THE UN I VE R S I T Y OF M I C H I G AN COLLEGE OF ENGINEERING Department of Engineering Mechanics Department of Mechanical Engineering Tire and Suspension Systems Research Group Technical Report No. 8 BENDING CHARACTERISTICS OF CORD-RUBBER LAMINATES o. r S. K. C-lark! Project Directors: -'S-.. K[<, i'Cla k and R. A. Dodge ORA Project 02957 administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR August 1961

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The Tire and Suspension Systems Research Group at The University of Michigan is sponsored by: FIRESTONE TIRE AND RUBBER COMPANY GENERAL TIRE AND RUBBER COMPANY B. F. GOODRICH TIRE COMPANY GOODYEAR TIRE AND RUBBER COMPANY UNITED STATE RUBBER COMPANY iii

TABLE OF CONTENTS Page LIST OF FIGURES vii NOMENCLATURE ix FOREWORD 1 SUMMARY 3 ANALYSIS OF BENDING STIFFNESS 5 EXPERIMENTAL MEASUREMENT OF BENDING STIFFNESS 11 EXPERIMENTAL INSTRUMENTATION 19 APPENDIX 25 ACKNOWLEDGMEJT 33 REFERENCES 35 DISTRIBUTION LIST 37:V

LIST OF FIGURES Figure Page 1. Schematic view of a small section of a two-ply laminate. 2. Schematic view of a finite cylindrical tube loaded by end moments. 11 3. Schematic view of a cylindrical tube under the action of radial line load applied on a circumference. 12 4. Tubular test specimen showing dimensions. 14 5. Typical line load vs. pressure curve for cylindrical tubes subjected to constant axial loads and variable internal pressure with a restraining cable on a circumference. 16 6. Bending stiffnessess D and DQ as predicted from Eqs. (10a) and (10b) vs. cord angle a, along with experimental values of D~ and Di. 17 7. Photograph of specimen in testing machine. 20 8, Schematic view of the measuring system and the constricting cable. 21 9. Photograph of the measuring system and the constricting cable. 22 10. Free-body diagram of a differential strip from a cylindrical shell under the action of forces and moments. 25 11. Free-body diagram showing the equilibrium between the constricting cable and the membrane hoop stresses in the cylindrical shell. 29 vii

NOMENCLATURE English Letters a Special case of v. A,B,C,D Constants of integration in the general solution of the differential equation of deflection of a beam on an elastic foundation. aij Constants associated with generalized Hooke's law using properties based on complete cord tension or compression. De,DoD Effective plate stiffnesses in bending.,ESY_,ErR Elastic constants resulting from inversion of Hooke's law when shear strain is zero.,Er,F Elastic constants for orthotropic laminates with cords completely in tension or compression when shear strain is zero. h Ply thickness of each lamination in an orthotropic laminate. k Negative inverse of the radius of curvature. kf Stiffness constant for an elastic foundation. m Bending moments per unit length applied to a specimen. M Total bending moment exerted on a specimen. n Half the total number of plies in a laminate, N Axial load per unit of circumference. p Total distributed load acting on the inside of a tubular specimen. Pi Internal pressure in the tubular cylinders. P Radial load per unit of circumference. Shear force exerted in the direction of the thickness of the wall of a tubular specimen. ix

NOMENCLATURE (Concluded) s Roots of the characteristic equation given by a beam on an elastic foundation, t Plate. thickness of an orthotropic laminate. w Displacement in the ~ direction. xy,z Orthogonal coordinates aligned along and normal to the cord direction, Greek Letters One-half the included angle between cords in adjoining plies in a two-ply laminate. Pv Real and imaginary parts of the roots of the characteristic equation. Strain, Q Circumferential angle included between two different radii of the cylindrical specimens. A constant, X = rkf/i D. Orthogonal coordinates aligned along and normal to the orthotropic axes of an orthotropic laminate, Stress, x: -:

FOREWORD The properties of orthotropic laminates subjected to local bending stresses about symmetric axes have not yet been discussed in this series of reports. It is necessary to consider bending so that a quantitative measure of bending stiffness for this type of structure may be obtained. This result should be useful in formulating a strain-energy function for a toroidal shell and also for understanding the basic reaction of a cordrubber laminate to bending stresses. This report is an attempt to apply the theory proposed in Ref. 1 to the bending problem.

SUMMARY The classical theory of pure bending is used in conjunction with the generalized Hooke's law to develop an expression for the stiffness of lamninated, orthotropic cord-rubber sheets which are subjected to bending deformations about symmetric axes of the structure. This stiffness may be predicted for a laminated structure with any number of plies and any angle between the cords of alternate plies if the four basic elastic moduli of a single ply are known. Tests were run to determine the validity o the expressions predicting stiffness. The agreement between experiment and theory was good. The stiffness was found to be dependent on all four elastic constants, with the effects of extensional moduli and cross modulus particularly large. 35

ANALYSIS OF BENDING STIFFNESS I Neutral Axis Fig. 1, Schematic view of a small section of a two-ply laminate, Consider a two-ply laminate, as shown in Fig. 1, in which the cords are either completely in tension or completely in compression. The angles a are between i, a symmetric axis of the laminate, and the cord direction in an individual ply, +c for the cords of one ply and -o for the cords of the other, h is the thickness of a ply and ~ is an orthogonal coordinate in the direction of the laminate's thickness. Assuming the cords all to have the same stiffness, say due to being preloaded into tension, let bending be about either of the symmetric axes t or T so that twist vanishes. From Ref. 1, the elastic law relating stress 5

and strain for each ply of Fig. 1 is: e = allao + a12a + a13a5, E = a2ls, + a22c + a23c'(iT) En = a3 lc- + a32Cr + a33aC~1 This is recognized as the generalized Hooke's Law, Due to the assumed symmetry of the deformation when bent about the 5 or rq axis, the shear strain cE due to twist must vanish. Hence, when ( and a are applied, Eqo (1) becomes: E = allo + al2ar + aI3Cyt 6~ = a21,l + a22Cr + a23oji (2) 0 = a31l + a32Cr + a33as where rl is the shearing stress necessary for strain compatibility between the plies, a~ acts on each ply and does not contribute directly to the bending moment about the neutral axis. The last of Eqs, (2) may be solved for all, and this value may be inserted into the first two of Eqso (2), giving a2 —-- + af- a13a23 8id = a321 - a+ 22 a23 These may be inverted to yield

(4) C = Eale + FPe where a11 -a all3122 - a2 a212 -12 a3= a3333 _2. a22 a23 a3 It might be noted here that the moduli ES, Eq, F are dependent on the Young's moduli EB and EBI and also on the cross modulus F.'* By substitution of Eqso (15) from Ref. 1 into Eqo (3) of this report, these relationships are found to be: - = BF;. - = _ (5a) F2 - E F2 - E F2 B E It is thus possible to predict ES, Er, and F knowing the elastic properties Ex, Ey, Fxy, and Gxy of a single sheet of orthotropic material, In formulating a stiffness constant, ithbe assumhtions of pure bending of thin plates will be used; that is, it will be assumed that planes remain plane

and perpendicular to the axes q and ~ and that displacements and displacement gradients are small, On the basis of these assumptions, the straindisplacement relations for a plate are 2w 2w = - 2 - k] Equations (4) may now be rewritten CT =- - (k + k1) (7) n = - ( lks + Tk- )With reference to Fig, 1, the moments per unit length me and m~ may be formed by multiplying the appropriate stresses by the lever arm, Cd:, and integrating from (-+nh) to (-nh) where 2n is the total number of plies and h is the thickness of each ply. This is a generalization to 2n plies with alternating angles of +ca or -.O Previously only a two-ply structure was considered, It is justified by the fact that the stress distribution in any ply is given by Eq. (2), within the confines of the assumptions in force here, Thus, +nh 2(nh)3 Inm = _ Ad-n - (Eekg +k) (nh) nh 3 (8)

These may be written me = - DekS - Dkq (9) mq = - Dlkq - DkS where *.-(nh)3 = I, (10a) _ 2 3 _ D3 = I(nh) =i, ( ob) D = F.-(nh) ='I (10c) represent the effective plate stiffnesses in beanding. With knowledge of thickness and elastic properties of a single sheet of orthotropic material, a bending-stiffness constant can thus be predicted for a laminate of these sheets. The fact might again be recalled that this relationship is valid only if the cords in a cord-rubber laminate are either in tension or compression. The case in which some cords are in compression and some in tension will be taken up in a later report.

EXPERINENTAL MEASUREMENT OF BENDING STIFFNESS Some effort was expended in planning a test which would be particularly applicable to the clear demonstration of local bending rigidity in a cordrubber laminate. Since only specimens of cylindrical shape were available, it seemed most direct to treat these as beams and to measure their deflection when loaded by end moments, such as shown in Fig. 2, However, this \t -:XJ Fig. 2, Schematic view of a finite cylindrical tube loaded by end moments. type of loading presented real difficulties in three areas: (a) It was not possible to preload the cords without inducing frictional end moments which would have been difficult to measure accurately. (b) The bending induced in the tube is in the form of a distribution of shell-membrane stresses around the tube, and is not composed of bending which is local in nature, so that the neutral axis lies inside the tube wall, between the two plies. It is desired to illustrate bending of this latter type, not the distribution of membrane forces, (c) Early buckling of the tube wall would most certainly occur, since it is very thin, 11

For these reasons it was decided to attempt to demonstrate bending phenomena by use of a somewhat more novel and complicated scheme in which a radial line load is applied around. the circumference of the tube by means of a constricting wire, such as is shown schematically in Fig. 35 This system also has the advantage of allowing the imposition of internal pressure and axial end loads in order to preload the cords, This problem is twodimensional in the coordinates 5 and 5 as the deformation is symmetrical about the axis of the tube. jr Fig. 3. Schematic view of a cylindrical tube under the action of radial line load applied on a circumference. The bending rigidity of'the tube walls may be derived from the solution for the radial deflection of the walls under this load, This problem is a relatively simple one in strength of materials, but-:ts solution is quite lengthy and, for that reason, only the results will be presented here, The solution itself is thoroughly discussed in the Appendix. YFrom the Appendix it may be shown that the bendinga stiffness is described, in a special case, by Eqso (11) 12

If N = 2 J/i-kf, (lla) then = __f (llb) in which N is the axial end load, P is the radial line load in pounds per inch, and Pi is the internal pressure. kf represents a foundation stiffness proportional to the modulus of the tube in the circumferential direction, Er, while d represents the bending rigidity in the longitudinal or axial direction, Equations (11) are valid only for a condition in which the axial load is set according to Eq. (lla), under which conditions the bending stiffness is given by Eq. (llb). From Eqs. (11) it is clear that the determination of bending stiffness De is an iterative or cyclic process. It may also be seen that bending stiffness is dependent both upon the stiffness kf and upon the fourth power of the radial load per unit pressure increase. This means that considerable care must be taken in experimental measurement since small fluctuations or errors in either measurement of the line load P or in measurement of the internal pressure could result in large variations of predicted bending stiffness, The specimens used for these experiments were previously described in Ref, 1. These are four-ply 6-in.-diameter tubular specimens, with alternate cord angles being equal but opposite as is the usual construction practice, and they are shown in Fig9 4. 13

Alternate Four- Ply Structure Plies Laid at Angle d to,36" I3 6 Fig. 4. Tubular test specimen showing dimensions. The experimental procedure which was found necessary here began with the calculation of a value Di for the particular specimen by use of Eqs. (10). This value was substituted into Eq. (lla), along with a measured value of the stiffness kf. The resulting axial load N was found. Next, calculations were made to determine the values of axial load and internal pressure which, used together, resulted in proper loading of the cords in the specimen by means of the point Mohr's circle criterion described in Ref. 2. The use of this Mohr's circle criterion also insured that angle changes would be quite small under the applied loads, thus removing the necessity for correcting data for angular change. Fortunately, the necessary value of end load N as predicted by Eqs. (11), when used in conjunction with the proper internal pressure as predicted from Mohr's circle as previously discussed, resulted in strains which were of the proper order of magnitude for the development of the full tension modulus of the textile cords. Table I shows

the strains used in the various experimental specimens described in this report. TABLE I a o0 15' 30~ 45 60 Cord Strain, 0.3 0.3 0,3 0o4 1.3 _per cent The specimens were placed in a standard tensile testing machine using grips previously shown in Ref. 1, and the proper internal pressure and axial load were applied. This accomplished loading of the cords into a state of tension strain, as shown by Table. I, as well as applying the necessary end load, as shown by Eq. (11e). At this point, a thin restraining wire was placed around the specimen with a small preload and its total length measured. Now internal pressure was caused to vary about this preset point. During this process a line load was generated in the wire due to expansion of the cylinder, and this load was measured by means of a load cell attached to the wire itself. A plot of the line load versus internal pressure, similar to that shown in Fig, 5 for a particular specimen, was then obtained for each specimen. The slope of this curve was then determined and used in Eqo (11b) for calculation of a new stiffness Db. Generally, this calculated stiffness was quite close to or identical with the stiffness originally predicted from calculation, and only one or two successive iterations of the experimental process were necessary to cause convergence. This type of experiment was 15

144 _ - 142 -- 140 / 138 2 * 134 - - - - - I IIIIIIII122 120 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37. 5 LINE LOAD- P # /IN. Fig. 5. Typical line load vs. pressure curve for cylindrical tubes subjected to constant axial loads and variable internal pressure with a restraining cable on a circumference. performed on two samples at each of the cord angles shown in Table I, so that a total of twelve experimental points was obtained. The results of these are shown in Fig. 6 compared with the predicted values of stiffness obtained from Eqs. (10). Note that D(a) = D(/2-a). Agreement between experiment and theory is relatively good, although a few substantial deviations from theory exist. In particular, the 45~ spec

o Experimental data for D 220 Theoretical curve from Eq (10) 200 180 160 140 20 r 100~6 40 20 ___ ___ 09 0 10 20 30 40 50 60 70 8C 9 a - DEGREES Fig. 6. Bending stiffnessess and 1% as predicted from Ecs. (10a) and (10b) vs. cord angle o,, along with experimental values of D5 and D

imens indicated bending stiffnesses somewhat lower than predicted by Eq. (10). In general, the fundamental difficulty in determining bending stiffness by the constriction of a cylindrical tube lies in the error propagation inherent in Eqo (11), where errors in measured quantities such as line loads or internal pressures are quadrupled due to the fourth-power dependence of bending stiffness on these line loads and pressures. In view of this admittedly bad sensitivity, it is considered significant that agreement between experiment and theory was so close, and it is felt that this provides sufficient evidence to show that bending stiffness may indeed be predicted from Eqs. (10) with some accuracy.

EXPERIMENTAL INSTRUMENTATION Since this is an unusual way of measuring bending stiffness, a description of the measuring techniques might be of interest. As mentioned earlier, the specimens were loaded in a standard Riehle screw-type tensile testing machine, Internal pressure was provided by bottled nitrogen, using both a 7-foot mercury manometer and a 150-pound Bourdon-tube pressure gauge in the lines For lower-pressure work the manometer was switched on, while for highpressure work it was cut out of the system. Axial loads ranged from 1800 to 2500 pounds, while internal pressure ranged from 10 in. Hg to 140 psi, depending on the specimens being tested, Specimens with low cord angle generally required larger end loads and lower pressures, while the 60~ specimens used very large internal pressures coupled with compressive end forces, The specimen in the testing machine is shown in Fig. 7. The constricting cable was 1/16-in. flexible aircraft control cable wound once around the specimen and connected through a turn-buckle on one end into a 500-lb Baldwin load cell, The other end of the cable was attached to the opposite end of the load cell, This load cell rode on a track so that translation of the specimen could occur without inducing load in the constricting wire. Using this mechanical arrangement, changes in diameter of the specimen were the only causes of load in the wire. Figure 8 is a sketch of this measuring system and the constricting wire, and Fig. 9 is a photograph of the arrangement. A scale attached to the cable 19

Fig. 7. Photograph of specimen in testing machine. 20

Specimen Restraining Wire Pulley Riders Vernier Load Cell Machinist's Scale Bearing ron Screw Turnhbuckle Fig. 8.Schematic view of the measuring system and the constricting cable.

ai-i~~~~~~~~~~~~~~~~i i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i _,:::.: —:-: — i::_ ~:-:-: -i —:::: ii!I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:: —-::iiiii -$ —Fig. 9. Photograph of the measuring system and the constricting cable.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:_-::-::::.:i:::::::" —'-::-::::: ~ —-—:'i- ii~~,: —

When cablhe

APPENDIX Consider an element of a plate with unit width as shown in Fig. 10, This might be a strip from a cylindrical shell if the ratio of plate thickness to radius is quite small. The moment equilibrium equation may be written M + dM- M + Nd~ - Qd = 0. M+dM Q+dQ Fig. 10. Free-body diagram of a differential strip from a cylindrical shell under the action of forces and moments, This may be differentiated twice. d2M + d2 dQ 0 dI2 dw2 dg It will be recalled from elementary bending theory that M =-E I d -De d22 da d Note that pd~ = kf~dS + Pid~ 25

where kf is the spring constant of the elastic foundation model and pi is the internal pressure on the cylinder. The assumption is made that the cylinder responds to a load as if it were a mass of springs. The equilibrium equation may be written d4 d D - N + kf = -Pi. (a) dS4 d,2 For the homogeneous problem let 5 = Ae. s is the solution of the equation, s4 _ N s2 + kf 0 DS Ig (b) I7 2 s = + - + ikf - Case I: N < 2 VkfSE The quantity under the inner square root is positive. s might thus be written as a complex number, S1, 2,34 = + V + i5, where vkf= +2 L D = kf N 4 = ( c) The general solution is - Ae'v ei + Be V~ eit + Cev e-i + De-v e-iB 26

or C= Cev~ + C2evj cos CO + LC3eV + C4e V]sin s. For the particular integral, the solution'= A is assumed and substituted into (a), from which A = -pi/kf* The complete general solution is written = P+ CjeV~ + C2e v cos B + [3e + C4e sin ~. The boundary conditions are now applied. At ~(co) the only deflection is due to internal pressure. (o) = Pi (dl) kf Also, the bending moment must vanish there: dAg: = 0. (d2) These conditions imply C1 = C3 = O. If only the positive half beam (0 > O) is considered, the other two boundary conditions are Q(O) _ P (d3) (where P is the line load applied at the origin). Also, the slope will be zero at the origin, so that (dt~ = 0. (d4) 27

Substitution into the solution gives P 1 P 1 C4 = C2 = 4 p( (2 + v2) 4 v( 2 + V2)15 and the elastic deflection line for Case I is + k2 -vkf 2 v kfe cos + sin (e) Case II: N > 2 Jkfr 5 of the previous discussion may be replaced by i", where - ~ N _ kf In this case, the constants C2 and C4 become P 1 P 1 C4 C= =4 D i(v2 -2) 4 v(v2 - v2)v and 2% - v - 1 The elastic line for Case II is thus written Pi + Pe-v [osh h +V sinhj. (g) Case III: N = 2 k In this case, v reduces to a and f vanishes. a =; = o, (h) 28

Substitution into the general solution yields the deflection line; = - Pi + P e- a (1 + at). (i) kf 4D as3 Little has previously been written about the relationship of the beam on the elastic foundation to cylindrical shells. Timoshenko (Ref. 3) shows that the two problems are identical since the shell walls act as an elastic foundation. For the tubular shell, the hoop strain ch is proportional to the radial displacement ~ if the radial strain is uniform across any cross section. Thus, r where r is the tube radius. A radial displacement thus generates a hoop strain related to the hoop stress by h = = hu E; ~h~r t r Et ahu = 4 where Thu is the hoop load per unit width. There must be a relationship between chu and the radial line load which caused it. Consider the equilibrium of the shell segment shown in Fig. 11. Pr sin GdG = 2hu Cr' = Pr[-cos 1o0 = 2Pr S ahu p _= hu Fig. 11. Free-body diagram showing the equilibrium between the constricting cable and the membrane hoop stresses in the cylindrical shell. 29

Thus the line load may be written P = X = 4 (j) r r2 This- relation expresses deflection in terms of the load. The proportionality factor is the radial stiffness, kf, of the structure. D5 is obtained by solving a deflection line equation for De when (0) is used as an experimentally determinable variable. For Cases I and II, _ D p- - Nkf (k) This expression might be useful in some cases, but in general the experiments could not be performed accurately enough to warrant its use. Subtraction of numbers nearly the same size magnifies errors, as does the process of raising to powers. Case III seemed to offer fewer problems and was finally accepted as the basis for experiments. Here it is necessary to satisfy the two equations N = (2 o () = Pi P kf 47D'a3 If the radius is held constant at zero deflection, C(o) = O, (1) and thus P. P Pi = kf 4D a3 Solving for De yields _ 2N pi 30

but, as this is Case III N = 2 Nk; (m) so, eliminating N, De = kf X-&-) (n) Thus, in running any experiments in Case III bending the relationships (1) and (m) were followed as closely as possible with the experimental variables being P and pi. 31

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ACKNOWLEDGMENT The instrumentation for this experiment was designed and constructed by Mr. N. L. Field. Mr. D. A. Dodge and Mr. Richard N. Dodge assisted in obtaining experimental data. 33

REFERENCES 1. S. K. Clark, The Plane Elastic Characteristics of Cord-Rubber Laminates, Univ. of Mich. Res. Inst. Technical Report 02957-3-T, Ann Arbor, Michigan. 2. R. N. Dodge, N. L. Field, and S. K. Clark, The Plane Elastic Characteristics of Cord-Rubber Laminates - II, Univ. of Mich. Res. Inst. Technical Report 02957-7-T, Ann Arbor, Michigan. 3. S. Timoshenko, Strength of Materials, Vol. II, D. Van Nostrand Co., New York, 1959.

DISTRIBUTION LIST No. of Copies The General Tire and Rubber Company 6 Akron, Ohio The Firestone Tire and Rubber Company 6 Akron, Ohio B.F. Goodrich Tire Company 6 Akron, Ohio Goodyear Tire and Rubber Company 6 Akron, Ohio United States Rubber Company 6 Detroit, Michigan So S. Attwood 1 R. Ao Dodge 1 The University of Michigan ORA File 1 SO Ko Clark 1 Project File 10 37

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