[1]United States Patent Office 3@552@175 P a t e n t e d J a n . 5 , 1 9 7 1 3,552,175 METHODS OF PERMANENTLY ELONGATING STRIP Marti.n G. Kinnavy, Warren, Ohio, assignor, by mesne assignments, of 40/90 interest to Richard W. Herr, 5 CortIand, Ohio; 15/90 interest to Salem Corporation, C:irnegie, Pa., a corporation of Pennsylvania; and 35/90 interest to a Trust whose trustees are Richard W. Herr and Jean L. Herr, both of Cortland, Ohio, The Union Savings & Trust Company, Waffen, Ohio, and John U. Anderson, Jr., Pittsburgh, Pa. 10 Filed Feb. 28, 1968, Ser. No. 708,947 Int. Cl. B21d 11102 U.S. Cl. 72-296 10 Claims 15 ABSTRACT OF THE DISCLOSURE A strip is permanently elongated by applying a tension force to it and constraining it while under tension to at least one bend and at least one bend reversal Which causes duced between a bend and a subsequent bend r4,versal in @which the magnitude of incremental elongation stram is related to the applied tension force and the applied bending strains. 25 nis invention relates to a method of permanently elongating sttip riiaterial to remove irregularities ifi shape..and to modify physical properties or structure. I. PROBLEM PRESENTED TO THE INVENTOR 30 The manufacture of metal strip leaves irregularities A common problem in the production of continuous metal strip is the irregularity of shape in the form of 35 wavy edges and bubbles between the edges. For many a a , material. with such irregularities is neither saleable nor useable. These irregularities commonly appear in strip produced by established methods and modifica- 40 tion of those methods is either not feasible or:too costly.- The. problem of @ strip irregularities, therefore must be solved in operations subsequent to the basic strip production.operations. H. PRIOR ART 45 Stretching process to remove irregularities in metal strip One of the earliest methods for removing ir regulariti4@s, froiii strip or plate was the simple jaw streich leveler whi@h 50 dates back to the nineteenth century. if one considers that a metal strip i,s composed of a plurality of metal fibers running in a plane in the longitudinal direction of a metal strip, then in the case of an elastic-perfectly plastic material, @ure stretching with the longest fiber elongated into 55 the yield zone will produce leveled strip; see definitions of leveled strip infra. If on the, other hand the @,ame strains are applied to a material which work-hardens, a somewhat differ6nt 6ff6ct takes place. Fibers which were short will remain short fibers in th@ "stretch-leveled str " The rea- 60 son for this is that after all the fibers 'were strained to the saiiie length .@ith each fiber undergoing a different individual str@in, the more highly strained material reco I vered @ -grea@ter amount up6n unloading. Thus ihe strip condition is only @artially cotrected. The consequence of this is 65 thai pure stret6h leveling (see de,finition of leveling infra) 2 works best on materials which work-harden only to a slight degree; that is, to materials which are nearly elastic-perfectly plastic. For this class of materials, however, stretch leveling is a very easily controlled process in that the only requirement is that the longest fiber be strained to at least the yield point. In view of the necessity to strain the shortest fibers well into the yield zone, the process of stretch leveling is not useful for materials inwhich the ultimate tensile strain is only nominally higher than the yield strain. Simple jaw stretcher leveling is applicable only to thestretching of material in sheet form and not to a continuous process. Nevertheless, the idea of simple stretching has been applied to stretching of strip continuously. In this process the essential components are a drag bridle, a drive bridle, a suitably powered mechanical drive system, and the associated controls. While this process, at first I ce, embodies the simplicity of the simple jaw stretcher, gian I close examination discloses serious, but not obviou.s, weak(1) It is necessary with continuous stretcher leveling, to provide sufficient tension capability in the power transmission system to produce the full yield tension in any material to be processed. This high tension requirement imposes the requirement of large expensive drive systems. (2) Basic kinematic analysis of the bridle-strip system shows that almost all of the strip elongation produced in the process must take place while the strip is in contact with the highest tension drag bridle roll. Tle consequence of this is that the roll covering, usually an elastomeric material, has an abbreviated life. Furthermore, because the strip approaches this roll while still in the elastic stress range and leaves the roll when in the plastic stress range after having been extended an ar@bitrary amount, there is large unavoidable velocity mismatch between the strip and the roll covering. (3) In many systems, it has been common to connect themechanical drive systems of the drag bridle system to the drive bridle system. The reason for this is to eliminate electrical motors and generators. This cost saving, however, is not without penalty. In a completely geared system one has no control over the distribution of the torque supplied to each drive and drag bridle roll because loads and torques always follow the stiffest possible paths. As a result of this, the manufacturers of this equipment have no control of torque distribution. The problem is further complicated by roll cover wear to the extent that even if only one cover wears, all roll covers must be reground. Furthermore, because any bridle roll will change tension, the strain in the strip is c6rrespondingly changed, and thereby produces an additional source of small velocity mismatch which effect cannot (in these systems) be minimized by adjustment of individual motor or generator units. Roller leveling process Another well-known method to level strip is the roller leveling process. This process is applicable to sheet leveling and to continuous strip leveling. It consists of passing the material between two sets of rolls placed relative to each other so that the strip is constrained to a tortuous path, thereby causing flexing of the material in alternate directions. This process is known to level strip. Generally, the work rolls are small in diameter and tend to diminish in size as the strip thickness to be processed is decreased. For example, one manufacturer uses Y8 inch diameter yielding and an incremental elongation strain that is pro- 2o nesses:
[2]325521175 3 rolls for a thickiness range of 0.010 inch to 0,050 inch and 11/8 inch diameter rolls for a thickness range of .015 inch to .064 inch. The use of small work rolls has necessitated that other sets of rolls of approximately twice the diameter of the work rolls be used in support of the work rolls to prevent excessive deflection; hence the so-called four-high leveler consists of two sets of work rolls plus two sets of backup rolls to support the work rolls. In rece-nt times, even four-high machines have had to become six-high machines because of the deflection problems. While backup rollers and backup rollers for backup rollers have, to a large degree, surmounted the deflection problems, they have introduced a whole new array of process deficiencies. One of the more serious defects is the so-called stripping problem-a direct effect of the backup roll. Another very serious defect is the control problem. It has been common practice to design into roller levelers vertical adjustability of the backup roll system so that tight portions of the strip might be subjected to more work in the process while loose portions of the strip are subjected to less working. Roller levelers are of two types, driven and non-driven or pull-through. A serious defect associated with the former type is the strip marking caused by the velocity mismatch between the strip and the driven rolls. The velocity mismatch arises as the strip is elongated in the process, and for which no provision is readily made in the roller drive system. Attempts to overcome this operational defect have been made by using adjustable slip clutches in the drive systems, as well as by resorting to other nonpositive drive means, e.g. hydraulic drives. The roller leveler with a skilled operator at the controls can produce acceptable strip from an appearance point of view. However, in many cases, leveled strip is subsequently slit into narrower coils. Because of the differential working of the rolls against the strip, different residual stress patterns exist across the width of the strip. Because of these differential residual stresses, there appears in the plane of the strip another system of stresses which when disturbed by a slitting operation produces objectionable edge bow in the slit coils. To overcome some of the limitations of roller leveling, many machine builders have incorporated tension bridles to the roller leveling process. This has produced an improved product and has allowed the processors to handle strip of lighter gauge than could be leveled without tension. A well-known process which uses high tension bridles is the Compagnie des Ateliers et Forges de la Loire (C.A.F.L. Process). Another tension roller leveling process known as the BNF process is described in Research Report A1241, dated June 1969 of the British Non-Ferrous Metals Research Association entitled, "The EIimination of Edge Bow in Slit Strip." Shortcomings of the prior art While all of the foregoing approaches to the leveling process bave had varying degrees of success, there remains with aIl of them serious drawbacks. In every instance in which work rolls are used for the purpose of flexing the strip, that is, in the roller leveling process, the tension roller leveling process, the C.A.F.L. process, and the BNF process, the work rolls are invariably srnall in diameter and require the use of backup rolls. With small work roll diameters, operating speeds are low, the design is complex because of the need for backup rolls, and there are greater wear and maintenance problems associated with frequent dressing of the work rolls and frequent work roll changes. Furthermore, the use of a backup roll system frequently causes strip marking. III. THE INV-ENTION PRESENTS A NEW APPROACH TO THE ELONGATION OF STRIP MATERIAL The present invention is a new process of permanently elongating metal strip. This can be used to remove strip 4 irregularities or to modify the physical properties of the strip among other uses, For any given set of physical properties of the strip material, one can assign combinations of tension and bending radii such that the desired strip elongation will be produced, and the desired effects, including levelng and modification of physical properties will result. It is possible that an arbitrary combination of tension and bending will produce some elongation. These arbitrary combinations, however, of tension and bending 10 will not produce the result sought whereas the new process will produce the desired result with great accuracy. -Use of the new process avoids major equipment problemsencountered in the present methods (prior art) for elongating strip. The advantages of the new process are: 15 low maintenance of equipment, low downtime of equipment, no strip marking caused by the differential velocity between driven rolls and strip, prolonged life of bridle roH covering, relatively low tension requirements, simplicity of control, ease of operation, and no restrictive speed 20 limit. Furthii-more because the new process relates tension with radius of curvature of bending in a prescribed manner, one is able to optimize tensions and roll diameter combinations to achieve the desired result. By doing this one is no longer limited to the use of small diam25 eter rolls. This can eliminate the design complexity of a backup roll system as well as the operational difficulties associated with such systems including but not limited to strip marking. Definitions 30 (1) Leveling.-Leveling is a process by which shape irregularities in metal strip are removed so that the strip appears uniform. Leveling removes wrinkles from strip but not gross irregularities such as coil set or roll set (defined infra). Leveling makes all fibers of equal 35 length in any given plane. The fibers, however, may vary in length from plane to plane throughout the thickness and the strip will still be leveled. The strip may have roll set or coil set and still be leveled. (2) Leveled strip.-If an area of metal strip without 40 roll or coil set were laid upon a flat table and its surface conforined everywhere with the surface of the table, it would be considered as flat or leveled strip. Irregular strip laid upon a flat table would not be everywhere in contact with the surface of the tabie. In this case some fibers 45 would be longer than other fibers. (3) Coil set or roll set.-A distinction must be made between local strip irregularities removable in a leveling process and the strip condition known as roll or coil set. If a length of strip assumes a longitudinal curvature 50 when it is free of restraint, it is said to have roll or coil set. It is possible for a length of strip to have roll set with or without irregularities. Strip with roll or coil set, but without irregularities, is considered herein to be leveled strip. A simple multi-roll flattener can remove 55 coil set from strip, but it generally is incapable of removing strip irregularities. If one considers strip to be comprised of a multitudinous array of fibers, each fiber in flat and leveled strip wil have the same length. Strip with coil or roll set will correspond to a linear variation 60 of fiber length throughout the thickness of the strip. If this pattern of variation is identical in a transverse direction, the strip is said to be leveled. If this variation changes in a transverse direction, the strip would have shape irregularities in addition to roll or coil set. Coil or roll 65 set is elated to what is called transverse bow by Poisson's Ratio, a physical property of the strip. I provide a method of permanently elongating strip which comprises, applying a tension force to the strip, and constraining the strip while under tension to at least 70 one bend and at least one bend reversal which causes yielding and an incremental elongation strain that is produced between a bend and a subsequent bend reversal in which the magnitude of incremental elongation strain is related to the applied tension force such that when the 75 yielding occurs on alternate sides of the strip on succes,
[3]3,552,175 5 sive bends the increment of 61ongation strain follows the equation t RI+R2_ R + R,, F Ae= -@2tr:y@ -i RIR2 yt) 5 in which F=tension force per unit width of strip ,T,=tensile yield stress t=strip thickness 10 e,@tensile yield strain Ae=inerement of elongation strain, and Rl, P'2=radi,i of successive bends I provide a method of permanently elongating strip which comprises, applying a tension force to the strip, and 15 constraining the sttip while under t6nsion t6 at least one bend and at least one bend reversal which causes yielding and an incremental elongation strain that is produced between a bond and a s@bsequent bend reversal in which the magnitude of incremental elongation strain 20 is related to the applied tension force such that,.when yielding occurs on at least one.side of the strip at a bend and on both sides of the strip at the - subsequent bend reversal the increment of elongation strain follows the equation ?.5 F R,+R, Ae 2o-y RIR2 Cy in which F=tension force per unit width of strip 30 v-,=tensille yield stress ey=tensile yield strain Ae@increment of elongation strain, and RI, Rs,,=radii of successive bends 35 Other details, objects atid advantages of the invention will become apparent as the followihg description of a present preferred method of practicing the invention proceeds.. -In the accompa4ying drawings I have@ illugtrated a 40 present preferred method@of practicing- the invention in which: FIG. I is a stress-strain diagram-, FId. 2 is an idealized stress-strain diagram; FIG. 3 is a diagram showing yielding on tension side 45, of the strip-known as Case I; FIG. 4 is a diagram showing yielding on both tdzision@ and compression sides of the strip-known as Case II; FIG. 5 is a diagram of strip:elongation upon bending strain reversal-Case II; 50 FIG.,6 is a diagrarxi of strip elongation upon bending strain reversal-Case I; @@ @ ; . . I : @ FIG. 7 is a diagram showing differeiit regions of elongation and bending strain parameteis; FLIG.; 8 is a schematic ' diagram., @of ap@aratus used in 5,5 exam les of, application. of @ method to stri r material; p p and F-IG.: 9- is a table:@showing. incremental elongation s.t.rain, for: Case-, IA, Case IB and Case H. Elongation of strip is@related to leveling. Thig rela@, 60 tionship can be shown by referring to an idealized@ stressstrain diagram for an elastic perfectly plastic material as shown in FIG. 1. Consider, for example, irregular strip free of coil set. If ihe strip is irregular, it has bubbles or waves and there will be a difference in fiber I.ength. If 65 a; length.6f uch strip is el6nkaied, the short fibers will be loaded in tension , first. ' As th6 olongation incr6@ses, longer fibers will begin to share the tension load. If the eloiagation is further continued until t.he, longest fiber is strained beyond the yi @eld point, the shortest fiber will be 70@ strained to, an even greater magnitude. Howov&r, each fiber so elongated will have the same stress. Ibis is illus@@ trated in FIG. I in which Point I shows the stress and strain for the longest fiber, and Poin,t 2 shows the stress and strain for the shoitesi fibe@. Now if the strip is un- 75 loaded, each fiber wiR relax the same amount because the path of the stress-strain points is parallel to the initial elastic loading curve. This means that recovery strain RI is equal to recovery strain 'R 2. Thus the final condi tion of the strip will be such that every fiber has the same length and the strip is said to be leveled. Consider a material in which work-hardening takes place after the yield point is reached. If such a material is elongated in a similar way, we will produce stressstr , ain Points 3 and 4, as shown. Now, if the load is relaxed, it can be seen that the recovery strain for the longest fiber R3 is less than the recovery strain for the shortest fiber 'R4. In this case the strip condition, although it can be improved, can only be partiallycorrected. Thus leveling by means of stretching or elongating the material works best on materials which do not exhibit a high degree of work-hardening. For materials with little or no work-hardening, leveling through stretching is a very easily controlled process in that the only requirement is that the longest fiber be strained at least to the yield point. Tbus the problem of leveling becomes the establishment of a suitable mechanical process whereby all fibers of the strip are stretched beyond the yield point. The amount of stretch or elongation necessary to level strip depends upon the initial fiber length variation throughout the strip. In the bending of strip, straight lines in an unloaded,- unstrained strip will remain straight upon the deflection which accompariie-s the application of loads. These straight lines, however, wifl become inclined toward each other. The consequence of this is that some fibers of the strip will elongate with respect to other fibers of the strip, and the relative elongation will be proportional to the distance from the unstrained fiber; that is, the fiber strain is proportional to the distance from tha unstrained, or neutral, fiber. This may also be stated in another way, namely; the fiber strain is a linear function of its distance from the neutral fiber. Consider, now, the idealized stress-strain diagram shown in FIG. 2 in which stress is the ordinate, and strain is the abscissa. In this diagram, Point I is the yield point in tension, and Point2 is the yield point in compression. The inagnitude of the tensile yield stress has been taken as equal to the magnitude of a compressive yield stress. The yield strains in both tension and compression have also been taken as equal. If tension is applied to a fiber, it will be accompanied by a strain such that allstress-strain points will lie upon the straight line 0-1 until the yield stress is reached. Thereafter, as the strain is increased, the stress will remain the same and all stress-strain -points after yielding will lie upon the line 1-5. if a. fiber has been strained suc-h that its stress and strain are given by Point 3 in FIG. 2, a further increase in strain produces no increase in stress. On the other hand, if the strain at Point 3 is reduced, the stress will decrease so that all stress-strain points upon continued decreasing strain will lie along the line 3-4 which line is parallel to the initial elastic stress-strain line. Thus Point 4 may have a positive strain and a negative or compressive stress. If a further strain decrease takes place, the stress-strain path will continue leftward along the line defined by 2-6. Thus, with the model shown, it is possible to develop an infinite number of continuous stress-strain paths bound by extensions of the tensile yield stress line 1-5 and thecompressive yield stress line 2-6. Any stress-strain path from one of these lines to the other will always be parallel to the - initial elastic curve. Consider typical linear strain distributions and their as,sociated stress distributions through the strip thickness. It i necessary to distinguish between stress distributions in which the yield stress appears on either one or both sides of the strip. If the yield stress is reached in tension alone, this is a Case I stress distribution. If the yield Wess value is reached on both tension and compression sides
[4]3,552,175 7 of the strip, it is called a Case II stress distribution. Refer to FIGS. 3 and 4. As a consequence of the linear strain distribution, any strain distribution may be considered to consist of an elongation component and a bending component. The 5 elongation component is the strain that exists at the middle of the strip; the bending component is zero at the middle of the strip and is maximum at the surface of the strip. Strain components are designated in terms of the yield strain ey. Thus acy is the extension component of the 10 strain and fley is the bending component of the strain. Two strain distributions are shown on the lefthand sides of FIGS. 3 and 4 in which the vertical line is taken as the zero strain axis, and strains to the right of this line are taken as tension strains. 15 The stress distributions corresponding to these strain distributions are obtained by utilizing the stress-strain diagram, FIG. 2, together with the strain distributions shown. These stress distributions are shown at the righthand sides of FIGS. 3 and 4. The force per unit width of strip corre- 20 sponding to the Case I stress distribution will be given by 4,3 25 A similar calculation for the Case @Il stress distribution will show that the tension force per unit width of the strip will be given by F= o-,t 30 The geometrical relation between bending strain and curvature is given by 35 fley = t 2R w here R is the radius of curvature of the neutral fiber of th e strip. 4 0 With these established preliminary considerations, the st rip subjected to tension in combination with a bending st rain will be examined when it is subjected to a reversal of the bending strain when the combination of tension and b ending strains lies within the teaching of this invention. 4 5 F IG. 5 shows a strain distribution by means of the line at the left side of the figure, this line labeled 1. The surfa ces of the strip are indicated by the horizontal lines at the top and bottom of the figure and the middle fiber of th e strip is so labeled. The initial strain distribution is 5 0 m ade up of the extension or elongation strain compon ent, aey=1.25 ey and the bending strain component, ,3 cy=2.50ey. The stress distribution which corresponds to th is strain distribution is obtained through use of the 5 5 st ress-strain diagram F-IG. 2, and is shown at the right of FIG. 5 by the series of straight lines 0-1-1*-I-l-O. T his stress distribution is in equilibrium with the force per u nit width of strip F=0.5 ayt. N ow upon reversal of the bending strain component, 6 0 th e slope of the strain distribution is simply reversed but si nce the applied load remains the same the new stress di stribution must be in equilibrium with the same force; n amely, F=0.5 a-yt. The constraints of the stress-strain r elations described earlier and illustrated in FIG. 2 must b e followed. The only strain distribution that has the 6 5 p rescribed bending strain component that obeys the stressst rain constraints described, and that transforms the initial st ress distribution into a new stress distribution in equilibri um with the applied tension force, is the strain distributi on labeled 2. The new stress distribution is given by 7 0 0- 2-2-2*-2-0. Upon inspection one will see that a point p a yt 2 above the rniddle fiber, a strain change of ey occurs, while 75 8 at Point 2* no strain change occurs. Point 2* is located at a distance F t I t o-yt 2 2,3 2 above the middle fiber. It now follows directly from geometry that the increment of elongation strain of the middle fiber, Ae is @Ae@( F 2,3-1),y oyi Use of the earlier stated geometrical relation between bending strain, strip thickness, and radius of curvature in this equation yields a more useful expression for the incremental strain produced during bending strain reversal under maintained tension, namely, F Ae@ Cy (Ro-ycy From this equation an expression for force required to produce a given increment of elongation strain may be obtained, nameiy, Ae F=Ro-yey -+l (ey or Ro-y2 F= (-y+ 1)E in which Ae 'Y=- Ey It is of significance that the increment of elongation strain is independent of strip thickness. This result is of utmost importance. However, as the material thickness is reduced, a given tension force causes increasing stresses. The particular case of strip thickness approaching zero indicates a tendency toward infinite stresses which cannot exist. The explanation of this apparent anomaiy lies in the range of validity of the above expression. A condition that must be met for this expression to apply is R4Ey<,_ F t o-yt or equivalently t> Rey+ F a-y The above teaching was directed to the case of a stressdistribution which has been designated a Case II stress distribution. The effect of a bending strain reversal when a Case I stress distribution exists in the strip will now be considered. In FIG. 6 a seiected extension strain component of &.ey=1.25 ey and a bending strain component of 3ey=1.50 ey are used. The reversal of the bending strain must be accomplished within the same constraints as before. In this case, it can be observed that the stress change at the surface initially at the tesnion yield stress is 2oy and that the stress and strain remain unchanged at a point a distance above the middle fiber given by V2 (1 F t ;Y-t The strain change at the surface initially strained above the yield strain in tension is 2ey 20 (,-F v o-yt
[5]3 552,175 9 10 From,geometry, the incremeiital elongation strain of the tical problems. With the substittition of these variables in middle fiber is the expression above, we have 3 F a>- Ae 120 V8# _;7yt)]r:y 4 5 In the equation which upon use of the earlier shown expression for bending strain becomes, Ro-y2 1=('Y+l) t 4t F the above limitation on a imposes a restriction on y, 1 -@yt cy 10 namely, y@:,0.5. ITC--Y -,v wi-Y Similarly for simple Case I stress distributi ons to transThis expression is valid for all surface strain changes form into simple Case I stress distributions upon the given by reversal of the bending strain, hereafter called a Case IA 15 stress distributi on, the requirem ent is 2.c. < 2ey O-yt which is equivalent to V 2@+ I F (Case IA) ->Jt - -@yt 20 In some cases, a simple Case I stress distribution is . transf ormed into a simple Case 11 stress distribution, hereOne may elect to subject the, strip while under tension to after called a Case IB stress distribution, and the condibending sitairis and bendijig strain rever@s'al@ 'siic'h that th6 radii of curvature corres@onding to those bending strains tion for this is that and bending strain reversals are unequal, in which case 25 ce > 3_ F the. increments of elongation strains D@oduced during bend@y-t ing gtrain reversals are (Case IB) F or equivalently, Ae--F-l@L,R,+R21 Cy -L2o-yfy R,R, 3 F 30 >--- -2 2Ro-yey when (Case 11) .2@y RIR2 <1 pThe above conditions have been illustrated graphically t RI+R2.- -aTy-tin FIG. 7. We have shown that for both Case II and 35 and Case IA the increment of elongation strain described earlier is given by Ae= i R,+R2_ 2tey gl+.R2 IF F W@-R2- : @- ;7yt Ae= 1 Cy (Ro-ycy when 49 and that for Case I-A the increment of elongation strain 2ey ;RIR@ F is given by .- - > 1--- t R@,+A@2- cyt t _A/@4JEt F Ae= 'y T'he stress distributions at -a bend "d at a bend reversal 45 IV.E-Y y(l yt)] c falling within th6 iOa6iiihg@of this invention are simple, or equivalently, that is, the only stresses below the yield stress in magiii" F )] Ey tude are confined to a region of the strip thickness and Ae= 20- 8# I-the stresses in this regi.on,vaf . y linearly and continuously ayt from the maximum tension I v . alue to the maximum com- 5,0 The inventiom.'can also.be understood by practical appression value. All stress distributions, however, that,.pre@"' - I ' ;plications of the method to strip material under certain vail between a given bend and a subsequent bend reversal are not simple. But withih theteaching herein, the process relates to the extensio@n'@of strip th@i occurs while subjected to,a tension force frpm a; simple. stress distribtition,: 55 whi'ch is associated with a bending strain, to, a subsequent " simple stress distribution, which is associated with a bending strain of opposite sig4. In these instaiices the previous strain history will not be evident in the stfess distribution; however the slope of, the. elastic portion of the'stre@s dis,- 60 tribution wfll not be the same as the slopp qf the.strain,, distribution. In order that stress distributions of the Case II type remain simple, it is necessary that Point 2* in FIG be above Point 1*. For this to be's'o 65 . ; F t ayt F t ayt @y-t a-yt 2 2#o-y > - or F 3 Rey. ayt -@2 T t It-has@lbeen!found.@convenient t6 use thd varidbles a and.,3 de @scribed,-,Oarli@r in.the@ a plication of the method:tb prac- 75 p conditions as shown by the following examples. The apparatus (referring to FIG. 8) used in these examples consisted essentially of a means to apply a tension force to the strip and a means to produce bending strains and bending strain reversals. The means. of applying tension f6rc6 was a hydraulic 6@linder whichwas connected to a for-c:e 96ge which wits connected to one end of'the strip. The other end of the strip was connected to a force gage which was connected by a bracket to a fixed building column. Bending strains were applied by means of rolls carried on a cart riding on a smooth track. A hydraullic power supply was used to provide a source of hydraulic pressure for the hydraulic cylinder. Strip was threaded over and und@r the rolls mounted on the cart and clamped as shown in FIG. 8. The hydraulic cylinder is used to remove slack from the strip; then a gage length is marked on the strip (50 inches). A suitable tension force is then applied to the strip by means 70,,of the hydraulic cylinder, the magnitude of the tension force being measured by means of load cells. The cart is then moved along the length of the strip, thereby subjecting the strip to an initial bending strain distribution; then to a bending strain reversal; then to a second bending strain reversal. The distance between the gage marks
[6]is measured after the passage of the cart along the strip and this measurement is compared with the original gage length. This comparison then yields the strip elongation; that is, final length-initial length percent elongatioli= initial length x 100 EXAMPLE 1 Material: Aluminum, 3105H25 Thickness: t, 0.0319 in. (measured) Yield stress: ay, 26,600 lb./in.2 Modulus of elasticity: E, 8,520,000 lb./in.2 Yield strain: ey, 0.00312 Yield tension: ayt, 850 lb./in. of width The diameter of the rolls on the cart was 6 inches; thus the bending strain pararneter is caiculated as follows: t (0.0319) 2Rey 6.0319 (.00312) = 1.695 In this example, a tension force of approximately 75 percent of the yield tension was used; i.e., F=637.5 lb./in. of strip width. To establish which Case in FIG. 7 is applicable, Table I (FIG. 9) has been developed. Thus, 0.295>(1-0.75) which shows a Case IA situation. First a is computed from the force equation. Thus, O-Yt Substitution of numerical values gives a= 1+ 1.695-@6.80@(0.25@) = 1.39 The increment of elongation strain is then yt) ey or Ae=[(2) (1.695) -V(8) (1.695) (0.25)]=1.55ey In the example fixture the initial bending strain is followed by two bending strain reversals. Thus from the initial bend there is a middle fiber strain of aey=1.39ey and from the two bending strain reversals, two increments of strain each of magnitude Ae=1.55ey to produce a total eiongation strain of e=[1.39+(2) (1.55)1(.00312) =0.0140 The example result showed that with a gage length of 50 inches an elongation of 0.782 inch was produced, which corresponds to a strain of e=0.782= 0.01564 @-o-- The difference is 0.01564-0.0140 or 0.00164 or 10.5%. EXAMPLE 2 Material: Aluminun, 5052H32 T'hickness: t, 0.040 inch Yield stress: ay, 27,600 lb./in.2 Modulus of elasticity: E, 9,550,000 lb./in.2 Yield strain: ey, 0.00288 Yield tension: cyt, 1105 lb./in. of width In this test the bending strain parameter is 0.040 (.00288) =2.30 0 = T6.0-40) 3;552;175 12 With an applied tension force of F=857 lb./in. of width, F 857 1 1 0.225 and 5 I 1_ F 0.1125 2 ( o-yt) = The quantity 1/2,6 is 0.2175 which means a Case IB situation. 'fbus a is computed as follows: 10 Oyt 1+2.30--,/ -.-2(0.225y= 1.86 15 The increment of elongation strain upon bending strain reversal is ( F _ 1)e Ae= _Y=( 857 4EY Ro-yc-y (3.02) (27600) (.00288) 20 =2.57Ey The total elongation strain is obtained as before e=,[ 1.86+(2)(2.57)1(.00288) =O.OZ16 25 The test result showed an elongation strain of 0.0222. The difference is 0.0006 or 2.7%. EXAMPLE 3 Material: Aluminum, 5052H32 30 Thickness: t, 0.0512 inch Yield stress: o-y 24,700 lb./in.2 Modulus of elasticity: E, 8,970,000 lb./in.2 Yield strain: ey, 0.00276 35 Yield tension: cyt, 1263 lb./in. of width The bending strain parameter is 0.0512 9=@@ (.00276)=3.06 (6.0512) 40 and 1/2#=0.1635 With a tension force of 653 lb./in. of strip width; F 653 1--=I--=0.483 45 o-yt 1263 and 1/!2 ( I - 0.242 a-yt 50 Since F 2 ayt 55 it follows from Table I that we have a Case II situation. Tbus a=- #= (0.517) (3-06) =1.58 O-Yt 60 and the increment of elongation strain upon bending strain reversal is Ae= (2a- 1) ey=2.16ey The total elongation strain is then 65 e=[1.58+(2)(2.16)1(.00276) =0.0163 The observed elongation strain in the experiment was 0.0164 which is different by only 0.6 percent. 70 While I have shown and described a present preferred embodiment of the invention and have irustrated present preferred methods of practicing the same, it is to be distinctly understood that the invention is not limited there-, to but may be practiced within the scope of the following 75 claims.
