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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM NO. 1188 THE ELASTOPLASTIC STABILITY OF PLATES* By A. A. Ilyushin In this article are developed the results of my work (reference 1) "The Stability of Plates and Shells beyond the Elastic Limit." A significant improvement is found in the derivation of the relations between the stress factors and the strains resulting from the instability of plates and shells. 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 buckling 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 TEE STRAINS IN TEE MIDDLE SURFACE On a moveable Darboux triheirci, 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,,, y, are smr.ll compared to X, Yy, 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 rill be ""Uprugoplasticheskaya Ustoichivost Plasteen." Prikladnaya Mat'ematika i 1Mekhanika X, 196, pp 623633. NACA TM No 1188 Sy xy + 2 (1.1) The state of strain of the element is determined by the components of the tensor of the strains exx, eyy. and exy' since the shears exz, eyz are small, but the relation to the strain ezz may be found from the condition of constant volume of the element, exx + eyy + ezz = 0 (1.2) The intensity of strain in this layer of the material is given by the formula ei =  e2 + ey2 + e ey + exy2 (1.3) In agreement with the laws for the elasticity and plasticity of materials the stresses and strains are connected by the relations 1 (.i 1 Ci i S X Yy = ex Sy 2 y 2 ei yy Xy 3e y Here ai = ai(ei) is determined for each material as a function of ei. The properties of this function are as follows. Within the elastic limit, that is, for ai a' where a' is a physical constant, Hooke's Law ai = Eei always holds. Beyond the elastic limit ai = @(ei) is a certain curve (fic. 2). If at a certain instant of time there occur infinitely small variations from the state of strain, that is, the quantities ex receive increments 5ex ., then the increments of stress below the elastic limit are given by the formulae (1.4) bysetting i = Eel. Beyond the elastic limit the increments of stress for 8ei > 0 are given by formulae (1.4) in accordance with the curve ai = 5(ei), but for bei < 0 inaccordance with the law of unloading ai = Eei. NACA TM No. 1188 The follows: and with critical there is close to problem of the stability of shells (plates) is stated as Given, a shell under given system of applied forces the states of stress and. strain known. Required, the value of the external forces "or which, at the same time, equilibrium with other possible states of strain infinitely the original state. Let the change in the first and second quadratic form of the middle surface of the shell relative to the given equilibrium position be characterized by the parameters E1 2, 253, and , X2, T where el', are length ratios and 2 is the shear in the middle surface in the x,y plane, and ., T are changes in curvature and twist. According to the Kirehoff hypothesis the increments in length and shear at distance z from the middle surface will be 6exx = E6 z 1 so y = E 2 y 2 ,e = ,< 2sr  .. We seek the stress increments corresponding to the strains (1.5). For this it is necessary to take the variations of reltiono (l.h), The variation of the intensity of strain may be found by making use of (1.5), but artean.rd we write for the voriation of the work of the internal forces ci ri, in teirm of the stroes coironentse, cri 5'i = "*' + Y y 50 Y 59+ (1.6) We introduce the nondimensional quantities X* =  i y h h Y i a1 (1.7) where h is the shell thiclnmcs. Then in agreement Trit, (I.1) X '. + y ,2 y + 3Xy2 = 1 From (1.6) and (1.5) we have S= e =  (1.5) (1,8) NACA TM No 1188 where E = XXE + Y*E + 2X *E xl y 2+ Y2 y 3 *= X + fyX2 X = X*X + Y *X + x 1 2 For the variation of formulae (1.4) we note that cia 8 ei 1' ei doci be. do i/ 1 in which by the properties of the curve cr = 4(ei), h We denote by zo = Zo* the coordinate of the the intensity of strain is unchanged (Bei = 0) d.uing It is clear that Zo _ S The variations of formulae (1.4) have the form /.S x Cr dCl dei S *X (z* z *) + x 0  d.0 0 ai da ei dei O layer for which instability. (1.10) ( eX z Sy = 1I S *y (Z* z*) Y daN_ del Xy* :(z* z *) + y All quantities entering into the except the strains and the curvatures state of stress of the shell (whose s" 2ai 3ei righthand are Imown, ability is (E3 T*z side of these equations since the original sou'1it) is supposed + y*T* S2X * + e af 57 5X =  y 1 NACA TM No. 1188 a E daI ai doi given. The quantit2eQ E, E' = , E"' = are shown in figure ? Sei : de. i as tangents of angles, Young's modulus being constant but E' and E" depending on the state of stress. f Before instability the shell may find itself wholly beyond the elastic limit, or it may.have elastic regions, elastoplastic regions, and purelyplastic regions. If the state of stress is ':onentless, then the region of elastoplastic strains, that is, the iogion where part of the shell thickness is elastic, part plastic, is absent. In this paper we confine ourselves to the detailed stability investigation of compressed plates in which the state of stress is alir2 .; momentless before instability. Hence we shall suppose that in the shells considered below the region of elastoplastic strain is missing, before buckling (this ossu.mtion is not essential). After instability the region of the shell where the stress was originally elastic vill be, generally speaking, elosticilly deformed, since the strain variations are assumed ilrfiiteslmal. Ths region of purely plastic strain will be, generally speaking, resolved after buckling into two one remaining purerl plastic, the other elasto plastic. Figure 3 shows a section norna? to a hell i7trih the three designated regions (the plastic region after instability : is shaded). Let the surface z = represent the boundar.j between the regions, one of which is elastic after instability, the other plastic. For its determination we shall suppose that in the Jclasionaastic zone, the plastic zone adjoins the shell surface = + s and the elastic zone which originates as a result of unloadlnr joinss the surface =  In the region of elastic strain and in the zone of unlcading (z zo) formulae (1.11) take the form 8Sx = E(E1 X1) Sy = E(c2 2z) 5X = E(c ) (1.12) In the region of plastic strain and in the Zone of 1.oading (z E o) of the elastoplastic region, these formulae may be presented in the formal1 1From (1.12) and (1.13) it is seen that the variations BS . on the boundary zo 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). ,A\CA TM No 1188 We proc moments arise 5Sx = (E E") SxX(z zo) + E'(E1 X1) 5Sy = (Et E") S *X(z zo) + E(E 2 ) (1 6X = (E E") X*X(z z),+ Tz) :eed to the derivation of expressions forforces and ing in the shell during instability. For their deter mination we have BT1 h I h iU. h M1 2 2 In the region of pure agreement with (1.13, E1 1 S (T, 895 E'h (1 2 and for the moments and. for the moments  M 1 .5M )  {SM2 1 m^ 3 4 ( M l 1 '6 2 X daz 5T2 x 2 h .h Xz dz SM2  2 Yy dz SYz daz h BS = h h 5H h y a .13) SX dz y d SX z dz Y sly plastic strains we obtain for the forces, in = l xlS *y = 22 %iSY*e l + X'Sx * = X, + X'S *X 1 S = E'h 3 YX .3 ) 4l. 2  85 = T + %$X *X 3D'3 . (1.14) (1.15) IIACA TI No. 1188 where E'h3 D' = _ 9 (1.16) E' E" E' In the region of purely elastic strains, formulae (1.1h) and (1.15) hold, crny E' = E" = E, X' = 0. Thus in the two regions, the forces are linear functions only of e1, 2 and 2 ., the middle surface shear, and the moments are linear functions only of the changes in curvature. In the region of elastoplastic strains, the stresses X . have different expressions for z > zo and. for z zo. Hence, the integrals in the expressions for the ''orces and moimenis must be split into two parts. For example, 851 h 3S z dz = S z dz + h'  tIs h SS.; z dz i 0 hh in which for the region zo ? z g we take 9"' rccor'llng to (1.12) and for the region z > z according to (1.13). As a result of these calculations we obtain for the forces S 1= = E + E') '+ (E * (E E')X 6TT2) +2,. + E E" S (1 z )2  1 5T\ =[E + Et + (E .E')z + o 2 z1 2 (E E')X * 2 2 E' E" 2 * + 2 S (1 Zo) 2 E (E E') 3  h SS =~E + El + (E E)zo + (E E')T E' E_ Xy ( 1 o9 : 2 X 1 S(1.17) 8 and for the moments rACA TM No 1188 1 M 3M2 = E + E' + + IE (( E"1 *) (2 + *) S)X* 2 . 2 = E + E' + (E E')o X* * (E  E')( 1 z42) 72 + (1 zo*)2(2 + zo*)S yX 0  E + E' + (E E')zo (E E')(1 z*23 3 E0 +E (1 zo*)2(2 + zo*)(yX* 2 (1.i8) The dependency between forces and strains is nonlinear, since z enters into the formula and from (1.10) it depends 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 zo* deending on both the changes in curvature X1, X2, T and on the strains el, 62', e3 be expressed only in the changes in curvature and the forces T1;, 6T2, 5S. Multiplying the first equation of (1.17) by ix*, the second by Y *, the third by 3X and adding, ye get Sx*ST1 + S *8T2 + 3Xy*8S EhX* (1.19) (E E')zo3 X (EE')(i z *2 12 H = h2 (l o*)2 + 4z o 4 NACA TM No. 1188 9 By introduction of the notation 5 for the ratio of the thick ness hp of the plastic layer to the thickness of the shell hp 1 zo h 2 (1.20) and solving equation (1.19) for w, we get E EE"(1 +) (1 )(1 + E E" l (1.21) where (p = x Sx*'T1 + Sy ST2 + 3Xy*5S 1  EhX;,: E E" SE da. dei =1E E (1.2?) Formula (1.17), (1.18) are appreciably simplified (otherwise conserving the principal complications) if we consider onl 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' = =E E Therefore, in the notation of have the form for the forces S 2 BT ) Eh 12 2) += h 3S 3 2+ x 2 3 3 2 y (1.20) the corresponding formulae Xh 2 2 (1.23) +h C! (1.23) i22 Y52 NACA TM No. 1188 for :the. moments : ; 3D(1 12 3D (M2 ,; l =  i 2  + S 2(3 2t)X S,,Sy* (3 .X 3 = I+ XX 3 + 2 )x 3 y where D is the usual stiffness for Poisson's ratio equal to 1/2. 2. THE STABILITY OF COI..RESSED PLATES: Denoting the bending. of. the plate during instability by,. w(x,y) and the displacements of points in the middle surface projected in the x,y directions by u(x,y), .v(x,y), respectively, we have expressions for the changes in curvature X, X, T, and the strains E1, E2, E3: 2 2w 2 6Y2 T7  ox oy 3 2 y (2.1) 1 x c ^. E1 ox The forces applied in themiddle surface may be written in the following form: Tli = haiXx* T2 = YciYy before instability S =hcr.X Y i J and their projection on the Zaxis after instability in the form T1X1 + T2X2 + 2ST = haiX (1.24) aVI\ ~ yl NACA TM No. 1188 Therefore, the condition of equilibrium of all forces applied to an element and projected on the zazis, gives 295M1 26H 2 + 2 + y26M 2 + haix', = 0 5y2 (2.2) The condition of equilibrium of the middle surface forces after instability will be + =0 5x by 2 + S  +  = 0 oy ox (2.3) Finally, the compatibility condition for the strains has the form E1 2 y2 x2 oy2 oix2 0 (2.4) The combination of differential equations (2.2), (2.3), and (2.4) is necessary and sufficient for the solution of the problem of stability, if the corresponding boundary conditions are set up. Indeed, according to (1.14), or to (1.21) and (1.20), the strains .l, 2', 3 may be expressed in terms of the forces F'T1, 58T, 8S and the curvatures (bending w), following which the moments 5M1, BM2, BH 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), (2.4) are of the type of equations in plane problems. In the region of purely plastic strain of the plate, (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 5M1, 5M2, SH from (1.15) into (2.2) gives a differential equation for w of the Bryan type: 4 hcri V X= D S'26 XX 1% 4x x 62 + 2X * x cy .2 + Y .;i XX NY (2.5) NACA M' No. 1188 where, in agreement with (l.9) and (2.l), x2 " ~~ 2X+ y ^'^r ' Y *_ * ^ (2.6) The two boundary conditions on w agree with the usual. boundaryconditions for the Bryan equation. '. , Solving equations (1.14) for the strains, w? get  (.1 T \ Eh EhE (2 S2) S1 ) j + x + Y (1 X)Eh SY *T + 3xy~ S *5T + S 2T + 3 Y*S \~ "  2 = 3S 3 Eh l S 6T + S *T + 31 S (1 x)Eh x 1 y 2 / Equations (2.3) are satisfied'if the stress function: ST1' 3 8T' .S .. 2Eh y2 Eh E x2 Eh F,. ,h y2 Eh 82 Eh .Ex; F' is introduced: (2.8) following which, analogous to (2.6): we denote t = sx Sx *2 Gy S 7 t2 (2.9) 3   Y a **o .r we obtain the compatibility condition for :strain in ih 4i /*rm S 3 S 3 * y ~ I > (a.7) 42 VhF x. Xt (2.10) (Sx : "'; ' TIACA TM No. 1188 In order to write the boundary conditions for this equation it is necessary to compute the variations of the normal force jTv,, and of the tangential force 5SV on a certain curvilineer contour in the middle surface of the plate. If the outward normal V and the tangent a to the contour constitute a coordinate system such that by rotation the positive direction of V coincides with that of y and the positive direction of a coincides with that of x and if the angle between the normal and the xaxis is denoted by a (fig. 4), then our quantities have the known expressions 5T + ST ST, 5T 2 2 STV = 1 + + STcos 2a + 8S sin 2a > (2.11) ST1 ST 2S, = sin 2a 5S cos 2a The purely plastic region of the plate may be bounded by a contour, part of which coincides with the boundary of the plate, the part adjoins the elastoplastic region. For the formulation of the stability problem in the first part, the boundary conditions have the form s5T = SS = 0 (2.12) and in the second nart STv, FS must be continuous. It is easy to show that during instability the entire "late may not remain in the pure'y plastic state; that is, an elasto pl.Estic region may come into belng. Indeed. going back we shall have the uniform boundary conditions (2.12) on all external edges of the plate. But the differential equations (2.3) and (2.4) for conditions (2.7) will be also linear and homogeneous and so will have the unique solution 5T1 = ST2 = S = 0 It follows from (2.7) that E = e2 = 3 = 0, from which on the basis of (1.9) and (1.10), z, = 0. But z = zo is the NACA TM No, 1188 boundary between the elastic and plastic zones through the thickness of the plate and the condition zo = 0 specifies that the middle surface is thisboundary. It follows that a given region of a plate is not purely plastic, but elastoplastic, which contradicts the assumption. During instability of a plate beyond the elastic limit it will either go completely over to the elastoplastic state or there will remain purely plastic regions in it, which are not diffused through out the plate. In the region of elastoplastic strains, equation (2.2) on the basis of expressions (1.24) may be presented in the form ha.i 2 2 x = x* + 2 2(3 2)x (2.13) D 4 x2 2x y + 2 7y in which, as in equations (2.5), (2.10), the operator in parenthesis acts like a multiplier on the quantity to its right. The condition of compatibility of strain (2.4) on the basis of (1.23) has the form VF = S S 3x (2.14) 2^o y y 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 (2.13) and (2.14), therefore they show compatibility; this quantity t is expressed by formula (1.21) in which the function p is, if use is made of the notation (2.9) S 2 t (2.15) h 1 \ X hlXX Equations (2.13), (2.14) 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 5Ty, 8Sy, the moments BMy, 5HV (where 8HR V is the rotational moment according to the boundary NIACA TM No. 1188 conditions of Kirchoff), the bending w and the slope of the tangent plane, there must also hold the condition hp= h 5= (2.16) From (1.21) for this condition we have Q = X and following which the remarked coincidence of the equations shown. t = Xh, 2 to easily The boundary conditions for equations (2.13), (2.14) on the elastoplastic pnrt of the contour, coinciding with the plate contour, yield the usual requirement STV = 8Sv = 0 and two conditions relating to the bending w. Condition (2.15) or t = 1 1 >h O (2.17) represents in itself the equation of the boundary between the purely plastic and the elastoplastic regions. The possibility of purely plastic regions arising at the same withthe elastoplastic regions follows from the fact that the value of t in agreement with (1.21) and (2.15) ma.y take on values not lying in the interval 1 C ;O. Certain exoi.wiles 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 sta.te. 3. EXAMPLES OF EXACT SOLUTIONS OF PROPLE14S IN THE STABILITY OF PLATES The integration of the system of differential equations (2.13) and (2.14) in the elastoplastic region, and of (2.5) and (2.10) in the plastic region with an undetermined boundary between them given INCA TM No. 1188 by (2.16), is fraught with significant mathematical difficulties. As was shown in 1, the stability problem simplifile when the variations of the forces in the middle surface are zero every where. In that case the relative thickness of the plastic layer is a known function of the coordinates, since from (1.22) p = 0 and consequently = . (3.1) If the state of stress of the plate before instability is uniform, the value of t will be constant, since in (1.22) dai will be the same for the whole plate. We call those solutions of stability problems anproximate, for which the variations 5T1, 1T2, 5S of the forces are identically zero. Thus, the equations (2.3) of equilibrium and the boundary conditions (2.12) are satisfied, but, except in special cases, the compatibility condition (2.4) is not satisfied. The simplicity of such a solution arises from the fact that in equation (2.13) the value of t 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 anisotropic plate. The exact solution of the system (2.13), (2.14) are undoubtedly of interest in their own 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 i ratess of arbitrary shape and the solution for a rectangular plate in the case when buckling into a cylindrical shape is possible. a. Stability of a Uniformly Compressed Plate of Arbitrary Shape (Fig. 4) In this case the state of stress of the plate before instability is uniform and given by the formulae Xx = Yy = ai, X = 0 (3.2) NACA TM No. 1188 where Ci is the compressive stress Falong the edge and is also the uniform stress intensity at any point in the plate. The resulting stresses according to (1.7) and (1.4) will be x* = Y =1 X = 0 Sx* = x Y 2 For the values of X and t we have the expressions from (2.6) and (2.9) ,.:= v t = 1 (3.4) 2 Equation (2.14) takes the form V t, = o (3.5) Neglecting the hnracnic function, we obtain a class of exact solutions t  as a result of which the value of .'p in (2.15) is expressed in terms of and from (1.2) we find =  = 1 = const. (3.7) The fundamental differential equation of stability (2.13) is now linear with constant coefficients and hns the simple form 1 + 2(3 2j vw + V2w = 0 (?.8) I D Its solution has been much studied for different shapes of plates and for different boundary conditions, although in connection with the elastic stability of comLpressed plates. NACA TM No. 1188 The value of t (3.7) is little different from the approximation (3.1), and characterizes the degree of deviation of the exact solution from.the approximate, In the general case we have from (3.5) t X8 x + (3.9) where ri is an arbitrary harmonic function. For continuous circular plates, for example, P is a constant. According to (2.15) and (1.21) we now have an expression for t in terms of X = 1 3\ + (3.10) 3% \ / 16X following which equation (2.13), having in the given case the form  (3 2 X + X = 0 (3.11) 4 D has only one unknown function X. By use of relations (3.4) it may be integrated once i \ 2 ha2 1 r(3 2) V +  = r (3.12) D 2 where 2 is a new harmonic function, also a constant for continuous circular plates, insofar as w and V2w must be finite in the middle. Equation (3.12), in view of (3.10), may be solved for V2w, after which the problem reduces to the integration of only one linear partial differential equation of the second order (for circular plates) V w = ,(w,P1rr2) The stress function F is now determined, in accordance with (3.9) and (3.4), from the Poisson differential equation NACA TM No. 1188 2 h = (2x+ rl) (3.13) As we see, the problem of the stability of circular plates may be solved in comparatively simple fashion through to the end. The details of a similar calculation will be clarified below for the example of a rectangular 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 o' length 7 is compressed in the xdirection, the width b in the yirection cannot change as a result of walls along the boundaries : = 0 and y = b. The plane x = 0 shown in figure 5, where C = 2c and L = .2; will evidently be a plane of esymetry of strains. We assume the buckling to result in a cylindri1caL. shape. In such a case,according to the conditions of the problem: we have for the stresses before instability (3.14) 9 i '3 x = Y =! 1 3S =0 /3 3 2 Y After buckling, w = w(x), E = e = 0. From equations (1.24) we have S3 = 0 1T = 8 oT Since, in accordance with the equations of equilibrium )T1 = const., and ST1 = 0 from the condition at the edge x = ?, then we have the case 5T1 = .To = 5S = 0. In consequence, the apnroximate solution, as was noted at the beginning of 3 here becomes exact. 20 '".",. TM No. 1188 The thickness ratio 5 for the plastic layer is a constant and is determined by formula (3.1). The stability equation (2.13) takes the form d4" hp d= 0 (3.15) ax4 D .x (3 2t) dax2 If the relative Karm`an modulus, expressed by dai K = de l )L (3.16) dai (1 + 1X)2 \. dei is introduced, then we get from (3.1) S, i.(l. k) (3.17) (2 ,/k)2 2 : following which we may simplify the expression for the parameter in equation (3.15) 2 hp hp (3.18) D [ 2(3 2)J Dk Since k = 1 up to the elastic limit, and k = 0 in a small area where there is flow of the material, and since the character istic value of the parameter 7 must be the same in elastic and in plastic problems, then it follows from (3.18) that the critical stress corresponding to the small area of flow, is zero, It is interesting to note that the Karman 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 4hp 3Dk NACA TM No. 1188 and consequently the critical stress is zero at the small area of flow. As seen from the preceding end following examples of exact solutions, the total loss of loadcarrying ability of a plate, predicted in the Karman problem, does not occur, generally speaking. This circumstance has already been noted (reference 1). c. The Stability of a Rectangular Plate Compressed In One Direction (Fig. 5) We shall suppose that the rectangular plate, sufficiently long in the ydirection and compressed only in the cdirection, buckles into a cylindrical shape. In this case ci 7 = r' Y = Xj = 0 S: ( .19) = 1 Sy X* = 1 YV Z* X y 0 Y 0 By the conditions of the problem, all section o the plJte y = const. remain pl.,ne after buckling and so we hve C = 0 =o = const. (3.20) on the basis of which from (1.24), SS = 0. Besides this, ST = 0 from the boundary condition at the edges x = + and consequently it follows from (2.3) that sT1 = 0 everywhere.2 Since there are no forces in the ydirection, we must use the condition /2 T dx = 0 (3.21) L 2 From the second equation of the system (2.14) wo have d2F T2 h 2 (.22) dx2 Eh 2 4 1 NACA TM No. 1188 since X =.Xl. It :isa not difficult to convince one s self that (3.22) is the integral of equation (2.14). The function c, by which is found the value of from (1.21), here has the form p8T2  2 + X2 2 (3.23) (1 Eh(1 X)h 4( x) The bending moment in any section is = D 2(3 2) X. (3.24) and so the boundary condition on the edges x = + I is X, = 0. 2 It is clear from (3.23) that X1 cannot be zero in the elasto plastic region since e2 / 0 (this follows from the constancy of 2 sign of 2 X1, positive along the entire plate, necessitating C2 0 to satisfy condition (3.22)). Thus the elastoplastic region does not go up to the edges of the plate and stops at the section x = + c. The region adjoining this to the edge will be 2 purely plastic. Indeed, since t2X is positive, then e2 is also positive. It follows from (2.7) that in the purely plastic and in the purely elastic regions the force 5T2 has the same sign as E2, that is, is a tensile force. But if, to the plate, compressed beyond the elastic limit in the xdirection, there is applied a tensile force in the ydirection, then the plate remains in the plastic state. One may convince one's self of this by formally calculating the value of 5e according to (1.8), which at the edges is equal to el, but the strain E, according to (2.7) is negative, and so the value of bei will be positive, that is, plastic strains before buckling remain plastic after buckling. From (1.21) and (3.23) we now have hX 1 1 = p(Q) = 4 + 8 3%t2 (3.25) 42 p(n) NACA TM No. 1188 23 From this we find the lower limit to the value of t ( > 0) S> \(3.26) The fundamental differential equation of stability (2.13) takes the form d2 3 ha, dT 1K 0 2 +f 1 = 0 (3.27) d:t _ By introduction of the notation Q() = 4 9X + 6~)3 = = (3.28) 2 2 we write equation (3.27) in the form + = 0 (3.29) d42 p P where n is the basic parameter determining the critical stress 2 i = (3.30) The integral of equation (3.29) may be obtained by quadratures. Through introduction of the notation PE() = l + 12 2 3 (3.31) (4 8i 3Xse)s we obtain as a result j T (a)2 = i2C r. ,/2 = c J , p. (3.32) NACA TM No. 1188 In the purely plasticregion we. have for the force' 8T2 and the moment 5M1, in agreement with the results of 2 and (3.19): XT ) Eh 4 3x '8M =  (3.33) The fundamental differential equation takes the form S2 4   + X = aS2 4 31 1 (3.34) The solution, satisfying the condition is written in the form, x 1 = 0 at the end 1 = C3E2 sin  <,A 3X in which as a result of symmetry we consider only deflections in the right half of the plate (x 0). For determination of the five undetermined constants namely, the three integration constants C1, C2; C3, the boundary coordinate a and the critical number i. we may, besides equation (3.21), write four more conditions: Conditions of symmetry (3.36) conditions at the boundary region S=a.* 5 =l (3.37) two continuity conditions, of moment and shear force, which in accordance with (3.25) and (3.35) take the form C3 sin = cos 3 45 3. h(4 3%) \/4 37 \/4 3x hP (1) hP2 (1) ( M (3.38) S= 1, (3.35) S= 0  = 0 7s TACA TM ITo. 1188 The constant e2 is not necessary and does not enter the conditions insofar as they are independent of and . By making use of the prescribed conditions and introducinG a new unimown t, the relative thickness of the plastic layer at x = 0, we get for the values of p and 1 a (the relative length of the purely plastic part) the following formule XM 4 %, L S= I a (3.39) .,.2(1 x) where L and M are the integrals I,'R() 1 + ,2 o) 1 .)2 &= ,R(I) ,/R( o) (() ,(1) ./(., ) ( P,) in which the value of r is determined br the rel.?t.on O cot2 2 4 L3 2(4 3 r) r() (1) (3.l1) /2(1 0) . As was already established., the value of 1 is positive, therefore the integral L must be positive: and ',or this it is required that 1 2 + X20 > 0, that is, (< 0(3.42) By considering the estimate (3.26), which is also reasonable for Io, we see that this quantity is contained within nrrrrw limits and close to the approximate value (3.1). It follows frcm this that the critical stress will differ only slightly from the approximate value. NACATM' No. 1188 d. Approximate Solutionof the Problem for a'Plate in a Uniform State of Stress Before Buckling In this case the stress components Xx Y and X and the stress intensity ai are constant everywhere; the quantity X will also be constant, and hence t by (3.1). 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 (2.13) takes the form stability equation (2.13) takes the form 3  k) XI*2 + 2 1F L hai  + ( k)Y *2 3(l k)XYy* 1 x cy ox287 Hx* 4 a2wi y+ y* v y .2 in which the generalized Karman.modulus is introduced in accordance with formulae (3.16) and (3.17), since the relation W2(3 2t) = 1 k holds. The coefficients in equation (3,43) are all positive, since the largest value of each of the quantities Xx*, Y is 2  and 1 > k > 0. \/3 S Hence, the problem may be solved as a linear differential equation of the Bryan type with constant coefficients, and in difficulty is little different from the corresponding elastic case. Translated by E. Z. Stowell National Advisory Committee for Aeronautics (3.43) 71 NACA TM No. 1188 27 PRFEJRNCES 1 Ilyushin, A. A : Ustoichivost Plast inik i Obolochek za Prerelom Uprugosti. Prikladnaya Mantra,?tika 1 Mlekhanika, H. S. 8, No. 5, 1944, pp. 337360. (Also available as ITACA TIM Ho. 1116.) 2. Ilyushin, A. A.: K Teoria Malikh Uprugoplasticheskikh Deformatsii. Priklacdnaya Mateniatika i IMekhanika, X, 1946, p. 347. NACA TM No. 1188 Figure 1. I I S I I Figure 2. pl ast c Figure 3. 30 NACA TM No. 1188 V 4_ 1=0 =0 Figure 4. Figure 5. OJ (11 0 0 pI o3 ol 0 4 2 lo d r r  10 K Pr e ii' * 4 (0 *0i 0 5tt III'i 0 H S4" 0m Cd 0 q0 (Dr 6H O S n E 4 <*  2 T m lq g H C2 H e1 4,? 40' H ^ K( ' p4^riu' .0 a 4 (W 0 i 0 P P. O u 4 d g P +P + co U4S1 4r l W l 00 0aP  P4 43 4) M 4 x l d 4 p R H 'd *d =P 4 PA 0C 0 0 4 ' ,o 0 H 0 A O mPr *.c j 0 >!arI $4 r E1 14 o 0e Cdo )s W4 Mo a 0 $4em e 4 4 o 0 i a bi4 o 1 a w pr d *r 0 *E = 0C In pD, o 4e ) Ww 40 o D 0 1Q 4 b 4 W4) 0 0 ( 0b P+ o P *d E H r 4 6 i 4 0+0 ) a r O 1 0 0B o 91 u H f a ,+ $4 . +A'd H 4 ) 4 e B  op a o o S 0 4 4)+ Sl0 Sma 0 P 10 r4 4 a > In r1 r 1D 4 0e r4 o w3 p. rt frl W q Q d 0P 04;P Cd p m *0 r3 044 P. e A, 0 4,2 4 .0 r 0 Sri S4 )O d P4, +o UNIVERSITY OF FLORIDA 3 1262 08106 659 8 
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