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0V j: 4 : ,2 NATIONAL ADVISORY CO1MMIT'IEE FCR AERONAUTICS TECHNICAL MEMORANtDUM 110. 1138 ON THE APPLICATION OF THE ENERGY METEOD TO STABILITY PROFLEB 1 By Karl Marguerre Since stability problems have come into the field of vision of engi neers, energy methods have proved to be one of the most powerful aids in mastering them. For finding the especially interesting critical leads special procedures have evolved that depart somewhat from those customary in the usual elasticity theory. A clarification of the connections seemed desirable, especially with regard to the postcritical region, for tho treatment of which these special methods are not suited as they are. The present investigation2 discusses this question complex (made important by shell construction in aircraft) especially in the classical example of the Euler strut, because in this case  since the basic fea tures are not hidden by difficulties of a mathematical nature the prob lem is especially clear. The present treatment differs from that appearing in the Z.f.a.M.M. (1938) under the title "Uber die Behandlung von Stabilitatsproblemen mit Hilfe der energetischen Methode" in that, in order to work out the basic ideas still more clearly, it dispenses with the investigation of behavior at "large" deflections and of the elastic foundation; in its place the present version gives an elaboration of the 6th section and (in its 7th and 8th secs.) a new example that shows the applicability of the general criterion to a stability problem that differs from that of Euler in many respects. "Uber die Anwendung der energetischen Methoie auf Stabilitats .probleme." Jahrb. 1938 D'L, pp. 252262. 2 In the paper investigations were continued at the instigation of Professor Trefftz (during his activity at the Deutschen Versuchsanstalt fur Luftfahrt). For a large part of the work (especially in secs. 4 and 6) the author is very grateful to his colleague, R. Kappus, for his close collaboration. 3 See the next to last paragraph ofthe Introduction. NACA TM No. 1138 SUMMARY In the two examples of the Euler strut and the slightly curved beam under transverse load it was shown that the difference between the sta bility problems and the problems of linearized elasticity theory rests upon the fact that in the stability problems the expression for the energy of deformation contains terms of higher than the second order in the dis placements. This idea makes it possible to establish the connection be tween the energy method in the special form most used. for stability in vestigations and the principle of virtual displacements in its general elasticity theoretical version; besides, it permits the investigation of elastic behavior beyond the critical deflection. INTRODUCTION Kirchoff's uniqueness law states: An elastic body can assume one and. only one equilibrium configuration under a given (sufficiently small) external loading. In the formulation from the energy point of view: the potential II of the inner and outer forces has one and only one extremal II = 0 (1.1) and the extremal is a minimum.1 The 'niauerese law holds without restrictions in the realm of line arized theory of elasticity, that is, as long as the stresses a, T can be expressed linearly in terms of the strains 7ik and the strains line arly in terms of the displacements u, v, w. Then the function II is of, at most, the second degree in the displacements, and geometrical considera tions Ehow directly that a "parabola" of the second degree (positive def inite quadratic form) can have one ani only one minimum (or in mechanical terms, equilibrium position). The situation changes, however, when struc tures are considered tho behavior of which can no longer be expressed with lFor the derivation of the principle of virtual displacements (equa tion (1.1)) for elastic equilibrium, see, for example, reference 1, pp. 70 ff. A careful foundation of the general theory of the behavior at the stability limit appears in reference 2, p. 160. (For further literature, see reference 1, pp. 277 ff., or reference 2). The investigation in section Stable and Unstable Equilibrium in particular make use of the Trefftz point of view. NACA TM No. 1138 sufficient accuracy by the linearied straindisplacement equations. Such are, particular, bodies for which one dimension is small compared to the others, structures in the form of shells, plates, or bars. For example, a rod can, without exceeding the proportional limit, undergo bending de flections several times greater than its thickness, and under these cir cumstances the quadratic part of the (transverse) displacements in the strain displacement equations is no longer small compared to the linear part. Then the energy of deformation of the potential II becomes of higher than the second degree in the displacements, and a parabola of higher order can naturally have several extremals (equilibrium positions). The problem of the theory of stability is usually considered to be the determination of that external load under which several neighboring equilibrium configurations are possible. The reason for limiting in vestigation to this "critical point" lies in the fact that the differential equations describing the elastic behavior in the postcritical region are, in general, no longer linear and an analytical treatment would therefore be difficult; while at the critical point itself the problem can still be "linearized".1 This purely practical viewpoint has, however, led to a certain (as is shown, from unfounded standpoint) systematic separation of the sta bility problem from the other problems cf the theory of elasticity, which finds its mathematical expression in a formulation of the principle of virtual displacements somewhat different from the usual one has also for convenience led to the formulation of a special principle. (For ex ample; see reference 3.) The principle of virtual displacements states that during a virtual (that is, geometrically possible) displacement from an equilibrium posi tion, the energy of deformation taken up by the elastic body is equal to the work done by the external forces. For the use of this principle in the theory of elasticity it is convenient to express this fact in the following way: An equilibrium state is distinguished by the fact that for every virtual displacement from that state the potential of the inner and outer forces II = A, + V Knowledge of the postcritical region was until now of secondary practical interest, because buckled structural elements were considered unpermissible. It has been only in recent years that in the shell con struction of aircraft critical loads have been permitted to be exceeded by large amounts unhesitatingly. NACA TM No. 1138 has a stationary value: l= 8(Ai + V) =Q0 (1.2) Therein the potential Ai of the inner forces is given by the energy of deformation (inner work), and the potential V of the external forces by the negative product of the external forces considered constant and the displacements of their points of application. In the region of appli .cability of the proportionality law numerically V = 2Aa, where Aa is the work done by the external forces as they increase from zero to their final values in passing through only equilibrium states. The prin ciple (1.2) can therefore be written conveniently also in the form. (For example, see reference 3.) 8(Ai Aa) = 0 (1.3) As against this there is often used as a "minimal principle" in stability theory the condition 6(Ai Aa) = 0 (1.4) The author intends to show in the present paper (in the classical example of the strut) that also stability investigations are best handled in connection with the single main equation (1.2), wherein terms must be retained of higher order in the deformationslogically only in the ex pression for the energy of deformation. This procedure is essential from the practical standpoint, if the relationships are tc be investigated. in the postoritical region, and desirable from the systematic standpoint, because it becomes clear in this manner that no additional principles are required. In particular, this consideration will clear up the apparent contradiction between equations (1.3) and (1.4). The calculation itself is carried out in the following manner: First, the expression for the energy of deformation Ai is set up, then the differential equations for the two components of displacement are derived from the condition 1n = 0 NACA TM No. 1138 5 and the question of bhe stability of the equilibrium position is answered by the restricted condition II = minimum (1.5) Then the same problem is treated with the help of a Ritz procedure; in this way the result of the stability consideration is brought out in an especially elegant manner. After a thorough discussion of the usual sta bility theory, it is shDwn in conclusion how the same considerations can serve for the treatment of the snapaction problem of a slightly curved beam. ENERGY OF DEFORMATION If the customary assumptions of the beam theory are retained that for small deflections of the beam the wcrk of stretching and work of bending are independent of each other and that the part of the work re sulting from the shear forces is mall compared with the other two parts  then the energy of deformation can easily be given. As a result of the assumption of small displacements without at first saying anything about the sizes of the displacements u and v (fig. 1) relative to each other the strains u_ and wv(or wx2) may be neglected in comparison with unity; that is, in a development of both quantities in powers of ux and wx only the lowest power need be retained. If, there fore, the square of a line element of the beam centerline before deforma tion is dx2 and after deformation S/ ",2 2] 2 S+ uY, + vx idx (2.1) (the subscripts on u an.i w indicate derivatives with respect to x), then the strain of the beam centerline (reference 1, p. 57) is S=/ + ux 1 (2.2) NACA TM No. 1138 From Hookers law the corresponding stress is = E ux 2 5/ G^E(^+2 ) (2.3) and therefore the energy of stretching is A =  2 o + 2 + _ J dx 2  The incremental strain ex due to bending is, according to the assumption that normal are preserved, : x = ZWxx (2.5) the bending energy is therefore given by S(w)2J . E A dx dy dz a vEf wdx (2.6) If El = u(0) u(Z) = u(I) is the distance of approach of the ends, P = pF = xF the compressive force, then (pF)(El) = V is the po tential of the external forces; the total potential IT (measured from the stressfree state as the zero position just as the displacements u, w, e !) is therefore II =  2 0u + W2 dx +  .2 ) 2 I w2 dx plt I 0 (2.7) The potential per unit length after division by EF can be written SFn i2 SEF3 21 S2 2 2 us + )dx + f / X2 aX R e 2 / 2Z xx E o (2.8) (2.4) NACA.TM 9p.1.138 S.From equation (2.8) and. the condition (i5)' II = minimum (2.9) there is obtained all information about the behavior of the strut at and beyond the stability limit. THE DIFFERENTIAL EQUATIONS FOR THE DISPLACEMENTS u, w Consider a rod the left end of which is (x = 0) is supported and the right end gives (x.= 1) is freely movable in a horizontal direc tion under a centrally placed compressive force P. (See fig. 1.) .As given (that is, as the independent variable of the problem) take either the horizontal displacement  or the compressive " of the righthand end stress I *u(M) = c1  According to the principle of virtual displacements, the displacements are to be varied under a constant load in a manner compatible with the geo metrical conditions. If at a boundary point the displacement (in the pres ent problem, for example, e) is prescribed, the point is held fixed during the variation, so that the work of the (unknown!) external forces .doee not remain in the calculation; if on the other hand, the force is given, then the (not fixed geometrically) end point is varied, and the work of the external load (in the present problem P~u) enters into the calculation. If in equation (2.8) the displacement u is varied (that is, if the elements of the rod are.given a virtual displacement in the axial direc tion, while the boundary point or correspondingly the load is held fixed) there follows from the rules of the calculus of variations / 2 o together with the boundary conditions (3.1) u(0) = 0 u(0) = 0 I wx u(M) = z I ux +  + 0 7 8 NACA !'A96. 1138 by varying v there is obtained' imindependentlyf whether. or p is considered as given): Ux + ux + T wxx+ 0 with the boundary conditions (3.2) w(O) vw() = xx(O) = wxx() = 0 'The exact integration of the system (3.'1), '(3.2) of nonlinear (!) a imultaneous'equations offers no difficulty here. " From (3.1) there follows' Ux + constant e fo PX 2 (3.3) u = u(s) oX x and with the use of (3.3), (3.2) becomes x + W + XX 0 (3.4) The solution of this linear equation with constant coefficients is w = f sinkx +.g cos kx + fix + g where k is the positive root of the quadratic equation 2 0 i 2 For the determination of the six constants of integration f, g,j fl, gi, u(0), eo there are the six boundary conditions (3.1) and (3.2): NACA TM No. 1138 g + g6 = 0 kg = 0 f sin k1 + g cos k1 + f1l + g, = 0 fk2 sin ki + gk2 cos kZ = 0 u(o) = 0 2  El = u(0) co Lx dx o 2 p Co + = 0 E It is found that fI = gS = g = u(0) = 0 and either f = 0, that is, w = 0 u = C u = EX p 0 E U/ X E f 0, w = f sin kx, k =  E2 2 o ~12  Ir f 27rx u = CX + g sin  There are evidently two kinds of equilibrium positions: the straight (f = 0) and the bent (f 0). The straight position is specified uniquely by either p or e, the bent by for the amplitude f of the deflec tion can be determined uniquely from the left one of equations (3.5) Of 2 ,= C E = (3.8) 4il2 0 (3.5) (3.6) (3.7) 10 NAGA TM No. 1138 not, hovweer, by p. From the right one of equations (3.5) there follows rather that for f / 0 'a completely determined "critical" value p = Ee p* (3.9) cannot be exceeded. Therefore p is unsuited for an independent parameter (the situation is different in the case of the corresponding plate problem) (reference 4, p. 124). From equation (3.8) it can be seen that f assumes real values only for e'> e*. In figures 2 and 3 the quantities f and .p are.plotted against e. The solid lines are for the solution (3.7), the dotted lines for equation (3.6). The result (3.9) that the load for the buckled strutdoes not in crease beyond p* even when the .critical end shortening has been con siderably exceeded is a result of the limitation to "small" deflections. For the present problem this limitation is not important because here it was only a question of seeing that as a result of the appearance of higher powers of w in equation (2.8) the elastic rod can assume several equi librium positions especially that the existence of a real multiplicity is bound up with the exceeding of a certain "critical" strain e = e*. STABLE AND UNSTABLE EQUILIBRIUM. In the "crd:nary" theory of elasticity it is necessary to consider only the condition IT = 0, that is, _TI = extremum (4.0) 'Reference 5, pp. 70 ff. Also the theory of the socalled exact differential equation of strut buckling EJ/p + Pw = 0 (reference 1, p. 280) shows that at large deflections (because of the in creased demand on bending energy) there is a very small increase in load. Froa the energy tandpoint, to be sure, this "exact" equation is not im portant, for if wj2 is taken as not small compared to 1 in the bond ing term (i, at is, the curvature l/o is used in place of wvx) it is. necessary to proceed in a corresponding manner with the stretching term (equation(2.2)) unless it is assumed that an incompressible strut exists from the start, as is done in the theory of the Euler elastica. NACA TM No. 1138 11 in the determination of equilibrium states, for the supplementary con dition 52II > 0 (mechanically: the stability of the equilibrium posi tion) is assured there because of the linearization (reference 1, pp. 71 72); in the present problem the minimal condition must be set up explicitly, since only through o1I = 0, 52II > 0, that is, II = minimum. (4.1) can the stable equilibrium positions be distinguished from other possi ble (the unstable) positions. The concept of stability is made precise here by the following convention.1 1. An equilibrium state is called stable if for every neighboring state the potential energy has a larger value." 2. An equilibrium state is called labile (unstable) if there is at least one neighboring state for which the potential energy is smaller. 3. A stability limit (that is, a neutral equilibrium state) is spoken of when there is at least one neighboring equilibrium state the potential energy of which is equal to but none having potential energy less than that of the given equilibrium state. Return to equations (2.8) or.(2.9): 2 1 21 p I= i + 1 dx +  / dx c = Iir.iLm. 21 \ / 2 E and perform e. variation; that is, replace u by u + Bu and w by w + 5w; there results, after arranging in powers of Su, 6w and stopping after terms of the second order, The following definition was given, in substance by E. Trefftz in cidentally to his DVL lecture. &ACA TM No. 1138 A A 1 2^ Ali = II (u + Su, w + w) (u,w) i + II + . 0 2z+8xd + ux. oUK 0(3ux 2\ ;!) vx wx + i2 b wxx_ ]dx dx + w 8u 6 w w dx 1z vt 1' a + 0 + ( )2 + d.; J )2 The condition that the terms of first order shall vanish leads to equa tions (3.1) and (3.2); the question of stability is answered by the terms of second order. By inserting for *u, w..the values obtained from mII = 0 there results for (3.6) (straight position) V. 21d +u d (4.21) 0 for (3.7) (bent position) + f cos. &wj *}S2 ,/ X + 2()2 2 1 2 +  cos 8vx 8ux dx + 5u dx 3 o I o (4.22) First the stability limit will be determined. According to the defi nition given above there must be at the stability limit a state for which the second variation vanishes but none for which it becomes negative: The value zero is therefore the smallest value that 8FII can assume at the limiting pointt' If, therefore, S2IT has certain continuity prop 6rties (the existence of which is obvious on mechanical grounds), then (52k) (52 1A ) o (82 ) \ )2 ( *2 C 2 dx i, 5u "^E [(U + w!__2X 7 2 NACA TM No. 1138 the "characteristic" value 2fI = 0 is at the same time an analytical minimum, compared with neighboring values, and the associated ("charac teristic") displacement system Su, Sw is determined from the condition /( ITI= (4.3) \ / The straight position (see equation (4.21)) thus reaches the sta tility limit when bu, Sw satisfy the two differential equations (6u) = 0 (SW) + (CV)x = 0 with the end conditions Bw(0) = Bw(1( = 8w(O) ) = wf(2) = Su(O) = 0 and (4.5) 5u(l) = 0 Bux(l) = 0 The solution of this eigenvalue problem reads in both cases Su = O, 5w = 5f sin (4.6) The amplitude &f p 0 remains undetermined and.from the second of equa tions (4.4) there is obtained for the critical value of C i2i2 crit =  that is exactly the expression crit = E* by which the branch point of the equilibrium was characterized; stability limit and branch point coin cide. The investigation proves to be somewhat more difficult for the bent position. The two minimal conditions read NACA I No. 1138  x u+ o 8 = 0 ('4.7) 25 3 + C( os 1 cos .6u 0 for the boundary condition there is retained 5w(O) = 5w(Z) = 8wxx(O) = 8wxx() = Bu(0) = 0 and, depending on whether or not the righthand boundary point is pre scribed or not, Bu(l) = 0 8U + cos 6) 0 (4.7) It is recognized immediately that for f = 0, hence at the beginning of buckling, the bar is in neutral equilibrium, for all conditions are satis fied by the solution (4.6). This result is trivial. It is not so directly obvious, however, that also for f 0. there is a variation that makes S271 a minimum; the homogeneous system of equations with homogeneous boundary conditions permits of a nonvanishing solution also for f f 0. In fact if the variation is soperformed that the second boundary con dition of (4.7') is satisfied there follows from the first of equations (4.7). bux = cos 8w (4.8) If this is put in the second of equations (4.7) the latter reduces to i .W + C 12xx= (+ z 2 x = 0 and this equation (together with its boundary conditions) is satisfied 1y 89 = 5f sin . For Bu there is obtained from (4.8) NACA TM No. 1138 Iftff i 2Ixx =u x + 1 sin 212 21 I in which 5f again represents the (not determinable by a system of homoge neous equations) arbitrary factor. The question of the stability of the equilibrium positions below and above the limit can now be answered. 1. Since the straight position (see equation (4.21)) is stable for very small e, from considerations of continuity it is so for E < e*. 2. For e > 6* the straight position represents an unstable equi librium state, for a variation 5w, namely, bw = 6f sin  can be given for which 52 1 <0. 3. For e > c* the bent rod is against the variation bw = 5f sin (. 1 (4.9) x f^f ( I 2nx> Bu = x + sin T 21s 21 T in neutral equilibrium.1 Since the variation (4.9) (and only this) makes 2e11 a minimum (of value zero) every other variation gives it a positive value. If in some way the special variation (4.9) is prevented, then the bent equilibrium position is stable. This holds especially in the impor tant case where not the force but the displacement of the end point is prescribed; for the displacement system (.4.9) is in fact excluded by the boundary condition Bu(I) = 0. At this point a result discussed later in section Connection between the Ordinary Investigation of Stability and the Procedure Presented Here, should be emphasized. The behavior of the rod beyond the stability limit is different depending upon whether the load or the displacement is 1This result is naturally connected with the assumption that the strut adheres strictly to the law of deformation established by the ex pression (2.7). NACA TM No. 1138 considered as the prescribed quantity. More noteworthily, however, the behavior at the stability limit is not affected by this difference. For, although there are the two different boundary conditions (4.5) bu(I) = 0 and 5ux(l) = 0 they both (together with the differential equation buxx = 0) lead to the same result Bu = 0 That is, it makes no difference whether a motion of the end points in the x direction is "permitted" or not: during the buckling they do not move. Thus the result is arrived at that the two mechanically entirely different problems: buckling under constant load and buckling under constant end shortening, lead to exactly the same critical state bu, 5w, ecrit, Parit. IITERPRFTATION OF THE RESULTS WITH AID OF THE RITZ METHOD The results of sections The Differential Equations for the Displace ments u, w and Stable and Unstable Equilibrium can be illustrated very elegantly if the variation problem is turned into an ordinary minimum problem by the use of the Ritz method. In the case of the Ritz method to be sure nothing certain can be said about the question of stability, since from the start only quite definite displacements ar8 corn.Eiered and thlerefor r.n. gnel c:n b. c:oncli. i .:'t the sln cf the second varia tion; nevertheless, with a Judiciously chosen deflection system the question can be answered with great probability or the answer made very plausible. In the present, especially simple case the earlier results are found again exactly. As a set of displacements satisfying all boundary conditions are chosen,the solutions of the differential equations (3.2) and (3.1) 'This is a peculiarity of the problem. In general, the critical load depends upon the boundary condjtiorn whether are prescribed forces or displacements, the system is supported, or guided, or built in, etc.; for the "minimum" variation, from which the critical load follows, differs according to the boundary conditions prescribed by the data of the problem. NACA TM No. 1138 =z t2 g2f2 w= f sin , u 2 + = + 2 Then the minimal condition S 1I if 4f4 S32 u( 2 2 1 2 2  If' S12 dx + 2 22 (Ce e*) + wxx2 dx E 0 p  e = minimum E furnishes an equation for the "free value" f as a function of E af 212 \ 1 2 0 (5.3) The relationship between load and end displacement is obtained by means of the stressstrain equation (2.3) from the second of equations (5.1); it becomes 2 2 p n f (5.4) There are again the two possible solutions f = O, p = Ee and Shence 2f p2 f V o, hence 2 = E*, p = E* The question of stability is answered (with the abovementioned limitation) by the second variation. As hitherto, two cases are distin guished: 1. If the force is prescribed, e therefore left open, there are two displacements to be varied, and the sign of the expression (5.1) (5.2) .(5.5) (5.6) NACA TM No. 1138 Af A A 2 f8 + (8E)2 2f+ ((07 must be investigated.. with the coefficients This expression is 2 2  E* e f2 212 f32 8E2 a quadratic form in 8f, 22 irf + 3  4z2 Since > 0, this expression is positive definite (that is, never negative) as long as the discriminant 2II A  f 2 ( 2:&>2 85 2 af Iae 2 FIT. f+ e+ sr (5.8) is greater than zero. This is certainly the case below the critical point (e < e*); here the system is therefore stable. On the other hand, A < 0 for f = 0 and e > e*; that is, the straight position is unstable above the critical point. Finally, the bent position is neutral, because by (5.6) the discriminant vanishes for f 0. This result agrees with that of the previous section and is an expression of the fact that a buckled strut in the elastic range can be bent arbitrarily farther without in 'creasing the load. 2. If the end displacement e is prescribed, only the quantity f need be varied and it is found that: S2 ( = 2 2 3 ) + f)322 2 2 2 *. + , (5.9) (5.7) NACA W No. 1138 From this relation it follows fimmeilatealy thatt. for e < e* 52A > 0 for' c'> e* and f j0 62 > 0 A for e > e* and f = 0 8t < 0 ' that is, the straight position is stable for s < c*, unstable for e > e*, the bent position, as soon as it is mechanically possible (hence for E > e*), alimys stable. Figure 4 shows the energy relationships in this second case (prescribed displacement c = ae* of the righthand end of the rod). Plotted.as 'i or Ai 1 4 ( a (al)+  2 <* 2 2 .as a function of f or = with or a= as a parameter. (The. pbtentfal of the external forces is not included because it is not affected by the minimal condition with respect to f.) It is seen that the straight position (f = 0) is an equilibrium position under all circumstances, for all curves start with a horizontal tangent. For a < 1 associated with f = 0 is a minimum, for a > 1 a maximum; the curves for a > 1 have further to the right also a minimum, whereby the bent position f / 0 is characterized as a (stable) equilibrium position. This figure shows especially well the "type" change of the curves in the transition from the subcritical to the supercritical region: the coin ciding 'of maximum and miniman for a = 1. It is also clear here that, although at the moment of transition the displacements are small, the behaviQr of the body at the stability limit is nevertheless determined b. y the "possibility" of greater deflections, expressed mathematically, bythe existence of the fterms of higher order in the expression for * S ' *NACA TM N6. 1138 COMI0ECTION BETWEEN TH ORDINARY INVESTIGATION OF STABILITY AND THE PROCEDURE PRESENTED HERE The ordinary stabiy theory is limited*.to an investigation of the critical point. It was seen that the critical point is characterized by two energy conditions'. The condition" . : * ''* *: :" 5&TI = 0  forer advariatfoh Su, v, Sw (6.1) characterizes it as an equilibrium position in general; the condition 52nI = 0 for a characteristic variation 5u, 5v, Sw (6.2) as the critical one. Or, the critical point is distinguished by the fact that.there a variation of the state of deformation can be made for which the potential II remains unchanged to terms of the second order. Now in .practical buckling problems it is usually a question of the transition fripm.avery simple (often independent of the coordinates) initial state of stress .to .a comparatively very complicated one. It is therefore cuditm ary to .specify the initial .stateof stress directly without recourse to the definition in terms of energy (6.1) and to proceed with the variation im mediately...in regard to the determination of the second state. There then ....remas as thesingle important condition the statement (6.1), which can beexpressed. in the form of a methodofprocedure as follows, for example: Consider a system of infinitesimal distortions superimposed upon the critical state of deformation, collect the parts of the potential energy II quadratic in the added displacements and set the sum equal to'zero. Thai such a procedure is at all possible rests upon the fact that as a result .of the "large" initial stresses two types of quadratic term arise: an'(alwayse positive) part that represents the work done by the stresses caused by the added displacements, and a second part that comes from th6 work done by the stresses already present upon the quadratic part of the added displacements. (See reference 2.)1 The fact that this statementofprocedure concerning the vanishing of the quadratic members is nothing more than the extended principle of virtual displacements ("extended" in the sense of the statement about S21I) does not come out clearly in the applications mostly for three reasons. IIn equation (4.21), for example, the last two terms represent the first type and the first term, the second type. 20 NACA TM No. 1138 1. Since a confusion with the very simple initial state is in general not to be feared, it is possible to dispense with the designation bu, Bv, Bw and write more briefly u, v, w for the added displacements. This manner of writing does not express the fact that the added displace ments are to be not only small in the sense of the general hypotheses of the theory of elasticity but also infinitesimal in the sense of the calcu lus of variations. 2. In close connection with the above, in considering the energy it is customary to start not with the total potential II but directly with the energy changes (appearing as the result of u, v, w) and to designate these changes by1 A, V instead of by 5A, 8V; the (extended) principle of virtual displacements becomes in this manner of writing A + V_= O, or even A A = Aa (6.3) since the potential difference V also represents the work of the external forces on the infinitesimal2 displacements u, v, w. Equations (6.3) can be put into words as follows: For the virtual displacement u, v, w, through which the original equilibriuiA configuration goes over into the neighboring ("buckled") configuration at the critical point.the internal energy .Ai taken up by the system is'equal to the work done by the external forces Aa taking into account the terms linear and quadratic in, u, v, w. By this formulation the two conditions (6.1) and (6.2) are combined into one; a procedure in which there is the danger of losing sight of the difference between the (holding for any equilibrium position) principle (6.1) and the (characterizing the critical position) extension (6.2). 3. As the proper equation for the determination of the critical system of virtual displacements u. v, w there follows (see sec. Stable and Unstable Equilibrium) from (5.2) and the added requirement 52 1 > 0 for all other .u, 5v, 5w the condition 85 21I) = 0 (6.4) To distinguish 'them from those used earlier,' the quantities usually written Ai, A, VA are designated by Af, Aa, V. 2The second form of the law (6.3) theriTore does not represent the special energy law Ai = Aa, by which is expressed the fact that for conservative systems.the external work introduced by the transition from the initial to the (not neighboring!) final state is stored up as elastic energy in the body. NACA TM No. 1138 If it is agreed to consider in A A,._ V on.y the (alone essential for the critical behavior) quadratic terms, the condition.(6.4) is written in the form S(Ai + v) = o (6.4,) (A  A) = 0 This form, which is only a natural consequence of the original agree ment to write u, v, v instead of 3u, Bv, Fw, makes it quite clear to what extent the simplified meaner of writing can lead to conceptual errors. For the statement (6.4'), aside from the deceptive formal agreement, has nothing to do with the principle of virtual displacement (1.2) or (1.3): The principle (1.2), in content the same as the energy law (see the Introduction), answers the question of the equilibrium positions under given loads (or edge displacements), and equation (1.3) is a special form of the same principle possible only in the realm of linearized elasticity theory besides being very inexpedient1; equation (6.4'), on the other hand, in content the same as the minimal condition (6.4) concerning the behavior of the quadratic terms at the stability limit, gives the second equilibrium position possible at the branch point and the soughtfor value of the load at which the equilibrium begins to be manyvalued. The difficulties so far discussed were difficulties in interpretation arising from the symbolism of writing. There is another, more factual circumstance that makes the question complex especially difficult to see through. It was seen in section,Stable and Unstable Equilibrium that for the rod t13re were two independent equations (4.4), with the likewise independent boundary conditions (4.5), for the two added displacements bu, 5w (which here would have been written u, w). From them it was concluded that u vanished identically. This result and correspondingly u = 0, v = 0 in the case of plates makes possible, when (as is tacitly done in the stability theory of a bar) it is presupposed as known, a treat ment of the problems of bar and plate stability deviating from the general methods of stability theory depicted above. Since, however, bars and plates are the most wellknown problems, being analytically the most tractable, frequently ideas that were developed there are erroneously brought into ISo, for instance, for the compressed strut below the critical point twice the external work can be written in three forms: E12, pC, p2/E  which is to be varied (with respect to ef)? The second form is meant, but as a result of writing 2Aa in the place of V that is no longer uniquely discernible. NACA TM No. 1138 more general stability problems. It is therefore necessary to examine more thoou.4ly the various special interpretations that can be given to the occurrences at the stability limit in the case of bars and plates. First of all outline the method by which it is necessary to proceed according to the directions formulated at the beginning of this section. If it is aseiuad a virtual displacement u, w at the critical point, then, as can easily be seen1 the strain of a fiber to terms of the second order is given by 2 VWX x = x +  z xx 2 Therefore the terms stress = x zwxxT of second order are: in the work done by the added / (ux ,xx) dx dy dz in the work done by the already present (critical) compressive stress p 2 Ip _ dx dy dz .'~ p (The external force the second order.) I EF f (f Ai =a(\ / does work Pu(I); this makes no contribution of After integration over y and z there results ^ i2 /Wxxc2  Sdx+ o' and the conditions (6.3) and (6.)1) become Aa = 0 I aP dx + 12 I w,2 dx jo  wx dx = minimum = 0 0 The term ux /2 goes out in the expansion of. the radical (see *1 xx equation (2.2)), and the expansion of the curvature v would give terms of the third order. SU2 (6.5) (6.5') Vy2 dx) NACA TM No. 1138 The expression coincides perfectly with the earlier expression (4.2); therefore the same differential equations and boundary conditions and es pecially the result us 0 ,aro obtained, entirely independently of whether a motion in the xdirectipn of the righthand end point during the buckling is permitted or prevented. This double result (that u = 0, and that the buckling is independent of the condition u(3) = 0 or ux(l) = 0) makes possible the two following "customary" interpretations of the buckling process. The first procedure consists in considering instead of the "natural" problem, buckling under fixed load, the problem of buckling under fixed end point and at the same time (what seems almost a natural consequence of this stipulation) assuming from the start the vanishing of u also in the interior. Hence there is superimposed upon the straight position w = 0 a purely transverse displacement as a variation, keeping in mind the presence of the still unknown longitudinal compressive stress Eeo. According to equation (2.2), as a consequence of the change in length connected with the transverse displacement, the following stretching energy is released EEo Ai = (Eo) Jdx = w2 dx at the same time a bending energy EJ A = T v 2 dx 1 2 2 must be added. This interplay between the two types of energy (and hence the two equilibrium positions) takes place when the values of Ai and A are numerically exactly equal; that is, when SAi =T fw J X dX w dz ) 0 (6.6) 0 0 From the additional condition &A_ = 0 there'follows as above the sine equation.for w. This procedure thus leads to the correct end result without, however, permitting. a guarantee of really having found the mini mal buckling load. For the "restraint" assumption u s 0 limits the number of possible variations', and that it leads to the correct buckling load for the rod (and plate) requires" at 'least a supplementary verifi cation. NACA TM No. 1138 More important because in a still more special way a peculiarity of the rol anid plate is another manner of thinking, which is almost universally made the basis of derivation of the buckling equations. With reference to the natural buckling process, the boundaries are considered as movable; however and this is the characteristic mark of this method  they cannot be allowed virtual displacement u (or u, v which, as has been observed, would subsequently become zero) but are given a displace ment that is of a higher order of smallness (compared with w). In the case of the beam it is customary to start this procedure with the assumption that no additional stretching energy is taken up during bending; that is, that the bent beam has the same length as the straight one; it follows therefrom that as a result of the bending tihe ends must approach each other by an amount u = ,wx2 dx (6.7) (which in fact is of the second order in w!), so that the external forces PoF do the work poFu = / w2 dx. Now by formulating the quality of inner and outer work (wherein by inner work is to be understood only the bending energy) I I EJ p oE 0 Ai = Aa or  2 X. dx 2 dx = 0 2 Jw 2 ,:, and assuming as above the minimal property of this expression, thiE pro cedure leads to equation (6.5') naturally likewise without the uterm.1 In the assumption (6.7) there is anr. inconsistency: It cannot be assumed a priori that a strut that changes its length elastically below the buckling limit suddenly ceaees to do so beyond it. (In reality it changes its length by quantities of higher orier.) It is more logical to consider a perfectly incompresoable rigid against extension but elastic in bending strut, for which the two hitherto independent dis placements u and w are related from the beginning by the (geometrical) assumption W2 ux + 0 (6.7') 2 (Continued on p. 26) NACA TM No. 1138 Since the procedure of equating the stretching energy to the external work cannot be used in the case of the plate, a special auxiliary idea has been used there in order to preserve the conceptually so similar idea, that the boundaries are to move. (See reference 6.) Without connecting the displacements u, v, w with each other nu merically proceed, in this method, from the assumption that u, v are of a higher order of smallness than vw; that is, consider u, v not as really independent virtual displacements but as connected with the transverse displacement w by the orderofmagnitude condition 0 ux s Wx2 (etc.) According to equation (2.2) there is obtained for the work done by the critical stresses ax, ay, T on the displacements u, v, w to terms of the "second order" 2 i = cxs fu x + dy + c oys vy 11(V + T lr jP uy + vx + vwwy ix 'dy + dx dy (6.81) the bending energy is, as always, given by Ai ) w)2 2(1l )(wxx w v dx dy 12(1 2i) J) (s = thickness) (6.82) (Continued from p. 25) Such a strut permits no deformation at all below the critical load; above it takes on only bending energy, which is furnished by the external work lpCF WX2dx Since up to the critical load no elastic deformation at all has taken place, the two laws Ai + V = 8(Ai + V) = 0 and S(Ai + V) = &5 (A2 + V) are here in content completely identical. =0 NACA TM No. 1138 SBoth parts together must be equal to the work Ag of the external force in the sense of equation (6.3). The external work can now (and this is the essence of Reissner's idea) be expressed generally in a very simple manner, if it is remenbemedthatthe straight position is an equilibrium position and that therefore in every virtual displacement A = Ai. On taking the special displacement u* = u, v* = v, w* = 0 (with the bound ary displacements uj* = UR V = vR), A i *; therefore A i = A f oxu + yVy + T (7 + V d id (6.8x) and now on collecting terms in (6.8), the u and vterms cancel out; there results the wellknown Bryan plate equation (reference 1, p. 293), exactly in the form obtained also under the assumption of purely trans verse displacements and Immovable boundaries. The advantage of this method is that it offers the possibility of formulating exactly the related presentation of a solution of the buckling proceap by a boundary displacement. Its disadvantage is a double one: The emphasizing of the boundary displacements gives the impression that the participation of the external vork is universally important in a * buckling process, which, as has been seen, is not so. But besides this it is important for the entire consideration, just as for that of Bryan, that u, v are of the second order with respect to w, which must be known somehow beforehand;' therefore the interpretation of the external work axux as a contribution of the second order is not transferable to more general buckling problems. (See reference 6J To summarize briefly the result of this section: In considering the critical point it iscustomary to dispense with the correct method of writing the virtual displacements Bu, 8v, 8w in favor of the more convenient u, v, w; thereby the connection between the customary stability criterion and the principle of virtual displacements is con cealed. To be added is that the stability problem of the rod and of the plate permits a special treatment which rests upon the fact that at the critical point the tangential displacements u, v and the normal 1The very obvious conclusion, that can just be seen from the form (2.2) of the strain that ux and Wx2 must be of the same order, is not tenable; for an equation of the type (2.2.) holds, for instance, also for the longitudinal, fibers of a cylinder, and yet here ux and wx can become comparable because the tangential displacement v, which is of the same order as w, is linearly coupled with u through the shear and the transverse contraction. 2NACAW No. 1138 displacement w are of different orders ^of'magnitude'. Since, however, the rod. or plate problem, 's the analytically simplest, is at the same time the. best known, the need. easily arises of transferring methods of thinking successful in these problems to more complicated problems, which, as was to be shown,, is not possible. DSE UCBSCELAG PROBLEM OF THE SLIGRIILY CURVED BEAM In section Stable and Unstable Equilibrium, it was shown that for the Euler strut the instability point (defined by 5(82II) = 0) coincided with the branch point of the equilibrium. Branching problems are, however, not the only kind of stability problem; a second'class, which is Just as suited to the energy definition of stability as are the brar.htirg problems, comprises the socalled Dirchschlag problems. In the Durchschlag problem the critical load is designated as that load under which an (infinitesimal) displacement of the point of applica tion of the load is possible without an increase in the load, for which  as in the branching problem there are therefore two (infinitesimally close) equilibrium positions. Above the critical point an increasing displacement is in general accompanied by a decreasing load the state is unstable, the system snapsp" into1 or falls into a stable configuration. *Prerequisite for such a phenomenon is a nonlinear relationship between force and displacement even in the stable region. The simplest Durchschlag problem is that of a slightly curved beam under a transverse load. (See reference 7.) If the ends of the initially curved beam are prevented from dis placing (fig. 5), then connected with the deflection caused by the transver3e force Q is a shortening of the axis of the arc, as the result of which a horizontal force R is made to act. Because the effect of this (very large) compressive force upon the equilibrium of forces in an element of the beam cannot be neglected, there arise phenomena related to the buckling process in the Euler strut, insta bility phenomena. Without carrying out all the details of the calculation (presented completely elsewhere, reference 8) the principal method of solution for this stability problem will be briefly outlined. %In a manner similar to that in which a strut compressed beyond the Euler limit at the least disturbance snaps or falls into the bent position. S28 NACA TM No. 1138 By a process that follows very closely that carried. Energy of Deformation, is obtained, with the notation of now taken positive downward), for the potential energy EF 2 40 S wx2 W ) out in section figure 5 (w 2 1 2 x f + i Wxx dx Qf (7.1) From UII EF F ( ux x2 Wxx) ^ WUtex + i 2 w'xxX d I dx Q~f there are obtained the two equilibrium conditions expressed in terms of the displacements: S + a;x W2 xx = 0 2 or, integrated once, (U 2 WX 2  WXw )= constant = h (7.2) and, with the use of equation (7.2) 1i xxxx + hwxx = hWxx (7.3) also the boundary conditions u(O) = u(M) = w(O) = w() = wx(0) = wx(l) = 0 Wxxx(/2 + 0) wxxx(l/2 0) = Q/EJ Suxdz + EF uz2 (7.4) NACA TM No. 1138 which together with the continuity requirements for u, ux W, w, WV at x = 1/2 give the 12' conditions that are necessary for the evaluation of the 2 x 6 constants of integration in the 2 regions x 3 1/2. The physical significance of the constant h can be recognized di rectly from (7.2): On the lefthand side is the stretching of the middle line of the arch; therefore, to a factor 1/EF h is equal to the hori zontal force H, and (7.2) expresses the equilibrium condition that E does not vary with x. Just as in the case of the Euler strut the constant of integration h = H/EF of the first equation enters into the second as the coefficient of the unknown v; that is, the system (7.2) and (7.3.) is nonlinear. (See equation (3.4) or (3.2)). Nevertheless, just as before the exact solution can be given in this simple case without difficulty in terms of the at first unknown 2 "12h H EJx2 & H*  i2*2 H* 12 rx for example, for W = fo sin : 2 3 ai a x Q sin a arx Ssin + (7.5) of0 _] 22aEJ cos  (x < 1/2) and from (7.2) by another integration taking into account the boundary conditions u(O) = u(l) = 0 there is obtained subsequently a transcen dental equation for the dependence of a upon Q and fo of the form: S(x f W) dx (7.6) EF I0 2 o NACA.TM No. U138 The determination of the critical load can be carried out in two. basically different ways. The first method (see C. B. Biezeno,reference 7, pp. 21 from the condition 8Q/8f = 0, wherein the relation between is established by equations (7.6) and (7.5) for x = 1/2 w(l/2) = f = fo Q2 + 2n~aEJ taIm sr tan  2 2 ff) proceeds Q and f (7.5') A second method proceeds by way of the energy criterion (4.3). For B2II is obtained from equation (7.1) after writing for brevity' 2W e + x + x  Wx _ 1= W (x,a) the expression .2 52I = EF u ()+ 2 (W x 5uxWx w (x, a)(&vx) fL( 2  Th atiua dslceetsytm 5u 5 b hch0I The particular displacement system Bu, 8w by which 52Il equal to zero is obtained from (521TI) = 0 that is, from the two homogeneous differential equations B u + W Y x Swx I=0 (7.7) +(7.8) (7.3) is Just made (7,9)  wx) Bux + i2Bxxx + W(a,x)Bwx = 0 and Therein v is at first according to (7.5) a function of the tcw parameters a and Q. Q for example, is considered as eliminated with the help of equations (7.6) and (7.51).  _ NACA TM No. 1138 with the homogeneous. boundary conditions . .. u() = u() = (O) = ) = (Z) = 8w(0) =8wv(l) = 0 The desired critical avalue is the lowest eigenvalue of the equations (7.9). APPROXIMATE DETERMINATION OF THE SNAP LOAD Because of the great mathematical difficulties that equations (7.9) present, the second method outlined is not suitable for an exact treat ment of the .problem but is well suited and therefore that procedure will be considered here to an approximate treatment by the method of Ritz or Galerkin. This procedure can be started at either of two points: either, make a Ritz approximation for 8w in (7.9), determine the correspond ing Su from the first of equations (7.9), and following Galerkin from the condition 8(52II) = 0 obtain a (transcendental) equation for the determination of a; or very much.more simply, if also necessarily with a corresponding loss in accuracy introduce at the start a Ritz approximation for w itself in place of (7.5) into the expression (7.1). It is well to use the second method but only indicate (reference 8) the course of the calculation. If again W = fo sin is chosen and as a Ritz expression xM 2irx w = f, sin + fa sin  (8.1) 2 2 then all boundary 'conditions are fulfilled, except for the one dis continuity requirement (7,4), the violation of which is however, un important. Further,by satisfying exactly equation (7.2) (obtained by variation with respect to u) and calculating the horizontal force E from (T.6), the integral in (7.1) can be evaluated and IT is obtained as a function of the amplitudes fo, f1, f2 or the dimension less parameters NACA TM No. 1138 7o = f 0  f2 A2 = i (i = radius of gyration) 1. in the simple form 4 t 4i4 n 1 EFi ",2 2+ 1622 + Qj 2 EJ 2 1 413 q  ?o C 2 ~/ b4 Q nafo H* (8.2) (8.2') The equilibrium equations read ( 2AI + (?o 1) 2 ) qo = 0 O E 2,72 (8.3) _N 22 = 321 2i hs02 ", = 0 They are in the two unknowns AN and N2 and of the third degree; nevertheless a complete discussion is possible without numerical calcu lation, because the second equation may be written as a product A2 ^ O 22  8) = 0 Therefore the cases can be .distinguished A 2=0 2X1 + (N 7) ( 2 )=q o and 7%a 0, that is, 2A2 = ?7o, .2/ 8 (8.5) 8No  67 = q o where  q 0A1M NACA TM Wo. 1138 the discussion of which no longer requires any labor. The first system provides a symmetrical deformation, the second a superposition of a sym metrical and an antisymmetrical deformation. The corresponding horizontal compressive forces become SH* NO / 2 2 (8.4') (8.5') The critical load q Q2 F II (X 3) 2 As long as crit is found from the condition +2 (5 ,? 8 ).+ f (2) = 0 ()Yk ~ ,. jy and. the discriminant ()2fi )2 7,2 X 2 '. are greater than zero, 52II as a positivedefinite quadratic form in 5xi and 8A2 cannot be zero for any combination of these two variables. Vanishing of the discriminant characterizes the pair of values Ax, 'M for which there is exactly one combination AI, 052 for which 8211 becomes zero but none for which it is less than zero. The condition 22 2 on 9 2 (8.7) gives, therefore, the stability limit and together with the two equations (8.3) determines the three unknowns xi, 72, and qcrit. In this case (8.7) reads (8.6) 2' 2 ( uN  K^2 34 ,2fA ) = 0 6Nd)2I/~ NACA TM No. 1138 4 [8 (Now  22 + (  + 22 16X22a 0 )2 = 0 which condition becomes A2 = 0 with 418 (Q 0 2 L2+ (^o )2 . and with "12 8 2  48 'o\ .2/2) = 0 There are therefore two sets of values for A;i, ?a, 1 qcrit = 2 + 0 and qcrit " 4\3/ 3 / (8.8) 72 = 0, 71 = No A/ 16, qcrit = 2 Po 02 16 by which a critical state of the elastic system is characterized. The physical significance of the relations (8.8), (8.9) and espe cially the (in this case unstable) behavior above the critical load vill not be pursued in detail (see reference 8). (See fig. 6.) There the load Q is plotted against the deflection f, of the point of application, 2 ./ X2 = 0, 7%i 1 7k)o .Xo2 4 ,/" (8.9) 2 ,2 = X0 o . with the initial amplitude fo as a parameter. For fo/i = 70 2 Q. increases monotonically with fl; instability is not possible. In the region under the critical load given by (8.8) there enters definite instability, increasing deflection f, without increase of load' .Accordingly the beam snaps under constant load until it finds a stable configura tioni at E7'. Since Az' > No the beam is now convex downward; it also can be seen clearly that the system nowmust be stable with respect to an increase in load; a further deflection results in a longitudinal pull. (See equation (8.4').) For AO "22 (8.9') the critical load is given by (8.9). Before the external load can assume the value (8.8), the longitudinal compression according to (8.5') reaches the value 4H*, that is, the second Euler load, under which the strut assumes the Sshape configuration 'g J'0. It snaps again into a stable position A?', this time, however, passing through an un symmetrical deformation. At the critical load there appears a branch ing of the elastic equilibrium; figure 6 shows the two branches of the Q f, curve, both of which however .and this is the noteworthy differ ence from the Euler problem are unstable. For further details see the publication referred to. Here it was just a question of.presenting the chain of ideas that led to the deter mination of the critical loads (8.8) and (8.9), in order to show the application of the general stability criterion (4,3) to a stability problem of an entirely different kind. That is, at first vibrates about Mz' as a stable equilibrium position. NACA TM No. 1138 1FERENCES 1. Handbuch der Physik. Bd. 6 (Berlin), pp. 70 ff., 1928.: 2. Trefftz, E.: Zur Theorie der Stabilitit. Z.f.a.M.M. Bd.13, 1933, p. 160. 3. Poschl, Z. B. Th.: Uber die Minimalprinzipe der Elastizitatstheorie. Bau Ing. Bd. 17, 1936, p. 160. 4. Marguerre, Karl: The Apparent Width of the Plate in Compression. NACA TM No. 833., 1937.  5. Marguerre, Karl: Uber die Behandlung von Stabilitateproblemen mit Hilfe der energetischen Methode. Zf.a.M.M. Bd. 38, 1938, pp. 70 ff. 6. Reissner, H.: Z.f.a.M.M. Bd. 5, 1925, p. 475.' 7.Biezeno, C. B.: Das Durchschlagen'eines schwach gekrummten Stabes. Z.f.a.M.M. Bd. 18, 1938, p. 21. Brazier, L. G.: The Flexure of Thin Cylindrical Shells and Other "Thin" Sections. R. & M. No. 1081, British A.R.C., 1927. Heck, 0. S.: The Stability of Orthotropic Elliptic Cylinders in Pure Bending. NACA TM No. 834, 1937. Weinel, E.: Ober Biegung und Stabilitat eines doppelt gekrumnten Plattenstreifeus. Z.f.a.M.M. Bd. 17, Dec. 1937, pp. 366369. 8. Marguerre, Karl: Die Durchschlagskraft eines schwach gebkr'smten Balkens. Sitzunieberichte der Berliner Mathemiatischen Gesellschaft. Bd. 37, June 1938, pp. 2240 NACA TM No. 1138 TRAiSLATOR 'S NOTES Equation 3.7, last ,pat  Trans, note: ?It appevrx that this equation, .ehuld ,be .. f2 2ax ;.f2n2 *u 6* in X* . x 8 1. 41 . Equations (7.5) and (7.5') Q} Q3 Trane' note: It appears .that. the term shouXd be ., Equation (8.8) r1 /ot It a" Trae.,*note., It appears that 2 + 0 4I.... : hod'be NO 3 2+ o , o , Page 24 Trans. note: It appears that Ai xx dx should be OV EJ fr 2, 0 Page 36 Trans. note: It appears that (8.4') should be (8.5'). NACA TM No. 1138 1.o  ~M )=CI F.igure 1. StrL.t under compression. '4 j / 0 2 3 0 1 2 3 Figure 2. Amplitude f against e. I 4 0 1 2 FiLcure 3. Load P against c. NACA TM No. 1138 2.5 2'. 0 0 .5 1 1.5 2 = f/2i Figure 4. Variation of energy of deformation with the amplitude f z 2it. Edge compression = a* as a parameter. Moment of inertia J, Section F. Radius of gyration i, Elasticity modulus E Figure 5. Slightly curved beam under transverse load Q. 'UACA :TM ITo. 101 7  51 / . 8 >/'=6 .. __  0 6" .\L 2 0 2 3 4 6 8 9 1 0 \ / 1 Figure 6. Variation of the load 0 with the displacement fl bf the point of application, parameter = initial amplitude fo. rn r4 H UNIVERSITY OF FLORIDA II 11 11 0 10 6 II1 2 .3 1262 08106 308 2 
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