mikm-wbi 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 
 
 No. 1138 
 
 ON THE APPLICATION OF THE ENERGY METHOD TO 
 STABILITY PROBLEMS 
 By Karl Marguerre 
 
 TRANSLATION 
 
 "tjber die Anwendung der energetischen Methode auf Stabilitatsprobleme' 
 
 Jahrb. 1938 DVL, pp.252 - 262 
 
 '^QE^ 
 
 Washington ^,^,vers1TY OF FLORIDA 
 October 1947 DOCUMENTS DEPARTMENT 
 
 120 MARSTON SCIENCE LIBRARY 
 
 P.O. BOX 117011 
 
 GAINESVILLE. FL 32611-7011 USA 
 
/OO 06^ Q^iZ inOYJ 
 
 riATIOKAL ADVISC5Y C0^ME1TEE FOR AERONAUTICS 
 
 TECBiniCAL MEMOR^lilD'UM ITO. 11 38 
 
 Cn TEE APPLICATION OF THE ErJERGY WJmOQ TO STABILITY PSCBLEMd^ 
 
 By Karl Marguerre 
 
 Since sta.l5illt3r problems have come into the field of vision of engi- 
 neers, energy method's have proved to he one of the most powerful aids in 
 mastering them. For -finding the especially interesting critical loads 
 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 the 
 treatment of which these special methods are not suited as they are. 
 
 The present investigation^ discusses this question ~ complex (made 
 important hy shell construction in aircraft) especially in the classical 
 example of the Eiiler 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 "tJber 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 Tth 
 ar.d 8th sees.) 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 Methode auf Stabilitats— 
 probieme." Jahrb. 1938 D'lTL, pp. 252-262. 
 
 ■ 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 sees, h and 
 6) the author is very grateful to his colleague, R, Kappus, for his close 
 collaboration. 
 
 3 - ■ ' 
 
 See the next to last paragraph of- the Introdu-ction. 
 
KACA TM No. 11 38 
 
 SLWIAEY 
 
 In the two examples of the Euler stmt and the slightly curved 'beam 
 ■under transverse load it vas shovna that the difference hetveen the sta— 
 hilitv problems and the prohlems of linearized elasticity theory rests 
 upon the fact that in the sta'bilitj' problerns 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 ener.^y 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. 
 
 IHTROLUCTICSf 
 
 Kirchoff *s uniqueness law states: An elastic body can assume one 
 and only one equilibrium configuration •under a given (sufficiently smsill) 
 extenjal loading. In the formulation from the energy point of view: the 
 potential U of the inner and outer forces has one and only one extremal 
 
 ?II = (1.1) 
 
 and the extremal is a minimum.^ 
 
 The rniquenesE 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 y^-^ 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 show directly that a "parabola" of the second degree (positive def- 
 inite quadratic form) can have one and only one minim-ori, (or fn mechanical 
 terns, equilibrium position) . The situation changes, however, when struc- 
 tures are considered tho behavior of which can no longer be expressed i-rith 
 
 -ror 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 fo-'-mdation of the general theory of the behavior at the 
 stability limit appears in reference 2, p. I60. (For further literature, 
 see reference 1, pp. 277 ff., or reference 2). The investigations in 
 section Stable and Unstable Equilibrium in particular make xise of the 
 Trefftz point of view. 
 
MCA ™ Ko. 1138 
 
 srif fie lent accuracy by the lineari?;ed strain-dieplacement equations. Such 
 ai"8, pai'ticulary, bodies for which one dimension is small compared to the 
 others, sti-'uctures in the forni of shells, plates, or bars. For example, 
 p, rod can, vrithout exceeding the proportionaJ. limit, undergo bending de- 
 flections several times greater than its thiclmess, and ijnder these cir- 
 cvmstances the quadratic part of the (transverse) displacements in the 
 strain displacement equations ,ia 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 severa.1 extrema3.s (equilibrium positions). 
 
 Tlie 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".^ * . . 
 
 This purely practical viewpoint has, however, led to a certain (as 
 is shoTm, from ijnfounded standpoint) systematic separation of the sta- 
 bility problem from the other problems of 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 dene 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 
 
 H = A^ + V 
 
 Knowledge of the postcritical region wa.s until now of secondary 
 practical interest, because buckled striicturaJ. elements were considered 
 unpermissible. It has been onlj-- in recent years that in the shell con- 
 struction of aircraft critical loads have been permitted to be exceeded 
 by large amounts unhesitatingly. 
 
k - MCA TM.No. 1138 
 
 has a statlonaiy value : • 
 
 ■ 5TI = 8(a^ + V) = 0. (1.2) 
 
 Therein the potential A^^ of the iruier forces is given hy the energy 
 
 of deformation (inner work), an.d the potential. V of the external forces 
 ty the negative product of the external forces considered constant and 
 the displacements cf their points of application. In the region of appli— 
 cahility 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 paesing through only etjuilihrium states. The prin- 
 ciple (1.2) can therefore he written conveniently also in the form. (Far 
 exsjtnple, see reference 3.) ■ . • 
 
 &(Ai - 2A^) = (1.3) 
 
 As against this there is often used as a "minimal principle" in stahility 
 theory the condition 
 
 6(Ai ~ A^) = (l.k) 
 
 The author intends to show in the present paper (in the classical 
 example of the strut) that also stahility investigations are "best hajidled 
 in connection with the single main equation (l.2), wherein terms muet he 
 retained of higher order in the deformations logically only in the ex- , 
 press ion for the energy cf deformation. This procedure is essential 
 from the practical standpoint, if the relationehlpe are tc be investigated. 
 in the poetcrltjcal 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 eqiTations (l.3) and (l.^). 
 
 The calculation itself is carried out in the following manner: 
 First, the expression for the energy of deformation Aj is set up, then 
 
 the differential equations for the two components of displacement are 
 derived from the condition ; 
 
 SH = 
 
MCA TM No. 1138 5 
 
 and the question of the etahility of the equilihrium position is anevered 
 hj the restricted condition 
 
 n = minlrnvm (1.5) 
 
 Then the same prohlam is treated vith the help of a Eitz procedure; in 
 this vay the result of the statilitv consideration is "brought out in an 
 especially elei^ant manner. After a thorough discussion of the usual sta— 
 hilitv theory^ it is shovn in conclusion how the same considerations can 
 serve for tlie treatment of the snap-action protlem of a slightly cLurved 
 ■beam. 
 
 EKSRQY 0? LEFORT/IATIOrT 
 
 If the customary asB''jmiptions of the heam theory are retained ~ that 
 for small deflections of the heam the work 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 small compared with the other two parts — 
 then the energy of deformation can easily "be given. 
 
 As a result of the assumption of small displacements — vlthout at first 
 saying anything a'bcut the sizes of the displacements u and w (fig. l) 
 relative to each other ~ the strains u^j. and Wj^(or . w^j.^) may "be neglected 
 
 in comparison with unity; that is, in a development .of "both quantities in 
 powers of Uj- and w^ only the lowest power need be retained. If, there- 
 fore, the sq_uare of a line element of the "beam centerline "before deforma- 
 tion is 
 
 dx^ 
 
 and after deformation 
 
 / 
 
 ( 
 
 
 ] + v^ j dx (2-1) 
 
 (the su"bacripts on u and w indicate derivatives with respect to x), 
 then the strain of the "beam centerline (reference 1, p. 57) is 
 
 /; ^ 7 „, 2 
 
 C" 
 
 =./(l + Uxf + wx -l^u^+-_- (2.2) 
 
 X -v V -^ ^ '^x / ^ ™x -^ ^x 
 
g- _ MCA TM No. 1138 
 
 From Hooke*s law the correeponding stress Is 
 
 and therefore the energy of stretching is 
 
 1 
 
 — W r I W; 
 
 2\ 2 
 
 ^ o 
 
 The incremental strain e^ due to tending is, according to the 
 assunrption that normale are preserved, 
 
 : e^ = -zvj^ (2.5) 
 
 the tending energy is therefore given "by 
 
 I 
 
 '^ = 1 fff G^^) ^ ^y <^^ - ? /^xx^<i^ ^2.6) 
 
 
 
 If eZ = u(C) — u(Z) = -u(l) is the distance of approach of the ends, 
 P = pF = - <7^F the compressive force, then - (pF)(eZ) = V is the po- 
 tential of the external forces; the total potential TI (measured from 
 the stress-free state as the zero position Just as the displacements 
 u, w, e I ) is therefore 
 
 
 Gx -^ ^) d^ + H I ^^2 dx - pP^Z (2.7) 
 
 The potential per unit length -- after division hy EF — can he written 
 
 Z Z 
 
 li = -5I = J. f f^^Y£y^^ll r 2d^^_E, (2.8) 
 EFZ 2Z J \^ 2 y 2Z j ^^ E 
 
!i/lCA, m Jfo..-il38 
 
 ■From equation (2.8) and^ the coniition (l,v5)'''- 
 
 H = minimum 
 
 (2.9) 
 
 there is ohtained all information about the 'behavior of the strut at and 
 "beyond the stability limit. 
 
 THE DIFFEEENTIAL EQUATIONS FOE THE DISPLACEMENTS u, V 
 
 Consider a rod the left end of which is (x = O) is supported atid 
 the right end giv^s {%,= l) is freely movable in a horizontal direc- 
 tion under a centrally placed compressive force P. (See fig. 1*) '4s 
 given (that is, as the independent variable of the problem) take either 
 the horizontal displacement | or the compressive ■ , . , 
 of the right-hand end ; stress 
 
 .u(l) = - el 
 
 P 
 
 According to the principle of virtual displacements, the displaqements 
 are to be varied under a constant load in a manner compatible with the geo- 
 metrical conditions. If at a boundary point the displacejnent (in the pres- 
 ent problem, for example^ e) is prescribed, the point is held fixed 
 during the variation, so that the work of the (unkno^Aml) external forces 
 .does 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 P6u) enters into the 
 calcuJLation. 
 
 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 
 
 together with the boimdary conditions 
 
 'u(O) = ■ 
 u(l) = -sZ 
 
 = 
 
 (3.1) 
 
 u(0)' = ■ 
 
 j , vx£ p 
 L ^ -^ "2- -^ E 
 
 = 
 
 x=l 
 
8 MCA '^-K'«. 1138' 
 
 ■by varying w there is obtained •( independently of whether e" 'or p is 
 considered as given): 
 
 ^ f wx^ ^ f ^^ \ ^2 ^ 
 
 with the toundary conditions C3f2) 
 
 v(0) = w(z) = w^(0) = Wj^Cz) = 
 
 " The exact integration of the system (3.l), (3-2) of nonlinear (?) 
 simultajieous equations offers no difficulty here. 
 
 From (3.1) there follows 
 
 \ix + ~~;:^ - constant = — Cq 
 u = u(®) - £oX - I -"^ dx 
 
 (3.3) 
 
 and with the use of (3.3)^ (3.2) hecomes 
 
 e 
 
 The solution of this linear equation with constant coefficients is 
 
 w = f sin kx + g cos kx + fiX + gi 
 
 where k is the positive root of the quadratic equation 
 
 i2 
 
 For the determination of the six constants of integration f, g, fi, gj., 
 u(0), £f^ there are the six boundary conditions (3.I) and (3.2): 
 
NACA TM Uo. 1138 
 
 g + gi = k^g = 
 
 f sin kZ + g cos kl + f iZ + gi = 
 fk^ sin kl + gk^ cos kl = 
 u(0) = 
 
 and 
 
 - €1 = u(0) - €0^ 
 
 'J r 
 
 djc 
 
 - e o + - = 
 
 E 
 
 (3.5) 
 
 It is found that 
 
 and either 
 
 f 1 = gi = g = Ti(0) = 
 
 f = 0, that is^ w = 
 
 ^o = ^ 
 
 u = — ex 
 
 _ P 
 
 ^0 ~ S 
 
 u = -^x 
 
 (3.6) 
 
 or 
 
 f 5^ 0, w = f sin kx/ k = - 
 
 . . .2 2 
 
 (3.7) 
 
 jr f 2«x 
 u = - ..X + g - Sin — 
 
 There are evidently two kinds of equilihrium positions: the straight 
 (f = 0) and the tent (f ^ O) . The straight position is specified uniquely 
 hy either p or g, the bent hy € — for the amplitude f of the deflec- 
 tion can he determined uniquely from the left one of equations (3.5) 
 
 
 e -e* 
 
 (3.8) 
 
10 KAGA TM -No. 1138 
 
 i 
 
 not, hoT'eyer, "bj p. FrcEi the right one of equations (3. 5) there follows 
 rather that for f ^ a completely determinel "critical" value 
 
 p = E€* = p* (3.9) 
 
 cannot he exceeded. Therefore p is unsuited for an independent 
 parameter (the situation ie different in the case of the corresponding 
 plate prohlem) (reference h, p. 12^. From equation (3.8) it can "be seen 
 that f apsuwes real values only for € > e'^. 
 
 In figures 2 and 3 the quantities f and p are .plotted against e. 
 The solid lines are for tlie solution (^.T), the dotted lines for equation 
 (3.6). 
 
 The result (3.9) - that the load for the huckled strut -does net in- 
 crease heycnd p* even when the critical end shortening has teen con— 
 siderahly exceeded -- is a result of the limitation to "small" deflections. 
 For the preeent prohlem thjs lim.itation If not important hecause hers 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— 
 lihriuDi positions — espec:5ally that the existence of a real multiplicity 
 is hound up with the exceeding of a certain "critical" strain € = e*. 
 
 STii^LE AKD MSTABLE EQUILIBErUM, 
 
 In the "ordinary" theory of elasticity it is necessary to consider 
 only the condition 
 
 511 = 0, that is, II = extremum (^.0) 
 
 ■'•Seference ?, v-^. 70 ff . Also the theory of the so-called exact 
 differential equation of strut huckling 
 
 Ej/o + ?w = 
 
 (reference 1, p. 230) sho^^^s that at Isrge deflections ("because of the in- 
 creased demand on oendlng energy) there is a very small increase in load. 
 From the er.srgy standpoint, to "be sure, this "exact" equetion is not lm~ ■ 
 portant, f c^ if w^^ is taken as not small compared to 1 in the bond- 
 ing term (x'jat is, the cm^vature l/p is used in place of w^x) it is. 
 necessary \o proceed in a corresponding manner with the stretching term 
 (equation (2. 2)) - unless it is assumed that an incompressible strut exists 
 from the stai^t, as is done in the theory of the Euler elastica. 
 
mCA TT^IJo- 1138 11 
 
 in the determination of equilitriimt Bts.tes, for the supplementary con- 
 dition S^H > (mechanically: the stability of the equilihriim posi- 
 tion) is assured there hecaus© of the lineiarization (reference 1, pp. 71-- 
 72); in the present prohlem the minimal, condition must he set up explicitly, 
 since only through 
 
 SH = 0, S^H > 0, that 'is,: H = minimum ■ (^i-.l) 
 
 can the stahle eguilihrium positions he distinguished from other possi- 
 ble (the unstable) positions. The concept of stability is- made precise 
 here lij the following convention.^ 
 
 i. iin oguilibrium state is called v^^table if for every neighboring 
 state the potential energy has a larger value." ... -j 
 
 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 
 potentiaJ. energy of which is equal to bixt ' none having potential energy 
 less than that of the given equilibrium, state. 
 
 Seturn to equations (2.8) or, (2.9)j 
 
 ■ I I 
 
 r/ „ 2\^ i2 r T^ » 
 
 ^ = rr / I ^x + --— ; ^ -i- -7 / %x • dx - -e = rJ^iM^M 
 21 ^ \ 2 / 21 -Q E 
 
 and perform a variation; that is, replace u by u -f 5u and v by 
 w + pw; there results, after arranging in powers of 5u, 6w and 
 stoppir^ after terms of the second ordei'. 
 
 The following definition wag given, in substance "oj E. Trefftz in- 
 cidentally to his DVL lecture. 
 
12 
 
 MCA TM No. 1138 
 
 A. A 
 
 AH = II (u + 5u, w + 5v) - H (u,w) = &II + K s n + • 
 
 I 
 
 1 
 
 ^^i * \ I i'^ * ^) ""^ "^ 
 
 * 7 I L("- ^ T") ": 
 
 6w^ + 1 fiT--. Sw, 
 
 X "X 
 
 XX "XX 
 
 dx 
 
 1 rV ^' 1- r 
 
 } r/ 
 
 iT / b 
 
 W-v 
 
 \ 
 
 21 J L\ ^ 2 
 
 o 
 
 \ 
 
 "x A ""X 
 
 + -^^ + w^" ;( Sw^.) + i( 5w^ j I dx 
 
 .n ^^ 
 
 
 n 
 
 'XX 
 
 The condition that the terras 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 
 Sn = there results for (3.6) (straight position) 
 
 2i^ ^ 
 
 / 
 
 2^ \ 
 
 5 n ;^ . [^^ hX = 
 
 1 
 I 
 
 / 
 
 ^^ 2 
 
 L V / , 
 
 + 1 
 
 '-, 2 
 
 Swxx 
 
 '■J 
 
 dx + - /(&0 ^ (^.2x) 
 
 for (3.7) (bent position) 
 
 J ^2. 1 
 
 1 ' !/ 
 
 
 5^11,1 = 6% ) = i / }( -e* + ~ f^ cos 
 
 2 '^^'^r^./V 
 
 X/' 
 
 
 dx 
 
 2irf 
 
 jrx 
 
 cos — SWjj. Suj. dx + — / I &Ujj y dx 
 
 (^^.22) 
 
 First the stability limit -will bo determined. According to the defi- 
 nition given above there must be at the stability limit a state for whioh 
 the second variation vanishes but none for vhich it becomes negative: 
 The value zero is therefore the smallest value that S^II can assume at 
 the limiting point. If, therefore, S^ri has certain continuity prop- 
 erties (the existence of which is obvious on mschanical grounds), then 
 
MCA TM No. 1138 13 
 
 the "characteristic ' value b^TL = ip at the same time an analytical 
 minimum, compared with neighhoring values, and the associated ("charac- 
 teristic")- displacement system 5u, 8w is determined from the condition 
 
 •5(5^0 ■)= .0 (1^.3) 
 
 The straight position (see equation (^.2i)) thus reaches the sta- 
 tility limit when 6u, 6v satisfy the two differential equations 
 
 (6u)„ = 
 
 'XX 
 
 (8w) ■ + — (dw)- ■ ■= 
 
 vlth the end conditions 
 
 8w(0) = 5w(Z-)- = Bw^QjCo) = t\tj^(l) = 5u(0) = 
 and 
 
 (h.k) 
 
 bud) = 
 
 b-uM) = 
 
 (i^.5) 
 
 The solution of this eigenvalize prohlem reads in hoth cases 
 
 Su = 0, 5w = 5f sin -~ ik.6) 
 
 The amplitude 5f j^ remains undetermined and. from the second of. equa- 
 tions {h.k) there is obtained for the critical value of • 
 
 i . . - 
 
 ^ - crit = --^ 
 
 that is exactly the expression ^crit = ~* ^^ 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 — 
 
ll^ NACA m No. 1138 
 
 T— ( 5vu, + — cos -—- oWy ; = 
 
 C'+.7) 
 
 Sx 
 
 t 6 If _j 
 
 XXX V ,2 
 
 for the "boundary condition there is retained 
 
 8w(0) = 5w(Z) = 6wxx(0) = 5V3^(1) = 5u(0) = 
 
 and, depending on whether or not the right-hand "boundary point is pre— 
 scrlhed or not. 
 
 &u(Z) = 
 
 ( 5ux + ^ cos ^ 6wx) , = (if. 7') 
 
 It is recognized lamediately that for f = 0, hence at the heglnning of 
 huckling, the "bar is in neutral equilihrium, for all conditions are satis- 
 fied "by the solution (^.6). This res^llt is trivial. It is not so directly 
 o'bvious, however, that also for f ^(^ there is a variation that laakes 
 
 8^11 a minimum; the homogeneous system of equations with homogeneous 
 "boundary conditions permits of a non-vanishing solution also for f 5^ 0. 
 In fact if the variation is so performed that the second "boundary con- 
 dition of (4.7') is satisfied there follows from the first of equations 
 
 5u_ = cos — Sw^. C+.S) 
 
 •*■ I I . -^ 
 
 If this is put in the second of equations (^.7) the latter reduces to 
 
 i^^xxxx -^ ^*S^xx = i^ (s^xxxx + ^ ^^xx J = ° 
 
 and this equation (together with its boundaury conditions) is satisfied by 
 5w = 5f sin — . For 5u there is obtained from (^.8) 
 
HACA TM Ko. II38 I5 
 
 — 5-1 X + — sin -—— ) 
 
 in which Sf again represents the (not determinable ty a system of homoge- 
 neous equations) arhitrsj^y factor. 
 
 The question of the stahility of the equilibrium positions helow and 
 above the .limit can now be answered. 
 
 1. Since the straight position (see equation (^.2i)) is stable for 
 very small e, from considerations of continuity it is so for e < e^. 
 
 2. For e > e* the straight position represents an unstable equi- 
 librium state, for a variation 6w, namely, 
 
 &w - 8f sin ^ . 
 
 can be given for which 5^11 <0. 
 
 3 • ?o^ e > e * the bent rod is against the variation 
 
 &w 5: 6f sin -y- ■ 
 
 \ . (^.9) 
 
 jt^fSf f'^ I 27rx 
 
 ovt .= .r- [ X + — Sin 
 
 21^ \ 2it I 
 
 In neutral equilibrium.^ Since the variation (^.9) (and only this) makes 
 6^11 a. minimum (of value zero) every other variation gives it a positive 
 value. If in some way the special variation (^.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 th© 
 boundary condition 5u(l) = 0. 
 
 At this point a result discussed later iii section Connection between- 
 the Ordinary Investigation of Stability and the Procedure Presented Here, 
 shoiild be emphasized. The behavior of the rod beyond the stability limit 
 is ■ different depending upon whether the load or the displacement is 
 
 This result is natxirally connected with the assumption that the 
 strut adheres strictly to the law of deformation established by the ex- 
 presBion (2.7)..' 
 
16 MCA TM No. 1138 
 
 considered as the prescrited quantity. More noteworthily, however, the 
 "behavior at the stability limit is not affected hy this difference. For, 
 although there are the two different houndary conditions (U.5) 
 
 . 5u(l) =0 
 and 
 
 &ux(l) = 
 
 they "both (together with the differential equation BOj^^r = O) lead to 
 the same result 
 
 Su = 
 
 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 resvilt 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 6u, Sw, ^ grit' ^crit. 
 
 IKTERPBETATION OF TBE RESULTS WITH AID GF THE EITZ ^!ETHOD 
 
 Tlie res-ults of sections The Differential Equa.tions for the Displace- 
 ments u, w and Stable and Unstable Equilibrixan can be illustrated very 
 elegantly if the variation problem is turned into an ordinary minijnum 
 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 p.rs cor.eidered and 
 therefore nothing general c:in be -concluded about the sign cf the second varia- 
 tion; nevertheless, with a Judiciously chosen deflection system the 
 question can be answered \-rlth. 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.I) 
 
 This is a peculiarity of the problem. In general., the critical load 
 depends upon the boundary cond-" ':ior3 — 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. 
 
 - 
 
MCA IM No. 1138 17 
 
 -en--^ (5.1) 
 
 
 - = ^=^f''^*1 
 
 Then the minimal 
 
 condition 
 
 A. 
 
 n = 
 
 I 2 
 
 ^ [ Tux ^ ^M dx + 
 21-/ \^ 2/ 
 - 
 
 = 
 
 2t - 7-5- (g " e*) + 
 
 321 kl 
 
 — ■ / Wy„^ dx - - e 
 o 
 
 ,2 p , 
 
 € = mlnlmijm 
 
 2 E 
 
 (5.2) 
 
 furnishes an equation for the "free value" f as a function of e. 
 
 arc f jr= Z' * . . TT^f 2 \ 
 
 
 €* - € + ^^ ) = (5.3) 
 
 3f 21^ V i^l 
 
 The relationship hetveen load and end displacement is obtained hy means 
 of the stress-strain equation (2.3) from the second of equations (5.l)j 
 it becomes 
 
 P Jt f / 1 > 
 
 i. = e - -r-^ (5.i^) 
 
 There are again the two possible solutions 
 
 f = 0, p = Ee ' (5.5) 
 
 and 
 
 f 5^ 0, hence — i- = e - e*, p = E€* (5.6) 
 
 The question of stability is answered (with the above-mentioned 
 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 
 
18 . MCA TM No. 1138 
 
 S^n = ^ (5f)^ .- 2 fe 5f5. -f fe (8e)^ (5.7) 
 
 must "be investigated. This expression is a quadratic form in 5f^ 8e 
 with the coefficients 
 
 a 2 
 
 * -e + 3 
 
 Ul2 
 
 
 af2 ~ 222 V^ 
 
 
 ■•■^^11 t^ 
 
 
 Bf^e 21^ 
 
 
 As 
 
 3.= 
 
 >.2 ^ 
 
 
 Since -■■ o > 0, this expression is positive definite (that is, 
 
 oe 
 
 never negative) as long as the discriminant 
 
 A = ^^-V^; =^(^-^-]^) (5.8) 
 
 is greater than zero. This is certainly the case belov the critical point 
 (e < e*); here the system is therefore stable. Cn the other hand, A< 
 for f = and f > 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 far f 7^ 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— 
 'Cji^eaaing the load. .•,•■; 
 
 2. If the end displacement e is prescribed, only the quantity f 
 need be varied and it is foimd that: ■' > 
 
 / 
 
 52n = fe (sf)^ = 2lu* -,.e + 3 ^1^ ) (5f )2 (5.9) 
 
 ^f- 2Z V Ifl^ / 
 
MCA TM No. 1138 19 
 
 From this relation Hfeildws^iSsniedf&tely that, •■- :.r; . -: , 
 
 for G < e* 5^ > 
 
 ']--■- ■::;-:■■ '■'- for <:> €^:': and f j^cO ,,5\6: >• G ; ^ . 
 
 -■■ - .. .■■-' ' - • ■ '•'' ''' z^ ' ■■■ ■:■ ■ ' ■ -'■■.■;■ 
 
 for 6 > 6* and f =:■ 5 U. .< . . •>. 
 
 that is, the straight position is staTjle for £ < c*, vmetatle for 
 e > €*, the tent position, as soon as it is mechanically possible (hence 
 for G > 6*)^ al-'.ra.ys stable. Figure 4 shows, the energy relationships in 
 this second case (prescribed displacement e = ae* of the right-hand end 
 
 of the rod).- -Plotted -as A^^ • ■-:^ ^ 
 
 or 
 
 ^i i,^_,/(,.,),£ 
 
 e*2 
 
 ^ :. . , •■ . / f \ V e ^ ■ "" ' ■ '■ 
 
 .as aftinction of f ( or | = — j -with e ! or a = — r } as a parameter. 
 ■ ■ " ■ \ 2i/ \ . e* y -^ - 
 
 (The_ potential of- the external forces is not included, beoatise it Is not 
 affected by the minimal condition vith reispect 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 = is a minimum, for a > 1 a maximum; the 
 curves for a > 1 have f^urther to the right also a minimum, whereby the 
 bent pc^sition f 5/ is characterized as a (stable) equilibriimi position. 
 This f igiire shows especially well the "type" change of the curves in the 
 transition from the sub— critical to the supercritical region: the coin- 
 ciding 'Of maximum and minimum for a = 1. It is also clear here that, 
 although at the moment of transition the displacements are email, the 
 behavior of the body at the stability limit is nevertheless determined 
 by the "possibility" of greater deflections, expressed mathematically, 
 by- the existence of the f— terms of higher order in the expression for 
 
20 "NACA 1M-W6, II38 
 
 COMECTION BETWEEN THE^ ' ORDIMEIY IITVESTICATION " OF STABI-LITy ' 
 Airo THE PEOCEDURE PRESENTED HERE 
 
 The ordinary statrlll'ty theory is limited to an investigation of the 
 critical point. It vas seen that the critical point is characterized 'by 
 two energy condition. The condition ' ' 
 
 ■':.: ■-• - , . :■■ bU ^ Q- ■ ■ ■ ■ -• • •- 
 
 ~f or ar^' variation • '6U; " Bv, 5v ' • " (6.I) 
 
 characterizes it as an equilihrium position in general; the condition 
 
 8^n = 
 
 for a characteristic variation 5u, 5v, 5w (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. Nov 
 in practical "buckling prohlems it is usiially a question of the transition 
 from a- very simple (often independent of the coordinates) initial state 
 of stresB to a comparatively very complicated one. It is therefore custom- 
 ary to specify the Initial, .state of stress directly without recourse to the 
 definition in terms of energy (6,l) and to proceed with the variation imr- 
 media,tely .in regard to the determination of the second state. There then 
 remains as the single iarportant condition the statement (6,1), which can 
 be ejcpressed. in the form of a me thod-of— procedure as follows, for example: 
 Consider a system of infinitesimal distortions superimposed upon the 
 critical state of deformation, collect the parts of the potential energy 
 n quadra-tic in the added displacements and set the sum equal to zero. 
 That such a procedure is at all possible rests upon the fact that as a 
 result o.f the "large" initial stresses two types of quadratic term arise: 
 an'(£ilways positive) part that represents the work done by the stresses 
 caused "by the added displacements, and a second part that comes from the 
 work done "by the stresses already present upon the quadratic part of the 
 added displacements. (See reference 2.)^ 
 
 The fact that this statement-of -procedure concerning the vanishing 
 of the quadratic members Is nothing more than the extended principle of 
 virtual displacements ("extended" in the sense of the statement ahout 
 S H) does not come out clearly in the applications mostly for three 
 reasons . 
 
 In equation (if.2i), for example, the last t\m terms represent the 
 first type and the first term, the second type. 
 
MCA TM No. 1138 21 
 
 1. Since a confusion with the very simple initial state is in general 
 not to te feared, it is possible to dispense with the designation 6u, 
 5v, 5w and write more hriefl.y u, v, w for the added displacements. 
 This manner of writing does not express the fact that the added displace- 
 ments are to he not only small j.n the sense of the general hypotheses of 
 the theory of elasticity hut 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 hut directly wilii 
 the energy changes (appearing as tlie result of u, v, w) and to designate 
 these changes hy^ A, _V instead of hy ^A, SV; the (extended) principle 
 of virtual displacements hecomes in this manner of writing 
 
 A^ + V'= 0, or even -Aj = A„ V (6.3) 
 
 —J, ■ T— '• —74. "^d 
 
 since the potential difference —X also represents the work of the 
 external forces on the infinitesimal displacements u, v, w. Equations 
 (6,3) can he put into, words ^ follows: For t^ie virtiial displacement u, 
 V, V, . through which the original equilihriuii conf igiuration goes over 
 into the neighbor Ing ("buckled") configuration at the critical point -the 
 internal energy . Aj^ • talcen up by the system is ' equal' to the work done hy 
 
 the external forces Ag, taking into accovint the terms linear and quadratic 
 In^ u, V, w. By this formulation the two conditions (6.I) 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.I) 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 (6.2) and the added requirement . 
 
 2 ' ••.-■• 
 
 6 n > for all other ^u, Sv, 5w . 
 
 tire condition 
 
 ■■■"/'f- ,, ■ •. ■ 5(5^11) = . {6.h) 
 
 To distinguish them' from those used earlier,' the quantities usually 
 wrlttep ■ A^, -Ag^, V'; are designated by Af;, A^^, V. 
 
 The. second form of the law (6,3) thei-^fore does not represent the 
 special energy law A^ = Ag^, 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. 
 
22 - • KACA TM Ifo. 1138 
 
 If it is agreed to consider in A^^, A^,, y_ only the (alone essential f or 
 the critical tehavior) quadratic terms, the condition •(6» is -written 
 in the form 
 
 S(A. + Y) = 
 ■ S(Ai - A^) = . . 
 
 This form, which is only a natua\'5l consequence of the original agree- 
 ment to write u, v, w instead of 5u, &v, ?5w, makes it quite clear to 
 what extent the simplified maniier of i^nriting can lead to conceptual errors. 
 For the statement (6.^'), aside from the deceptive formal agreement, has 
 nothing to do with the principle of virtual displacemente (l.2) or (1.3)5 
 The principle (l.2), in content the same an 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 inexpedient^; equation (6.^1-'), on the other hard, 
 in content the same as the minimal condition (6.'+) concerning the behavior 
 of the quadratic terms at the stability limit, gives the second equilibrium 
 position possible at the branch point and the sought-for value of the load 
 at which the equilibrium begins to be many— va.lued . 
 
 The difficulties so far discussed ■'.^re difficulties in interpretation 
 arising from the symbolism of •\'?ritinc. 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 fcr 
 the rod ttare were two independent equations (^,^), with tlie likewise 
 independent boundary conditions (*i-.5)j for the two added displacements 
 6u, 6w (which here would have been iinritten u, w) . From them it was 
 concluded that li vanished identically. This result — and correspondingly 
 u = 0, V = in the case of plates — makes possible, when (as is tacitly 
 done in the stability theory of a bar) it is presupposed as Imo'tvn, a treat- 
 ment of the problems of bar and plate btability deviating from the general 
 methods of stability theory depicted above. Since, however, bars and plates 
 are the most well-known problems, being analytically the most tractable, 
 frequently ideas that were developed there are erroneously brought into 
 
 ^So, for instance, for the compressed strut below the critical point 
 t-VTice the external work can be written in three forma: E"^, pc, P^/E ~ 
 which is to be varied (with respect to e.')? The' second form is meant, 
 but as a result of writing 2Ag^ in the place of -^T that is no longer 
 
 uniquely discernible. 
 
MCA TMUo. 1138 23 
 
 more genera.1. stability problerae. It ia therefore necessary to examine 
 • more tlio:rou:<hJ.y the various — special ~ interpretations that can "be given 
 to the occuiTeuces at the stability limit in the case of hars and plates. 
 
 First of eJ.1 outline the method hy which it is necessary to proceed 
 according to, the directions formulated at the beginning of this section. 
 If it is asRuaied a virti,aal displacement u, w at the critical point, 
 then, as can easily be seen^ the strain of a fiber to terms of the second 
 order is given by 
 
 .2 
 
 ., ■ ■ ■ Vx _- ■ 
 
 . ■ ^x = "x + *r"' ^''^x = ^x ' ^"^xx 
 
 Therefoz-e the terms of second order are: in the work done by the added 
 
 stress • 
 
 pr'p 2 2 
 
 /// ^ (u^ ^--r^-x) dx dy dz 
 
 jj- 
 
 in the work done by the already present (critical) compressive stress p 
 
 - / / / p dx dy dz 
 
 J.J J 2 
 
 (The external force does work - Pu(l); this makes no contribution of 
 the second order. ) After integration over y and z there results 
 
 I I I 
 
 Ai = y( j ^^ '^^ + i^ J ^xx^ -^ " E J ^x^ ^) -^a = (6.5) 
 o • o-' . .0 ' ■ 
 
 and the conditions (6.3) and (6,^') become 
 
 ■'■■. ^ / Ujj^ dx + i^ / v^^ <^ ~ 2 / "^x^ ^ ~ minimum = (6.5*) 
 
 The term u^. /2 goes out in the expansion of, the radical (see 
 
 1 ; ■ ■ Wyx 
 
 equation (2.2)), and the expansion of the curvature — = 7 ::: :::: — ^n 
 
 would give terms of the third order. 
 
2h MCA OM Ko. 1138 
 
 The expression coincides perfectly with the earlier expression (^.2i); 
 therefore the ssjne differential equations and boundary conditions and es- 
 pecially the result u = aro obtained entirely independently of whether 
 a motion in the x-directipn of the right-hand 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(l) = or u^(^) = O) makes 
 
 possible the two following "customarj'"" intei-pre tat ions 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 silso in 
 the interior. Hence there is superimposed upon the straight position 
 w = a purely transverse displacement as a variation, keeping in mind 
 the presence of the still unknown longitudinal compressive stress -EeQ. 
 
 According to equation (2.2), as a consequence of the change in length 
 connected with the transverse displacement, the following stretching 
 energy is released 
 
 I I 
 
 Ai = J (-EF€^) e^djc : 5— J w/ dx 
 
 o 
 
 at the same time a bending energy 
 
 EJ 
 
 2 
 
 % = -T , Mxx ^ 
 
 '^ 
 
 must be added. This interplay between the two types of energy _(and hence 
 the taro equilibrium positions) takes place ■vAen the values of A^ and %:, 
 
 are numerically exactly equal; that is, when 
 
 X X 
 
 h = T l-^ J ^xx^ ^ - ^0/ ^x^ ^ ) = (6.6) 
 o o 
 
 EF 
 
 From the additional condition 8Ai = there follows as above the sine 
 equation, for w. This procedure thus leads to the correct end result 
 without, however, permitting. a guarantee cf really having found the mini- 
 mal buckling load. For the "restraint" assumption u = limits the 
 mjmber of possible variations, and that it leads to the correct buckling 
 load for the rod (and plate) requires at least a supplementary verifi- 
 cation. 
 
'haca tm ko. 1138 .25 
 
 More iHrportant - "because in a still more special way a peculiarity 
 of the rod aiid plate — is another manner of thinking, which ia almost 
 uniTersally made the "basis of deriTation of the "buckling equations. "With 
 reference to the natural "buckling process, the houn.daries are considered as 
 mova"ble; however - and this is the characteristic Eiark 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 hJ-gher 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 dra-ing 
 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 the ends must 
 approach each other by an amount '■ 
 
 I 
 
 111 = --^J wx^ ^- (6-7) 
 
 (which in fact is of the second order in w!), so that the external forces 
 
 Po^ r 
 
 do the work - Pq^^I ~ ~Z' / ""x^ '^^ ^°^ ^7 formulating the equality 
 
 of inner and outer work (wherein by inner work is to be understood only 
 the bending energy) 
 
 I I. 
 
 EJ p EFeo r 
 A^ = Ag_ or — / w^^j^ dx - — — / w^.^ dx - 
 
 . 2 ■ Q 2 ~^. 
 
 and assuming as above the minimal property of thi? expression, this pro- 
 cedure leads to equation (6.5') - naturally likewise without the u-term,^ 
 
 In the assumption (6.7) there is an inconsistency: It cannot be 
 assumed a priori that a strut that changes its length elastically below 
 the buckling limit suddenly ceases to do so beyond it. (in reality it 
 changes its length by quantities of higher order.) It is more logical 
 to consider a perfectly incompressible - rigid against extension but 
 elastic in bending - strut, for which the two hitherto independent dis- 
 placements u an.d w are related from the beginning by. the (geometrical) 
 assumption ...• . 
 
 Wy 
 
 ^'■x + -g- = (6.7») 
 
 (Continued on p. 26) 
 
2g NACA TM No. 1138 
 
 Since the procedure of equating the stretching energy to the external 
 work cannot he 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 w; that is, consider u, v not 
 as really independent virtual displacements hut as connected with the 
 transverse displacement w hy the order- of- magnitude condition 
 
 Vl^ IV \T^ (etc.) 
 
 According to equation (2.2) there is chtained fcr the vork done "by the 
 
 .cal stresses 
 
 the "second order" 
 
 critical stresses a^, a , t on the displacements u, v, w to terms of 
 
 2 , 2 
 
 Ai = c^s 11 (ux + ~j d.x dj + OjB Jj (^vy + — J dx dy 
 
 + rs I i ( % + ■''x ■•" ^x^y ) djc dy ■ ■ • (6.8i) 
 
 the tending energy is, as always, given "by 
 
 ^" ^ 12(1 ■ v^) jJ I'' ~ ^^-"'V^ ""yy " V;j^ ^ ^^-^^^ 
 
 (s = thiclcness) 
 
 (Continued from p. 95) 
 
 Such a strut permits no deformation at all below the critical load; ahove 
 it takes on only bending energy, which is furnished hy the externa], work 
 
 1 r' 
 
 Since up to the critical load no elastic deformation at all has taken place, 
 the two laws 
 
 Ai + V = b{A^ + 1) = ^ and 5"(Ai + V) = 5 
 are here in content completely identical. 
 
 b (Ai + V) I = 
 
NACA TM Ifo. Xi38 27 
 
 ■ Both parts together must be eq,ual to the Vorlr Ag of the external force 
 
 in the sens.e. of eq^uation (6.3). The external work can now (and this is 
 the essence of Reissner's idea) he expressed generally in a ^^ery siinple 
 manner, if it is remeAeied thatthe s.traight position is an equilihriim 
 position and that therefore in every virtual displacement A^ = A^. On 
 
 taking the special displacement u'*^ = u, V* = v, w* = (with the bound- 
 ary displacements tl* ="u^,' V|,* > y^) , A^ = A^*j therefore 
 
 and" now on collecting terms in (6.8), the u~ and v-terms cancel out; . 
 there resiilts the well-known Bryan plate e^quation (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 
 process "bj a boundary displacement. Its dleadvantiage is a double one: 
 The emphasizing of the boundary, displacements g^-ves the impression that 
 the participation of the external \7ork is universally important in a 
 btickling 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 knovm 
 somehow beforehand; ^ therefore the interpretation of the external work 
 Cj-Uj. ... as a contribution of the second order is not transferable to 
 
 more general buckling problems. (See reference 6.) 
 
 To summarize briefly the resiXLt of this section: In considering 
 the critical point it is- customary to dispense with the correct method 
 of tn:'! ting the virtual displacements 5u, 6v, 6w in favor of the more 
 convenient u, v, w; thereby the connection between the customary 
 atabj.lity criterion and the principle of vii'tual displacements is con- 
 cealed. To. be added is that the stability problem of the rod and of the 
 plate permits a special treatment vrhich rests upon the. fact that at the 
 critical point the tangential displacements ii, v and the normal 
 
 ■ . The very obvious oonclvision, that can Just be seen from the form 
 (2.2) of the strain that vl-^ and w^^^ must be of the same order, is 
 
 not tenable; for an equation of the typ© (2.2.) holds, for instance, also 
 for the longitudinal fibers of a cylinder, and yet here u^^ and w^^ 
 
 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. 
 
28 HACA TM No. II38 
 
 displacement w are of different orders of magnitude. Since, however, 
 tlie rod or plate prol^lem, 'as the analytically simplest, is at the same 
 time- the test laiown, the need easily arises of 'transfeiring methods of 
 thinking successful in these prohlems to more complicated problems, which, 
 as ira,s to he shown,, is not possible. 
 
 IBS DURCHSCELAG - PROBLEM (F THE SLIGHTLY CURVED BEAM 
 
 . ■ In section Stable and Unstable Eqxiilibrium, it was shown that for 
 the EiAler strut the instability point (defined by 5(5'^II) = 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 a.e are the brar-ohing 
 problems, comprises the so-called Durchschlag 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 (inf initeeimally 
 close) equilibrium positions. Above the critical point an increasing 
 displacement is in general accompanied by a decreasing load - the state 
 is unstable, the system "snaps" into-'- or falls into a stable, configuration. 
 ■Prerequisite for such a phenomenon is a nonlinear relationship between ■ 
 force and displacement even in the s -table 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 
 ■transverse force Q is a shortening of the axis of the arc, as the 
 result of which a horizontal force H is made to act. Because the 
 effect of this (very large) conipressive force upon the equilibrium of 
 forces in an element of the beam cannot be neglected, there arise 
 phenomena related to the buclcling process in the Euler strut, iiista- 
 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 s-fcrut compressed beyond 
 the Euler limit at the least dis-burbance snaps or falls into the bent 
 position. 
 
MCA TM Kq. 1138 ^ 29 
 
 By a process that follows very closely that carried out in section 
 Energy of Peformation, is ohtained, vith the notation of figure 5 (w 
 now taken positive downward), for the potential energy 
 
 m 
 
 n = — ■ 
 2 
 
 I 
 
 W^j \ 2 2 
 
 dx " Of (7,1) 
 
 From 
 
 SIT ^ EF J^^u^^"^ -. w^w^y &u^dx 
 
 + ^ / [(^ + ~- - Wx^x)(^x - ^x/^x + i^^xx^^xx J ^^ - ^Sf 
 
 o 
 
 there are obtained the two equilibrium conditions expressed in terms of 
 the displacements: 
 
 S / WjS ' \ 
 
 or, integrated once, 
 
 V "x ■*■ ~^ ^x^x)~ constant = - h (7.2) 
 
 and, with the use of equation (7.2) 
 
 i\xvx + l^^xx = h¥_ (7.3) 
 
 also the "boundary conditions 
 
 u(0) = u(z) = w(0) = w(Z) = v^(0) = V:^(l) = 
 
 Vy^{l/2 + 0) - W3^(l/2 - 0) = q/EJ (7.4) 
 
30 MCA TM No. 1138 
 
 which together with the continuity reguirements for 
 
 u, u^, V, w^, v^ 
 
 at X = Z/2 give the 12 conditions that are necessary for the evaliiation 
 of the 2x6 constants of integration in the 2 regions x < l/2. 
 
 The physical significance of the constant h can he recognized di- 
 rectly from (7.2): On the left-hand 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 equilihrium condition that H 
 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 w; that is,_ the system (7.2) and (7.3) is nonlinear, 
 (See equation (3.^) or (3.2)). Nevertheless, Just as hefore the exact 
 solution can he given in this simple case without difficulty in terms of 
 the at first unknown 
 
 • . . .. 2 -Z^h H EJir^ 
 
 a = 
 
 H* = 
 
 nx 
 
 for example, for W = f^ sin 
 
 L 
 2 3 OUTX 
 
 QI sin-p ^^ 
 
 ^ = ^o "i — - ^^ -- + 7T-^r — ^ — r ^"^'^^ 
 
 a - 1 I 2n'^arEJ cos -^ I 
 
 (x < 1/2) 
 
 and from (7.2) hy another integration taking into account the "boundary 
 conditions u(0) = u(Z) =0 there is obtained subsequently a transcen- 
 dental equation for the dependence of a upon Q and f^ of the form: 
 
 ^ = \ J ( w.w. - ^ W (7.6) 
 
. .KftCA.a-M No. 1.138 
 
 31 
 
 The determination of the critical load can be carried out in txro- 
 "basically different ways. 
 
 The first method, (see C. B. Biezeno, reference 1, pp. 21 ff) proceeds 
 from the condition SQ/3f = 0, wherein the relation between Q and f 
 is established by equations (7.6) and (7.5) for x = l/2 
 
 w(z/2) = f = f 
 
 a 
 
 QI 
 
 + — 
 
 ° a^ - 1 Qn^a^SJ 
 
 
 iraX 
 2 2 / 
 
 (7.5') 
 
 A second method proceeds by way of the energy criterion (^.3). 
 For S^n is obtained from equation (7.1) after writing for brevity^ 
 
 <^ + ^-Vx^x)+(^x-^x) j=^-[ 
 
 H / . 
 
 Vy - w. 
 
 EF \ ^ 
 
 X 
 
 2 n 
 
 = '-Lj(x,a) (7.7) 
 
 ■fee expression 
 
 . " , I 
 
 B^n = EF 
 
 |l5u^) +2lw. 
 
 W3,J5U3,8w^-<o(x,a)(5v^) + i t^5w^) 
 
 dx 
 
 (7.3) 
 
 The particular displacement system 5u, 5w by which 5*^11 is just made 
 equal to zero is obtained from 
 
 6(S^II) = 
 
 that is, from the two homogeneous differential equations 
 
 S r 
 
 S%x + 3- [1, % - ^x j ^^x 
 
 = 
 
 and 
 
 Sx 
 
 ( wx - W35 j 5ux . ••■ ^ j i^wxxx + a)(a,x)5wx I = 
 
 (7.9) 
 
 Therein w is at first according to (7.5) a function of the two 
 parameters a and Q. Q for example, is considered as eliminated with 
 ■the help of equations (7*6) and (7.5'). 
 
32 . MCA TM No. 1138 
 
 \Tith the homogensous, 'boundary conditions 
 
 5u(0) = 6u(Z) = ?>w(0) = Sw(Z) = Sw^(O) = ^^XK^''-} = 
 
 The desired critical a-value is the lowest eigenvalue of the equations 
 (7.9). 
 
 APPROXIMATE EETEEMHTATION 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 hut is well suited - and therefore that procedure 
 vrill "be considered here — to an approximate treatment "by the method of 
 Eitz or Galerkin. ' ' .' 
 
 This procedure can he started at either of two points: either, 
 maie a Eitz approximation for 5w in (7.9), determine the correspond- 
 ing 5u from the first of equations (7.9), and following Galerkin from 
 the condition S(5 H) = 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 Eitz 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 
 
 itx 
 
 W = f „ sin 
 
 I 
 
 is chosen and as a Eitz axpression 
 
 jrx 2jrx ,„ . 
 
 ■ w = f 1 sin — + f 2 sin (8.1) 
 
 L L 
 
 then all boundary conditions are fulfilled, except for the one dis- 
 continuity requirement (7.^), 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 
 H from (7.6), the integral in (7.1) can be evaluated and II is 
 obtained as a function of the amplitudes f^, fi, fz or the dimension- 
 less parameters 
 
KACA TM Ho. 1138 33 
 
 f f f 
 
 Ao = -^, >vi = -— , ?\2 = -- (i = radius of gyration) 
 i i i 
 
 in the simple form 
 
 4 2 2 
 
 where 3 2 
 
 <i = r~ -4T3 ~ = '~^"~ rr^ - ~f^ zr ^0.2^ 
 
 ^o Jr i IF n"f o EJ^r n f o H* 
 
 The equilihrium equations read 
 
 an 
 
 I Z' ^1^ gX 
 
 ^^- = 2?^i + (>^o - ^1) (^^0^1 " -~ - ^2 J- qho =^ 
 i^ H 327V2 - 4^2 Ao^i - -J^ - 2>^2^V 
 
 (8.3) 
 
 Th^ are in the two unknowns A^ and ^z ^^^ of the third degree; 
 nevertheless a complete discussion is possiMe without numerical calcu- 
 lation, "because the second equation may be written as a product 
 
 ^2(Vi--~-2V-8) = 
 
 Therefore the cases can te. -.distinguished 
 
 ^2 = 
 
 27.1 + {\ - AJ fvi - -2~ ) = <l^o 
 
 and 7^2 ?^ 0, that is, " ' 
 
 27\2^ = "Xo^i -K^/2 - 8 
 
 (8» 
 
 (8.5) 
 
3k MCA* TM No. 1138 
 
 the discussion of which no longer requires any laTjor. The first system 
 provides a symmetrical deformation, the second a superposition of a sym- 
 metrical and an antisycmetrical deformation. The corresponding horizontal 
 compressive forces "become 
 
 ^^,^l£:-.zbm (8.!,.) 
 
 Eu = H^ -^-^ ~ ^ = in* (8.5M 
 
 The critical load q^rit ^^ found from the condition 
 
 5^11 = — ^ (5AJ^ + 2 — — - {^\ b\),+ -— ib\f = (8.6) 
 
 b H 
 As long as - — — and the discriminant 
 
 A. 
 
 are greater than zero, 8^11 as a positive-definite quadratic form in 
 5^1 and SAo cannot he zero for any comhination of these two variahles, 
 A''anishing of the discriminant characterizes the pair of values A^, ^g 
 for which there is exactly one combination Sh^, oAg for which 6211 
 hecomes zero hut none for which it is less than zero. The condition 
 
 ^2^ -..2^ ^ -^2'^ -^2 
 
 h n S n /^ a n ' 
 
 2 2 y , = . (8.7) 
 
 gives, therefore, the stahility limit and together with the two equations 
 (8.3) determines the three unknoTTns Ai, ^2, and q^vit* -^^ this case 
 (8.7) reads 
 
mcA m No. 1138 
 
 35 
 
 k\8 ~(\o-hx ----- 2^2^)+ i^Tv/j [2 + (-Ko -y^if - (^o>^i 
 
 2/ 
 
 + 2^2^ I - iShsH-h^ - >^i)2 = 
 
 vhich condition becomes 
 
 >.2 = 
 
 vith 
 
 
 \ -] r ? / 
 
 2 /. 
 
 I = 
 
 and with 
 
 
 " hafw - V/2 -s) = 
 
 There are therefore two sets of values for Tvi, >2, and q^j-it 
 
 >^2 = 0, 7^1 = ?\o - — r. y ^0 " - ^; <lcrit "^ ^ \ \ 
 
 V ^ Q \ J '^ 
 
 + .iV V.:i_^N'^/^ 
 
 (8.S) 
 
 ^2 = 0, ^1 = "Xq - -Ao^- 16, 
 
 Icrit 
 
 = 2 - I- v^"^o2 - 16 
 '^o 
 
 (8.9) 
 
 "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 ahove the critical load will 
 not he pursued in detail (see reference 8). (See fig, 6.) There the 
 load Q is plotted against the deflection fi of the point of application. 
 
36 'MCA TM No. 1138 
 
 with the initial amplitude f^ as a parameter. For fg/i = T^q - 2 Q 
 increases mono tonically with fx; instability is iaot possible. In the 
 region ■ 
 
 2 < >^o ■S ^22' ^" ■ (8.80 
 
 under the critical load given by (8.8) there enters definite instability^ 
 increasing deflection fi without increase of load* Accordingly the 
 beam sns-ps — under constant load ~ until it finds a stable conf igura'- 
 tion-"- at ^1'. Since ^^i' > 7\q the beam is now convex downward; it also 
 can be seen clearly that the system now must be stable with respect to 
 an increase in load; a further deflection results in a longitudinal 
 pull. (See equation (8.^').) 
 
 For 
 
 >Vo^,/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 1+H*, that is, the second Euler load, under which the 
 strut assumes the S-shape configuration Tvg jf'O. It snaps again into 
 a stable position Ai', this time, however, passing through an un— 
 S3niimetrical deformation. At the critical load there appears a branch- 
 ing of the elastic equilibrium; figure 6 shows the two bremches of the 
 Q - f 1 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 (^,3) to a stability 
 ■problem of an entirely different kind. 
 
 That is^ at first vibrates about ?vi' as a stable equilibrium 
 position. 
 
KACA m No. 1138 37 
 
 HEFEIEKCES 
 
 1. toadbuch der Physik. Bd. 6 (Berlin), pp. 70 ff., 1928.;: 
 
 2. Trefftz, E.: Zur Theorie der StabUitat. Z.f.a.M.-M, -Bdv^lS, 1933, 
 
 p. 160. 
 
 3. Pbschl, Z. B. Th.: Ufa er die Minimalprinzipe der Elastizitatstheorie. 
 
 Bau - Ing. Bd. 17, 1936, p. I60. 
 
 h. Marguerre, Karl: The Apparent Width of the Plate in Compression, 
 
 MCA m Wo. 833., 1937, 
 
 5. ferguerre, Karl; Uber die Behandlung von Stabilitatsproblemen mit 
 
 Hilfe der energetischeA Methode. Z.f .a.M.M. Bd. 38, 1938, 
 
 pp. 70 ff . 
 
 6. Reissner, H. : Z.f .a.M.M, Bd. 5, 1925, p. ^75.' '' 
 
 7. Biezeno, C. B.: Das Durchechlagen'eines schwach gekrummten Stabes, 
 
 Z.f. a.M.M. Bd. I8, 1938, p. 21. 
 
 Brazier, L. G. : The Flexure of Thin Cylindrical Shells and Other 
 "Thin" Sections. E. & M. Ko. IO8I, British A.R.C., 1927. 
 
 Heck, 0. S.: The Stability of Orthotropic Elliptic Cylinders in Piire 
 Bending. NACA W Wo. 83^, 1937. 
 
 Welnel, E.; ttber Biegimg und Stahilitat eines doppelt gekrunmten 
 Flattenstreifeus. Z.f. a.M.M. Bd. 17, Eec. 1937, pp. 366-369. 
 
 8. Marg-uerre, Karl: Die DurchschlagBkraft einee schvach gekr"ijnmten 
 
 Balkens. Sitzungeberichte der Berliner Mathematischen Gesellschaft, 
 Bd. 37, June 1938, pp. 22-40 
 
38 ' WAGA TM No. 1138 
 
 TRAITSLATOR 'S HGTES 
 
 Eq-uation 3.7, last peurt : ' . , .- . ■ .-. 
 
 Trahe^, • note : ; It appears that this equation, should "be . . . 
 
 .: ... . ':> • n f 2 • 2nx . ■ f 2^2 . . 
 
 Us- fc"% sin • ^— r X ■ ■" . . . • 
 
 Equations (7.5) and (7.5') " 
 
 Trans';- note: It appears.: that, the term ■- — . should he — :;r-:i — , 
 
 . ; . . . ' • 2jt2a*EJ 2n^a EJ 
 
 Equation (8.8) 
 
 1 /A/-l^\'3/2 • ■ 
 Trans, note: ■ It appears that 2 + --- ' ■ j should "be 
 
 2-^ V^'Ao^-^ 
 o 
 
 Page 2h 
 
 ■ ' ■ 'EJ r" 2 
 
 Trans . note : It appears that A^^ "^ "p" / '"'^xx ^ should he 
 
 o 
 
 . . . ■''':''!.■. .V EJ r „ 
 
 o 
 Page 36 
 Trans, note: It appears that (8.^+') should he (8.5*). 
 
MCA TM Sfo. 1138 
 
 39 
 
 z w 
 
 ^Vl)=^i 
 
 ligax*e 1.- Strat under compression. 
 
 ii 
 
 it 
 
 1 
 
 e/ e* 
 
 a 
 
 0X23 
 Figure 2.- Amplitude f against e. 
 
 1 2 
 
 Jigure 3.- Load P against e. 
 
il-0 
 
 MCA TM Ko. 1138 
 
 Figure 4.- Variation of energy of deformation 
 
 with the amplitude f =, 2i|. Edge 
 compression e = ae* as a parameter . 
 
 Mom.ent of inertia J, Section I 
 
 Radius of gyration i, Elasticity modulus E 
 
 ar^ 
 
 figure 5.- Slightly curved team -under 
 transverse load Q. 
 
K4CA I'M Ho. 1138 
 
 41 
 
 o 
 
 03 
 OJ* 
 
 II 
 
 1 2 
 
 3 4 5 6 
 fl/i=Al 
 
 Pigore 5.- Variation of the load Q with the 
 
 displacement fx »f the point of 
 application, Parameter = initial amplitude 
 
 fo- 
 
CO 
 
 O 
 
 ft 
 
 n 
 
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