the bending of plate using a three-roll pyramid type plate bending machine

http://sdj.sagepub.com/Design

The Journal of Strain Analysis for Engineering

http://sdj.sagepub.com/content/1/5/398The online version of this article can be found at:

DOI: 10.1243/03093247V015398

1966 1: 398The Journal of Strain Analysis for Engineering DesignM B Bassett and W Johnson

The bending of plate using a three-roll pyramid type plate bending machine

Published by:

http://www.sagepublications.com

On behalf of:

Institution of Mechanical Engineers

can be found at:The Journal of Strain Analysis for Engineering DesignAdditional services and information for

http://sdj.sagepub.com/cgi/alertsEmail Alerts:

http://sdj.sagepub.com/subscriptionsSubscriptions:

http://www.sagepub.com/journalsReprints.navReprints:

http://www.sagepub.com/journalsPermissions.navPermissions:

http://sdj.sagepub.com/content/1/5/398.refs.htmlCitations:

What is This?

- Oct 1, 1966Version of Record >>

at UNIV OF OKLAHOMA LIBRARIES on October 28, 2014sdj.sagepub.comDownloaded from at UNIV OF OKLAHOMA LIBRARIES on October 28, 2014sdj.sagepub.comDownloaded from

http://sdj.sagepub.com/

http://sdj.sagepub.com/content/1/5/398

http://www.sagepublications.com

http://www.imeche.org/home

http://sdj.sagepub.com/cgi/alerts

http://sdj.sagepub.com/subscriptions

http://www.sagepub.com/journalsReprints.nav

http://www.sagepub.com/journalsPermissions.nav

http://sdj.sagepub.com/content/1/5/398.refs.html

http://sdj.sagepub.com/content/1/5/398.full.pdf

http://online.sagepub.com/site/sphelp/vorhelp.xhtml



THE BENDING OF PLATE USING A THREE= ROLL PYRAMID TYPE PLATE BENDING MACHINE

M. B. BASSETT Department of Mechanical Engineering, Lanchester College of Technology, Coventry*

w. JOHNSON Department of Mechanical Engineering, University of Manchester Institute of Science and Technology

An experimental three-roll pyramid type plate bending machine was constructed and measurements were made of the upper roll vertical force, the driving torque and the relaxed radius of curvature for a given dis- position of the top and bottom rolls. Two methods of analysing the plate bending process in the machine are given: (1) a geometrical analysis based on the assumption that the deflected form of the plate is in the arc of a circle and (2) a load analysis using the uniaxial stress-strain relationship for the material. The experimental results for an aluminium alloy (N.P.8) were best accounted for by the geometrical analysis whilst the load analysis gave disappointing predictions.

In Part 2 experiments are reported on the bending of mild steel. In most of the experiments the two lower rolls were driven, but to determine the effect of plate roll friction, some were carried out in which the lower exit roll only was driven. A discussion of the various theoretical analyses which have been proposed is given and they are compared with the results of the authors’ experiments.

INTRODUCTION THE PLATE BENDING MACHINE Was first brought into Use around 1830, its development being related to that of the cylindrical boiler (as opposed to the earlier ‘onion’ shaped type boiler) and to the increasing availability of rolled plate. Although two types of machine have evolved, the ‘pinch’ and ‘pyramid’ types, which are shown in Fig. la, no significant alterations to the original ideas have emerged since the earlier machines. Only modifications, such as hydraulic activation, have been introduced.

The range of this paper has been restricted to pyramid type machines which are the most common. In this type of machine one or both lower rolls may be driven, although to minimize slipping of the workpiece over the rolls it is better to drive both rolls. When bending shapes other than plates, grooved rolls are sometimes used to support the section during bending. For machines bending narrow sections, cantilevered rolls may be used while for plate the rolls are supported at each end. In the latter case there must be some method of removing material which has been bent into a full circle. For machines with very long rolls intermediate support rolls are required to avoid the production of barrel shaped cylinders. However, in this paper the authors have concentrated on determining the forces on the rolls and not on the roll design details.

Anticlastic curvature can affect the shape of the cross- section of the bent workpiece but usually its effect is neglected. The setting of the roll parameter k for a required radius of curvature (Fig. lb) is determined by trial and error and this is likely to remain so in the future. However, this parameter is of use in design calculations The M S . of this paper was first received at the Institution of Mechani-

cal Engineers on 19th April 1966 and in its revised form, as accepted by the Council for publication, on 8thJuly 1966. 33 Formerly research student at University of Manchester Institute of Science and Technology.

and so has been derived below. A third factor to be considered is the slipping of the workpiece over the rolls which imposes a limit on the radius of curvature produced in one pass. Normally several passes would be required to produce a small radius of curvature. Since the largest forces would occur if a given radius was produced in one pass, the analysis only considers one-pass bending.

The experimental work reported in Part 1 of the paper was performed on aluminium specimens. To determine the general validity of the presented analyses, a series of experiments using mild steel specimens was also under- taken. The latter set of experiments are reported in Part 2 together with comparisons with other workers’ analyses and the results of experiments when only one roll is driven.

Notation A b d e

K k

f

51

I M N.A. P R Ri

Constant. Width of bar or plate.

Engineering strain. One half the bottom roll centre distance. Anticlastic bending parameter. Distance between the top and bottom roll centres,

Distance between the maximum top and bottom

Horizontal supported length of the plate or bar. Bending moment. Neutral axis. The vertical force acting on the rolls. Radius of curvature. Internal radius of curvature of the plate when

minus k.

Fig. lb(ii).

roll centres (Fig. lb(i)).

under load (geometrical analysis).

398 J O U R N A L O F S T R A I N A N A L Y S I S V O L I N O 5 1966

at UNIV OF OKLAHOMA LIBRARIES on October 28, 2014sdj.sagepub.comDownloaded from


THE BENDING OF PLATE USING A THREE-ROLL PYRAMID TYPE PLATE BENDING MACHINE

Upper roll radius. Lower roll radius. Driving torque. Thickness of plate or bar. Speed of plate or bar. Work done. Equation of shape of section. Maximum deflection. One half the angle of contact of the plate. Coefficient of friction.

PLATE

a

GAUGE AND PRESSURE ROLL - BENDING ROLL

FIXED

PINCH TYPE ROLLS

ADJUSTABLE ROLL m

PYRAMID TYPE ROLLS

Fig. l a . The two types of plate bending machines

V Poisson's ratio. p

0 Nominal stress.

Radius of curvature of the neutral plane of the plate.

Subscripts Average. Back roll. Compressive. Front roll. Lower roll. Plastic. Relaxed. Tensile stress or top roll. Yield stress.

All units are expressed in lbf-in-sec units.

Fig. l b . The limits of parameter k

Part 1: Analysis and experiments with aluminium THEORY OF PLASTIC BENDING

Fig. 2a(i) shows the cross-section of a prismatic bar of a material which is isotropic, homogeneous and whose tensile and compressive properties are shown in Fig. 2b. The bar is regarded as being loaded in such a manner that it is subjected to a pure bending moment in a vertical plane through the section centroid. On the assumption that plane sections remain plane, which implies a linear strain distribution, see Fig. 2 4 9 , the bending moment carried by the section when the radius of curvature of the beam is p is found to be

M = P2 /oe'Bt(Ylft(e)e de+ P 2 ioe0 Bc(r)fc(e>e de (1)

where 1 et+ec P t

. . . . . ( 2 ) _ - --

J O U R N A L O F STRAIN A N A L Y S l S VOL I N O I 1965

The position of the neutral plane is fixed by the necessity for the sum of the tensile forces above it to equal the sum of the compressive forces below it. This requires that

joet Bt(YIft(4 de = joeo Bc(Y)fc(e) de ' (34 where p,(y) and P c ( y ) are equations of shape. For a beam of rectangular section of width b,

M = bp2 joe' ft(e)e de+ bp2 Joe' fc(e)e de where

IOet f t ( 4 de = 6" f c ( 4 de * - (3b)

The position of the neutral plane is defined by equation (3a) or (3b) and is the same as the centroidal axis only when f t (e) = fc(e), i.e. when the tensile and compressive stress- strain characteristics are identical. For real materials t h i s

399 at UNIV OF OKLAHOMA LIBRARIES on October 28, 2014sdj.sagepub.comDownloaded from


M. B. BASSETT AND W. JOHNSON

4

DERIVED GRAPHICALLY

OF BEAM STRAIN STRESS I DI ST R I BUT ION DI ST R I BUT I ON

II m a

Q Il Fig. 2. General stress-strain, relationships and stress-strain

distribution during bending

is generally not so and the neutral plane shifts as the loading increases so that equations (3a) or (3b) are satisfied. For a rectangular section beam the movement of the neutral plane is not large, so that if it may be assumed that the neutral plane coincides with the centroidal axis, equation ( 1 ) may be reduced to

' M = 2bp2fa1 f(e)e de . . . (4) where e, = t/2p.

Two alternative assumptions may be made regarding the function f (e ) . If the compressive properties are not known it may be taken that f (e) = fc(e) = ft(e), or a better assumption is to allow for the difference in tensile and compressive properties by writing f (e ) = +[ fc(e)+ft(e)].

To determine M, equation (4) may be solved graphically or incrementally, since the ef(e) versus e curve is very nearly a straight line, as was observed by Williams (I)*. Alternatively, a stress-strain relationship may be assumed and an analytical expression derived. Fig. 3 shows the moment-tensile strain curve derived graphically, and it is compared with that derived after fitting equations of the form u = Y+ He and u = Ae" and integrating. The graphs shown are for the 0.756 in material, whose stress-strain characteristic is reproduced in Fig. 10. In both cases it was assumed that f t (e) = f e ( e ) . The agreement is seen to be reasonable over the range of about 1 to 6 per cent strain. * References are given in Appendix 3.

FOR t = 0 . 7 5 6 1 n

M A T H E M A T I C A L DERIVATION FROM u = A c " ----- FROM u = Y + H e .-.-.-

e-par cant

Fig. 3. Comparison of moment-strain characteristics obtained graphically and by assuming u = Y+ He or u = Ae"

The above theory is suitable for the plastic bending of bars, i.e. to a section whose width is comparable to its depth. However, if the bending of plates is considered, the problem is one of plane strain rather than plane stress, because the material is restrained in the transverse direction over a large part of its width. For the elastic bending of plates the transverse restraint is such as to stiffen the plate and in effect the second moment of area is multiplied byafactor of 1/(1-v2), where v is Poisson's ratio-thus increasing M by about 1 1 per cent for a given radius of curvature if Y = 0.34 and 33 per cent if v = 0.5. For the plastic bending of plates the authors suggest the use of the equation

2b M = "Joe A f(e)e de . . * (6)

where A = [1-{+m+(l-m)v}2] and m = [1-(2u,p/tE)]. The derivation of these expressions is to be found in Appendix 2.

At this point it should be stated that it is not obvious whether a section is a plate or a bar. It is suggested that a section should be called a plate when there is an appreciable central restrained region within which bending in the transverse direction does not occur, anticlastic curvature being confined to the edges. Searle (3) and Ashwell (4) have shown p a t the parameter K = b2/pt is indicative of the degree of anticlastic curvature which occurs in elastic bending. The transition in behaviour from bar to plate is

400 JOURNAL OF S T R A I N A N A L Y S I S V O L I N O 5 1966




gradual, however, and while Searle expects the change from bar to plate bending to occur when K > 6 Ashwell predicts that suppression of anticlas tic curvature will com- mence when K > 100. In the following experimental work, values of K = 9 were the maximum experienced and anticlastic curvature was clearly present.

The loading system imposed on the plate while passing through the plate bending machine is such that transverse loads are applied and shearing stresses exist. However, in most cases the effect of the shearing stresses may be neglected since they are small compared with the bending stresses, Drucker (5). Rate of strain effects in the process may be neglected since the rate of working is approximately of the same order as a tensile or compressive test. Also, total plasticity laws may be used in any analysis since the loading path is radial.

The plate or bar will undergo elastic springback after the maximum bending moment has been applied. The relaxed radius, after springback, of the plate may be determined from the following equation (6)

1 1 1 1 M - * (7) Pr PP ~e PP EI '

DISCUSSION OF THE PROBLEM The loading system and the deflected form of the plate in the pyramid machine used in this work is shown in Fig. 4. The bending history for any plate element is as follows. It enters having an infinite radius of curvature, is progressively bent elastically and at some point yielding of the outer fibres occurs so that plastic bending com- mences. The radius of curvature then decreases until the maximum bending moment is experienced when the radius of curvature is pp. After this point the portion of the plate considered gradually undergoes elastic springback to emerge with a radius of curvature pr. Since the plate enters with zero curvature and emerges with curvature l /pr the tangent angles of the plate at entry and exit will be different, i.e. elf # Olb. It is to be noted that contact with the top roll occurs on the entry side of the vertical centre line of the machine.

The loading system caused by deforming the plate is complex. The configuration shown in Fig. 4 is for a

A \ I MOTION PLATE MOTION

*I 4 Fig. 4. Forces on a plate during bending

machine with both bottom rolls driven. Hence driving forces Pfb and P,, occur at the bottom rolls and the top roll bearings' friction resistances P,, must be overcome. Since the lever arm of the bending moment due to the frictional forces is small, it is assumed that these may be neglected.

A further complication arises from the fact that as the maximum deflection of the plate increases the support length, I = &+Zb decreases. Another effect of having rounded supports is that an end thrust occurs, which not only tends to increase the bending moment but causes an axial compressive force. It is proposed to neglect the compressive force and thus to assume that the compressive stresses set up are small relative to the bending stresses. However, the bending couple caused by the hoop thrusts may easily be incorporated Xdesired.

GEOMETRICAL ANALYSIS T o calculate roll torque, roll force and other process details a simplified loading system is assumed. This system is shown in Fig. 5 and the assumption that the deflected form of the plate is an arc of a circle is made. The radius of this circle is assumed to be pp. We write R1 = p p - $ but if t is small compared to pp then Rl -h pp.

Considering Fig. 5 for a plate of unit width perpen- dicular to the plane of the paper,

F

k = r,-XJ+XE . . . . . (loa)

k = r,-RR,+d(R,+t+rl)2-f2 . (lob) and hence

or by re-arranging and simplifying

. . (11) (k- rJ2 + f 2 - ( t + rJ2

2(r, + t+rl- 4 Rl =

DIAGRAM F O R GEOMETRICAL ANALYSIS NOTE LENGTH

Fig. 5. Assumed deflected form of the plate duting bending

J O U R N A L O F S T R A I N A N A L Y S I S VOL I NO 5 1966 401




Two factors of importance may be deduced using Fig. 5 which are not based on the assumption of circular arc bending. These are the maximum and minimum separa- tions of the roll centres, 1 and &, which are shown in Fig. lb. The maximum separation of the roll centres occurs when no bending (including elastic bending) occurs. The minimum separation occurs when the minimum radial distance between the top and bottom roll faces is equal to the plate thickness.

We have

R=r ,+r ,+ t . . . . * (12) and

= d(r ,+r l+ t )a - fa . . . (13) Now, P = 2P, cos 6 and from equation (9),

The sidethrust P, = PI sin 6, which with equations (8) and (14) gives

Pf * J P - . . (15) - 2 d ( R , + rl+ t)2 -72

By considering the forces acting about point F (Fig. 5), and assuming that the maximum bending moment is constant across the cross-section,

M = +PJ C O ~ e+P,s or

where 6 is the greatest deflection of the plate below its support points, i.e. 6 = GF.

M = SPl++PS tan6 . . . (16)

From the theorem of crossed chords, (HG)2 = (2R1-GF)GF or (+l) , = 6(2R,-6)

so that assuming 6 is small 1 2 a = -

8 4 . . . (17)

Length 1 is a variable, depends on R, and 1 = 2f-2r, sin 6

Thus substituting into equation (16) A P tan 6

M = 4-P(2f-2r1 sin 6+- (2f-2r1 sin ~ 9 ) ~ (18) 16R1 where sin 6 and cos 6 are obtained from equations (8) and

If 6 is small, which is the case in most practical cir- (9).

cumstances, then M = a ~ l = 4p(2j-2rI sin 6) . (19)

The internal resistance of the bar to bending is

where t t e l = - =

2PP 2@1+3t) Equations (20) may be used with equations (18) or (19) to determine P. When bending thin bars the weight of the upper roll should be taken into consideration so that the vertical thrust on the top roll housing would be P minus the weight of the upper roll.

After the load is removed elastic springback occurs such that

Consider a small element of the bar of initial radius of curvature R, and length dl along the neutral plane with some bending moment M, applied, as shown in Fig. 6b(i). Let the bending moment increase to MI such that the radius of curvature becomes R1, see Fig. 6b(ii). Assuming that the average bending moment Ma = +(Ml+ M,) then

W.D. = 1" Ma d6 = Ma(61-62) %

W.D. = Ma --- per unit length G1 l2 The work done in bending the plate is expended plastic-

ally and any elastic springback is assumed to embody re- coverable work. Therefore, equation (22) may be modified to

W.D. = M a --- ( P i P:) = 2

from equation (7). Now p e = EI/M and p p occurs when M = M and thus Ma = M. Hence per unit length

M Pr

W.D. =- . . . * (23)

and the hp to drive the machine

. . , . (24a) - MV --

55OPr The horsepower equation may also be re-written using

w as the angular velocity of the rolls, and assuming no slip occurs so that w = v/r, , and

Tv hp = - . . . (24b) 550r, '

Equating these last two expressions

LOAD ANALYSIS The method by which an analysis of this kind proceeds assumes a stress-strain relationship, u = Ae" or a = Y+

i I I

Fk. 6. General explanatory diagrams

402 J O U R N A L O F S T R A I N A N A L Y S I S V O L I N O 5 1966




Cu 1 Mg

0.04 4.55 0.07 4.43 0.05 4.88

---

He, may be found in Appendix 1. As in elastic bending theory, it is assumed that

Si

0.11 0.12 0.18

This assumption is only applicable if p is reasonably large.

DESCRIPTION OF APPARATUS AND SPECIMENS

The experimental rig is shown in Fig. 7. The two lower rolls were driven through a double worm reduction box and a helical gear train, at 8.29 rev/min. The lower rolls had equal diameters of 4.431 in and were placed at equal distances on either side of the top roll (5,999 in centres).

The top roll diameter differed from that of the lower rolls and was 4.9196 in. All rolls had a working length of 9 in and were designed to have a maximum central deflection of 0.0015 in; the maximum design loading was not in fact encountered. The rolls were made of a good quality alloy steel, being hardened and tempered prior to final grinding.

The distance between the upper and lower roll centres was measured by inserting packing slips and shims under the upper roll bearing housings. The setting was made firm by the use of clamping screws, which could be locked, forcing the housings of the upper roll bearings on to the shims. A load cell was inserted under each clamping screw and the output signal was recorded on a U.V. recorder. A torque bar was utilized to record the driving torque. The input to and the signal from the unit were communicated by a mercury bath commutator. The output signal was also recorded on the U.V. recorder. The torque bar was rotated at the same speed as the lower rolls.

The torque bar measured the frictional resistances of the six roll bearings, the lower rolls driving gears and one support bearing. Since the torque to cause bending only was required it was necessary to obtain a calibration of frictional resistance versus upper roll load. All strain gauge devices had fully active bridges and the galvano-

meters of the U.V. recorder were calibrated by using standard testing machines. A device was incorporated to measure the angular velocity of the lower rolls drive shaft during bending.

The radius of curvature of the specimens after bending was measured by a two inch gauge length circlometer with a one ten-thousanth of an inch dial gauge.

The specimens reported on here* were prepared from three thicknesses of N.P.8 aluminium alloy, nominally &, + and 3 in thick. Table 1 shows their chemical analyses and Table 2 their thickness variations. All specimens were nominally 4+ in wide with their length in the direction of rolling. It was determined experimentally that specimens

Table I . Chemical composition of specimens

t, in

0.756 0.491 0.263

Spec. B.S.

0.1* 3.5- 0.6* ~ 5.5 ~

- Fe -

0.37 0-35 0.34

0.7*

- Mn

0.72 0.39 0.42

1 .o*

-

- Ni

0.01 0.01 0.01

-

-

- Zn

0.10 0.05 0.10

0.1*

-

Note: All constituents per cent. * Maximum.

Table 2. Specimen thickness variations

- Cr

0.12 0.04 0.03

0.5*

-

- Ti -

0.06 0-06 0.06

0.2*

Nominal 1 Maximum I Minimum I Mean

0.265 0.260 0.263 0.493 0.489 0.491

~ 0.759 I 0.750 I 0.756

t .t a

All dimensions in inches.

* Specimens of two other nominal thicknesses, 4 and Q in, were also rested. As the correlation between experimental results and theoretical predictions was similar in all cases, information concerning these specimens has been omitted for clarity.

Fig. 7. General arrangement of experimental machine: A-motor; B-gearbox; C-U.V. recorder; D-mercury bath commutator; E-tachometer; F-load cell; G-rolls

JOURNAL OF STRAIN ANALYSIS VOL I NO 5 1966 403 at UNIV OF OKLAHOMA LIBRARIES on October 28, 2014sdj.sagepub.comDownloaded from



0.756 80.6 0.188 7.25 26.5 3.14 9.8 9.9 3.5

25.0

26.8

48.0

12 in long could be used and that a zone of constant curvature approximately 3 in long was then produced. The leading edge of each specimen was tapered for 3 in and on those specimens used for bending to smaller radii of curvature this edge was also carefully bent.

Since the plate from which the specimens were machined was delivered in three pieces per thickness and was not marked to show the original continuity, all specimens were taken at random. A total of four tensile specimens were cut from two sheets, three being used to derive the tensile stress-strain properties and one being used as a trial specimen. The results of these tensile tests are reproduced in Figs 8, 9 and 10 and are shown with the moment-strain plot derived by graphical integration after assuming fc(e) = ft(e). Compression specimens were tested for the +in plate and the results are shown plotted in Fig. 10. A summary of the physical properties is shown in Table 3.

B.S. - - - - -

10.0 10.0 3.5

- ~ ~ G T I

38.0m

tr0.263 in E ~ 1 0 . 6 Ibf/in2x lo6

I I 1 1 I 9 0

Fig. 8. Stress-strain and moment-strain relations, t = 0.263 in

1.0 2.0 30 40 .-per cent

404

4 0

30

2 3 f5.x x x N C

8 0

51 IC

C

Table 3. Physical constants for the specimens

Specimen t , in Brinell Hardness No. : Indexn* . constant A*, 104 lbf/inZ: Constant Yt, lo3 lbf/ina . Constiyt Ht, lo5 lbf/in2. E tension, lo8 lbf/ina E compression, loB lbf/if;" G, loe lbf/inz . . Limit of proportionality, lo3 lbf/inz .

0.1 per cent proof stress, 1031bf/inz .

Ultimate tensile stress, 1031bf/ina .

0.263

0.204 5.76

3.71

75.0

18.0

10.6 - -

17.6

18.8

40.8

0,491

0.209 6.61 23.0 3.00 9.6

76.6

- -

21.2

23.6

-

* D = Ae". t o = Y+He. m = minimum.

TESTING PROCEDURE The main series of tests on the three-roll bending machine was aimed at determining the radius produced, the vertical force on the upper roll and the torque required for a given upper/lower centre height distance parameter d . It was found that the speed variation of the driven rolls was insignificant, being less than 2 per cent except when specimens slipped over the roll surfaces.

The upper/lower rolls' centre height was set by means of the setting blocks and shims for a given d, the machine was started and zero readings of the load cells and torque bar were recorded. The loading screws were then tightened to force the top roll on to the setting blocks; the total pre- load varied with the plate thickness from 200 to 500 lbf. The specimen was then inserted between the rolls and 'squared' visually before being released and allowed to proceed between the rolls. For each value of d at least three specimens were tested and the results averaged. The results of these tests are shown in Figs 11 to 19 by open circles. (The units, where not indicated, are in inches and pound force.)

At the smaller radii of curvature, the specimens after bending exhibited marked anticlastic surfaces and it was clearly seen that during bending the specimen was not in

I I I I I I 1.0 20 30 4.0 5.0

*-per cent


JOURNAL OF STRAIN ANALYSIS VOL I NO 5 1966




4 -

a-per cent


L

,=0.7561n FROM U='fic 2 - I I I

complete line contact with the back lower roll and the upper roll. The upper roll contact was symmetrical about the centre line of the specimen but extended over only a fraction of the specimen width, as shown in Fig. 22, while contact on the back lower roll was limited to two bands at each edge of the specimen. Transverse profiles of the outside of the bend of certain specimens are shown in Fig. 20.

DISCUSSION OF RESULTS The principal experiments consisted of measuring the relaxed radius of curvature, the vertical upper roll force and the torque required to bend the specimen for a given roll setting d. Since the parameter d is of minor interest in practice, the results are shown plotted against the logarithm of the relaxed radius of curvature pr. Because the speed of the rolls was substantially constant the horsepower was proportional to the torque and plots of these are therefore omitted. The speed remained nearly constant during testing because the machine was over-powered and the gearbox tended to steady any fluctuations in loading. The maximum horsepower used in bending was 0.85 with the gearbox absorbing 2 of the motor's rated 5 hp. In all the single pass tests the maximum load and torque occurred just after the specimen entered. After this initial peak the

n

0 1.2

P J 0.8

- ~ 0 , 2 6 3 in ' 0.8 I .o 1.2 1.4 I 4

'09 Pr

Fig. I I . Experimental and theoretical results, P/log pr, t = 0.263 in

5.0

3.7 5

3

i

X

2.5 .E

P -

1.75

3

load and torque dropped sharply and then climbed to a steady value. The initial peak was minimized if bending by several passes was attempted and this would seem to indicate that had the specimens had pre-formed ends the initial peak would not have occurred.

The results of the upper roll vertical force versus log pr are shown in Figs 11,12 and 13. They show a considerable

GEOMETRICAL 6

n

0 - 4 0

x

d. 2t

I z0.491 in

0 o : e ' I . '0' 1.2 ' 1.4 ' ' 1. 6 I.8 ' A '09 R

Fig. 12. Experimental and theoretical results, Pllog pr, t = 0.491 in

241 ALLOWING FOR SIDE THRUST 20

" 16

I2

EOMETRICAL

0 X

- --I--- - .

FROM d=Ae"

J O U R N A L O F S T R A I N A N A L Y S I S V O L I N O 5 1966 3

405 at UNIV OF OKLAHOMA LIBRARIES on October 28, 2014sdj.sagepub.comDownloaded from



amount of scatter particularly with the thinnest material. This may be attributed to the difficulty in measuring such small loads with such a massive rig. The general scatter may be caused by the bearing housing tending to wedge itself due to the sidethrust, although care was taken to ensure that the sliding bearing housings were free before pre-loading. Some of the scatter may also be due to the initial entry conditions. It is possible that specimens may have been subjected to a longitudinal compressive or tensile force depending on whether the initial entry was caused by a pull from the back roll or a push by the front roll.

When producing bars having a relatively large radius of curvature a significant error could be caused by the variations in thickness of the specimen. A small variation in the value of t when d is small makes a large difference in the resulting relaxed radius of curvature. It is also when deal- ing with the larger radii of curvature that circlometer inaccuracies are at their greatest. The surface finish also becomes significant with these specimens and any variation in physical properties has a more pronounced effect.

The geometrical analysis is seen to agree well with the experimental results. The shape of the plot is correct but in general lies some 10 per cent higher than the theoretical points. This is probably because only the tensile stress- strain plot was used in deriving the moment curvature relationship; the tensile stress for aluminium is generally larger for the same strain than that for compression, as shown in Fig. 10. Results of a geometrical analysis when it is assumed thatf(e) = +[fc(e)+ft(e)] are shown in Fig. 13 for the 0-756 in material. This latter analysis is seen to give a better fit with the experimental results than that derived by assumingf,(e) = ft(e). Also shown in this figure is the plot when the sidethrust is introduced into the geometrical analysis. A small improvement in fit is evident but is too slight to be worth including in the theory considering the extra computation involved. The predictions due to the loading analysis are reasonable for the larger radii but tend to give low results for the smaller radii.

The plots of the logarithm of the torque versus log pr are shown in Figs 14,15 and 16. Scatter is seen to occur in these plots but not as much as in the case of the upper roll force plots, which seems to support the idea that some jamming of the upper roll bearing housings may have occurred. The causes of this scatter are probably similar to those advanced earlier.

26

24

2.2 m

s x )

1.6

1.4

109 P,

Fig. 14. Experimental and theoretical results, log Tllog pr, t = 0.263 in

The plots of log ( l / d ) versus log pr are shown in Figs 17, 18 and 19. The experimental plots exhibit far less scatter than is seen in the previous plots. The worst area for scatter is the portion of the graph where pr is large, which is to be expected as explained earlier. Both the geometrical and load analyses are seen to give poor correlation in this plot. In general, the shapes predicted by the two analyses are similar but the curves are translated. Fig. 19 shows a plot of log ( l / d ) versus log pp, where pp is the plastic bending radius, i.e. before elastic springback occurs. The use of the theory of elastic springback is verified since a com- pletely different curve is found.

Since the torque and upper roll force predicted are satisfactory, correction factors for d must be introduced.

+- . I 4 1.4 1.6 1.0 1.2

2.21 I t I 0 1'4 I I , I I ,

109 P,


GEOMETRICAL

FROM Q=Ac*

1 1 I I \ , , , , I I 0 . 9 1.1 1.3 1.5 I .7 I .9

109 P,


5 b OI - EOMETRICAL m , , , , , , r=0.263in

109 P,

Fig. 17. Experimental and theoretical results, log (l/d)/log pr, t = 0.263 in

0.3

0. ,9 1 . 1 I .3 I .5 I .7

406 JOURNAL OF STRAIN ANALYSIS V O L I NO 5 1966




Using log pr = 1.4 as an average value of pr, the factors are shown in Table 4. Consideration of Tables 3 and 4 shows that neither the physical properties nor the thickness would seem to have much influence on this factor. Apart from the results of the 0.756 in material the variation is small. Averaging all such correction factors* a value of * The average of 0.687 was derived by including values of the correc-

tion factors for the unpublished 0.363 and 0.618 in thick aluminium specimens and the 0.503 in mild steel specimens reported in Part 2: these values were 0.645, 0.682 and 0.700 respectively.

Table 4. Parameter d correction factors

Thickness in . 0.263 0.756 Correction factor for h : ! 0.676 1 :% 1 0.793

O * I W I = , 0.491in , , , , , ,

0.3 0.7 0.9 1 . 1 1.3 1.5 1.7 1.9

109 P,

Fig. 18. Experimental and theoretical results, log (I/d)/log ,or, t = 0.491 in

0.687 is obtiined and the effect of multiplying d by this value is shown in the figures. It is seen that, apart from the 0.756 in material, the geometrical analysis then gives close agreement with the experimental points. This correction factor is obviously dependent on pr and it is reasonable to suppose that this dependence is attributable to the variation in the support length. It is proposed to round off this correction factor to 0.7.

The photograph shown in Fig. 20 is of a grid (which was scribed on a specimen before bending), after the specimen had been bent to a radius of curvature of about 9 in. The radial lines are seen still to be straight. This justifies the assumption that plane sections remain plane and that the shearing stresses due to bending may be neglected.

The specimens used by the authors were not plates but

I , , 1 . 1 1.3 1.5 1.7 1.9

109 P,

Fig. 19. Experimental and theoretical results, log (l/d)/log pr, t = 0.756 in

Fig. 20. Photograph of grid on a 0.756-in aluminium specimen after bending

JOURNAL OF S T R A I N A N A L Y S I S VOL I NO 5 1966 407




wide bars, if the definition of a plate as a section exhibiting no anticlastic curvature except at the edges is accepted. Anticlastic bending occurred to some extent in all cases. The effect of this phenomenon was to change the loading system on the rolls as shown in Fig. 21. It is seen in Fig. 22 that plastic distortion occurred at the outer edges of the plate on the outside of the bend.

? PER UNIT LENGTH

U

LOADING ON TOP AND BOTTOM ROLLS WITH NO ANTICLASTIC CURVATURE

TOP ROLL PER UNIT

LENGTH b - P' 2

- $' BOTTOM ROLL

+ , - - - --I=-- I J

LOAD I NG WITH ANTIC LAST I C C U RVAT U RE

Fig. 21. Effect of anticlastic curvature on roll loading I

The assumption that the deflected form of the plate is an arc of a circle leads to better results than does the assumption that the loading system is as shown in Fig. 6a. It is suggested that this may be due to the following factors. At least one half of the plate is at some radius between p p and pr in magnitude and the remainder of the plate is between co and p p in magnitude. Also, it can be seen from Fig. 3 that the expression u = Ae" does not give a good fit to the stress-strain characteristics of the material until the strain is about 1 per cent. In assuming the loading system shown in Fig. 6a much of the beam has a maximum strain of less than 1 per cent.

CONCLUSIONS From the experimental work described the following conclusions may be drawn.

(1) (a) The geometrical analysis may be used to predict the upper roll vertical force, the torque required for bending and the relaxed radius of curvature for a given roll setting to the first degree of approximation by considering the tensile stress-strain properties of the material only.

(b) Better approximations are found if the stress used to determine the internal bending resistance is taken to be

(c) T o determine the upper/lower rolls centre distance from the maximum possible d, the derived value of d must be multiplied by a constant of 0.7.

(d) The shearing stresses due to bending may be neglected.

(e) Plastic anticlastic curvature appears at values of K = 9, where K = b2/prt, and causes a re-distribution of loading on the rolls. (f) The formulae are restricted to bending to a given

radius of curvature in one pass.

f(4 = +[fc(e)+ft(e)l.

HEIGHT ABOVE

0.5 1.0 1.5, DISTANCE FROM OUTER EDGE-in

Fig. 22. Anticlastic surfaces of specimens after bending, outside of bend

408 JOURNAL OF STRAIN ANALYSIS VOL I NO 5 1966




(g) The theory of plastic bending is only partially verified.

(2) While the approximations u = Y+He and u = Ae' may be used to calculate the internal resistance to bending over certain ranges, the deflections calculated from these assumptions are likely to produce large errors.

Summary for the geometrical analysis For a given R,,

Part 2: Experiments using mild steel P R E V I O U S ANALYSES

Whilst other workers have given analyses and empirical methods for determining the force on the upper roll and the horsepower required when bending plates (7)-(13), as far as is known no experimental work which simulates the practical operation and which would facilitate assess- ment of their relative merits has been made and published. The authors have therefore compared their experimental results with the theoretical predictions of other workers as well as with their own.

In 1932 Knight (7) published a semi-empirical method for determining the forces on the rolls of a pyramid type bending machine when bending mild steel plate. The assumption of perfect plasticity was made, but he remarked that the true modulus of the section would lie between that for the elastic and that for the perfectly plastic cases. His analysis was geometrical in nature, since deflection into a circular arc was assumed. The thickness of the plate was assumed negligibly small compared with the radius of curvature of the plate. On the basis of these assumptions Knight derived the equation

ubt2(r+ rl) 2fr . . * (27) P =

where r = p,+Jt, and 2f is the lower roll centre distance. The experimental work of Knight was performed by

removing the rolls from a standard machine and mounting them in a specially constructed rig. The plate was placed across the bottom rolls and the top roll was loaded by adding weights to a cradle. Before measuring the radius of curvature the load was removed. The exact method of obtaining the radius is obscure; Knight refers to a rubbing of the deformed plate from which the radius was found. However, since the radius varies over the length of the plate by virtue of the loading system it is difficult to see how the minimum radius of curvature could be accurately

-'= P&2 I t

Fig. 23. Knight's experimental results, reproduced from reference (7)

4M P = -, I where M = +(RJ = #(pP-+t)

k = t;-Ri+ 1/(R,+rl+t)2--f2 d = 0.7(k-k) = 0.7(r,+r,+t-K) 1 1 M

Pr PP EI -= _-_

Mr T=- Pr

Mv hp = - 550

and some theoretical comparisons found. Knight tested five plates of various thicknesses and widths of nominally the same mild steel, some initially flat and others curved. He observed that the accuracy of his experiments was not high but stated that they were accurate enough for machine design calculations in which a safety factor of four was incorporated.

From these experiments Knight derived the curve reproduced in Fig. 23; no actual experimental points were given. Knight attempted to overcome the shortcomings involved in assuming a perfectly plastic material by using a modified yield stress u in equation (27). To determine the force P required on the upper roll to bend a plate of thickness t to a required radius of curvature r the r/t value was calculated and a value of u was then put into equation (27). The number of passes to achieve the required radius of curvature was considered immaterial. The curve in Fig. 23 is restricted to mild steels of yield stress 16-22 tonf/in2 and ultimate tensile strength of 31-32 tonf/in2. Knight also used his method to calculate the loads required for four-point loading systems. No formulae for the torque required for bending or for the necessary roll setting to produce a given radius of curvature were quoted.

A paper by Oehler (8) is a summary of the five different analyses of Flender (9), Geiger (xo), Oehler (11), Zurcher (IZ), and Geleji (13). Flender (9) neglected all frictional effects and assumed bending into a circular arc. The maximum bending moment was considered to be equal to Plu, where PI and u are as defined in Fig. 5. An elastic-perfectly plastic stress distribution was assumed for larger radii of curvature, and a perfectly plastic distribution for radii of curvature of less than 100 times the thickness. Flender wrote at length about the design of the rolls; his derived equations are:

1 P = - 2b uu cos e --- 'fk)] for r , > loot U

bt2 2u P = - uU cos 8 for rm < l O O t

and bt2u, 700, [" - 3;2ebr,) P = -+- bt 4u 3u

for r < 25t where u, is the ultimate tensile strength and r , is the radius of curvature of the centroidal axis. Flender did not derive equations for the torque, work done, or power required during the bending process.

409 J O U R N A L OF S T R A I N A N A L Y S I S V O L I N O 5 1966




The exact assumptions and analytical method of Geiger (10) are not known by the authors but bending into a circular arc was clearly assumed. Oehler states that the formulae derived were

and } (29) bt2 2r,e W.D. = - u y ~ l [1--1] 1 4rm

where Zl is as shown in Fig. 5, Part 1. (An expression for the horsepower was derived by Geiger but contained dimensional errors and was not included by Oehler.) The above equations suggest that a perfectly plastic stress distribution was assumed but that it was modified by the addition of a second term. The symbol e is stated to be equal to t/2rm by Oehler; however, substitution of this into equation (28) makes the second term equal to 1/3 and equation (29) then gives the work done as zero. Oehler lists three e symbols: the one defined above, eb the strain at the ultimate tensile strength, and e, the strain at the yield stress uy. As the first term of equation (4) is a term due to perfect plasticity it would seem reasonable to modify it by a function of the elastic properties and the authors have therefore identified E as Oehler’s ee = uy/E.

Geiger also derived some equations concerning the friction due to the top roll bearings and to bottowrolllplate friction. While these equations are valid, the question of assigning the magnitude of the various coefficients of friction arises. The roll/plate coefficient, in particular, is an extremely difficult one to ascertain.

Ziircher (12) assumed a perfectly plastic material and used an external bending moment of 4PZ. The derived equations were :

P = z j u y bt2 [ 1 + - ;:I . . . (30)

Plr, 8rm

W.D. = - . . . . . (31)

and

where r, is the external bending radius, Z, is the curved length between roll contact points and v the speed of the plate. Ziircher also considered the necessary starting torque, the efficiencies of motors and wormgear drives, multipass bending and cases when the bottom rolls are equidistant from the top roll centre.

The exact assumptions made by Geleji (13) are also not clear. The analysis assumes circular arc bending and it would appear that a stress distribution of the elastic- perfectly plastic type was used. The derived equations are:

p = - [ 3t2- (““F)”] - . . . (33) 61

W.D. = - [ 3tz- f;q2] - . . (34) 12 and

hp = (12)(550R,) uybv b t z - fq2] . (35)

In 1952 Oehler (8) published an analysis of a pyramid type plate bending machine. (It is also briefly discussed in his later paper (t), but without reference to his initial assumptions and method of derivation.) The method is semi-empirical, as certain constants were determined by experiment. Oehler’s equations are

clbt2u, p = - 1 . ’ .

where c1 0.7 or c1 = (1+4tZ) for cl/ t < 15, and

The constant c2 allows for rolling friction and for the inefficiency of the gear components of the motor. By considering cz = 1.5, a value of 6 was determined for the expression 2nc2. The horsepower was given as

where r is the time for one pass in seconds. The summarizing paper of Oehler of 1962 (8) is of

necessity curtailed but in each instance certain important pieces of information are, unfortunately, omitted. The effect of elastic springback is not explicitly mentioned and no attempt is made to incorporate it even indirectly. Since the analyses were made primarily to provide machine design information, the actual radius produced was probably considered of little consequence. No attempt to differentiate between bar or plate bending was made, although anticlastic curvature is briefly discussed. In all the analyses referred to the formulae seem to be based on elementary bar theory. The parameter K, the vertical centre distance between the upper and lower rolls, is ignored. All the analyses seem to be for mild steel plate although it is not stated explicitly. Also, no mention is made of any difference between the tensile and compressive stress-strain properties, but as mild steel often has similar properties in tension and compression this neglect may be justified when the analyses are applied to the bending of mild steel. The interest in roll design and drive efficiencies is to be expected. However, the incorporation of roll design and gearbox efficiency factors in an analysis of this description is not of fundamental value. Oehler uses a value of 1.5 for his constant c2 in equation (37). The constant is incorporated to allow for gearbox inefficiencies. Since gearbox inefficiencies may range from about 30 to 90 per cent, depending on the type of gearing and loading, very large errors could occur by assuming Oehler’s value for this constant.

EXPERIMENTAL RESULTS The method of testing using the experimental three-roll pyramid type plate bending machine is to be found in Part 1. The mild steel specimens on which tests were con- ducted were in the black condition with poor surface finish. To improve as far as possible the accuracy of the measured relaxed radius of curvature, circlometers of gauge lengths 2,3 and 4 in and specimens of lengths 12,15 and 18 in were used, the longer specimens being used when it was the intention to bend only to the larger radii of curvature. All specimens had their leading edges tapered, and their corners removed, to permit entry.

410 J O U R N A L O F S T R A I N A N A L Y S I S VOL I N O 5 1966




Fig. 24. Stress-strain and moment-strain relationships for the mild steel specimens

The average tensile stress-strain characteristics of the specimens are shown in Fig. 24; the characteristic shown is based on an average of four test pieces taken from three different plates. The moment-strain relationship, derived graphically from the tensile properties only, is also shown.

The chemical analysis of the mild steel was 0.24 per cent C, 0.03 per cent Si, 0.041 per cent S, 0.035 per cent P, 0.95 per cent Mn, 0.03 per cent Ni, 0.03 per cent A1 and nil Cr.

The thickness of the plates varied between 0.499 in and 0.507 in with a mean of4503 in. The specimens to be tested were taken at random from the supplied material on all occasions since no information concerning the continuity of the specimens was known.

The principal experimental results are shown in Figs 25, 26, 27 by open circles. These plots show how the upper roll vertical force, the logarithm of the torque required to bend the specimen, and the logarithm of l/d varied with the relaxed radius of curvature pr. Variation of the bottom roll angular speed was negligible throughout the tests, except when slipping of the specimen over the rolls occurred. The predictions of the geometrical analysis are shown by the thick line and those of the load analysis, in which it is assumed that p = Aen, are shown chain dotted.

Fig 4 shows the forces on a specimen as it passes through a bending machine with both bottom rolls driven. If the lower exit roll only is driven then the friction force P,, is reversed in sign and changed in magnitude. Consequently,

if the friction forces have a significant effect on the bending process an appreciable difference should be found. Figs 25-27 show the results of bending tests, indicated by black squares, in which only the exit roll was driven. No significant difference is seen between the results.

DISCUSSION OF EXPERIMENTAL RESULTS The upper roll vertical force versus log pr plot, Fig. 25, shows a considerable amount of scatter, as did the similar data from the aluminium tests. The geometrical analysis is seen to give low results: this may be due to the yield stress in compression being larger than in tension, a feature that is sometimes found in mild steel. The geometrical analysis predicts values about 15 per cent lower than those found experimentally. Fig. 26 shows that the torque predicted by the geometrical analysis is also low. In the case of torque the discrepancy is greater at large values of pr which may be due to the retention of the upper yield point. The log ( l /d ) versus log pr plot, Fig. 27, also shows a deviation at large values of pr. The load analysis predicts slightly low values for the P versus log pr results and progressively lower values are predicted as pr decreases; the torque and log ( l / d ) versus log pr plots are unsatisfactory.

In Fig. 27 the effect of the factor 0.687, whose derivation is discussed in Part 1, is shown. The full line was obtained from the unmodified geometrical analysis and

J O U R N A L OF S T R A I N A N A L Y S I S VOL I N O 5 1966 41 1 at UNIV OF OKLAHOMA LIBRARIES on October 28, 2014sdj.sagepub.comDownloaded from



0

the chain dotted line was derived by multiplying d by 0.687. It will be seen that a much better correlation is obtained. (The correction factor was actually rounded to 0.7.)

The results of driving only the lower exit or back roll are shown in Figs 25-27. No significant differences in behaviour can be seen up to the point at which the experiments were terminated. (Slipping of the plate over the rolls occurs more frequently with this system than with a two-roll drive and it was because of the resulting scoring of the rolls that the experiments were terminated.) It is therefore concluded that the effect of the roll plate friction forces during bending may be neglected.

In order to attempt to assess the accuracy of the analyses of Geiger, Zurcher, Oehler, Geleji and Flender it was con-

- r =0.5031n

I I I t t , I r I

I GEOMETRICAL

TWO ROLLS FROM d = A C " ' BENDING WITH O

1 = 0 . 5 0 3 In , ONE ROLL 't DRIVEN

O1 0:8 ' IlO ' 112 ' ll4 ' 1:6 ' ll8 ' 210 109 P,

Fig. 25. Experimental and theoretical results, Pllog pr

BENDING WITH 0 ~ ~ ~ E ~ L L S . ONE ROLL DRIVEN

EOMETRICAL

109 Pr

Fig. 26. Experimental and theoretical results, log Tllog pr

J n.o I jwp GEOMETRICAL

,- ncuniue w i T u .. TWO ROLLS

0 .9 1 . 1 1.3 1.5 1.7 1.9 109 P,

Fig. 27. Experimental and theoretical results, log (1ld)llog Pr

sidered necessary to assume that the radii of curvature they referred to was the plastic bending radius, i.e. before elastic springback. No remark is made by Oehler (8) concerning this parameter even though large differences between pp and pr occur when pr is large. The authors calculated the relaxed radii of curvature by assuming M = aP1 in the theory of elastic springback irrespective of any other author's expression for the applied bending moment. The comparisons are illustrated in Figs 28 and 29 for the mild steel specimens; for further information, comparisons are made in Figs 30 and 31 for the 0.756 in aluminium specimens discussed in Part 1.

For the aluminium alloy the authors' geometrical analysis is seen to give the best correlation for the torque and upper roll vertical force for a given k. The predictions of Geiger, Zurcher and Geleji are low, and that of Oehler high. The loads predicted by Flender and Geleji and the torque values of Zurcher are far from the experimental results.

For the mild steel, Oehler's equation predicting the upper roll vertical force gives good correlation though this equation contains an empirical constant. The forces predicted by Geiger, Ziircher and the authors are low. The method proposed by Knight, which is based on experimental data, is also low. While this may be due to the authors' specimens having different stress-strain characteristics from those of Knight, it is more likely that the error lies in the manner in which Knight determined the

2.41 I I I

0.9 1 . 1 1.3 1.5 1.7 1.9 109 P,

Fig. 28. Comparison of experimental results with the analyses of previous authors for mild steel, log Tllog pr

FLENDER I I / I ,

I2 n

P - 8 P- I

x

a

412 J O U R N A L OF S T R A I N A N A L Y S I S VOL I N O 5 1966




4 - I ' / ' I-+

0.7561n I I I O I I 8 k f i I

radius of curvature. For the torque required the predictions of Geleji and the authors are satisfactory while Oehler's is low. The equations for deriving the upper roll vertical force of Flender and Geleji and that for predicting the torque required for bending of Zurcher are again in error.

In general Oehler's equations are the most satisfactory of the previous workers although his predicted torque values are low. In using Oehler's equation for predicting the torque required his constant c2 was omitted. This constant was introduced by him partly to allow for gearbox inefficiencies, a correction which was not necessary in the authors' case.

For the two dissimilar metals tested the authors' geometrical analysis appears to give the most reliable predictions, and it is hoped that this may therefore be applied with some confidence to the bending of a wide range of engineering materials.

2.4

CONCLUSIONS From the experimental results presented the following conclusions may be drawn.

(1) (a) The geometrical analysis of the authors predicts to the first degree of approximation for mild steel, the upper roll vertical force, the torque and the roll

- f=0 .756in -+ I I 1

I

3.b ASSETT AND JOHNSON

32

h rn 0 - 2.8

setting required to bend plate to a given radius of curvature.

(b) The general applicability ofthe geometrical method is made clear for use in connection with any metal.

(2) The assumption that the roll/plate friction forces may be neglected in formulating an analysis is upheld.

ACKNOWLEDGEMENTS The authors wish to thank Professor F. Koenigsberger for advice and discussions concerning the design of the machine in its early stages and Mr J. Flowett, Mr G. Robinson, Mr J. Grady and Mr D. Graham for their very helpful technical assistance. Dr M. B. Bassett was awarded an Aluminium Scholarship given by the Alu- minium Federation (administered by the Institution of Mechanical Engineers) for which he wishes to express his appreciation. He would also like to thank Mr J. D. Wallis for help during the preparation of the manuscript. The authors are also grateful to B.I.S.R.A. through Mr T. Johnson for financial help in purchasing equipment and material.

APPENDIX 1

A N A L Y S I S B A S E D O N T H E L O A D I N G S Y S T E M O F T H E P L A T E

Let u = Ae" and ft(e) = f&), so that MI = j::, f(e)y dy per unit

width of plate with e = y/p. Hence A I n 2

MI = - Y where Z - - P" n - (n+2) Px Py 2 2 M0 = ----tan ed From Fig. 6a

Assuming tan 0, is small

may be integrated to determine 6 = 9, thus 1 ( l l n + 2) 5 = = (l/n+l) B (f) [l-ml

The total deflection under the load is d = R--K = 6+rl(l-cos e,)

where

Now

eo =- (IlnB+ 1) (f)"'"' l)

1 = 2f-2rl sin 8,' or I = 2f-2ri sin - [(l/nB+l) (f)"'"'')]

This equation may be solved by Newton's approximation

where fo(l) = 2f-I-2rl sin - [(l/nB+l) (f)(li"+l)l (tin + 1) B

and yo([) = -Z-rlB (;)'In cos [- ( L ) ] ( l /n+l ) 2 Hence for a given P values of d, pr and T may be found since

g = E 4

AI, 'In ; PP = (3)

A similar procedure to the above may be obtained by assuming u = Y + He and with M = - Px/2 - Py/2 tan 8,' i.e. allowing for the bending component of sidethrust, and by solving using Lap- lace transforms. However, it may be shown that the results have no real validity since the term Y, in (I = Y + He, introduces a small negative radius of curvature with no load. The stress distribution is perfectly plastic with a superimposed elastic condition.

413 J O U R N A L O F S T R A I N A N A L Y S I S VOL I N O 5 1966



A P P E N D I X 2


and hence

DERIVATION FOR THE EXPRESSION POR THE B E N D I N G OF PLATES

The Prandtl-Reuss equations may be expressed in terms of the principal stresses and strains as three equations of the type

Now for plate bending u3 = 0 and ea = 0, where suffixes 1,2 and 3 denote the longitudinal, width and depth components respectively. As ua = a constant x ul the loading path is radial and a total stress- strain law may be used. Therefore, to a 6rst degree of approximation, we may re-write the equations as el = B(u1-~u2) , ez =O = B(u2 -fq) and e3 = B( -$u1 -)u2), where B is a constant.

Hence, el = B ~ ~ ( 1 - a ) . Now for uniaxial tension u = f ( e l ) and therefore, for plate bending, u = f ( e ) / ( l -a) which is a similar expression to that used in elastic theory.

The bending of partially plastic plate In general not all of the cross-section will be plastic when under- going plastic bending, i.e. the central fibres will still be elastic. To allow for the portion of the beam which is still elastic let

2b M = ;;i p2 IOe f(e)e de . . . (39)

where A = ( l - ~ , ) ~ and Y , is the effective Poisson’s ratio. Let

v,=+m+(l-m)v . . . . (40) where m is 0 for complete elasticity and 1 for complete plasticity.

If an element of a beam is bent plastically its strain distribution is asshown in Fig. 2a(ii) where e, is the yield point strain, i.e. e , = uy/E.

2 i e From similar triangles - = 2 or t , = o,f- t te E2i

Now m = - 2tp and t p = ) t - t , t

m = ( I-- ’;‘‘) . . . . . (41)

A P P E N D I X 3

REFERENCES

(I) WILLIAMS, H. A. ‘Pure bending in the plastic range’, 3. Aero. Sci. 1947 14,457.

(2) TIMOSHENKO, S. Strength of materials, Part XI 1962, 76 (Van Nostrand, New York).

Q) SEARLE, G. F. C. ‘Experimental elasticity’ 1933,40 (Camb. Univ. Press).

(4) ASHWELL, D. G. ‘The anticlastic curvature of rectangular beams and plates’,J. Roy. Aero. SOC. 1950 54,708.

(5) DRUCKER, D. C. ‘Effect of shear on the plastic bending of beams’, Trans. Am. SOC. mech. Engrs, J . appl. Mech. 1956 78,509.

(6) JOHNSON, W. and MELLOR, P. B. Plasticity for mechanical engineers 1962, 98 (Van Nostrand, New York).

(7) KNIGHT, A. W. ‘Permanent bending of cold steel’, Mechani- cal World 1932 92, 1.

(8) OEHLER, G. ‘Die Rechnerische Ermittlung der Anstellkraft, der Umformarbeit und der Leistung des Walzen- Hauptantriebsmotors von Walzenrundbiegemaschinen’, Industrie Anzeiger 1962 79, 228.

(9) FLENDER, H. ‘Theoretische Grundlagen zur Berechnung der Walzen bei Dreiwalzen-Blechbiegemaschinen’, Werk- zeugmaschine 1934 38 (Nos 12 and 14), from [8].

(10) GEIGER, W. R. ‘Das Runden von Blechen’, Blech 1954 1, 615, from [8].

(11) OEHLER, G. ‘Blechrundbiegemaschinen’, Progressus 1952 4, 53, from [S].

(12) Z~RCHER, F. W. ‘Das Biegen von Blechen a d Walzen- Biegemaschinen’, Industrie. Anzeiger 1954 76,1957, from

(13) GELEJI, G. Bildsame Formung der Metalle and Versuch 1960 662 (Akademie-Verlag, Berlin), from [S].

181.

414 J O U R N A L O P S T R A I N A N A L Y S I S VOL I N O 5 1966



the bending of plate using a three-roll pyramid type plate bending machine

Documents