An Immediate Formula for the Radius of Curvature of A Bimetallic Strip

Transcription

1 An Immediate Formula for the Radius of Curvature of A Bimetallic Strip G. D. Angel School of Engineering and Technology, University of Hertfordshire G. Haritos School of Engineering and Technology, University of Hertfordshire Abstract An alternative formula has been derived to enable a close prediction of the radius of curvature of a thin bimetallic strip that at initial ambient temperature, is both flat and straight, but at above ambient temperature, forms into an arc of a circle. The formula enables the evaluation of the radius of curvature of the strip as a function of heating or cooling. A formula for calculating the radius of curvature of a bimetallic strip already exists, and was produced by Timoshenko in his paper on Bimetal Thermostats. The formula by Timoshenko has been vigorously proven, tried and tested and accepted in countless papers and journals since its original publication. The formula introduced by this work, very closely approximates to the Timoshenko formula for equal thicknesses of the two mating metals within the bimetallic. The drawbacks of the Timoshenko formula are that it is both complex and unwieldy to use, and requires some form of electronic spreadsheet to enable its evaluation. The formula put forward here is both simple and quick to use, making it more immediate. For the correlation of the new formula, Timoshenko s formula is used as a datum, or benchmark. From the simulation a good overall correlation was shown to exist between the Timoshenko generated values and the values generated by the new formula put forward here. Key words: Timoshenko, bimetallic strip, radius of curvature, formula, thin. 1. Introduction The original Timoshenko [1] bimetallic bending formula was published in 1925 and since then it has been applied in by multitudes of engineers and scientist and referred to in many papers such as by Krulevitch [2], Prasad [3] and books Kanthal [4]. Whilst it has been proven and accepted to be the formula to evaluate the hot radius of curvature of an initially flat bimetallic strip, it is an unwieldy and a complex formula to evaluate. This work introduces a new simpler, quicker method of evaluating the radius of curvature of a bimetallic strip from an initially flat ambient condition that has been uniformly heated. When a bimetallic strip is uniformly heated along its entire length, it will bend or deform into an arc of a circle with a radius of curvature, the value of which, is dependent on the geometry and metal components making up the strip. As will be seen later on, the nature of the bend as a function of temperature change from ambient is characteristically asymptotic. The new formula introduced here, closely approximates to the Timoshenko formula with the exception of accommodating the change in the thicknesses of the two mating metals making up the bimetallic strip. It is important to note that in the majority of applications of bimetallic strip, the ratio of the thickness of the two constitute metals is normally one to one, i.e. of equal thickness. This comes about due to the way that the bimetallic strip is manufactured. The dominant method in the mass production of commercially available bimetallic strip involves either hot or cold rolling the two separate metals under intense pressures to produce interstitial bonding of the atoms at the bi- material interface Uhlig [5]. Under such conditions, it is expensive, because of setup costs, to make special separate metal thicknesses unless specifically required. Moreover, there is no data to support that different thicknesses of the bimetals in the strip would have any performance benefit over equal thickness bimetal strip. Cladding of metals Haga[6] is used to provide a product with a less expensive base metal that benefit from a thinner skin for decorative and or surface protection purposes, although clad metals are technically bimetallic metals, clad bimetallic strip is not being used for its functional bending qualities. Therefore the need to cater for separate material thicknesses is not required for most applications where the bimetallic bending qualities of a bimetallic strip are being exploited. 1312

2 2. Timoshenko Formula From the Timoshenko [1], the radius of curvature of a bimetallic strip is given by: eqn.(1) Where ρ is the radius of curvature, total thickness of the strip. are the individual material thicknesses. is the ratio of thicknesses. is the ratio of the Young s Moduli. are the hot and cold temperatures states., is the linear Modulus of the two materials. & are the coefficients of linear thermal expansion for the two metals. is assumed to be numerically larger than Fig.1 shows a bimetallic strip in two states of heating, at state 1, at ambient temperature, the strip will be flat with no discernible radius of curvature R. At state 2, uniformly distributed heating will cause the strip to form into a radius of curvature. Note that has a numerically higher coefficient of linear thermal expansion and thus naturally wants to extend further than the side with The differences, leads to internal stresses, forces and moments at the material interface, resulting in the bending as shown at state 2. It is commonly known that the internal force developed within a metal bar by heating or cooling, can be written as follows: eqn.(2) Where F is the force (N). α is the coefficient of linear expansion of the metal (. ΔT is the temperature change of the metal from ambient (K). A is the cross-sectional area of the bar( ). E is the Youngs modulus of the material of the bar ( ). Eqn.(2) can be re-written in terms of the stress, since, thus the internal stress due to heating: By substitution of eqn.(3) where y is assumed to be the distance from bi- metal interface to the outer edge, this is also equal to half the total thickness of the bimetallic strip. From the well-known simple bending equation substituting and re-arranging using the first two terms of the simple bending equation, thus ; eqn.(4) Where: R is the radius of curvature of the bimetallic strip to the bimetallic joint center line (mm). t is the total thickness of the bimetallic strip(mm). with the average coefficient of linear expansion of both metals ( ). Thus eqn.(5) This is a first order estimate of the radius of curvature of the strip as a function of change in temperature that approximates to the Timoshenko formula, see Fig.2. Fig.1 Bimetallic strip in two states of heating 3. Derivation of the Approximation Formula The derivation is based upon the amalgamation of two well established formulae, with the addition of new correction relationships that are a combination of the ratios, sums, and quotients of the coefficient of linear expansion of the metals. Where possible, the nomenclature employed in the Timoshenko formula will be used in the new formula. The simple derivation resulting in eqn.5 and shown in Fig.2, provides a rough or first order estimate of the radius of curvature as a function of temperature change in the strip. It should be noted that although eqn.5 is an approximation to the Timoshenko formula, the accuracy of eqn.5 tends to improve as the temperature increases, or as the expression becomes more asymptotic. Furthermore, the nature the first order derivation, closes resembles the Timoshenko formula and follows a similar trajectory, that of an asymptotic curve. 1313

3 Thus that expands to Combining and re-arranging and reducing, thus: eqn.(7) Fig. 2 Comparison of Timoshenko ρ vs. R Using a similar approach to Timoshenko where the ratios of the thicknesses of the metals and the sum of the thicknesses in the metals, play an influential part of the derivation. In the derivation put forward here, the ratios of the coefficients of linear expansions, sum and differences are used as a correction factor with similar effect that ultimately modifies the first order expression eqn.5, to a close approximation of the Timoshenko formula. The rationale for using the coefficients of linear expansions to correct the first order curve, is based upon the fact that in this derivation, the thicknesses of the two constitute metals are assumed equal for reasons explained earlier. From Fig.2, it can be seen that the radius of curvature of the approximate curve R, is slightly larger numerically, than the Timoshenko line. Thus a possible correction factor needs to multiply R by a number slightly less than unity. Introducing a proportional correction factor reduces the value of R to a very close approximation of Timoshenko ρ. Thus letting and this provides an initial lowering of R. and also letting and be the proportional difference in the coefficients of linear thermal expansions, then the proportional correction influence on R is: = eqn.(6) Eqn.6 is a very effective modifier when multiplied by the eqn.5 see Fig. 3 for the very close approximation of. At the same time, the correction must take into account the rapid change of curvature in the lower temperature range of 20 C to 80 C. A further correction factor is achieved if the ratios of the sums and differences are also considered, thus letting: and and combining the ratios. Fig.3 showing effects of the correction coefficients Therefore adding derived components, to eqn.5, enhances the accuracy of eqn.5 to that of Timoshenko. Thus the radius of curvature R, as a function of temperature is given by: eqn.(8) From eqn.8 it can be seen that in this derivation, there is no requirement to know E, the Young s modulus of the two separate metals of the bimetallic strip. Also eqn.8 can be also be expressed directly as: eqn.(9) Where t is total thickness of the bimetallic strip (m). is the change in temperature of the strip from ambient ( C ). R is the radius of curvature of the strip (m). is the correction factors multiplied by ( C ). Eqn.10 was used to generate data in the simulation. comparison to eqn.1 by Timoshenko. 4. Simulation Data For the generation of comparison data, parameters & were varied in the simulation; = 0.4, 0.8, 1.2, 1.6, 2, 4, 6, 8 and 10 mm, being the total thickness of strip. It should be noted that in most applications of bimetallic strip, the total thickness is usually quite thin, up to 1mm thick for switching applications [4]. 1314

4 5. Simulation Data For the proof of the correlation between the new formula proposed in this paper, and Timoshenko s original formula, two separate simulations were performed. The first simulation was based around a bespoke Bimetallic strip SBC206-1 from Shivalik [7] ; 100mm long x 5mm wide x 0.4mm thick, these were the starting values of the first simulation set. For simulation set 1, the following data was assumed; The strip thickness t, was varied from 0.4mm thick to 10mm thick. = 213 ( ) ; Young s modulus of Steel side of the bimetallic strip. = 145 ( ) ; Young s modulus of Invar 36 side of the bimetallic strip. = ( ) coefficient of linear thermal expansion for Steel side of the strip. = ( ) coefficient of linear thermal expansion for Invar 36 side of the strip. both equal, total thickness of the strip. = (20 C) assumed ambient temperature constant throughout the simulation. Input variable temperature ( C). change in temperature, ( C). = radius of curvature evaluated by eqn.(1) Timoshenko formula (m). R = radius of curvature evaluated by eqn.(10) proposed new formula (m). For the second simulation, a combination of different materials within the bimetallic strip and also a variety of thicknesses of strip were used. It should be noted that the material combinations put forward in the second simulation may not be practical for the manufacture of bimetallic strip by modern mass production methods of cold pressure rolling, but they can be produced by other, older fabrication methods such as by riveting the two metals together. The simulation data in set 2 have been included in a random fashion to test the robustness of the new formula. The second simulation data set is as shown in Table 1. Note that Invar 36 is used as the common mating material since it possesses a very low coefficient of linear expansion as compared with all other engineering materials. As per simulation 1 data set, the ambient temperature is assumed to be = 20 C. Material Mix Specificaton Thicknesses Coefficient of Linear Expansion Youngs Modulus Reference Invar 36 B mm 1.45 x 10-6m /m K Gpa [8] Aluminium EN AW 1050A H mm 23.5 x 10-6m /m K 69 GPa [9] Invar 36 B mm 1.45 x 10-6m /m K Gpa [8] Nickel Silver BS2870 NS mm 13 x 10-6m /m K 118 GPa [10] Invar 36 B mm 1.45 x 10-6m /m K Gpa [8] Brass BS 2870 CZ mm 18.7 x 10-6m /m K 111GPa [11] Invar 36 B mm 1.45 x 10-6m /m K Gpa [8] Mild Steel BS EN1 A 0.8mm 11 x 10-6m /m K 195 GPa [12] Invar 36 B mm 1.45 x 10-6m /m K Gpa [8] Copper BS 2879 C mm 16.6 x 10-6m /m K 117 GPa [13] Invar 36 B mm 1.45 x 10-6m /m K Gpa [8] Stainless Steel AISI BS mm 17.3 x 10-6m /m K 213Gpa [14] Table 1 Simulation set 2 6. Simulation Results The two formulae of eqn.1 and eqn.10 were used to generate data values of and R respectively for both simulation sets. The radii of curvature for simulation set 1 were plotted against the change of temperature for each thickness of bimetallic strip, see Fig

5 Fig.6 Timoshenko Comparison Invar vs. Nickel silver 0.6mm thick Fig.4 Timoshenko Comparison range 0.4mm to 10mm For simulation set 2, see Fig. s 5,6,7,8,9,10. Fig.7 Timoshenko Comparison Invar vs. Brass 1.3mm thick Fig.5 Timoshenko Comparison Invar vs. Aluminium 0.508mm thick Fig.8 Timoshenko Comparison Invar vs. Mild Steel 1.6mm thick 1316

6 Fig.9 Timoshenko Comparison Invar vs. Copper 2.6 mm thick Fig.10 Timoshenko Comparison Invar vs. St. Steel 0.8 mm thick From Fig. 4 it is evident that for the 153 overall data points from simulation set 1, yielding nine different thicknesses of the strip, an excellent correlation between the two formulae has resulted. From simulation set 2, comparing the six different materials types and thickness combinations yielding 66 data points, a very good overall correlation between the new formula and Timoshenko was shown to exist. 7. Discussion of results From the 9 data tables generated from simulation 1, the overall average error was 0.64%. The break down in average error for each thickness was as follows: 0.4mm thick = average 0.73%. 0.8mm thick = average 0.62%. 1.2mm thick = 9 average 0.51%. 1.6mm thick = average 0.64%. 2.0 mm thick = average 0.88%. 4.0mm thick = average 0.62%. 6.0mm thick = average 0.40% 8.0mm thick = 0 average 0.60% 10mm thick = 0 average 0.60% From the breakdown of average error it is evident that the error fluctuates slightly as a function of the thickness of the strip. The maximum fluctuation of error lies between the 2mm thick and 6mm thick test strips, was only 0.48%. From simulation set 2, the error breakdown was as follows: 0.508mm thick = average 1.01% Invar vs. Aluminium 0.6mm thick = average 0.54% Invar vs. Nickel Silver 1.3mm thick = average 0.75% Invar vs. Brass 1.6mm thick = average1.58% Invar vs. Mild Steel 2.6mm thick = average 0.4% Invar vs. Copper 0.8mm thick = average 0.52% Invar vs. St. Steel The maximum fluctuation of error of the function was 1.18% which occurred between the 1.6mm and 2.6mm simulation data. It should be noted that the second test was simultaneously subjecting the formula to all changes of the data, i.e. different thicknesses, different Young s modulus, and different coefficients of linear expansions. The average error in simulation set 2 was 0.8% and the maximum fluctuation error was 1.18%. From simulation set 1 the average error was 0.622% and the maximum fluctuation error was 0.48%. The derivation has shown, and the correlation of the new formula to the Timoshenko formula has proven, that the values of Young s Modulus for each metal within the bimetallic strip are not required in the evaluation of the new formula. This is very useful since it is not always quick and easy to find the Young s modulus of the metals, and the value as used in the Timoshenko formula, takes the average of both Young s moduli which can only be an approximation at best. It should also be noted that this work assumes that Young s modulus for a bimetallic strip is the average of the two constitute metals making up the strip, as per the original Timoshenko formula. 8. Conclusions The results prove an acceptable overall maximum error of 1.18%, and an overall average error of 0.64%. Thus it has been demonstrated that the formula put forward here can be a useful, quick, easier alternative to Timoshenko s radius of curvature formula for close estimates of the radius of curvature as a function of temperature change. Furthermore, it has been shown that the new formula works without the requirement of first knowing the Young s moduli of the two metals within the bimetallic strip. Most usefully, the formula presented in this work can be evaluated without the need of an electronic spread sheet or program as is required with the more complex Timoshenko s formula, but can be easily used on a hand held calculator at a fraction of the time. 1317

7 9. References [1] Timoshenko, S., Analysis of Bi-metal Thermostats. JOSA, (3): p [2] Krulevitch P, J.G.C., Curvature of a Cantilever Beam Subjected to an Equi-Biaxial Bending Moment, in Materials Research Society Conference [3] Prasad, K., Principle and Properties of Thermoststat. Journal of Materials, [4] Kanthal, Kanthal Thermostatic Bimetal Handbook. 2008, Box 502, SE Hallstahammar, Sweden: Kanthal [5] Uhlig, W., et al, Thermostatic Metal, Manufacture and Application. 2nd, revised ed. 2007, Hammerplatz 1, D-08280,Aue/Sachsen: Auerhammer Metallwerk GMBH [6] Haga, T., Clad strip casting by a twin roll caster. World Academy of Materials and Manufacturing Engineering, (2): p [7] Shivalik,S.Bimetallic strip supplier. 2013; Available from: Accessed online Sept [8 alloys.net /invar_nickel_iron_alloy.html Physical_properties, [9] /downloads/1050a.pdf, [10] Zn/CuNi10Zn27.pdf [11] [12] mild steel, [13] Bronze_CW024A-C106_122.ashx [14]Munday. A,J, and Farrar.R.A An Engineering Data book, MacMillan, London, ISBN Appendix Data tables Simulation set 1 t2+t1=0.4mm m m t2+t1=0.8mm m m Tc( C) Th( C) ΔT Rt Rg Rt/Rg Tc( C) Th( C) ΔT Rt Rg Rt/Rg t2+t1=1.2mm m m t2+t1=1.6mm m m Tc( C) Th( C) ΔT Rt Rg Rt/Rg Tc( C) Th( C) ΔT Rt Rg Rt/Rg t2+t1=2.0mm m m t2+t1=4.0mm m m Tc( C) Th( C) ΔT Rt Rg Rt/Rg Tc( C) Th( C) ΔT Rt Rg Rt/Rg

8 t2+t1=6.0mm m m t2+t1= 8.0mm m m Tc( C) Th( C) Δ t2+t1= 10.0mm Tc( C) Th( C) Δ m m Tc( C) Th( C) Δ Simulation Set 2; 0.5mm Simulation Set 2; 0.6mm Invar 36 vs. Aluminium Invar 36 vs. Nickel silver Simulation Set 2; 2.6mm Simulation Set 2; 0.8mm Invar 36 vs. Copper Invar 36 vs. St. Steel Simulation Set 2; 1.3mm Simulation Set 2; 1.6mm Invar 36 vs. Brass Invar 36 vs. Mild Steel 1319