The application of RV Southwells' relaxation methodsto the solution of problems in torsion of prismatic bars

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Authors Leitner, Murray Irving, 1922-

Publisher The University of Arizona.

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THE APPLICATION OF R. V. SOUTHWELL'S RELAXATION METHODS TO THE SOLUTION OF PROBLEM IN TORSION

OF PRISMATIC BARS

by

Hurray I* Leitner

A Thesis submitted to the faculty of the Department of Mathematics

in partial fulfillment of the requirements for the degree ofMASTER OF SCIENCE

in the Graduate College, University of Arizona

19U9

ApprovedDirector of Thesis /<mte z

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V Photo^elastxc Mbthods and Re suitscooooooooooooooooooo© 3 <4Mathematical Solution for

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INTBODUCTIOH. ' '

•Exjact' solutions '-of bouiidai r-v&Lue' problems- arising in ite physicaldBienees-are" often so difficult or illusive as'to reamin undiseoyered for■ ■

. very long periods of time® - In/-.general ;when ^propriate differential: :equations have been formulated, and partidular/'Selutions for .them have been: disc or ereds,' the probability - is very small that one-will discover how ;to . •

construct exact solutions satisfying the bouhdary conditions» Before :methods for obtaining approximate solutions were available and whenever.,

approximate solutions had been deemed;unsatisfactory3, it. had been thepractice to solve certain purely mathematical problems involving.boundary

conditions relatively easy to satisfy but of very little.interest to

physicists and engineers® • : ; - - - . ...For approximate solutions». -however, the boundary shape often is of -

no great difficulty®' This is notably true when .Relaxation Methods are

useds. .a problem .-so solved for one shape could be solved for any shape®It is true that fee results, are not exact because the formulated dif=»

ferential': equations, are replaced by; difference, equations whose solutions-

may be duly, -approximate solutions of the differential equations® Theaccuracy of the results depends upon the labor and time expended and not

• ©n "any higher mathematics used® v ■ - ... . .. . : -Bouthwellj, the• inventor of.:Relaxation;Methodss statess^^

; No problem is regarded as solved until5 for the actual confutation , no more than the first four rules of arithmetic are required® fee initial formulation of the problem is regarded as outside the task

. ■ R® Vo: Southwell, Relaxation Methods:,In-Theoretical Physios, (Oxford! ,G® Cumberlege, 19h6) Frontispieceo

. of the computer$ who is concerned only to solve a specified equa- . tioti and to satisfy a specified boundary condition! but to this

task Relaxation Methods require him to bring only a slight know=. ledge of mathematics and with their, aid he can obtain solutions - . having all the accuracy-that is warranted by the physical datae

• Much of our discussion follows, rather closely- that of Southwell in- his explanation of Relaxation Methodso There are5, however j. several points which require clarification in order to make the subject more easily un= • derstaridable'and -we have undertaken to present these« making such changes, as seemed.desirable®' ■ - ' - . - ' . "

The purpose of this thesis is to apply the Relaxation Methods to certain torsion problems and compare the results with those obtained by Photo=elastie experiments® Some of the numerical results are believed to beVnewd‘ The application of Photo=8lastie methods to the torsion pro­blems presented in this paper have not been carried out to the knowledge .

of the authoro . -Indebtedness is acknowledged to Dr© W® If® Denton for his assistance

and - to, the Departmeht of Mechanical Engineering' for its cooperation in - preparing the Photo=ela.stie equipment® - "

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. ' - • ■■■ CH&PTEE I ' \

. HISTORY OF RELAXATION liETHODS ' ,

For a number of years. engineers have been seeking a new type, of mathematics with which they 'might solve problems which worthodox mathematical methods18 have been unable-to-solves One such problem ap­

peared in the stresses, of a braced framework^ the solution of which led

■ to the invention of Relaxation.Methods in-1935-®.- ' "■ ' ''. Southwell considered a framework having elastic members and frie=

tipnless joints6 Rigid constraints were imposed at the joints such.that

the movements of the joints were prevented but the ends of the- members were left free to turn® All joints were constrained in this manner and specified forces applied® Since the 'constraints were rigid they sus=>tained the whole force at every joint® . - . . .... . / ’-; •-

Then one constraint was relaxed so that, one joint could move

slowly(2;). through a specified distance in a specified direction® If -the

initial ..force on the. constraint had &.'component in the direction' of travelv this component of the force was relieved and the strain energy .; stored in the members © All joints but one having remained fixed, it was possible to calculate how much force, was transferred® - - : '

Another.constraint was 'relaxed-and a.similar operation performed®

Systematically proceedings with each constraint the amount of total

R® .Fo;.Southwell, ,$S tress Calculation in Frameworks^, Proceedings of. - the.Royal Societyj,. Series Aa QU (1935) p®. ; . ■ :

(2 )so. that equilibrium -is maintained and vibrations are not . eicitedo

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strain energy was inereased with each step until the forces remaining on the constraints were so small as to be negligible. Since each Con­straint was “relaxed1* in turn, the name "Relaxation Methods", was given to the procedure^ '■ ■ • . • . \

Several people "began working with Southwell on the application of Relaxation Methods to engineering problems» The problem of the Deflec­tion of Beams under Transverse loading was solved in 1937 by Kw NoVBe

■ Bradfield^ in collaboration with Southwell® The .idea of Finite

Difference" Relations•was. introduced- in 'this' problem* . , t■ -V. "'i ■ :• : ■■ 7 ■ ■:In the same year A® Black'-5! and Southwell applied the Re­

laxation Methods to the solution of problems in Surveying and in elec- ; .trieal networks® . They also‘extended, the Relaxation -Methods for use."in gyrostatie. systemse This. extension was used in the solution of problems involving two independent. variables by Chris topherson^^ amd Southwell

early, in, 19RSo " " . • . /.• . ' Until 19W Southwell and his associates worked on problems of. ageneral nature® in I9RO3, with the advent of war in Britian/ Relaxation r. Methods were put to work solving problems closely related to the war -1

^ N«• Be Bradfield & Rs V® Southwellg "The Deflection - of Beams Under . Transverse Loading" s. Proceedings of - the Royal Societys‘ Series A, G1XI(1937) ,p. l55o , ‘ ‘ * - - .

Sec® ZoR.. ■ " \ y " V " . " V .

N« Black and R© Souttrairellg'*5Basic Theory 19.th Application to .\ - Surveying • and ■ to .Electrical ■ Networks g. Proceedings of. the 'Royal :. Society a. Series A» CL3Q.V (1938) p® ..RR7° 'f - ■

^ De Go Christopher son R® - Southwell . "Problems Involving "Two In- - dependent Variables"# Proceedings of ..the- Royal 'Society- Series A* ;cmni (1938) p0 ' - . - \ - v : ' . / . " • -

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effort© The papers produced by Southwell and his fellow workers from • IpitO to 19kh were treated as secret and were not given to open publiea= tioh until the autumn of 19k$o^^ Many'of these papers dealt with pro­blems involved in aircraft, construction and the solving of these problems saved many hours which trial tests, would have requiredo ' •

The post-war period had also produced many papers on stress and .

strain as well as papers on the solution of problems in tors ion $ tension fluid motion and percolation® It is now evident that any problem which .

is reducible to a system of linears , simultaneous, equations in a finite number of unknown variables can be solved, to the satisfaction of the

engineer, with the use of Relaxation Methods© . . . h -

(7) ■ ’ -R«, ¥©. Southwell,.;-.Relaxation Methods in Theoretical Physics, (Oxfordj - G» Cumberleges(1955)"p©~239o ‘

. . . ■ CHAFFER II . . •pmiiiEARr coNSiDBR^rioNS / ... ' -

2oI Definition' of a function from the Mathematical and Experimental . ; 'standpplnto ••¥e' say that a quantity , is a function of two in­

dependent variables x' atid/.y, when with any pair of values x and y there corresponds a definite'value for s© This dependence can,be stated either mathematically in the form of an equation^ for example5, Z ax bsqr oy tk ;

dr geometrically by 'a'suffaee such that the. height of any point P - above some zero plane represents Z and the independent variables x and y are ■ measured along two fixed directions in the sero' plane,, - 1 - '

The ordinary, relief'- map. ;ls a familiar examples “sea lev@l“ deter-1 \ ■ : mining the zero plane and the height above sea level representing Z as a function of the latitude and longitude® - .

• 2©2 Determination of a function© It is necessary to emphasise

that the determination of a function from the coordinates of points ob­tained by making measurements on a surface can neither be exact nor com­plete©" Exact data cannot be obtained because all measurements are.liable

to error: complete data would require a measurement for evexy point inthe region, a task requiring an infinite time© This is true even though the desired ablution is a function of a single variable as in the deflec­tion of a loaded beam® . 5 - .

This being truej, it is evident that the determination of a wanted.• function has-a different meaning in practical: work than in orthodox, mathematics3, where a solution .determines .the-wanted functions at every

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point in the >egion0 Rather than attempt the latter it will suffice to determine values of the function with a reasonable degree of accuracy at a number of points which is large enough to define the trend .of the func-

Uon . : v ■■■. ■:. , : ■■ ■ ' ' - r}2<s3- The Relaxation Net \ .. :

i. 2o31 The Solution of Problemse -The solution of problems in ..tillsthesis Will keep in mind, the practical"standpoint^ i.e* the problem willbe considered as solved "when- the' wanted function has been determined at". -

a number, of points large enough to allow: constant value, contour lines tobe. drawno ' To' facilitate the drawing of countour lines $. points are arranged .on a regular 18net86 so that on straight lines, drawn in various directionsthere ;-will ■ be a series of computed values equally spaced Every mesh, of:the net should be similar to every other mesh and special interest isplaced bn meshes formed by regular polygonso ;

There are two regular polygons which satisfy the above requirements %namely, the square and the equilateral triangle® These are indicated ;in Fig© 1 together with the number (N) of meshes which adjoin at any one

point and the number ( K) of sides in any one, mesh© - The . length of a niesli

side is denoted by “a1-8© : .V ■ :V;, ■ ,The: meshes above are termed '"relaxation nets'% the sides are termed ■ •

88strings* and. the junction of the strings ’’nodal'points18 or ’’nodes’8 ©

-Usually, by converting the governing equation into ’’non-dimensional” fma :we-, can made the mesh side 16a” representable by a pure. number, independent .of-the units, which is some ratio of the.mesh side to the whole boundary©

>. ■ %outhweli-'indicates these polygons, tte square, equilateral - triangle..»and hexagoh, but thepe is some-doubt in the. author’s mind as to- the

" latter©: - '..V- - ' . .. -

(a) N * Ij. k « I4. (b) N = 6 k * 3

Fig, 1

2,32 Advance to a finer net. When values are computed at points

on a given net and found to be widely spread it is possible to find values at points between the original nodes fcy the process of advancing to a finer net. While the values of *N" and "K11 are retained, the value of "a" is changed,

(2 )An example of this idea is shown in Fig. 2. ' Fig. 2c was taken from a newspaper and an estimate made of the fraction of each elemental square which was inked. Then from the record of these estimates a diagram was constructed, showing the squares and the fraction of each square inked. Another diagram was constructed in which the size of tne square was changed

as in Fig. 2b. Four of the elemental squares were combined and the inked fraction of the new square recorded. So the four uppermost elemental squares were combined into one having 39 parts in U00 inked. This process was repeated to get the diagram shown in Fig. 2a where the uppermost square contains 16 of the elemental squares.

Considering all these diagrams in order from left to right, the nature of the "advance” can be seen. Fig* 2a (the coarsest net) contains only the distribution of light and shade and Fig. 2b shows only the main_

Worked by Mr. A. N. Black •

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points of that distribution* However, there is enough detail in Fig* 2c to make any further advance unnecessary*

F3.g. 2a Fig. 2b Fig. 2c

62 10

20h Finite Differ®in@@ ApproxlBia.tioBo Bather than attempt the solution of the differential equations' formulated for some of the pro= blems w@ shall replace them with finite difference approximationso Let. us first examine the formulae for approximate differentiation^

Giveny 0 e@ "S- e , x * c3

y ® f(o) ® e©'■y a f(a) = e© e, a * eai

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-(2o2il)y .= X{gaj = e© » 2c& a ^ itc a

plying equation (Z-fi) for and Cg. we get

Jo 2jj ya2ae

S l'Since ^ - c, * 2c cby substituting for c% and eg we get

anddy „ -3y0 * lari - yg + dx 2a

2(jo - Sjj. ♦ y£)x 2a2 .. .

?! (”3y0 1% - yz * 2yo

Is (-% * yg) ' -.|i<"iyo * % ^ % * % 0 i , (ye .• 3yJ

% ^ 2y» (201|2)

V

^^The results of actual physical problems justifies the. use of the polynomial representation of the function©

Now from the equation for 5 - .we can get ii ts.and-substituting: for o ' we get: ,.

■ Y a •' " (2-eU3)Only three poizits were necessary to-find an approximation for ;

To find,,the third derivative four points would be required and our initial equation would become:' : •- ■ " . - . . . ; ■ ;

Y = .56 -> c,% -v c2k2 C3Proceeding as before .we solve for c0, c,, c2 and c3 and make the

pr oper:'. substitutions a . The resTiLts . are as " followss

Idx/o1

Idx

*■ _ i (” lly0 iSy, ' 9j%

i (— 2yc - 3ji '*■ % . ™ .y3 )

3% * Syg ,)l.dx/2a .31Wo/dy)

a

Vdx/3a- 3!^ 2yo + 9y ^18% + 11%)

= 2y0 - 3yi * liy2 - y32/d ■To ■” 2y, * y2

- 2y,, > y3

< £ zIdx2 y0 "* Wi - '5y2-;>'2y3.;

( 2 M

To * 3tx- - 3yz j3 (4)

^^The coefficients of in a range of n points follow thecoefficients: of the Binomial ■ Expansions " -

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’ • The aeeuraQr with which we can estimate . etco.. depends up©n. the number of points in the range of x at which y is specified^ • Referring to Fig© 3 we - can find, approximate values for tiie derivatives by taking the difference between values at two points and dividing by the distance b@=> '

\ tween the points© ■. ••■r- . f, - ■ . - ,- ■ f . • . .

n

8

m I,-; o /' / f

-

■ , i' 1r

2T: ; .

. . , ' ; . : Fig© 3 _ , ■ '. . .

■ It is possible to approximate the first, derivative at the point 0 by taking values of the function at the. points 1 and 3 and dividing by ; the distance between them, i®.e© 2ae ■ . -

The ordinary differentials will be replaced by partial derivatives., : in oztier to extend - the results to functions of,/two variables© Thus we have

• ; ; let us introduce; the,points I8 and-38 for reference .points althoughthe wanted function is not. specified at these points 9 The second derive .. ative. can be found by. talcing the difference between first derivatives at two points and dividing by the -distance between the points© . '

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M l •* fel-iP HY — - Wa VJq - 1a . a,

, <7© / a, a. , _

: ;■ = (2.W)

■ . • By taking the additional points I and III it is now possible to.find the third and fourth -derivatives® / ■

ld )s - 2 3( - 2.W, + 2 w3 - VU )y - ‘ . . . . z- (2 6-1*9)

. U^l ' + 6Wo>?W5 + yjjsj)

(2els.7) and (2 ©1*8) can be .compared to similar results found in- (2»1*2) and (2©1*3)© (201*9) can. be ccmpared to the results found in a- range . of five points G That is in the. ease of . : '

,y « ce + <,« -+ + csu •-«. c x . .

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. . ■ -. ' • . cmpteh i n : : ,

‘ BELMXMKM METHODS APPLIED TO .PLPJE POTENTIilL PROBLEl© . <' ' 3@1" The plane harmonic equation and its finite^difference ap^ •

'• : proxiraatioiio We . have alrea^r shoim in 2 <,4 how partial dif­ferential equations can be replaced tgr approximate (finite=differenee)

V .relations imposed at certain (nodal) points ;on a neto vie now apply the; same te’eatment to the 11 two-dimensions! Laplace equation’®

~ : (3»11).and to the: more general JtPoisson equation18 . v . ■ _ - . .. -

. -... .. ^ Z(x-.y) = o. . .■ ■ • ' . ; (3©12)in fiiieh w denotes the wanted function and Z(xsy) denotes some specified function' of x. and y0 . ; . •

A finite difference approximation to. the operator v which governs .- . the wanted f unction in the ...Laplace equation (loll), can tie ..obtained from •;

equation (2® 1*3) and i’igo 3® If we take (2ok3) once with respect to thehorizontal axis, and .once with respect to the vertical axis "we: obtain -

'' . V V V- , (3® 13)as a suitable approximation for use with a net of square mesh® ■" - -

• ; The same result can be obtained by another approach which is'; ' lengthy and compile ated but. which can be generalized to apply to any \ •

of the nets in this paperQ For the net with square mesh (M s i*) one

Southwell^ op® - cits p® 23s"' . ' ' - . . . . .

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obtains - w» = ^ (vlw)e ^aN * O.lk)

where E(w) stands for the sum of the values assumed by the wanted a.wfunctions at N points equally spaced on a circle of radius a with center at the given point 0*

For the net having triangular meshes (N * 6) this becomes

irJZ M - w 0 - [ V W M j = (3.15)where the summation symbol has the same connotation as in (3.11|).

Equation (3*12), therefore, has as its finite-difference approxi­

mation . _X , , ,xif E M - Wo -V = o (3.16)a.

where the net is of square mesh and (3)- Wo + [vzZ(^y)0] ~ o (3*17)

3,6

for the net of triangular mesh*

3.2 A corresponding "net analogue1* of the finite-difference approx­imation* The Poisson equation (3*12) can be interpreted as

governing the small transverse displacement of a transversely loaded mem­

brane and equations (3*16) and (3*17) as governing the displacements of

a net*

Suppose that the meshes in Fig* 1 represent real nets of mesh side a in which every string exerts a tension T and so, by symmetry, every node is in equilibrium when the net is flat* Let w stand fcr the trans­verse displacement from the flat configuration which occurs as a result

( ^ E q u a t i o n (3*lU) also applies to the mesh for which N * 3*

^From Equation (3*12)*

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of transverse loading. In the deflected net the string joining any two adjacent nodes, say 0 and 1, will exert transverse force on 0 equal to

T sinwhich becomes

t(-w ‘

when W| is very small and it follows from the definition of 53(w)a.N

that5. = x C Z Z M - N(w)„] (3.21)

measures the total force exerted on 0 by the N strings radiating from it.

If we now denote by 1?q the external load applied at 0, and if we

putF- * F- 1 F. (3.22)

then F° °is the condition for equilibrium at t he point 0.

This can be identified with (3.16) and (3.17) provided that

P. = N = 4 1F. = N = fc J v3,?'

The factor T which appears both in (3.21) and (3.23) may be divided a

out after the substitutions are made in (3.22) so that the final form for

equilibrium-t- F0 - o

becomes (w) - + alZ(xy)<. = oh (3.2k)for the net of square mesh (N ■* 4) and

Z6(w) ~ fcw0 + G [irZ + I? VzZ(*>v\l -° for the net of triangular mesh (N = 6).

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3>3 Concepts of; the Relaxation Methodso Having this mechanicalpicture of tiie finite=*difference:relations (3*16) and (3® 17)' w@ can-

visualise' our problems as that- of determining fee behavior of a eh ©sennet th 'specified .transverse.Sa'ding® .- We: can utilises as - deemed n.eo*” .essaxgri, . the - devices of individual m d block relaxation presently to bedescribedo. ■ - ; - ‘ : : , - -

■ Equation (3s22): eicpresses mathemtieally fee method used to solve

the- problem, of .the" loaded framework and c m be'applied, to the nets ' -Pqstands; for the total force on. the constraint at 0© Since the external

force Fw is knowis we can, by changing the displacement (w)s bring the

Fq bqual.'-tS zero®-''.-From ' ’ , . . ' ’ '- f; > ( w ) - cv-M0 . - (3.31)

we can determine-the-effect of a unit .operation that is* .the imposition-of . a unit displacement, at fee given node© ' . -

For all values of M the effect of a unit increase in Wq is .an ,

increment--V./ . -. - ‘. : . - - . . .: ■ ' : ; . A Fo A Eo . - • , : y : .

: sine®, is specified; and - invariant f or any given net®- - But .from - .-, ': • ;.-. ' : • - • f o - | l , C < i 0 ) . - . M i w ) 0 - . . - ■ (3©31)

a unit .increase in Wq gives, rise, to an increment .- ; - -._ A F o , ~ N .u/y /: ;; .; f

therefore ' - A F« - -N . ' -'■■■■ / -. ,From (3®31) -it is seen that for any of the nodes immediately,

surrounding;0 fee'increment , . . . •

V : A M - . AFk - A Fa A %

' Each stringj, in effect, transfers a unit fer ee from 0 to its other, end point as a result of a unit increase made.to. w' at 0© (Fig© k)

; , Fig® ii© Relaxation Patterns . . • ' . - '

• The -values of Fqs that, 'is/'the right number of (3®'22)s before

equilibrium- has been attained measures the loads not yet accounted: for^' i©e© residual fO’ces, and shall be termed residuals© When all the re­siduals equal zero the entire net-is; in equilibrium©

- 3©U The torsion problem of Saiht°¥isnant© to equation of type -'(39.22.) is presented In Saint =Venant8s theory of the stresses'induced by. '.- tidisting' Couples in .'straight; bars of - ‘non-cireular cross-section© The . '

shearing- Stresses acting on cross-sections are expressed-in terms of a' •

^stress function16 which has a constant value on the boundary and satisfies

the'condition^^' -• ' ’ * ' - .. ' .- rz- .- : ' ; ; C3«la)

at every ."internal point,© Comparing .; (3shl) .wiW- (3©12) it is'_ seen to be -.’.

a special ease of that equation with ■ % independent of x and j and every­where having. tiB value 2<3

Using this relation with equations (3 =>23) and" (3e2!},) we get

Miether B S .;3s li or 6D Hie , is, now. sa non-dimensional number j. s;oae

eonvenient-fractional part of the boundary^ s r® • .

3o5 %e application of Saint^Venant^s. torsion principle to a bar- . of triangular .ero§s°sectiohe To make the problem definite

most' have -spee'lfie: - values at all points, ©n the boundaryo In the

torsion, problem for. a-"solid section ^w18, must, have a constant value along . the boundary and since we are seeking constant value contour lines no generality is lost by making this value gero©

■ One of the simplest .forms of boundaries to use for an example is .

that of the.equilateral triangle0 The. computations are not complicated

and it has a knownj, exact solution available- for - comparison®’ j The function- . - - . : . i. .

- ti'- - .. ' (5) ■■ .

w 4-: i Xusv - 3y - i 4-3v - i) (3 e 51)satisfies in the form

; ::: : ;and vmishea when'- .' ..; . ., v" ..

C ■ V V-' - M/=t' zk . :V-': ' -Vor ' %'/ ; : - - ' . ' ; / - ' :' V-:;; ;■ - f- . . . : ' ' - . ' .

-Wat is;: on 'every. s ideof length 1 "unit of .the. equilateral triangle

■it7Southwellg• op© Pitp* p© h9o ."'o'- . .. . - ..

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shown in Fig. 5.

Fig. 5The shape of the boundary suggests the use of a net having triangular

meshes (N * 6). In the coarsest net which is convenient 11 a1* has the value 1/3 as shown in Fig. 6. From (3#U2)

F0 » ^ N a1 = -£- M = 6 , a - y .

There is only one internal node 0 and it is the only point at which

(3.22) must be satisfied. The equilibrium equation for node 0 is Re = Cv>) - 6 wd + - o

where Rq indicates the residual force at 0. Since each "w" on the boundary is zero we get

~ 7s"

as the displacement necessary to liquidate Tq, that is, to leave no re­sidual force. Comparing this with (3*5>l) we see that it is an exact

solution, at the point (0,0)#

If we are to determine "w" at other points in the section we must use the device of "advance to a finer net".(^) Consider the triangular mesh

^See Sec. 2I3.

0 a b in Fige, 6 and. let its centroid be designated by 6® ^inee 0S a and b are grouped symetrieally around c at a distance ' . / . / . :,

the equation (3®lj.2): "will giye" a value-:for: the externally applied .force Fq© - in this case.-N- » a I',*': Ve have from (3=142)' - *'

we get ' - ~' ig ~ ; ,

From (3e22.) at .the point e we' get . ■ ■ ■ . . ;

18 -T 71"•which wiH - vanish if -wc > l/27®- ■ ' • \ ;•■ ■ • ' -■ • • ; ;, ■ . ■ Since a and b might be taken to -be any two adjoining nodes of the

nodes surrounding 0 then from consideration of symmetry it is seen that w must have the same values at .d> e$ f5 g and h as at c0 Using similar

reasoning the value for "h# at k 'i and m' can be found, to be -. • . ""

, ' - - ' H H r . ; . . . ; - / ;. y ■We now have values of !tw!t which may be used as., s tarting aSSuaptions •

in relation to a triangular net of mesh side, . - ... -. ;

\ : v ,From (3©22) with given the "value 6^) we. can deduce the initial values of the residual forces and. these can be liquidated as before®

■-Howeverjj at d/ £.$ hj, k: X. and m,we cannot use N a 6 for %% do'not have values for w at "six •surrounding nodes®■ To provide these, we. can . ' ' ,

v?/lhe» the values .at six' stirroundihg nodes'.' are: khowho

27

17,

Big. 6

(22)

proceed to a third and still finer net in which 15aw has a value 1/9 as shorn in Fig® 7o The points os cs d and k correspond to similarly named points in Fig® 6® '

Since the points j and g are .equally distant from the boundaryj, we •will assume a linear change from d to ko The assumed values for "ww at

these points become

W Y = /ft' and vos »

Assuming the change from P to 0 to be in the ratio 5s3^1 the value for. at x becomes - .

To eliminate fractions and facilitate- computation a multiplier 1*86 is used® (Fig® 7b)' - ' . ' ' - - -

Tq order to calculate the external -force Fo to bring the point.0into equilibrium we use ■ .

fo + C - o.This becomes . • - ■ ,

Z C ) -t- F0 - o „a,s ~Since j, from symmetry^ all the wes are equal to wx we get

6 w % - <o XiJ0 -o

- 6(z?) + po- 0therefore. F. = 18.

w ;From symmetry3 points adjoining the line of center have similar points equally distant on the opposite side of the line of center (Fig® 6)o - . . .

£ —

(a) No multiplier (b) Multiplier U86 (c) Multiplier 1*86

Fig. 7

(23)

From Fig© 7b we can write the residual equations arri. evaluate the residuals at various points <, These are .

R* ™ wc * wy > 2w k * w0 6w k 18 o' 0

Ec is 2w* * 2wy * 2wz == 6we # 18 - 12R 2w ]£. * 2wy = 6wd * 18 s 12 ' .

Ry » * wc "*■ ws wA <=» 6wy * 18 > 0R% - wK * wz -i* wc * wy <=» 6wz 18 ^ 0Rte 2wz “ 6wk * 18 - 12.

.We now proceed to eliminate the residual forces by making changes in the w8s and applying (3o32)e Relaxing first at one of the points with the highest residual (the point k) we have

A w K »- 2and , v ' . . .Rk -12 ~ &C-Z) 5- e. '

The notations are made as shown in ,Fig® . 7bo ■ The increment of «2-.■in 1% affects the residual forces at the other internal nodes connected . to k by an amount =2 for eaeh0 Since and its synraetrically- plaeed point are the only internal nodes connected to k it is only necessary to show the change at z® We next relax at the point e® If we make

A Wg 1 * 2then ■ |?C = - it - G(z) e a,and we proceed as before® The third relaxation takes place at the point

d® If we make A - - 2 ■

and its effect is to make all the residuals equal to zero® •'. . -Contour lines of constant, value are then drawn to show regions of . ,

-18stresS'=eoncentrati'©nsm as. .shown" in. Fig® ?c@ : '

3s-6. Discussion of Block Relaxations0 In the' example of the equi=>

lateral- triangle the basic operation entailing the us©;-of the individual' relaxations led quickly to complete liquidation of residual. forces®

Normally the convergence of the computations is much less rapid and we . therefore introduce the idea of “block relaxation®11 © ■ ' ^

Let us connect a group of arbitrarily chosen nodes to a flat, rigid place (of.negligible weight) in such a manner that they shall all,undergo the same, deflection when the plate is.: deflected® - If the net is flat be=

fore the deflection of the plate then at all ..points within the area, of - .

the plate the net remains flat® However, should the various points connected to the plate have initial displacements (Fig® 8e) then the

relation of. the internal nodes to adjacent internal nodes remains the ■ same, and therefore the residuals, are unchmged®: , : v " ■> . • . ;

Moreover, if the nodes outside- the. plate are constrained then the ’ only strings whose ends undergo a difference- in displacements are those. ,;

strings (indicated by the heavy lines in Fig® - 8) which,connect the- nodes on the boundary of ‘.the'plate to adjoining nodes not connected to the

plate® For a unit increase each string exerts a force at the fixed end and'-1 at the end which is moved, according to (3«32)0 ' . , - ■

It is possible now to liquidate the resultant of the residual

Fig* 5a

Fig. 8b

Fig. 9a

2-J4 iJ 1 23 212 Z 0 2,2 -2

2, 3,-/ J j z d -/ 0 -2 a -r z

Z -o' 2 5,2 a a a(-y 2

2,% 3 V .? V r S,-3 $ 5 -2 2,2

z z 4 X 2 X -2 X -2 2.2 2 V 2 -b

Fig. 9b

2.8] ,Z 33

, +.0rJL J*/-z

/ /

v./

/.% ^ 2-1-, /*,-.% 23

forces in any block or section of the net ® If the block is connected to the outside nodes by Mnw stringsthen a unit Increase in the displacement

: of the block" will transfer ®ntt units of the resultant force from the platee ■ j’igo 8 shows the effect of. a unit increase, in. the .displacement of the , ■

shaded areas® ./ r="; - \ ' ’ . ■ ■

The .shaded area may. be. chosen in'my way® To exemplifylet us . .

consider the shaded areas id ;Figo 9&s the *ole of. which is denoted by .:,I and the central striped area is indicated by Ho -Tor I the initial -

value of the resultant residual force is 52 and the number of strings ■ . '

n ;* 263, tiier ef ore an appropriate value for its increase-in "V is 2 units o'. For II the resultant, residual force is 18 and n 18 s"o; tiie -appropriate"."

. increase in 'V is 1 unit® When these two block relaxations have been com­pleted the residual forces at points in the entire area are as shown in

Fig® 9b® % applying pdint^relaxations the residual forces may be brought-to . the values shown to the right of the nodal points in Fig® 9Cj, when the -

.. displacement increments are as shown to the left of-the nodal points® It.

■ is possible to. make these .values as accurate as desired by making- the

residual forces', as small as;neeessarye .In this case thh smallest": Aw.-;

taken was'Ool giving:residual' forces, no- greater than 0®2® 1

• - ■ 3©7 The Solution of the torsion problem for a solid bar of rectan-° - ": . . gular erogs^seetiongA- " let us iate the "rectangle as shown - in

Plate 1 such that its dimensions are D -and . From ■ symmetry, with-respect. . ; .. : - : : .' - - -3. ; - . ' ' . \to. the given lines of center it will be necessary only to solve for one of the .four symmetric quadrants0 We will, therefore work only in the section "

(9)The diagrams referred" to in this section will be found in Plates 1«=8 . ; at the,end of the booka v , ,, : . : . ' ■ " , . -

designated X©■ From the shape of the boundary the net’ of square mesh (N s ii) is appropriate® Let us identify points^ not on the boundary#

.regularly spaeed at intervals Ip by the subscripts shown in Plate t©

-. - If is first necessary to obtain one or .two. points at which aninitial value, of 'V may be found#, .just as done in the solution of the equilateral triangleo■ By taking a = 1 for' the coarsest net# initial

/ 1:'.. /. , 3..';..- ; . , . : '■values for wy, and may be, found from their equilibrium equations©

-. . ' ■ ihce a ,1#. from.F ‘2a .:-we;g@t F:®-2© ■ Using this in the residual , ' 3' .. • j •■ 9

equations and finding equilibrium wss, we get ;

R44 = ■+ w84 ~ "4 VJ/,4 4- - O

. Rg4 * 2_WW + VUg^ - a-XMgtj, 4- >■ o .

Solving (3»?1) we get ' '

. . . : VU84-2 # - . '

as the necessary values® :1© eliminate fractions and facilitate conpu- ‘ tations we use a-multiplier ..?200o ^^ Accordingly ■.

: ; = W44 - , , ^$4 A (143 - ■ h.

are t aken as initial values as shown in . Plate.3®: ' ‘ '' ■■ ... W.th these values as a basis. we proceed to a finer’ net a - 1©,

’ ■, ’■ ' - ■ - . , • ■ ‘S '

\19) in^e gor the finest net used, a ®-1; and F “ 1 • we shall use 7200V /I; S r: ’ : 72. ; : ••■ ’• to provide integers, with an error no greater than 1/2% from'the

corresponding fraction® ’ ’

(3®71)

(27)

This net has nodal points at

22 2V 26 -2*2 hU 1*662 ' 61* ' 66 ' ■82 ~ 82* : 86

and points symmetrically placed with Respect to lines of center a

We can set up the equilibrium equations'to find unitial values of

at the nodal points in the ..net of mesh side a = l<y From ¥ - 2a we

get F s 1 and since we are using a multiplier of 7200 the value of F m 2*00 iF . * '

is used in the equations0: These equations are

t ' # 1*00 " 0 t Wife .<*» 2*w24 2*00 ®. 0

Rzt s '2w -%(, -« 2*wx& 4- 2*00 ® 0R42. q Wy, - ^ ^ 900 ^'0

® •t w a. * W(,4 + w46) = 2*w 4 ' + 2*00 53 0R4& s Wt.4, t 2w^ 4- - 2*WzM,.' * 2*00 s 0Rfez s w a. + 4- Wgz, «= 2*w6Z “> 1*00 - 0 (3o72)

Rfo-f » ^'#62 ^ w 8 V .» w 66 => 2*w 641 * 2*00 " 0

R m , ^ W^6 *5-' 2w 64: t :w86; =-.itw66 *.2*00 " 0

R 82 * 2w6Z * w 8tf - 2*wgz * 2*00 - 0

R 84 ^ 2w 64. * w ez * w e6 - l*wa¥ + 2*00 0

R86 s 2w6o. * 2w8¥ - 2*w8& * 2*00 = 0

Solving these twelve simultaneous equations for exact values of the

Wj_j is too lengthy by orthodox methods ' thereforearbitrary initial values

are assigned to .the displacements at the nodes and residuals calculated*

the-w's and •A"w8s are recorded, to the left of the • nodes. and residuals,

to the right® - ' " .' V-' .. ' ' -t'.It was seen in the solution of the equilateral triangle that by ,r

. assuming .successive .value's;'in some ratio we."get an.initial value closer.:. „

to. the final value than ty assuming a linear change c In the fine nets this device will .aid- only ■ .slightly but. in the coarser nets it saves a-'consider­able.- amount'of time® ' ■ ' —. ■ J ' . ■ ■ . , ■ ■ . .

•. Assuming- the ratio of. the "change in ,8wi! from the boundary to 2h te . the change '‘w*. from 24 to to foe approximately 7 to 5® an initial value

of 525 is given; for' the! w'at. 21®. A similarly assumed ratio of 3 to 1. is 'taken to find the w at 61}% and its .initial value is found to. foe 1088® The

same values are given to the points k2 and t,6 respectively0 The values for

: the other points shown in. Plate y. are f ound by solving - equilibrium equa­tions for. the necessary' wl.So ."- For example W26 can be found from the equa­tion, for R26 for, it has oruly one unknown# . ' - , . ...

. , 4 ■+-\xJttb + -yoo - 4 - o. "-'z'(52,0 .+ rodA -V4'oo - 4 . .

- . ; ' ' ' . .: ' ' ■ - .. ■ , ' - - to the closest integer® " . , . ; ' ' . :

- It is'important-to realize that no matter what initial values "are . takens the residual. forces.:will bring the w6s to their correct values. , ';and errors. will elimihatetianselves whenever residual .forces are rer- ‘ . .

(29):

caleulatedo ; r ; . ' ' ' - ■

The residual forces were computed from (3®72) and the values noted to the right of the nodal points^ By individual relaxation the values of the residual forces are reduced to the (in generdLv) smaller values shown above the initial values© The- final values of “w11 for these residual

vv- : : \ ■ ; .. ■. ■' : : ' . V : ■.forces are shown above the ■original' values»k _ . .■ ■Proceeding to a 'still- finer net- in which--a. s 1 we find that (7200)F

- 100® Oince we are . in a fairly fine ; net a linear change; in between points previously evaluated is assumed© •‘■his gives the values for w's • ;

shown in Plate 5 to the left of the nodal points from which the residuals

are. calculated,, .Since comparatively large positive and negative residuals. • are found- the computations converge, rapidly at f irsts causing a rapid ate— duction in residuals© Each time an advance is made to'a new working sheet the residual, forces are recalculated to check: on. possible errors and to allow for the fact that points on the lines of center are doubly affected

by change in adjoining points© ^or example, a -unit change in 15 also means a unit change in its symmetric point, therefore -the change of the residual force at 16 is two units©;; . . - - . .

. ; ■ Proceeding thW, the residual,forces - are - gradually reduced until .the .: values shown in Plate 6 are reached® Although each residual force.is less;- - than, d the resultant residual force is 91$: ihe idea of block, relaxation

is used to reduce this residual force© : v- ; . ■ ; . ■ .‘ Consider the ttbloek16 bounded, by' column and row with subscript 1 .

(Plate 2)-© This is joined to. the boundary by It strings and has a ..

This, is the standard procedureg ' displacements and changes in. dist . placements shown to .the left and residual forces shown to the fight® ;

- resultant residual;f cr ce; of 91> An. increase of 6 units in', the displace- : ment of .that; plate or block #111 take-Sit units. from, the resultant residual

" -forces" Now consider the block bounded by.a line joining joints 22 and 26 and by a line joining 22: and 82®" his area has a resultant residual force

. of 65 units and is.; joined to the outer area by 12 ■ strings, therefore A w ;

for that area is 5; units® ' Similarly for the areas having 33, kky'BS. and 66 for comer nodes® The total displacement of each point is then W c o r d e d . giving :the values shown in Plate 7© A few pdint-relaxatiqns then.gives the . final valuesr shown in Plate 8© The resultant residual force is now only 2 units and no individual residual force is greater than 2 - units in magnitude® The problem;is them completed by".drawing ths constant value contour lines . .

(Plate 8), , . : = : ; . ■ ' ' . ; ' ; - "

(3D

CHAPTER IV

EXTENSION OF THE RELAXATION METHODS

ii.l Treatment of irregular stars♦ In the problems previously treated as examples nodal points fell directly on the boundary# In the

problem of the circular flywheel and, indeed, in every problem involving

circular sections this does not happen* It is therefore necessary to in­troduce the device of "the treatment of irregular stars"*

A discussion of the device is given by Southwell for a triangle with

circular sections removed* However for a symmetric section of the cir­cular flywheel this device is used as follows* Consider the point A in Fig* 10 belowi

A

Since the strings joining A to adjoining nodes or boundaries are not all of the same length it is necessary to change formula (3*31) to apply to "stars" of unequal strings. This is a problem in interpolation.

^Southwell, op* cit* p* 68*

(32)

an elaborate discussion of "Which is not "worthwhile as we are in general

working with a method which in itself eliminates errors. Therefore we

shall use a modification of the formula consistent with our concept of

an actual net in which a force is transmitted by each string.

In the loaded net the force exerted by any string is its tension T

multiplied by the tangent of the angle caused by the displacement of its

ends and so, for given displacements is inversely proportional to its

length. Therefore in place of

A F - - N

ZX F. ~ & A - ... = F,

if the N strings radiating from a displaced node 0 have lengths x^a,

X2a, x^a,...,](%& we have

A F ' ~ (x; -V ^ "x;at each such node which moved and ZXF = 'xj > ■>

at "the N surrounding nodes, respectively 1, 2...,N. The relaxation

pattern is modified accordingly, so that at A it becomes that shown in

Fig. 11, since the string connecting A to the

Fig* 11boundary is only .856 units in length. Similarly the relaxation patterns

(33)

at the other darkened, nodes can be found*. ' .

i4.o2 Solution of the torsion problem for a circular flywheels It is seen in tai that a problem dealing with circular sectionfeamot be

solved as those previously treated therefore the Hirregular star11 method

is. applied*, ' ' • " ; /. . “- . The initial value for w at the center was found, in the same manner

as' in the triangle problem® M initial value 1 was given to a (compared •

with the radius) and Fq found from Fe = ^ H a 2- :

This gave a value for Fq of 3 which was substituted into the equilibrium equation . — . ■. ^ = E ~ £ w* -t- F„ - o•'' , - v . .• ■ v-. :■ Since each w was' Q, this'became- • ' , ,

G V0o a 3 vJo ~ "zT. A multiplier 3200^2 was used which gave' an initial value for :Wq:

of i600o' We proceeded as before and assumed certain values at the various'• nodes® It .must be.remembered that the•values adjust themselves to the

correct .ones® - 1 • : , ... .. ....: . The; relaxation was accomplished as in the previous problems and the

final results are shown in Fig® 12®' fhe reader can verify the results -

and check the residuals at each node to see.the accuracy of the results®-- .

^ 'Since -in'the "finest net a ■ "1 . and .-pQ .»■. 1

ors'\o>n rvo\a\Qliyi 10 Ka<2.lC\rcu. Var

VAtLV-tvoXver 32.00

or

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i-jrS -2/T27 /V<9<» liiP/331 e*>° /z/«/ 3807oo

a

uu;

CHAPTER V

PHOTO-ELASTIC METHODS AND RESULTS

5#1 The reason for Photo-elastic experiments# Although the re­

sults of the problems presented in this thesis can be checked by sub­jecting a prismatic bar to torsion until it fails, a quick check on the

results can be made with plastic models subjected to torsion. Of course,

certain physical constants of the material will cause different strength

results but the general trend of stress lines are the same. In the Photo­

elastic tests the actual lines of stress can be seen as they travel through

the model until failure results.

The Department of Mechanical Engineering requested the use of Photo­

elastic methods in order that the testing machinery could be properly set

up and best testing methods devised.

5.2 The Photo-elastic machine. T'he equipment used in the Photo­

elastic experiments is as shown in Fig. 13• The light source "a” can be

either monochromatic for black and white pictures or white light for colored

pictures*

h d e e 6 c b a

Fig. 13

' / At’ Mb.M‘ there:.ii3 a. distilled- water, condensing 2ens> and;-an iris."

diaphragm is located, at. ;!®c!3| these two being arranged on a. single movable

trolley at a pre-determineddistance apart® Another paii* of trollies :

each carry a eollimator .lens. (d) and a polariaing lens (e) also pre-set ' to .eliminate adjustinge ' ihe frame for holding the model is. at Wfw ail'd

the camera is at whSU /. ..The position of the condensing lens and iris diaphragm' is - adjusted' /

to give a? clear iwell^foeused light® The col'limator- lenses are: ihen ad== justed bo that parallel pays - of light are produced (dotted lines )0 The axis of .polarization is determined by rotating the polarizing' lenses.' un= ' - til no .light passes to the view plate; of the' camerao- Ihe, model to- be -

tested is placed in the holchng frame and subjected to loads® ?he stresses,

induced interfere with the polarization and cause stress lines to appear@

\£®3 ^he torsion device & The testing device of the 'Mechanical. ■gineering Departmmt is one for tension and ebmpi^ssion® ^t *as, there-

fore> necessary to devise some means of producing tersion on the models» - • After several trials this model was produced* .M- square bar was-, taken and

the edge, ground to l/# round® The ends were flattened^ drilled and• • ' • . r . . (2) ■ t-tapped so that two aluminum rods could be screwed into them© . The

opposite- .ends of the rods were also, joined - and then the entire device was:

joined to the ‘loading spring on the frame© (Jig©- lit) . . . . :

.'(l)sharp. edges• and corners "tend to produce excess .'Stress concentration©

(" Rods; at each -end,of' center, piece keep the torque. evenly distributed. ■through the thickness of the model© "... . ; ... - "

Fig© llj. loysion device

$ok Reparation of modelsa The models used in the torsion test were made from lArblett , a, rather- hard plastic mater 16 I'hey were first machined to ah approximate sise and finished by hand so as to fit snugly into the holding frames© The surfaces were ground and polished with

jeweler1s rouge to eliminate surface scatches o Stresses induced by

machinery were ’removed by annealing the. models in an electric oven at

J,80°F for 3 hours© '

5o5 Photographs :of...results© It was found that best results were

obtained with the Eastman Kodak Camera.No© 33A with a lens opening fell

and a shutter speed of .l/2'5 of a second© Ortho-press Safety film was used and the developing and printing were done according to the technique

by Frochto^ ■

5©6 Analysis of results@ A close survey of the photographs-.-at the

end of this thesis does convey the general impression that the models behave in accordance with the results found mathematically © It must be

. I© Rroohtj, Photo°elasticitya IIfl (liley & Sonss 19lpL) p© 392@

(37)

remembered that/ the mathematical results are for the eross=seetidn of a twisted'solid prismatie bar while-in the Photo-elastic examples it was '

necessaiy-to remove "the center, section to allow for the torsion-device® let us examine the Photo-elastio results and compare them with

the resulte'found :lr Relaxation Methodse ; The ;flrst example used, was

that of the triangular bar0■ The Photo=elastie results are shown in

Plates In" Plates 9=11 the torque, was gradually increased until

a-definite, stress pattern appeared® Tlie reason-for the clover=like

arrangement of stress lines is that a square bar was used for providing the torque causing .an undesirable force normal, to the bar’s surface© ;

Although the. rounded edges produce .less stress: coneentrations than sharp. . edges wouldg there is still a tendency for stress lines to converge at

eornerS®^^.'In Plate 12 'the load was reduced-, to see if the stress lines"

would be relieved, then it was built up again as shown in Plate 13© The model fractured along the narrow edge between the white arrows shown in

..Plate lit© ^hese compare favorably with, the-mathematical results© .*. If the results of the" other tests are examined similar conclusions ■

are reached® ■ rectangular model failed between the white arrcs® shown •-in Plate 17© V The fracture' continued across the model and prevented its staying In the holding frame© '

5o7 Conclusions© It is seen that the Relaxation Methods of

Southwell are capable of providing solutions which compare favorably

(n)Theoreticaliy a circular bar would be ideal but in practice it cannot- be used, because of slippage© An elliptical bar has been suggested- for future tests® ' . " ' '. '

(38)

with the results of physical experiments e In the other types of pro-? blems (heat-distributionj, percolation^ .etea) physical tests .also show

; that the mathematical, reshlts are qualitatively eor-reet® k quantitative • .. analysis of the.Photo=alastie results would, require more precise machinery

than, is1 available.in our labartory® • . "• .. . '

BOOKS

Froehtr 1© Ph-oto°elastioitys' IIa Hew Yorks Wiley and Sons,, 1%1Southwells Re Vo Relaxation Methods in Theoretical Physicso - Oxfordg

Go Cumberlegej, 19^6

Blacky Ao Mo and Southwell^ Ro Vo !iBasio Theory vath Application toSurveying and to Electrical Networks88j,J?roeeedings of the Royal ' Society^ Series Aa CEUV (1938)y po'hh7v

Bradfieldy K0 No Eo and Southwell^ R0 V® !8The Deflection of Beams under Transverse Loading” 9 Proceedings of ..the Royal Society^ Series &9 . CHI (1937), P» 155© .

Christopheraoiiy Do Go and Southwells R0 Vo 66Problems Involving TwoIndependent Variables”s I¥oeeedings of,the Royal Societyfl Series A$ CHVII .(1938 ).s p& 3X7©

Southwelly R© V® “Stress Calculation in Frameworks811 s l^oceedings of the Royal Society,, Series GDI (193$)# p» 56© .

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