Bending of Circular Plates W ith Large Deflection

APM-56-12Bending of C ircular Plates W ith L argeDeflectionBy STEWART W AY,1 EAST PITTSBURGH, PA.I n t h e d e s ig n o f t h i n p la te s b e n t b y la t e r a l lo a d in g ,fo r m u la s b a s e d o n t h e K ir c h h o f f th e o r y w h ic h n e g le c tss tr e tc h in g a n d s h e a r in g i n t h e m id d le s u rfa c e a r e q u it es a tis fa c to ry , p r o v id in g t h e d e fle c tio n s a re s m a ll c o m p a r e dt o t h e th ic k n e s s . I f d e fle c tio n s a re o f t h e s a m e o rd e r ast h e th ic k n e s s , t h e K ir c h h o f f th e o r y m a y y ie ld re s u ltsw h ic h a re c o n s id e r a b ly i n e rro r a n d th e re fo re a m o r erig o ro u s th e o r y w h ic h ta k e s a c c o u n t o f d e fo r m a tio n s i nt h e m id d le s u rfa c e s h o u ld b e a p p lie d . T h e f u n d a m e n t a le q u a tio n s fo r t h e m o r e e x ac t th e o r y a re k n o w n ,2 a n da p p r o x im a te s o lu tio n s 3 h a v e b e e n d e v e lo p e d fo r t h e caseo f a c ir c u la r p la te .T h is p a p e r gives t h e g e n e ra l s o lu t io n o f t h e f u n d a m e n t a le q u a tio n s fo r t h e case o f a c ir c u la r p la t e b e n t t o a fig u r eo f r e v o lu tio n . P a r tic u la r s o lu t io n s a r e f o u n d w h ic hs a tis fy o n e o f t h e tw o b o u n d a r y c o n d it io n s , a n d stressesa n d d e fle c tio n s are c a lc u la te d f r o m th e s e s o lu tio n s . B yin te r p o la t io n , t h e stresses a n d d e fle c tio n s a re t h e n f o u n dfo r p la te s s a tis fy in g b o t h b o u n d a r y c o n d it io n s . T h ed e fle c tio n s are c o m p a r e d w it h e x p e r im e n t a l r e s u lts a n dw it h t h e a p p r o x im a te f o r m u la s . I t is f o u n d t h a t th e s ed e fle c tio n s agree clo s e ly w i t h t h e e x p e r im e n t a l r e s u ltsa n d a ls o w it h th o s e o b ta in e d b y t h e a p p r o x im a te m e t h o d so f A . N a d a i a n d S . T im o s h e n k o , as s h o w n i n F ig s . 8 a n d 10.D = Eh3/12(l — ft2) = flexural rigidity of the plateWr, mt = radial and circumferential bending momentsper unit of lengthp = radial displacement of points in the middle surfaceDisplacements and stresses are assumed to be functions of r only;that is, they have radial symmetry.Loads and deflections will be taken as positive when downwardand moments as positive when they tend to make the plateconcave upward.2 E x p b e s s io n s f o b S t r e s s e s a n d M o m e n t s i n T e b m s o pD i s p la c e m e n t sThree assumptions are made as follows:(1) The material is homogeneous, isotropic, and obeysHooke’s law(2) The curvature of a meridian may be replaced by d2w/dr2(3) Straight lines initially normal to the middle surfaceremain straight and normal to that surface after bending.It follows from the third assumption that if the radial andvertical displacements of every point in the middle surface areknown the displacements of all points of the plate can be found.The radial extension in the middle surface isFUNDAMENTAL EQUATIONScO N SID ER an initially flat circular plate of uniform thicknessand let1 N o m e n c la t t jb ea — radiush = thicknessp = load intensity, assumed uniformo>, a,', o>" = radial stressesat, a t , at" = circumferential stresses2 = distance from middle surface, downward directionpositiver = distance from axis of symmetry to a point in theplate before deflectionw = vertical displacement of points of the middlesurface relative to the center of the middlesurfacetp = dw/drE — Young’s modulusm = Poisson’s ratio1Research Laboratories, Westinghouse Electric & ManufacturingCo., East Pittsburgh, Pa.1Due to von K&rm&n, Enzyk. d. Math. Wiss., vol. 4, art. 27.See also A. Nadai, “Die Elastische Platten,” Springer, Berlin, 1925,p. 284.3Solutions such as that by A. Nadai and discussed in this paper.Contributed by the Applied Mechanics Division and presented atthe Annual Meeting, New York, N. Y .t December 4 to 8, 1933, ofThe A m e b ic a n S o c ie t y o f M e c h a n i c a l E n g i n e e b s .This paperwas prepared by the author as a thesis for a Doctor’s degree at theUniversity of Michigan.N o te : Statements and opinions advanced in papers are to beunderstood as individual expressions of their authors, and not thoseof the Society.and the circumferential extension isA t a distance z from the middle surface there are the additionalextensions due to bendingThe middle surface extensions give rise to the membranestresses a,' and at! which by Hooke’s law are as follows:627The extensions due to bending give rise to the following bendingstresses

628 TRANSACTIONS OF THE AM ERICAN SOCIETY OF MECHANICAL ENGINEERSSumming the moments of o>" and at" from z = — h/2 to zh/ 2 the following are obtained:It will be noted that if deflections are small compared to thethickness the term involving ^ could be dropped from Equation[la] and it would not appear in Equation [10]. Equations [9]and [10] would then show that the membrane stresses wereconstant, and Equation [8] would become a Bessel equation.4 F u n d a m e n t a l E q u a t io n s i n D im e n s io n l e s s F o r mHereafter a," and at' will be used to denote the outer-surfacebending stresses which are given byThe new variables q, S,', St', Sr", St", and u are now introducedand are defined as follows:A useful relation between p and the membrane stresses is givenby Hooke’s law and isAlso, the following dimensionless expressions3 R e l a t i o n s B e t w e e n

APPLIED MECHANICS APM-56-12 629G E N E R A L SOLUTION A N D PART IC U LAR SOLUTIONS6 T h e G e n e r a l S o l u t io nA solution for Equations [12], [13], and [14] is now desired.Either by successive differentiation of these equations or by aconsideration6 of the basic assumptions of the theory of elasticity,it can be shown that all the derivatives of the displacementsp and w will exist at all points in the plate. The displacements,extensions, and stresses may therefore be expanded in series ofpositive powers of u. Since S,' is a symmetrical7function of uit can be expanded in a series of even powers of u and since

APPLIED MECHANICS APM-56-12 631The results for plates o to / are given in Table3. The stresses at the center are given and,as has been noted, they depend on Bo and C1.Stresses are multiplied by u02, for it can be•shown that w/h and uo2

632 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERSThe first is equivalent to Equation [8] and the second isEquation [9] with a,' and at' replaced by their values in termsof p and tp.Nadai8 assumes this expression for ipTherefore, a fairly accurate method of finding a*/Eh* wasthought to be to determine Lim.(p — >■0)wo/ph by experimentat small pressures. For the various plates values of a*/Eh*were found as follows:and placing it in the second Equation [31], solves for p. Theresulting expression for p and expression [32] are then placedin the first Equation [31 ] which is solved for p. The constantsc and n are then determined so as to make p as nearly a constantas possible. Nadai gives the following equation for the maximumdeflection:This is obtained when n = 0.25. In Fig. 8 a curve is shownbased on this formula and beside it a curve based on the powerseries solution discussed in this paper. The agreement is seento be close.The approximate method of Timoshenko10 is simpler. Heassumes that the radial displacement is given byand that, for purposes of finding ci and C2, the deflection is ofthe same form as in the elementary theory. He then minimizesthe potential energy of stretching to determine ci and c2, andarrives at the following equation for the deflection:F i g . 9M e t h o d o f (.’l a m p in g E d g e s o f C ir c u l a r P la t e sThe values of wo/h and pa*/Eh* obtained in the various testsare recorded in Table 4.A curve for Wo/h based on this formula also is shown in Fig. 8.This approximation it will be seen errs on the safe side, whilethe formula [33] errs on the unsafe side.In Fig. 8 the curve obtained from Hencky’s membrane theory6is shown also.E X P E R IM E N T A L RESULTS11 A p p a r a t u s f o r T e s t in gExperiments were made on circular plates of duralumin of41/ 2-in. diameter and various thicknesses. All plates wereclamped at their edges and the method of clamping is shown inFig. 9. The compartment beneath the plate was filled withwater and the pressure controlled by means of a static head ofmercury. An Ames dial directly above the center of the platerecorded the deflections.Four plates of the following thicknesses were tested:Plate 1........................................ h = 0.064 in.Plate 2 ........................................ h = 0.052 in.Plate 3 ........................................ h = 0 .0449 in.Plate 4 ........................................ h = 0.032 in.In all cases it was assumed that the value of Poisson’s ratio was0.35.12 R e s u l t s o p T e s t sAs it was desirable to know the value of pa1/Eh* for eachmeasured deflection, a*/Eh* had to be found for each plate. Nowfor very small deflectionsT ABLE 4Plate 1Plate 2Plate 3 Plate 4wo/h quo* wo/h quo* wo/h quo4 wo/h quo*0 0 0 0 0.002 0.01 0 00.075 0.437 0.067 0.42 0.162 0.98 0.387 2.510.153 0.939 0.138 0.84 0.376 2.36 0.527 3.580.234 1.440 0.229 1.40 0.550 3.75 0.-649 4.640.308 1.842 0.333 2.09 0.697 5.13 0.793 6.240.377 2.443 0.432 2.79 0.826 6.51 0.980 8.910.445 2.946 0.529 3.49 0.931 7.89 1.135 11.580.507 3.447 0.611 4.19 1.022 9.27 1.263 14.250.687 4.89 1.108 10.66 1.375 16.920.752 5.590.821 6.280.883 6.980.937 7.6813 C o m p a r is o n W it h T h e o r yIn Fig. 10 the deflection curves found from the experimentshave been plotted together with the theoretical curve for n =0.35.It will be noted that the difference between the values ofW o /h by theory and experiment is seldom more than 5 per cent.It will also be noted that the experimental curves always lieabove the theoretical curve. This latter fact would indicatethat part of the discrepancy is due to slipping and rotating atthe edges of the test plate. The fact that there is no apparentrelation between the plate thickness and the amount of disagreementshows, however, that edge slippage is not the onlyexplanation. A much more likely source of error in the experimentalcurve lies in the determination of a*/Eh*. To expecta determination of this quantity within 4 per cent accuracywould be very optimistic. On the whole, therefore, it must beconcluded that the agreement found between experiment andtheory is not at all bad.8 A. Nadai, “ Die Elastische Platten,” Springer, Berlin, 1925, p.288. Nadai actually uses the Equations [31] in dimensionless form.10 S. Timoshenko, “Vibration Problems in Engineering,” D.Van Nostrand Co., New York, 1928, p. 317.14 S u g g e s t io n s f o r D e s i g n o f P l a t e sFor clamped circular plates with uniform loads the elementarytheory is applicable for maximum deflections less than 0.4 of

APPLIED MECHANICS APM-56-12 633the thickness. For such deflections the error in the deflectiondue to neglect of the middle surface stresses will be less than10 per cent. (See Fig. 6.) For larger deflections the middlesurface stresses should be considered, unless very rough resultsare permissible. The simplest method for the practical designerwho wants to take account of the membrane stresses inhis plate is to have at hand a set of curves such as those in Figs.6 and 7. As reasonably close agreement was found betweenthe power series solution and the approximate solution by Nadai,the curves in his book “Die Elastische Platten” 9 (p. 288) shouldWe find that the form of the basic equations is unaltered:The general solution will contain two undetermined constants,fixing the values of which will determine $, s/, and s

634 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERSa subject which is of much more practical importance than theattention it has hitherto received might indicate. In the caseof the bending of beams, a large-deflection theory is needed onlywhen the deflections are so great that the slope of the deflectioncurve is of the order of magnitude of one, that is, when the deflectionsare of the order of magnitude of the length of the beam.Such deflections are almost never experienced in practise, andengineers are thus prone to think of large-deflection theories asof academic interest only. But the situation is entirely differentwith flat or curved plates which are bent in such a way thatstretching or compressing of the plate must accompany theflexure, as is the case when a developable surface is bent to a nondevelopableone. Then the simple theory of small deflectionsbecomes inaccurate when the deflections are of the order ofmagnitude of the thickness, as is vividly illustrated by the resultsgiven in this paper. Such deflections are frequently met within practise so that researches of this nature have great practicalvalue.The author develops an “exact” theory for his case and comparesit with two previous more approximate large-deflectiontheories. The comparison shows that as far as deflections areconcerned the approximate theories are sufficiently accurate forpractical purposes. This is important, as it is advisable to haveas many checks as possible on our approximate methods. Howeverthe author fails to give any comparison between the maximumstresses in the plate, as given by the exact and approximatemethods, although such a comparison should be quite easy forhim to make and would add greatly to the value of the paper.General experience has been that such approximate theories areless accurate in regard to stresses than deflections.The writer believes that it will be necessary to use large deflectiontheories to satisfactorily explain and study a number ofimportant phenomena in the buckling of flat and curved sheets.Thus it is well known that the ultimate strength of a flat sheetwith clamped or supported edges, under edge compression, isusually much greater than the stability lim it given by the wellknowntheories of Bryan and Timoshenko, based on small-deflectiontheory. Dr. von K£rmdn has explained most of this differenceas an edge effect, due to the edges being artificially preventedfrom buckling. But there is an additional strengthening effecthaving no connection with this. This might be called a “diskwheel”effect as it is due to the bulging out of the sheet to somethinglike the shape and hence something of the strength of adisk wheel. This effect can only be studied by a large-deflectiontheory. It has practical importance in the case of very thinplates or plates of a material with a high elastic lim it. Also, inthe buckling of thin cylinders or curved sheets under axial load,experiments show many features which have never been satisfactorilyexplained by any theory based on small deflections.The writer has found that all these discrepancies can be explainedby considering initial deviations from cylindrical shapeand using large deflection theory.H. H e n c k y .13 The paper by Mr. Way is very valuablebecause the exact solution of his problem is very difficult andtedious to obtain and is nevertheless needed for a judgment onapproximative methods. He discusses the methods of Nadaiand Timoshenko and finds them fairly satisfactory.I t is perhaps of interest to remark that these methods areonly special cases of a more general method which was developed18 years ago by G. B. Galerkin, of Leningrad. The methodof Galerkin is one of unlimited adaptability to the precisionrequired in any special case and deserves therefore a greateramount of publicity than it has had.The problem in question is a two-dimensional one, the two11Lisbon, N. H. Mem. A.8.M.E.dimensions being the displacement w perpendicular to the plateand the displacement p in the direction of the radius of the circularplate. Consequently we will have two equations of equilibriumto satisfy which we will call symbolically E(w) = 0 and E(p) = 0.The first step is to substitute two integrations taken over thesurface of the plate without changing anything essential.This can be done by introducing two arbitrary variationsS and Sp corresponding to the displacements w and p and writing:with the integrations taken over the whole area of the plate.U ntil now we have not introduced anything of an approximatecharacter. Galerkin substitutes for w and p two series of arbitraryfunctions and extends the variations only over the arbitraryconstants to be determined in the case in question. Consequently,using the fact that the variations are absolutely arbitrary,the two integrations yield just the needed number ofequations to determine the constants. Naturally the functionshave to be chosen so that the boundary conditions are satisfied.The advantage of this method lies in the fact that by increasingthe number of constants the method can be employed even ifthe exact solution is impossible to obtain as the case of a rectangularplate.E. 0 . W a t e r s . 14 The author should be highly commendedfor giving to the engineering profession a simple, workablesolution for a rather difficult problem. It is true that the generalequations have been known for some time in differential form,but it is doubtful whether engineers have ever made much, ifany, use of them. Now, thanks to the author’s labor of settingup the power series and solving the coefficients, together withhis ingenious method of adjusting the results so as to fit theprescribed boundary conditions, it is possible to find the maximumdeflection and the important stresses in a circular plateclamped at the edges, simply by referring to the proper curve inFig. 6 or Fig. 7.It is interesting to note how nearly accurate are the two approximatesolutions mentioned by the author, as far as maximumdeflection is concerned. In this connection, it should be pointedout that the solution by Timoshenko, Equation [35], was evidentlyworked out for m = 0.3. If m is changed to 0.25, thecurve for this approximate solution would shift slightly to theleft, in the manner indicated in Fig. 6, and the error on the sideof safety would be greater. This, however, makes practicallyno difference as long as the proper value of m2is used in the lastterm of Equation [35], as was apparently done in the paper.The writer would like to place on record another approximatesolution (not original) which gives practically the same maximumdeflection as the Nadai and Timoshenko formulas, and in additiongives membrane stresses at the center and edge that agreesubstantially with the exact method. By multiplying equation[8] by r, differentiating and then dividing by r, the fundamentalequation of equilibrium is obtained:After performing the differentiation indicated in the third term,and using the relation between o>' and n ' given by Equation [9],we have:14Associate Professor of Mechanical Engineering, Yale University,New Haven, Conn. Assoc-Mem. A.S.M .E.

A PPLIED M ECHANICS APM-56-12 635where R, and Rt are the radial and tangential, radii of curvatureof the middle surface. If there were no membrane stresses,the third term would vanish, the external work would beor dr, and the equivalent strain energy of bending would besecond givesWhen these are equated and reduced to the same form as Equations[33] or [35], and n is given the value 0.25, we havewrdr. Load is proportionalto deflection. Now, if membrane stresses are considered, and weneglect all displacements except those normal to the middlesurface, e/w = 1/R, and the strain energy of stretching may beexpressed by multiplying the third term above by i/iui andintegrating over the entire plate. The total external work isno longer rpi ICaJ 0From Equation [9]wrdr, but, since the strain energy of stretchingis only a small part of the total, about 20 per cent when thedeflection at the center equals the plate thickness, we mayassume as an approximation that the straight line relation stillholds, givingThen we may assume a value for w in terms of r and a constant,perform the indicated operations, and solve for the constant orat least get a relation between it and p. The simplest assumptionmembrane stresses are neglected.Equation [10]as this is the expression for w when theLetting r = aS, we have fromFrom this relation, values for oy' and at' may be obtained:Furthermore, when S is given the values 0 and 1 in Equations[44] and [45], and n = 0.3, we have the membrane stress at thecenterand the radial membrane stress at the edgeAll three of these equations agree well with the exact curves inFigs. 7 and 8, up to deflections of the order of w0/h = 1.25;both [46] and [47] err on the side of safety, and [48] is alwaysless than [47] and is therefore not critical.When we compare the bending stresses of the approximatemethod the agreement is very poor, and the reason is not hardto find. The approximate method gives the same bendingstress as the linear theory, when expressed in terms of maximumdeflection, because the approximate method assumes the samegeometrical form for the bent plate as does the linear theory.Actually, the bent plate will tend to assume the form of a thinmembrane, with very sharp curvature at the edge and fairlyuniform curvature everywhere else. In consequence of this wehave, by the exact method, extra large bending stresses at theedge and smaller bending stresses at the center. In this feature,if in no other, the exact solution developed by the author isclearly far superior to any approximate method which assumesa form for the bent plate similar to that given by the lineartheory.A close approximation to the true bending stress at the rimmay be obtained by taking a slightly different form for the bentsurface, say w = m>0(1 —