m«^^-M 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 
 
 No. 1188 
 
 THE ELASTO -PLASTIC STABILITY OF PLATES 
 
 By A. A. ILyTishin 
 
 TRANSLATION 
 
 "Uprugo-plasticheskaya Ustoichivost Plasteen" 
 Prikladnaya Matematika i Mekhanika X, 1946 
 
 UNjMERSiTYOF FLORIDA 
 
 fPOUMBfl-SDePARTMENT 
 
 <3AINESVIU£,FL326t1-70l1uSA 
 
 Washington 
 December 1947 
 
I <j l^ ' ' 4S- \^L^ I ^ ^ 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDIM NO. II88 
 
 THE ELASTO-PLASriC aTABILITY OF PLATES* 
 By A. A. Ilyushln 
 
 In this article are developed the res-alts of my work (reference l) 
 "The Stahility of Plates and Shells "beyond the Elastic Limit," A 
 significant improvement is foimd in the derivation of the relations 
 "between the stress factors and the strains resiilting from the 
 instahility of plates and sheila. In a strict analysis the problem 
 reduces to the solution of two simultaneous nonlinear partial differ- 
 ential equations of the fourth order in the deflection and stress 
 function, and in the approximate analysis to a single linear equa- 
 tion of the Bryan type. Solutions are given for the special cases 
 of a rectangular plate huclcling into a cylindrical form, and of an 
 arbitrarily shaped plate under uniform compression. These solutions 
 indicate that the accuracy obtained by the approximate method is 
 satisfactory, 
 
 1. EXPRESSIONS FOR THE FORCES AND MOMENTS IN TERMS 
 OF THE STRAINS IN THE MIDDLE SURFACE 
 
 On a movePoble Darboux trihedron, relative to which we shall 
 study the element of the shell, we choose the xy plane to be 
 tangent to the middle surface, and the x and y directions 
 along orthogonal curves (fig. 1) . 
 
 The state of stress of the element is determined by the 
 tensor of the stress S. Its -components Z„, Z_. Z„, are 
 
 small compared to X , Y , and X , that is, each layer of the 
 
 shell element parallel to the middle surface is in a state of 
 plane stress. The intensity of stress in this layer will be 
 
 •"'"Uprugo— plasticheskaya Ustoichivost Plasteen," Prikladnaya 
 Matematika i Mekhanlka X, I9U6, pp 623-638. 
 
NACA TM No II88 
 
 The state of strain of the element is determined "by the 
 comDonents of the tensor of the strains e^„, e,„^, and e^„, 
 
 since the shears ^xz' ®"-z ^^^ small;, hut the relation to the 
 strain e^^ may he found from the condition of constant volume 
 of the element^ 
 
 ©xx + ®,yy + ®zz " ° ^^"^^ 
 
 The intensity of strain in this layer of the material is given 
 hy the formula 
 
 P / , 
 
 V 3 
 
 In agreement with the laws for tha elasticity and plasticity 
 of materials the stresses and strains are connected by the relations 
 
 S^ a X _ i Y = — e^^ S,, = Y, -Lt =: — e_ X^. = e^,^ (l.l^) 
 
 e 
 
 1 
 
 y y 2 ^ e^ yy y 39^ ^y 
 
 Here o^ = crj^(e.) is determined for each material as a function 
 of e^. The properties of this function are as follows. Within 
 the elastic limits that is^ for a. < ct' where a' is a physical 
 
 constant, Hooke's Law a^ = Ee^ always holds. Beyond the elastic 
 
 limit CT^ = 'l>(e;j^) is a certain curve (fin. 2). If at a certain 
 
 Instant of time there occur infinitely small variations from the 
 state of strain, that is, the quantities e . . . receive 
 
 increments Se^^-jj . . ., then the increments of stress helow the 
 
 elastic limit are given hy the formulae (l,U) by setting o^ = Ee^. 
 
 Beyond the elastic limit the increments of stress for ^e^ > are 
 
 given by formulae (l.^) in accordance with the curve a^ = ^{e^), 
 
 but for 56;!^ < in'-accordance with the law of unloading o^ = Ee^j^. 
 
MCA TM No. 1188 3 
 
 The problem of the stability of shells (plates) Is stattd as 
 follows: G-lveiv a shell vinder a given Bystfm of applied forces 
 and \rith the states of stress and strain known. Required, the 
 critical valxie of the external forces .■*'or which, tit the same time, 
 there is equilihrium vrith other possible states of strain infinitely 
 close to the original state. 
 
 Let the change in the first and second quadratic form of the 
 middle surface of the shell relative to th« given ©qullibriirai 
 position be characterized by the parameters e e , 2e_, and 
 
 X ^ D 
 
 y^, \f T where <?,, e^ are length ratios and 2s ^ is the 
 
 shear in the middle surface in the x,y plane, and X, , Xp, t 
 
 ar© changes in curvature and twist. According to the Klrchoff 
 hypothesis the increments in length and shear ab a distance z 
 from the middle surface will be 
 
 We seek the stress increments corrf spending to tho strains (.l.'^). 
 For this it is necessary to take the variations of relations (l,'-!-), 
 Th* variation of the intensity of strain may be formd by making use 
 of (1.5)^ but aften/ard we write for the variation of the work of the 
 internal forces g^ Ss- , in terms of the stress componexits, 
 
 a,_ 5c, = X^5o^ . Yy5eyy -. X^e^ (1.6) 
 
 We introduce the nondimensional quantities 
 _^ Y * - _: 
 
 where h is the shell thickness. Then in agreement with (l.l) 
 
 T * - '_Z Y * - -1 T * - ^ Y 4.:- - !i ^' Y -» - i V T-;:- - ~ T fl 7) 
 
 y -^ y ■ J 
 From (1.6) and (1.5) w© lia-'^e 
 
 56, = e - T^y^ = ? - zX (1,8) 
 
where 
 
 MCA TM Ho 1188 
 
 c - T *f, J. Y '-c + PX -^e X-;<- - T -"-X * 4. Y -;f-X -K- + ox *T-K- 
 
 X = X -"-X + Y ^X + 2X *T 
 
 ^1 y 2 '^ 
 
 >(l.9) 
 
 For the variation of formulae (l.^) we note that 
 
 da. 
 
 e^ \Qi dcu 
 
 !5e. 
 
 do. 
 
 in which hy the properties of the curve a. = ^{e-)., — > > 0. 
 
 We denote "by 
 
 h 
 
 Si ■ de. 
 
 Zo^ — the coordinate of the layer for which 
 
 the intensity of strain is unchanged (Se^^ = O) dtrrlng instability. 
 It is clear that 
 
 o X o 
 
 (1.10) 
 
 The variations of formulae (l.'-') have the form 
 
 fa_. dff . 
 i - 
 
 V 
 
 6S. 
 
 
 a. da. 
 
 1 1 
 
 y \e,. de 
 
 r (1.11) 
 
 5X = 
 y 
 
 
 da 
 
 . i 7 * 
 
 dOi/ > 
 
 
 All quantities entering into the right-hand side of these equations 
 except the strains arid the curvatures are known, since the original 
 state of stress of the shell (whose stahility is soufdit) is supposed 
 
MCA TM No . 1188 
 
 
 given. The auantltle? E, E* = —r, E"' = - — are sham-: In figure 2 
 
 ■ ■ ■- ei ■. ;"'P-e.j[ ' , ■ 
 
 as tangents of angles. Young's modulus "being constant "but E' and 
 E" depending on the state of stress. 
 
 '''".. t 
 
 Before Instabilitj'- the shell may find itself wholly "beyond the 
 elastic limits or it may, have elastic regions, elasto—plastic regions, 
 and purely plastic regions.. If the state of stress is ^ao'r:lentless, 
 then the region of elasto— plastic strains, that is,' the rogi op, where 
 part of the shell thickness is elastic, part plastic, ie ahsent. . In 
 this paper we confine ourselves to the detailed sta"bility investigation 
 of" compressed plates in which the state of stress is alvmys momentless 
 "before insta'bility. Hence we shall suppose that in the shells 
 considered helow the region of elasto— plastic strain" Is missing, 
 "before "biickling (this assumption is not essentleil). 
 
 After insta'bility the region of the shell where the stress was 
 originally elastic xx^ill "be, gene rails'- speaking, elastlcally deformed, 
 since the strain variations are assumod infinitesimal. The region of 
 piirely plastic strain will "be, generally speaking, resolved after 
 "buckling into two — one retiaining purely plastic, the other elasto— 
 plastic. Figure 3 shows a section norinal to a shell xrltli bhe three 
 designated regions (the plastic region after insta'bi.lity is shaded). 
 
 Let the surface z - Zq represent the "boundary "between the 
 
 regions, one of which is elastic after insta'bility, the other plastic. 
 For its determination we shall suppose that in the elastp-plastic 
 zone, the plastic zone adjoins the shell surface z - -f- ~ and the 
 
 r'. 
 
 elastic zone which originates as a result of tinloading adjoins the 
 surface z = — ^. • i . _ 
 
 In the region of elastic strain and in the sone of voilcading 
 (z< Sq) formulae (l.ll) take the form 
 
 683^ = E(6^ - X^z) 5Sy = E(ep - X^z) BX^ = | E(e^ ~ r z) (1.12) 
 
 In the region of plastic strain exid in the zone of loading (z> 1^) 
 of the elasto— plastic region, these formulae may he presented in the 
 form^ • \ 
 
 "Trom (1.12) and (1.13) it is seen that the variations SS . . , 
 
 on the houndary z^ are continuous in the case where the original state 
 of stress corresponds to the "beginning of flow, and equally so when, as 
 a result of variation, the state of stress changes in proportion to the 
 original state (reference 2) . 
 
imCA TM No 1188 
 
 5S^ = (E' - E") S^^X(z _ Zq) + E'Ce-^'- X^s) 
 
 5Sy = (S« - E") Sy^X(z - z^) + E»(-'^ - y.^z) ! (1.13) 
 
 6X = (E» - E") Xy*X(z _ z^) + I E»(?3 - Tz) 
 
 We proceed to the derivation of expressions for forces and 
 
 moments arising in the shell dtiring instability . For their deter- 
 mination we have 
 
 / 2 ■ / 2 12 
 
 5T-, = / ^ 5Z dz 5T = ; , SY dz 53 = / , 5X dz 
 1 / h X 2 1^- y / -'1 7 
 
 U 2 
 
 P / 11 
 
 u 9 
 
 h 
 
 p2- 
 
 5M^ = 
 
 h. 
 2 
 
 5X,.z dz BM, = 
 
 ^h h • 
 
 /2 /'o 
 
 5Y,.z dz 5H = / , 6X z dz 
 n J 
 
 o 
 
 5 / 1-, J 
 
 /._li 
 
 "J-S 
 
 Xn the region of purely -plastic strain s we obtain for the forces ;, in 
 agreement with (1,13): 
 
 E»h V 1 2 2/ 
 E'h ^ ^ 2 1/ 
 
 e^ — 
 
 
 ^'V^^ 
 
 1 p 
 
 E»h -'' 
 
 x.;x - 
 
 .7 
 
 (1.1^). 
 
 and for the moments 
 
 A (s,^ . i 5M,) = - 
 
 BD 
 
 3D 
 
 7 (s^-2 - I 5Mi) = - X^ 
 
 x^>x»s^-x 
 
 ± m 
 
 
 
 3D' 
 
 Xg -<- x'Sy^x 
 
 
 ^ T + X'X *X (1.15) 
 
MCA TM Wo. 1188 
 
 vhere 
 
 D' = 
 
 tv3 
 
 E»h- 
 
 , E« - E" 
 , = 
 
 E« 
 
 (1.16) 
 
 In the region of purely elastic atralne . formalae (l.lii) and 
 (1.15) hold, onl.7 E' = E" = E, \« = 0. 
 
 Thus in the two regions, the forces are linear functions only 
 of e,, e , and 2e , the middle surface shear, and the moments 
 
 are linear functions only of the changes in curvature. 
 
 In th e region of elasto— plas tic st rain s^ the stresses 5X . . 
 
 have different expressions for z > z^ and for z ^ z.^. Hence, 
 
 the Integrals in the expressions for the •^'"orces and moments must 
 be split into two parts. For example, 
 
 .h ' h 
 
 &S„ z dz + 
 
 BS . z dz 
 
 ,, -0 
 
 in which for the region z^ "^ z ^ we take SS^ according to 
 
 (1.12) and for the region - > z > z^, according to (I.I3). As 
 a restilt of these calculations we obtain for the forces 
 
 1 - zJ^2 
 
 i (5^1 - 1 5^2) 
 
 E + E' + (E - E') Zq* 
 ^LuIL S * (1 - z^*) 
 
 ^1-^ 
 
 2— (E -En\* 
 
 2 -.^ 
 
 /sTg - |:5Tt_J =rE + E» + (E - EMzo'' | ^g + ^ " g""^' ^^ " ^'^V^^ 
 
 E* - E" 
 
 S '-(1 - ^o*) 
 
 2y* 
 
 ^'^'^^X 
 
 y 
 
 >a. 
 
 (1.17) 
 
 5S = 
 
 E + E' + (E - E»)z„* 
 
 ■ 1 - Zo*^ 
 S + T^- (E-E')T^- 
 
 , 5i^ X / (1 - z .)2 X* 
 
8 
 
 IIACA TM No 1188 
 
 and for the moments 
 
 ?^ 
 
 i at,) = - 
 
 E + E' + (E - E')Zq*3 
 
 X-=^-|(E -EM(1 - ^o""^)^^ 
 
 , IL^ (1 _ z^.)2 (2 + z^^O S/X* 
 
 i|(5M2-|5M^=- 
 
 E + E' + (E - E')z ^-^ 
 
 X * -- 2(E - E')(l - z *^Y 
 
 > 
 
 + i! 21 (1 _ Zq*)2(2 + Zo*)S^-:^X^^ 
 
 i2 5H = 
 h2 
 
 2 
 
 3 
 
 E + E' + (E - E')z^-3 
 
 L 
 
 (E -E«)(l - V^)e, 
 
 E« - E" 
 
 (1 - z,-"-)^(2 + z*)X*X^ 
 
 O "y 
 
 (1 .18) 
 
 The dependency "between forces and strr'li'is is nonlinear, since 
 z * enters into the formula and from (l.lO) it depende on the strains. 
 From this fact proceed all the difficulties of solution of problems 
 in shell stability beyond the elastic limit. 
 
 • Further, it is essential that the ordinate 2 * depending on 
 both the changes in curvature X^ , X , . t and on the strains e,, 
 
 ^nt ' ^o3 "be expressed only in the changes in c.urvature and the 
 forces 5T'-]_, 5T2, 53. Miiltiplying the first eq.ua,tion of (l.l?) 
 t'y ^x'% "f^^® second by YJ^, the third by 3X, * and adding, we 
 
 y y • : 
 
 get . 
 
 X(l - zo*)^ + )+z^->^ - k 
 
 S„*5Tt + S *ST„ + 3X *&S 
 
 = 
 
 EhX-!!- 
 
 (1.19) 
 
MCA TM No. 1188 
 
 By Introduction of the notation t, for the ratio of the thick- 
 ness h^ of the plastic layer to the thickness of the shell 
 
 h„ 1 - z^-^ 
 t -E 
 
 h = 2 
 
 and solving egLuation (I.I9) for t, we get 
 
 (1.20) 
 
 ^ _ E -\,/eE"(1 + cp) 1 - ^(1 - X.)(l + cp") (121)^ 
 E - E" . ' X 
 
 where 
 
 cp = 
 
 ^ Sx*&T^ + Sy^ST^ + 3X,.*5S 
 
 J 
 
 1 -X 
 
 EhX-;^ 
 
 dd. 
 
 •ra T?" de^ 
 
 ^ X = 2-:^J_ = 1 i (1.22) 
 
 . E E 
 
 Formulae (I.I7), (I.18) are apprecia"bly simplified (otherwise 
 conserving the principal complications), if we consider Only the 
 ■beginning of flow, that is^ we suppose that the shell material ■ 
 "before instability exceeds the elastic .limit very slightly. -In 
 this case 
 
 E' = E X» = X = 
 
 E - E" 
 E 
 
 Therefore, in the notation of (1.20) the corresponding formulae 
 have the form for the forces , _ 
 
 1- (sTi - i 5Tp^ = S + ^ V^^- 
 Eh \- 2 2y 1 2. X. ,. 
 
 •Eh V 2 2 1/. 2 2 y ^ 
 
 i-- (sTp _| 6T1 )= ?^ + ^ S^*^-^X 
 
 ' 1 C.C! 2 ^ , Xh y ^f.2%. 
 
 (1.23) 
 
10 
 
 MCA TM Ho. 1188 
 
 .for the moments 
 3D 
 
 3D 
 
 (sMi - i mA = -\ + XS^^^2(3 _ 2^)v 
 
 SM. 
 
 - I 8M^V -X^ 4-^ ^Sy*C-(3 ~ 25)-X } 
 
 3D 3 y 
 
 (1.2U) 
 
 where D is the usual stiffness for Poisson's ratio equal to 1/2. 
 
 2. THE STABILITY OF COMPRESSED PLATES 
 
 -Denoting the tending of the plate during instability hy v{x,y) 
 and the displacements of points in the middle surface projected in 
 the x,j directions hy u(x^y)^ T(x,y), respectively^ we have 
 expressions for the changes in ciirvature X X^, t^ and the 
 
 strains 6-|_, ep, e 
 
 
 >2 
 X - C' w 
 
 2 --2 
 oy^ 
 
 Bx Sy 
 
 1 Sx 
 
 
 3 2 \5y Bxy 
 
 > 
 
 (2.1) 
 
 The forces applied in the middle surface "before instahility 
 may "be written in the following fo?:Tn: 
 
 T-L = hcTj^^X^-^ '^2 " '^^i^ * ^ ='^^iV 
 
 and their projection on the Zr-axis after instahility in the form 
 
 ^l\ "^ "^2^2 "^ ^^"^ "" ^°^'^ 
 
MCA TM Wo . 1188 11 
 
 Therefore, the condition of equilibrium of all forces applied to an 
 element and projected on the z— axis, gives 
 
 ~ + 2 + S + hCTj:X = (2.2) 
 
 The condition of equilihritim of the middle surface forces after 
 insta,l3ility will be 
 
 1^1^ = -^+1^ = (2.3) 
 
 ox dy oj ^x 
 
 Finally, the couipatibility condition for the etraine has the 
 form 
 
 ^2 ^2^ ,^p^ 
 
 __i + .-^ _ 2 — 2.-= (^.l^) 
 
 The combination of differential equations (2.2), (2.3), and {2.k) 
 
 is necessary and siJfficient for the solution of the problem of 
 
 stability, if the corresponding boundary conditions are set up. 
 
 Indeed, according to (l.li)-), or to (l.2i)-) and (1.20), the strains 
 
 e, , e-, e may be .expressed in terms of the forces ST-,^ 5T^, SS 
 X d 3 
 
 and the curvatures X (bending w) , following which the moments 
 SMj, SM2, SH are functions of these same four arguments. Thus 
 
 the problem reduces itself to four differential equations with four 
 unknown functions, of which (2.2) is of the Bryan type, and (2.3), 
 {2.k) are of the type of equations in plane problems. 
 
 In the region of purely plastic strain of the pl.ate, (that is, 
 such that the whole thickness, plastic before instability, remains 
 plastic after instability) , the system of differential equations 
 is resolved into two. For simplicity we consider only the case 
 of the beginning of flow. Substitution of the values of bM-^, 5M , 
 
 5H from (1.15) into (2.2) gives a differential equation for w 
 of the Bryan type: 
 
 vV - ^X= 161 X^* -. 2-ii-X/ -H ^ Y*]XX (2.5) 
 
 ^ n5x2 ^ ^Sy ^ By2 ' 
 
12 
 
 UACA TM No . 1188 
 
 where, in agreement with (l. 9) and- (2.1) ^ 
 
 ^ -s. 2 ^ -v -v y ..^..>P- 
 
 (2.6) 
 
 The two "bovmcLary conditions on w agree with the usual . 
 "boundary condit long for the Bryan equation. 
 
 Solving equations (l.l^) for the strains, w© get 
 
 1 L 
 
 1 Sh \ 1 2 2; 
 
 
 (i-x)Eh V ^ y ? -^ y y 
 
 xs * 
 
 'p = — (^^2 - ^ 5^1] + ■— ^^ (V'S^l + S,,-&T^ + 3X/SSp> (2. 
 
 2 Eh \ '■- / (1 - X)Eh \-^ ■" •' 2 J y 
 
 i-x)Eh\^ ^ y 2 -^y y 
 
 3^3 
 2e., = - — + 
 
 7) 
 
 Eh (; 
 
 Equations (2.3) axe satisfied 'if the stress fujictlon"' F'"' is introduced: 
 
 5^1 S% 
 
 5T 
 
 2 ^^F SS, .' <f 
 
 F 
 
 Eh ^y2 Eh Sx2 Sh ...53.55- 
 
 (2.8) 
 
 follpwing ^■rhich, analogous to (2.6) we denote 
 
 .- . .. S% . ,. '^^F ^ „ S2f 
 
 ^ N 2 y >■ 2 y > -N 
 
 (2.9) 
 
 we ohtain the compatibility condition for strain in the fonri 
 
 vV =-f^S * + ^' 
 
 ,Sy' 
 
 2 3C 
 
 2 S^ 
 
 s - - 3 
 
 \ 
 
 ■\ 2 y 
 
 X--^ 
 
 \ xt 
 
 cix 5y 
 
 /r 
 
 (2.10) 
 
MCA TM Ho. 1188 
 
 13 
 
 In order to -JTrite the toxmdary conditions for this equation, 
 it is necessary to compute the variations of the norcial, force 8T^ , 
 and of the tangential force 63^ on a certain curvilinear contour 
 in the middle surface of the plate . ' 
 
 If the outvard normal V and the tangent s to the contour 
 constitute a coordinate system such that hy rotation the positive 
 direction of V coincides with that of y and the positive 
 direction of e coincides -tfith that of x and if the angle 
 het-ween the normal and the x— axis is denoted hy a (fig. h) , 
 then oxw quantities have the known expressions 
 
 &T-, + 5T^ 5T-, - 5T^ 
 
 5Tv = 
 
 ■1 
 
 ~ cos 2a + 8S sin 2a 
 
 6Tn 
 
 8T, 
 
 5Sv = 
 
 sin 2a — 53 cos 2a 
 
 (2.11) 
 
 The purely plastic region of the plate may he hounded hy a 
 cent our J part of which coincides with, the houndary of the plate ^ . 
 the part adjoins the elasto— plastic region. For the formulation 
 of the stability prohlem in the first part, the houndary conditions 
 have the form 
 
 STy = 5S^ = 
 
 (2.12) 
 
 and in the second part 5T, 
 
 V 
 
 must he continuous. 
 
 It is easy to show tha.t during, instahility the entire plate 
 may not remain in the purely plastic state; that is, an elasto- 
 plastic region may come into heing. Indeed, going hack we shall 
 have the unifoim houndary conditions (2.12) on all external edges 
 of the plate. But the differential equations (2.3) and (2.U) for 
 conditions (2.7) will he also linear and homogeneous and so will 
 have the unique solution . ■ 
 
 6T. = 5T^ 
 
 5S ■= 
 
 It follows from- (2.7) that ^-^ =^2 " ^^, " ^' ^^°^ ^^^°^ °^ 
 the hasis of (I.9) and (l.lO), Zq = 0. But 2=2^ is the 
 
li^ MCA TM No, 1188 
 
 boimdary "bet-ween the elastic and plastic zones through the thickness 
 of the plate and the condition z^ = specifies that the middle 
 
 surface is this hoimdary. It follows that a given region of a plat© 
 is not purely plastic, 'but elasto— plastic, which contradicts the 
 as SLim.pt ion. 
 
 During instahility of a plate "beyond the elastic limit it will 
 either go completely over to the elasto— plastic state or there will 
 remain purel;/ plastic regions in it, which are not diffused through- 
 out the plate . 
 
 In the region of elasto— plastic strains, equation (2.2) on the 
 hasis of expressions (I.2U) may he presented in the form 
 
 . ^v - ^ X = 1 (^ X/- -H 2.-^1 X/ H- ^ Y -^ XC2(3 - 2^.)X (2.13) 
 
 in which, as in equations (2.5), (2.10), the operator in parenthesis 
 acts like a muJ.tip] ier on the quantity to its right. 
 
 The condition of compatihility of strain (2.f<-) on the "basis of 
 (1.23) has the form 
 
 (2.li^) 
 
 where the stress function F is determined "by formulae (2.8). The 
 value of ^, the ratio of the thickness of the plastic layer to 
 the plate thickness, enters into equations (2.I3) and (2.lU), 
 therefore they show compati'bility; this quantity ^ is expressed 
 "by formula (l.2l) in which the fimction c is, if use is made of 
 the notation (2.9) 
 
 cp = S —^ — t . • (2.15) 
 
 h 1 _ X X 
 
 Equations (2.I3), (2.IU) agree with the corresponding equations 
 (2.5) and (2.10) at the boundary of the purely plastic and the elasto- 
 plastic regions. Indeed, at this "boundary, "besides continuity in 
 the values of the forces 5Tv, &Sv, the moments BMy , 5Hy ' 
 (where 5Hy' is the rotational moment according to the "boundary 
 
HACA TM No. 1188 15 
 
 conditions of Klrchoff), the tending w and the slope of the tangent 
 plane, there must also hold the condition 
 
 h^ = h 
 
 (2.16) 
 C = 1 
 
 From (1.21) for this condition we have cp = -X and t = - ~ Xli, 
 
 2 
 
 following which the remarked coincidence of the equations is easily 
 
 shown. 
 
 The houndary conditions for equations (2.13), (2.1i<-) on the 
 elasto— plastic part of the contour, coinciding with the plate conto\ir, 
 yield the usual requirement 5Ty = SSy = and two conditions relating 
 to the tending w. 
 
 Condition (2.15) or 
 
 t = - :^ - ^ -m ■ (2.17) 
 
 represents in itself the equation of the boundary "between the purely 
 plastic and the elasto— plastic regions. 
 
 The possihility of purely plastic regions arising at the same 
 with" "the elasto— plastic regions follows from the fact that the 
 value of ti in agi-eement with (l.2l) and (2.I5) may take on values 
 not lying in the interval 1 ^ ^ ^0. Certain examples are given 
 "below of exact solutions of the stability of plates,, and, in 
 particular, the problem of the compressed plate freely supported 
 along two sides; the edges of the plate near the free supports, 
 after instability, remain in the purely plastic state . 
 
 3. EXAMPLES OF EXP.CT SOLUTIONS OF PROELE]\K 
 IN THE STABILiry OF PLATES 
 
 The integration of the system of differential equations (2.13) 
 and (2.1ii-) in the elasto-plastic region, and of (2.5) and (2.10) in 
 the plastic region with an undetermined boundary between them given 
 
16 NACA TM No. II88 
 
 ty (2.16), is fraught with significant mathematical difficulties. 
 As was shown in 1^ the stability problem simplifies when the 
 variations of the forces in the middle surface are zero every- 
 where. In that case the relative thickness C of the plastic 
 layer is a knoim function of the coordinates, since from (1.22) 
 cp = and consequently 
 
 If the state of stress of the plate before instability is 
 uniform, the value of t, will be constant, since in (1.22) 
 
 ■^-^ will be the same for the who3.e -niate . 
 
 We call those solutions of stability problems aTjproximate 
 for which the variations 57-, , ^T^, 5S of the forces are 
 
 identically zero. Thus, the equations (?.3) of eqtiilibrium and 
 the boundary conditions (2.12) are satisfied, but, except in 
 special cases, the compatibility condition (2.^) is not satisfied. 
 The simplicity of such a solution arises from the fact that in 
 equation (2.I3) the value of ^ is known and given by formula 
 (3.1), as a result of which this equation becomes linear with 
 constant or variable coefficients. It closely resembles the 
 equation for the elastic stability of an anisotrot)ic plate. 
 
 The exact solution^ of the system (2.I3), (2.1ii) sxe undoubtedly 
 
 of interest in their otvtl right, but for us they have significance 
 
 because they can be made use of to estimate the degree of exactness 
 of approximate solutions. 
 
 We discuss a certain class of exact solutions of stability 
 problems for uniformly compressed pj.ates of arbitrary shape and the 
 solution for a recta.ngular plate in the case when buclcLing into a 
 cylindrical shape is possible. 
 
 a. Stability of a Uniformly Compressed 
 Plate of Arbitrary Shape (Fig. k) 
 
 In this case the state of stress of the plate before instability 
 is uniform and given by the formulae 
 
 X^=Y =-a., ^j=0 ■ (3.2) 
 
MCA TM No, 1188 
 
 17 
 
 where o^ is the compressive stress along the edge and is also the 
 uniform stress intensity at any point in the plate. The resulting 
 stresses according to (1.7) and {l.h) will he 
 
 V = V = ''- V = ° ^x" = V 
 
 1 
 
 2 
 
 (3.3) 
 
 For the yalues of X and t we have the expressions from (2.6) 
 end (2.9) 
 
 X = -Vw t = - - V 
 2 
 
 i^^ 
 
 (3.U) 
 
 Equation (2.1^^-) takes the form 
 
 S^\t-f^t,\]= 
 
 <)■ 
 
 (3.5) 
 
 Neglecting the hanscnic fvmction, we obtain a class of exact solutions 
 
 4- _ Xh c, 2w 
 
 as a result of which the value of cp in (2.I5) is expressed in terms 
 of ^, and from (1.2) we find. 
 
 3X 
 
 -/^- 
 
 3X 
 1^ 
 
 = consi; , 
 
 (3.7) 
 
 The fundamental differential equation of stability (2.13) is now 
 linear with constant coefficients and has the simple form 
 
 1 -^ 'i^{3 - 20 
 
 V V + — - V w = 
 D 
 
 (3.8) 
 
 Its solution has been much studied for different shapes of 
 plates and for different boundar3'' conditions, although in connection 
 with the elastic stability of compressed plates. 
 
18 
 
 MCA TM No. 1188 
 
 The value of ^ (3.7) is little different from the p.pproxlmation 
 (3.1)., and cheracterizes the degree of deviation of the exact solution 
 from the approximate . 
 
 In the general case we have from (3.5) 
 
 . = f (sS. . r,) 
 
 (3.9) 
 
 where F-^ is an arlDitrary harmonic function. For continuous circular 
 plates, for example, F^ is a constant. According to (2.15) ejid (l.2l) 
 
 we now have an expression for t in terras of X 
 
 ^ = — 1 1 
 
 (3.10) 
 
 following which equation (2.13)^ having in the given case the form 
 
 ^- 
 
 1 -^ r(3 -2^ 
 
 X + — i :( = 
 D 
 
 (3.11) 
 
 has only one unknovm function X. By use of relations (3.^) it may 
 he integrated once 
 
 |i-~r(3-20 
 
 I 4- 
 
 ha. 
 
 V w + 
 
 — i w == r 
 
 L o 
 
 (3.12) 
 
 where T is a new harmonic function^ also a constant for continuous 
 
 circLilar plates, insofar as w and V^w must he finite in the middle. 
 Equation (3.12), in view of (3,10), may be solved for V^w, after 
 which the prohlem reduces to the integration of only one linear partial 
 differential equation of the second order (for circular plates) 
 
 V"w = 'i'(w,r^,r2) 
 
 The stress function F is now determined, in accord^jice with (3-9) 
 and (3.'+)i from the Poisson differential equation 
 
MeA.TiM No, 1188 . 19 
 
 A=:-^U^^+r^) . (3.13) 
 
 As we see J the proMeci of the stability of circTilar plates may 
 he solved In comparatively simple fashion. thrpu^i to the end. The 
 details of a similar calculation will he clarified helow for the 
 example of a rectangiilar plate compressed in one direction. 
 
 "b. Stability of a Rectangular Plate Under the 
 
 Condition of Plane Strain (Fig,.. -5) 
 
 Such a case occurs if ^ when a rectangular plate of length I is 
 compressed in the x— c!.irectlon, the width b in the y— direction cannot 
 change as a result of walls along the boundaries y = and y = b. 
 The plane x = shown in figujre 5> where C = 2c and L = 22 j- -will 
 evidently be a plane of sjTumetry of strains . 
 
 We assume the buckling to result in a cylindrical shape. In such 
 a case, according to the conditions of the problem,^ we have for the 
 stresses before instability 
 
 i X^ T,. ^ CT. = .11. 
 
 2 ^ y i 2 
 
 ^x = -P ^v = ^ ^ ^y = ° ^i = -TT ^ 
 
 > (3.11^) 
 
 X * = — Y ^ = 4= S * = - -^^ 3-^ = 
 
 ^ ^/3 y \/3 ^ 2 y 
 
 After buckling, w = w(x), sg = e^ " ^' 
 From .equ8,t ions (1.2^) we have 
 
 SS = ST„ = i 8T-, , 
 2 2 1 
 
 Since, in accordance with the equations of equilibrium-, ^Tj^ = const, 
 and STt = from the condition at the edge x --- -;, then we have 
 
 the case 5T-, = 5Tp = 5S = 0. In consequence ^ the ap-nroximate 
 solution, as was noted at the beginning of 3, here becomes exact. 
 
20 I'A.OA TM IIo. 1188 
 
 The thickness ratio t, for the plastic layer is a constant 
 and is determined by formula (3.I). The stability equation (9.13) 
 takes the form 
 
 ^^— ^2 T^=0 (3.15) 
 
 dx^ d[i -H^{3 - 2OI dx^ 
 
 If the relative Karraan modulus, expressed hj^ 
 
 dcr. ■ ■ 
 
 K = -7 ^,__. = ^(^ I_^.. (3.16) 
 
 ■f -m " • """" 
 
 7 
 
 is introduced, then wo get from (3.I) 
 
 (2 - 7k)^ 2 
 
 (3.17) 
 
 following which we may simplify the expression for the parameter in 
 equation (3.15) 
 
 ■ ^2._^ hP ^^llP (3.18) 
 
 d[i -U2(3 _ 2^) I Dk 
 
 Since k = 1 u;^ to the elastic limit, and k ~ in a small 
 area where there is flow of the material^ and since ti-}© character- 
 istic value of the parameter 7 must he the sojne in elastic and in 
 plastic problems, then it follows from (3.I8) that the critical 
 stress corresponding to the small area of flow, is zero. 
 
 It is interesting to note that the Karaan problem may be 
 considered as a limiting case of the stability of a rectangular 
 plate compressed in one direction, of small width b^ for which 
 the parameter 7 will have the expression 
 
 2 ^ ]ihP 
 3Dk 
 
MCA TM No. 1188 21 
 
 and consequently the critical stress is zero at the small area of 
 flow. As seen from the preceding and following examples of exact 
 solutions, the total loss of load-carrying ability of a plate , 
 predicted in the Karman problem, does not occur, generally speaking 
 This circumstance has already been noted (reference l). 
 
 c. The Stability of a Rectangular Plate Compressed 
 
 In One Direction (Fig, 5) 
 
 We shall suppose that the rectangvilar plate, sufficiently long 
 in the y-direction and compressed only in the x-direction, buckles 
 into a csdindrlcal shape . In this case 
 
 X = _cr Y,, = X,, = 
 
 X , 1 y y 
 
 s^* = -1 s -;^ = i X * = -1 Y * = 2 ^^ =-- 
 
 X y 2 3C J 7 
 
 (3.19) 
 
 By the conditions of the problem, all sections o'i^ the plate 
 y = const. rema.in plane after buckling and so we have 
 
 6=0 e^ = const. (3.20) 
 
 on 
 
 the basis of which from (1.2^), 5S = 0. Besides this, 5T = 
 
 7 
 from the boundary condition at the edges x = t — and consequently 
 
 it follows from (2.3) that &T-, = everywhere. 
 
 Since there ai^e no forces in the y— direction, we must use the 
 condition 
 
 / - 
 
 / 2 5T„ dx = (3.21) 
 
 /_ 1 2 
 
 From the second equation of the system (2-.1U) we have 
 
 ^.^^II.^, -Z^^Sx (^.22) 
 
 ^yP- Eh 2 i<- 1 ■ 
 
22 NACA TMlIo. Hi 
 
 since "^ = -X-j^. It is not difflciilt to convince one's self that 
 
 (3.22) is the integral of equation {?..lk). The ftmetion cp , hy ■ 
 which is foiind the value of C from- (l.2l)^ here has the fom 
 
 cp = i-^ = L + J:_i (3.23) 
 
 (1 - X)Eh^-<^ (1 - \)hX^ kil - X) 
 
 The hending moment in any section is 
 
 5M3 = -D 
 
 Q 
 
 1 _f xr(3 -2U 
 
 y^^ (3.2if) 
 
 and so the "boimdary condition on the edges x = t — is Xt = 0. 
 
 2 
 
 It is clear from (3.23) that X cannot he zero in the elasto— 
 
 plastic region since e^ ^ (this follows from the constancy of 
 
 2 
 sign of ^ X-j^, positive along the entire plate necessitating 
 
 ^2 ^ to satisfy condition (3.22)). Thus the elasto— plastic 
 
 region does not go up to the edges of the plate end stops at the 
 
 section x = — — . The region adjoining this to the edge will he 
 
 o 
 purely plastic. Indeed, since t''X is positive^ then e is also 
 
 positive. It follows from (2,7) that in the purely plastic and in 
 the purely elastic regions the force 5Tp has the same sign as 
 
 62^ that is, is a tensile force. But if, to the plate, coEipressed 
 
 heyond. the elastic limit in the x-direction, there is applied a 
 tensile force in the y— direction, then the plate remains in the 
 plastic state. One may convince one's self of this by formally 
 calc-ijlating the value of os^^ according to (I.8), which at the 
 edges is equal to 6]_, but the strain ?-, according to (2.7) 
 
 is negative, and so the value of be^ will be positive, that is, 
 plastic strains before buckling remain plastic after buckling, 
 
 ' From (1.21) and (3.23) ws now have 
 
 hX T 
 
 ~ = — , viO == - U + 8^ - 3>4^ (3.25) 
 
 ^'2 p(0 
 
MCA TM No. 1188 
 
 23 
 
 From thla we find the lower limit to the value of t (v > 0) 
 
 ^^5>5:(^ 
 
 v'-? 
 
 (3.26) 
 
 The fundamental differential equation of stability (2.I3) takes the 
 form 
 
 d2 
 
 d::2 
 
 3 ^e2 
 
 1 _^XG^(3 -2t) 
 
 ha. 
 
 -J 
 
 1 D '^ 
 
 = 
 
 (3.27) 
 
 By introduction of the notation . 
 
 Q(0 = k - 9Xi'^ + 6?.^3 I ^ 2x 
 we write equation (3.27) in the form 
 
 *'- n h! = 
 
 c 
 
 d| 
 
 2 P p 
 
 (3.28) 
 
 (3.29) 
 
 where n is the "basic parameter determining the critical stress 
 
 ^^ = -^ (3.30) 
 
 D 
 
 The integral of equation (3.29) may he ohtained hy quadratures. 
 Through introduction of the notation 
 
 E(0 = 2 
 
 Ij. - 12^ + 12^2 _ 3^^ 3 
 {h - 8^+ 3^t^)^ 
 
 (3.31) 
 
 we obtain as a resvilt 
 ,2 /.x2 
 
 '0)(3)-^'<'i-'^'-.J^^ 
 
 =2- 
 
 P dE 
 
 J ^A^- 
 
 (3.32) 
 
 E 
 
2k . MCA TM Ko. II88 
 
 In the purely plastic region we have for the force- 5T^ and the 
 moment 6M-j_, in agreement with the results of 2 and (3.I9): 
 
 M^ = -d/i -^ X^ (3.33) 
 
 ^^2 lj.(l _ X) 
 
 — = -^ 5M, 
 
 The ftindamental differential equation talres the form 
 
 2„ 
 
 i_l + _±i_. X =0 " (3.3i^) 
 
 d|2 1+ _ 3x ^ 
 
 The solution, satisfying the condition X-j_ = at the end | = 1, 
 is written in the form 
 
 X-, = Co?, sin — __ ' , (3.3:5) 
 
 ^ \/k - 3X 
 
 in which as a restilt of a^.Tanetry we consider only deflections in the 
 right half of the plate (x > 0) . 
 
 For determination of the five undetermined constants namely, 
 the three inte'gration constants C-, , Cg . C^, the "boundary 
 
 coordinate a and the critical nuraher |i. we roay, hesides 
 equation (3. 21), write four more conditions: Conditions of symmetry 
 
 1=0 TT = (3.36) 
 
 de 
 conditions at the "boundary region 
 
 ■ I = a ^"=1 (3.37) 
 
 two continuity conditions, of moment and shear force, which in 
 accordance with (3.25) and (3.35) take the form 
 
 „ . ^( 1 -g) - '• '^^ 
 C-, sm --^ - 
 
 ■" \Jh - 3>^ h(U - 3X) K/k - 3X \J 
 
 , ' 3_,_. cos '-^^.r,^ = 'K^ (f) (3.38) 
 
MCA TM No. 1188 2.'^y 
 
 The constant e is not necessary and does not enter the 
 
 <- Y 
 
 &T 1 
 conditions insofar as they are independent of — and —=•. 
 
 60 ^2 
 
 By, making use of the prescrihed conditions and introducing 
 a new unknown; t,^, the relative thickness of the plastic layer. . 
 
 at X = 0, we get for the- values of \x and 1 — a (the relative 
 length of the purely plastic part) the following formulae 
 
 X.M ., 1-!- _ 3X L (^ ._,r,\ 
 
 J2{1 -X) ^ ^ 
 
 whore L and M ' aro the integrals 
 
 . = f^^ -±^4.j^ ^, u-r^'"' (iri2!jL, (3..0, 
 
 / 
 
 '/e(i)' \m^^) -.^0 ' ja(i) yH(y;:r:. E(V) 
 
 in which the value of C is determined liy the relation 
 cot^ /_4k.=^3k_ lV 2(1^- - 3^) iE(^o) - ^0-) 
 
 .\/2(l-X) / ■ .1'° 
 
 (3. in) 
 
 As was already established, the value of 1 — o, is positive, 
 therefore the integral L must he positive, pnd for tliis it is 
 required that . 1 - 2t + XC^ > 0, that is,. 
 
 : 1 - x/rr 
 
 \ 
 
 ^ X- V.X-. ^3^j^2j 
 
 'O 
 
 By considering the estimate (3.26), which is also reasonable for 
 t^, we .see that this quantity is contained within narrow limits and 
 close to the approximate value (3.I). It follows from this that the 
 critical stress will differ only slightly from the approximate value. 
 
26 
 
 MCA'TM No. 1188 
 
 d'. Approximate Solution of 'the Problem for a Plate 
 in a Uniform State of Stress Before B-ackling ■ 
 
 and- X„ and the 
 
 'X- -J' T-y. 
 
 In this case the stress components X„' Y. 
 stress intensity a. are constant everywhere; the quantity \ 
 will also be constant, and hence ^ by (3.I). 
 
 The X and y axes in a given case may be so chosen that 
 the X stress is zero (principal axes of stress) . The fundamental 
 
 stability equation (2.I3) takes the fomi 
 
 0I S^w ! -5 
 
 1 - J(l - k)X^-2 _Z+2|l-4(l- k)X^-V'^ 
 
 1+' 
 
 o w 
 
 1 _o(i _k)Y^ 
 
 ...2 
 
 y 
 
 hK 
 
 J oy 
 
 ho, 
 D 
 
 h% 
 
 T -;<• ^ 4. Y ^<- 
 
 (3.1*3) 
 
 in which the generalized Earman ■ modulus is introduced in accordance 
 with formiilae (3.I6) and (3.1?)^ since the relation 
 
 hH3 -2i) =1 
 
 holds . 
 
 The coefficients in equation (3A3) sltq all positivej since 
 the largest value of each of the quajitities X.^-*^, Y * is 
 
 -^ y 
 
 2 
 
 — r and 1 > \z > 0. 
 
 v'3 
 
 Hence ;, the problem may be solved as a linear dif i'erential 
 equation of the Bryan type with constant coefficients, and in 
 difficulty is little different from the corrosponding elastic 
 case . 
 
 Translated by E. Z. Stowell 
 National Advisory Committee 
 for Aeronautics 
 
MCA TM No. 1188 ' 27 
 
 REFEESNCES 
 
 1, Ilyushirij A. A. : Ustoichivost Plastinok i OlDolochek za Predelom 
 
 Upriigosti. Prikladiiaya Mat.en13.tika i Meklianika, N. S. 8, No. 5j 
 19Uij-, pp. 337-360. (Also availalDle as MCA TM No. III6.) 
 
 2. Ilyiishin, A. A.: K Teoria Malikli Uprxigo— plasticheskikh Deformatsil. 
 
 Prlkladnaya Matematika i Meklianika;, X, l'^k-6, p, 3^+7. 
 
NACA TM No. 1188 
 
 29 
 
 Figure 1. 
 
 Figure 2, 
 
 Plastic zo"« ♦ Elasto-piastlc zone 
 
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 Figure 3. 
 
30 
 
 NACA TM No. 1188 
 
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