuming a compressive load F = 103 N, this quantity is ergy vastly greater than FH2 = 103×6.12 = 3.7×104 Jm. The 1 load would have to be increased by more than four orders U = πc 1 − α2 RA2 . (27) c,tot 11 H of magnitude, or the height increased by more than two orders of magnitude, to cause buckling. Comparing this expression to Eq. (24), we can express c11 by IV. ELASTIC BEHAVIOR OF SIMILAR H c = k , (28) HELICAL STRUCTURES IN NATURE 11 2πR(1 − α2) Every helical structure, from the coiled telephone cord where k is given by Eq. (25) and, for the sake of simplic- in Fig. 1(a) to the spiral staircase in Fig. 1(b) and ity, we use R = R . o to submicron-sized α-helices found in proteins, can be According to Eq. (1), assuming that load-induced mapped topologically onto a helical coil. The helix we changes of the stringer length L can be neglected, any describe here, which turns out very rigid, consists of two change in height H would cause a reduction of the radius helical coils with different radii, separated by a constant R and the circumference 2πR. We obtain distance. This particular design could clearly be utilized δ(2πR) H2 δH (H/R)2 δH to form man-made nanostructures that will be very rigid. = − = − 2πR (4π)2R2 H 16π2 H It is tempting to explore whether any existing struc- δH tures in Nature may look similarly and behave in a similar = −α , (29) manner. Among the biomolecules that immediately come H to mind is the double-stranded DNA that, coincidentally, thus defining the Poisson ratio is also left-handed. DNA, however, does not fulfill the constant-step-width assumption, since the bases from the (H/R)2 α = . (30) two strands are non-covalently bound in pairs, forming 16π2 a “breathing” rather than a rigid unit. Another system 10 According to Eq. (A9) of the Appendix, the total bending known for its toughness, collagen , has only some inter- energy is given by strand covalent bonding, but not at every step. More- over, its tripe-helix structure differs from the model we 3 2 discuss. After a long search, we believe there are no real 5 2 R 4 A Ub,tot = 4π c11A = 4π Dt , (31) counterparts in Nature of the structure we describe, at H H3 least not among biomolecules. 3 where Dt = πc11R is the flexural rigidity of the tube. The reduction in the height of the tube due to bending is given by V. DISCUSSION

" #1/2 Z H du 2 2π2A2 Our main objective was to elucidate the origin of the δH = dx 1 + z − H ≈ (32) previously unexplained high rigidity of the unsupported 0 dx H spiral Loretto staircase by developing a suitable formal- ism. Our numerical results should be taken as rough es- to lowest order in A, and the work done by the external timates. We expect the local axial deflections δh of this load is thus staircase caused by load to be significantly larger than the 2π2FA2 estimated values presented above. The estimated value F δH = . (33) H of the effective spring constant of the staircase helix is 5 likely to be reduced significantly by defects and human- APPENDIX made joints in this all-wooden structure. Further reduction would come from considering other deformation A. Deformation Energy due to Compression and modes including lateral compression or stretching of the Bending wooden steps and, to some degree, bending. Elastic response to shear stress in the stringers should significantly As introduced in the main text, any helical structure contribute to the spring constant especially in low-pitch may be mapped onto an elastic tube of radius R and spirals, with the coiled telephone cord as an intuitive ex- height H that could be either compressed axially or bent. ample. Even though the spiral staircase of Fig. 1(b) is We will consider the tube aligned along the x-direction a high-pitch spiral, allowing for shear deformations in and the tube material to be described by the 2D elastic the stringers should further reduce its effective force con- 7 constant c11 and the Poisson ratio α. We will consider stant. Even if all these factors combined should decrease the local distortions to be described by the force constant by 1-2 orders of magnitude, we may still expect a maximum local axial deflection δh of not x u = (−A) (A1) more than 1−2 cm in case that each of the 33 steps were x H loaded by the weight of a person. As expanded above, since the height is significantly larger than the radius, in the case of axial compression, and bending should not play a significant role as a possible h x π i response to applied load. We also note that at a later uz = A sin 2π − + 1 (A2) stage, the staircase had been augmented by a railing, H 2 shown in Fig. A1(a) in the Appendix. This railing does in the case of bending, where A denotes the amplitude of not affect the elastic response of the staircase under load. the distortion. As mentioned above, we have not found any asymmet- According to Eq. (A2) of Reference [7], the compres- ric double-helix structure in Nature that is rigid and does sion energy per length is given by not stretch much. Should such a structure exist, its stiffness should benefit from a constant separation between 1 du 2 U = 2πRc 1 − α2 x the helical coils. c 2 11 dx A 2 = πc 1 − α2 R . (A3) 11 H VI. SUMMARY AND CONCLUSIONS

In summary, we have used continuum elasticity theory to describe the elastic behavior of helical coils with (a) (b) an asymmetric double-helix structure and have identi- ﬁed conditions, under which they become very rigid. Theoretical insight gained for macro-structures including a stretched telephone cord and an unsupported helical staircase is universal and of interest for the elastic behavior of helical structures on the micro- and nanometer scale.

ACKNOWLEDGMENTS

A.G.E. acknowledges ﬁnancial support by the South African National Research Foundation Grant No. 80798. D.T. acknowledge ﬁnancial support by the NSF/AFOSR EFRI 2-DARE grant number EFMA-1433459 and the Figure S1 hospitality of the University of the Witwatersrand, South FIG. A1. Retouched photographs, with the background dig- Africa, where this research was performed. We thank itally removed, of the spiral staircase in the Loretto Chapel, Garrett B. King for his assistance with the literature Santa Fe, New Mexico. (a) Current view of the staircase in- search for rigid helical structures in Nature and their cluding the railing, which had been added long after construc- schematic graphical representation and thank Dan Liu tion. (b) Likely view of the staircase as constructed, with no for useful discussions. railing. 6

The total compression energy of the tube of height H which leads to is then 1 2π 4 x π U = πc R3A2 sin2 2π − . (A8) b 2 11 H H 2 2 2 1 Uc,tot = UcH = πc11 1 − α RA . (A4) H The total bending energy is obtained by integrating Ub in Eq. (A8) along the entire height H of the bent tube, yielding According to Eq. (A15) of Reference [7], the bending energy per length is given by R 3 U = 4π5c A2 . (A9) b,tot 11 H

2 1 d2u Finally, assuming the same distortion amplitude A for U = (πc R3 + πDR) z . (A5) b 2 11 dx2 bending and compression, we can determine the ratio of the compression and the bending energy

2 2 Since the ﬂexural rigidity D of the wall “material” Uc,tot 1 − α H = 4 (A10) vanishes due to the separation between adjacent helix Ub,tot 4π R strands, we obtain that is independent of the amplitude A and the elastic constant c11. We see that for H >> R, Uc,tot >> Ub,tot, 1 d2u 2 indicating that bending is energetically more aﬀordable U = πc R3 z . (A6) b 2 11 dx2 and thus dominates. The opposite situation occurs for H << R, when axial compression dominates.

Using the expression in Eq. (A2) for the bending defor- B. Photographs of the Staircase mation, we obtain Retouched photographs of the spiral staircase in the d2u 2 2π 4 x π Loretto Chapel, Santa Fe, New Mexico, are presented in z = A2 sin2 2π − , (A7) Fig. A1. dx2 H H 2

∗ [email protected] 6 Dan Liu, Arthur G. Every, and David Tománek,“Con- 1 Bill Brokaw, Loretto Chapel: The Miraculous Staircase tinuum approach for long-wavelength acoustic phonons (The Creative Company, Lawrenceburg, Indiana, USA, in quasi-two-dimensional structures,” Phys. Rev. B 94, 2002). 165432 (2016). 2 A. E. H. Love, A Treatise on the Mathematical Theory of 7 Dan Liu, Arthur G. Every, and David Tománek,“Long- Elasticity (Cambridge University Press, Cambridge, UK, wavelength deformations and vibrational modes in empty 1927). and liquid-filled microtubules and nanotubes: A theoreti- 3 D Fakhreddine, T Mohamed, A Said, D Abderrazek, and cal study,” Phys. Rev. B 95, 205407 (2017). H Mohamed, “Finite element method for the stress analysis 8 Leonhard Euler, De curvis elasticis, Methodus inveniendi of isotropic cylindrical helical spring,” Europ. J. Mech. A- lineas curvas maximi minimive proprietate gaudentes Solids 24, 1068–1078 (2005). (Lausanne and Geneva, Switzerland, 1744). 4 L. M. Zubov, “The problem of the equilibrium of a he- 9 Stephen P. Timoshenko and James M. Gere, Theory of lical spring in the non-linear three-dimensional theory of Elastic Stability, 2nd ed. (McGraw-Hill, New York, 1961). elasticity,” J. Appl. Math. Mech. 71, 519–526 (2007). 10 Rita Berisio, Luigi Vitagliano, Lelio Mazzarella, and Adri- 5 D. Tomanek, W. Zhong, and E. Krastev, “Stability of mul- ana Zagari, “Crystal structure of the collagen triple he- tishell fullerenes,” Phys. Rev. B 48, 15461–15464 (1993). lix model [(Pro-Pro-Gly)10]3,” Protein Sci. 11, 262–270 (2002).