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Large deflections and imperfection sensitivity of structural systems Stephenson, Norman G 1971

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LARGE DEFLECTIONS AND IMPERFECTION of STRUCTURAL SYSTEMS SENSITIVITY by NORMAN G. STEPHENSON B.A.Sc. Univers ity of B r i t i s h Columbia Vancouver, B r i t i s h Columbia, Canada, June 1968 A. THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the department of CIVIL ENGINEERING We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1971 i i . In presenting this thesis in partial fulfilment of the require-ments for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Civil Engineering The University of British Columbia Vancouver 8, Canada Date July 1971 1 ABSTRACT The sensitivity of a structural system to init ial imperfections is known to be largely determined by the post-buckling behaviour of the ideal system and further the rate at which the load capacity of the ideal system changes near the crit ical load. A general non-linear stiffness matrix capable of predicting the post-buckling behaviour of structural systems is developed and tested. A procedure is formulated for predicting the stable equilibrium zones of structural systems and examples are investigated to verify the correctness of the method. iv. TABLE OF CONTENTS Chapter Page I. INTRODUCTION 1 II. BENDING STRAIN ENERGY 3 1. Co-ordinate System 3 2. Displacement Field 3 3. Evaluation of Strain Energy 5 I I I . TOTAL POTENTIAL ENERGY 9 1. Strain Energy 9 2. Potential of Loads 9 3. Total Potential Energy 9 IV. DERIVATION OF ELEMENT STIFFNESS 12 1. Force-Displacement Equation for an Element 12 2. Linearization of Force-Displacement Equation 12 3. Evaluation of Equation (4.3) 13 4. Check of kp and V* 17 V. FORMULATION OF PROBLEM 18 1. General Case 18 2. Special Case of a Symmetric System 23 VI. DERIVATION OF EQUATIONS 24 1. Total Potential Energy 24 2. Potential of Loads 27 3. Strain Energy 29 TABLE OF CONTENTS - (Cont'd) Chapter Page V I I . EXAMPLES 35 1. E u l e r Column 35 2. Pitc h e d Frame 37 3. Right-Angle Knee Frame 39 V I I I . CONCLUSION 44 BIBLIOGRAPHY 45 v i . LIST OF FIGURES Figure Page 1. Co-ordinate System 4 2. Generalized Co-ordinates 11 3. Generalized Configuration Space 21 4. Equations (6.11) Evaluated 33 5. Plot of _L ^ " t f vs C t / 34 6. Euler Column 36 7. Pitched Frame 38 8. Graph of vs -e- 40 9. Right-Angle Knee Frame 41 NOTATION The s p e c i f i c usage and meaning of symbols i s defined i n the text where they are introduced. ACKNOWLEDGEMENT I am most grateful for the guidance and encouragement given by my supervisor, Dr. N. D. Nathan. Financial assistance from the National Research Council of Canada is thankfully acknowledged. 1. CHAPTER I. INTRODUCTION The relevance of a theoretically-determined critical load for any structure depends upon the sensitivity of that structure to imperfection both in geometry and in loading. The present work studies the problem of structural instability, in the context of non-linear elasticity where the non-linearities are kinematic and arise from finite displacements, with a particular view to predicting the behaviour of the imperfect structure on the basis of the post-buckling behaviour of an ideal structure. As discussed by Koiter (1, 2), Budiansky and Hutchinson (3) and Hutchinson and Amizigo (4) the real structure will be imperfection sensitive and the critical load will never, in fact, be reached i f the load capacity of the ideal structure decreases after the bifurcation point. But, i f the load capacity of the ideal structure continues to increase after bifurcation, the real structure will be insensitive to imperfections, and more important, the theoretical critical load will be an acceptable indication of the point at which the displacements become large or the structure unstable. Roorda (5), on the basis of Koiter's work, developed curves defining a stable equilibrium zone and conducted qualitative checks of them. Britvec and Chilver (6) studied experimentally the buckling behaviour of triangulated continuous plane frames and concluded that an unstable equi-librium path may exist in the initial post-buckling range. 2. The purpose of th i s thesis i s to develop a s t i f f ne s s matrix that contains s u f f i c i e n t non- l inear i ty to r e f l e c t these phenomena and to formulate a procedure for determining the stable equi l ibr ium zones fo r any s t ructure. CHAPTER I I . BENDING STRAIN ENERGY 3. 1. Co-ordinate System The co-ordinate system chosen i s shown i n Figure 1. The X and " Y are Cartesian reference frame for an element. S S , and s s 5 are t o t a l nodal rotations (positive counter-clockwise) of the element from i t s o r i g i n a l s t r a i g h t position to i t s displaced position and SSg. i s the t o t a l l a t e r a l displacement of the centroidal axis at one end measured perpendicular to the or i g i n a l s t r a i g h t element. 2. Displacement F i e l d The usual Hermitian polynomials are used fo r the transverse displacements of the centroidal axis of the element. The following i s the equation of t h i s curve: " Y - L C x - t x 1 -vx 1) s m SS, + ( i x 1 - 2.x5) s<su + L ( - x t + x i ) «Si'*x s s 5 (2.1) I f we now introduce the symbols Y } * f° r the old Y , x and Y * x * new dimensionless co-ordinates are defined Y = } X =» — equation (2.1) may be written more conveniently as: " Y = (X-£X l+X B) -siwSS, + (^X^-IX*) S5X + (-X t4X 3) U (2.2) The slope of the centroidal axis i s the spatial derivative ^_ ^x of this curve, obtained from (2.2) as Z^f = (\-Ax+3X. ,-)'siv>«SS, * ( 6 X - 6 X1 ^ «S5, . ^ C ' CO-ORDINATE SYSTEM FIGURE 1. 5. clearly -^>f Fx as required. 3x X * I It will later be shown that it is desirable to consider the element to be inextensible, thus an expression for X c a n D e determined from: X = X - 1 f fy?}Z <^X (2.3) where the first term is the co-ordinate of the undisplaced point on the centroidal axis of the element and the second term expresses the dis-placement due to the shortening of the projected length of the element which would occur during bending. From equations (2.2) and (2.3) X. can be found as: X = X - j (x - 4x l t u x 3 - u 4 + | X s ) s\n* *ss, - JL / 4 X * - 3 X 4 + 2. X 5 ) *iV. SS, - £. C x1" - «. x 4 + z. x* - 2. x5" ^  SS, S\v> . -+ ( X 1 - J i X 1 + 3 . X * - 3 . X M - S ^ - S S . ^ i ^ - S S , -V ^ ( | x } - i x 4 -* 2. X s ) ^ <x5 (2.4) 3. Evaluation of Strain Energy It is shown in Engineering Beam Theory that when curvature due to shear is neglected, the bending moment M and the curvature */R of the displaced centroidal axis of a straight elastic beam are related by (v\ = up to the elastic limit, where E. is Young's modulus and I is the moment of i nerti a. Substitution into the usual expression, XX - j gjJLx ^ s (bending strain energy for pure bending in one principal plane) yields XX = ^ ^^/^Rr)^^ • N°w' ^  P n m e s denote derivatives with respect to arc length s , have TI - i c; £x[(X")1+cx")1] <u ( 2 . 5 ) which is valid for arbitrarily large displacements provided that yielding does not occur (7). The fact that the element is assumed to be inextensible can be used to replace arc length s with x , where we obtain * - i c "[('SoM^n (2-6> It is pertinent to mention that the requirement that the element be inextensible is a reasonable restriction for most structures. The derivatives in equation (2.6) can be evaluated as follows: d^X. s - 1 (-4 + t l X - 3tX1"+ 18X J) •siv^'SS, * - L« (^x- 6x l -v 4 x* ) SS J" - JL ( 4 x - t S X ^ - v \ft ) s i ^ ^ S S j - .4 ( l - VOX -t-ZIXX-»71X*.)SSi sihSS, •V J_ / l - a ^ x - l - S A X 1 - ! ^ * ) s \'v> SS, -S » *-\ SS , + J> ( 4x - >5Xl + ^  x 3) ss,. -s\ VN S S 3 (2.7) - - i ( a - 3 x ) s\v> ss, d x . . v * £ t ( I - IX) SSz. - ( i - 3 x ) S\v> S S 3 (2.8) If define 0 , = - h. o 5 } = A ( i - % x ) 04 - i ( 1-IIX + \8X1-3X^) 0 6 = " I ( I X - 3 x S 9 x J ) 0-, = - ^ ( \ - l o X + H X l - \ l x 3 ) 08 - | i (» - H X + I 8 X J ) <(>^  ~ ( 4 X - I S X 1 " * 1 1 X J ) then equation (2.6) can be rewritten as "U. = Ex(^ | ^ 0 ( l S \ "vx t "«, -v 0J- ssj- + 0 3 L •+- 2. 0, 0 x S i VN SS , ' SS t •v 2- 0, 0 3 S\^SS,» Sv^SSj + £ ^flJj SS^  - I^wSS, + '0 + -si^SS, S i ~\ S* S» w SS «S \ *N * SS S \ v\ * SS 5VA S5 SS sst S \ V N «SS -ss + 2. 4^. 07 + 04- 08 + 04 03 + 4* •V 2- 05 0£ S v v v 1 SS + a 0? ^ sst3 •v a 05 G * 9 ssj + 0/ si^* ss3 + 2. 0< 03 siw* S53 • SS\ S \ v ^ ss. s\ ^ SS, S \ vv S5 ( "S \ w S S j S i v>, 5 S . S w \ SS Sn' v % t 5 S 8. + E. ^3 S S * ' s U « , + 2. 0 9 0 9 S'NVN S S , . s C - x ^ S S j - s s z * 0* *C ' ^ l ^ 3 1 cW (2.9) 9. CHAPTER III. TOTAL POTENTIAL ENERGY 1. Strain Energy The total strain energy is simply the strain energy of bending previously derived since the element is assumed to be inextensible and shear strains are suppressed in conformance with the usual Engineering Beam Theory. 2. Potential of Loads We suppose that the total system has n degrees of freedom expressed as r-t ( i i \} n ) in a set of global co-ordinates, which is common to the system. Due to the assumption of inextensibility, the Vj do not include axial degrees of freedom, and the potential of the axial loads will be introduced separately. 3. Total Potential Energy The total potential energy is defined to be "V -where -A. is potential of external loads andU is strain energy, giving for the complete system or 2 (3.1) where: F = vector of element axial forces (positive if compression) v\ = number of external generalized co-ordinates N = number of elements vr» = number of internal generalized co-ordinates 10. vectors of external generalized co-ordinates and forces respectively $ = vectors of internal generalized co-ordinates and forces respectively a = transformation matrix giving S = a £ We note that the elements of a are constant and unique for a given structural system, as a result of the definitions of the generalized co-ordinates: and f (ss) = f(£%)= transformation matrix giving S S = & (see Figure 2) \ o o o O v o - \ o o \ o displacement due to axial shortening of the projected length of the element which would occur during bending. From the previous equation (2.4) i t is evident that 47ss) = k s'wx^ss, v ± ss,1 - \s is Z. u. s ^1 ss »s - J_ ss, • «s\ ^ ss - _L ssA • s i vv sss l o — L_ ^ \ SS , • S "\ vs. SS -3o (3.2) is one component of •£(ss) . 11. \ GENERALIZED CO-ORDINATES FIGURE 2. 12. CHAPTER IV. DERIVATION OF ELEMENT STIFFNESS 1. Force-Displacement Equation for an Element The total potential for an element can be written as follows: V = - s s T g - P £(«s_0 + TI (sj) where P is positive for axial compression. For equilibrium we require that X^- - O thus obtaining }V = o = - $ - P (*j) -V (*Q ^ ss. " > *5 ) s s where ^ = generalized external forces for an element. Hence the force-displacement equation is f j = - P > - f + S~\X(Ss) } s j }ss (4.1) 2. Linearization of Force-Displacement Equation Most methods of solution require that equation (4.1) be linearized for arbitrary displaced configurations. This linearization can be accomplished by the use of a Taylor series expansion. Expanding f>; (ss) in a Taylor series about ss •= s_s° we obtain - ^ i ( « ° ) - * ii = 2.-, U ss • ^ ii (sj) ^ S S ss +• ••• (4.2) ss where it is seen that the second and subsequent terms can be neglected if we take Ass sufficiently small. Hence we have, retaining only the first term 13. & $ = U A ss A. ss ss" or where ss" (4.3) 3. Evaluation of Equation (4.3) It can be seen that the first term in matrix form is - P \j£c&) = - p wri tten where <- IS +- 1= 3o ( " s V v ^ s S , •S \ v \ SS j • S\w SS j - J_ to C O S « , - L 3o c o s S S 5 . cc-s SS, 4 -6 S L . - 1 IO C o s SS j \S r — s \ »-\ Z SS ^  +" COS**SS 3) •v v_ 3o s U ss"t • S i ^  SS j IO si v \ ss, (4.4) 14. w h e r e u II The s e c o n d t e r m 3 XX. (ss) >ss l w r i t t e n i n m a t r i x f o r m i s SS u k WL 15 k,, - E.T ) - 4 • s i ^ ss + _ E T _ ^ - i s " 4 1 0 5 l_ I . t cos ss , — 2. • s't^ SS •S\^v ss + 6 • s i^ ss t • ss,.  2.S  6 • s i v>1 SS , • S i ^  *" SS , +• T 68 • S »v. fc SS , • c o •s*" 55", + L l G ( - S ^ 1 ss -V C 0 S l S S , ) • SS t • SS,. L . l _ + 12. ( - S ^ 1 SS, -v c o s S s , ) • 5 » ^ 1 SS3 ~ 3feO ( - SS,) • S l w S S , • SS j - n o S t < A SS, • c o i l S S , • 5 £ i * ( - s; w*" ss,) • s \ ^  ss, • Sl'vx SS SS, . co j 1 SS, -V \*32. • si«^ SS-j • S iw - 4 8 ( - S»w>- SS,") • S£» . S»V SSj - 48 • SS X • S> A SSj • c o s * SS, u + n o - ss,. • sSi . ss,. - s i ^ ss, + c$0 - (- SSX^ . SS x . s i v v S S j - S \^SS, + 2 . 4 • SS i . s « *N 1 SS 3 • S \ wx SS , u - 31 . si^^SS. • S » S S j • S i SS , ^  k.. = E X B  ^ - 6 c o s ss,"^  E X \ o s t i l 4 3 L • SS_,. . S i v x S S , • c o $ S S , s m ss, c o s SS, - 48 • — 8 i-o S \ w S S • s s , . ^ \ *\ SS" , cos sy SS t . COS S S , + \ 8 0 • SS t • s v w SS l • co s SS , - 2. A- • 5 V v N t S 5 J • cos SS ^ 2. • C O S SS, • C O S SS j ^» + EI 5" 2.4- • s i v \ SS3 • SS, • cos SS} IOS L. I cos SS 3 • S\~» SS, • cos SS, - 4 8 • S S T . • cos SS, • SS, • cos SS \_ + 9 0 • SSt • SS,. • cos ss 3 • cosSS, l _ u - 4 - 8 • SSr . sCwx SS5 • cos SS 3 • cos SS + 3 6 • COS SS3 • s'\^* SS3 • cc»s SS, j -V E X 5 2.V4 • S \ V - N SS, I O S \J I -V 3 8 8 8 • SSi • SS,. L. u + C.V4, • S W X S S j - I f c l O • SS, - S**vx ss, u I S O • S tvx S S , • S » V N S S J - \Gc.O • SS,. • sx'vxSS^ ^ E I ( - 6 • c o s SS , " l + EX 1 - 2.4- • S tv, SS, - cos SS j 'OS L 1 L -V 452. • SS» • S»~ SS^  • Cos SS ^ •* ISO • SS_t - s.i^ SS • cos SS , u - 8 \ 0 • SS_t « SS*, . cos SS5 l_ L. — 4-8 • s\x>. SS ,, • S » w ss, - cos ss 3 - 3 6>0 - si^-^SS^ • c o s S ^ j 16. — 4 . s i s * S S 3 — 2. • s \ A SS, • s > w SS 3 + 6 - S S i - s i v N S S 3 ^ . + S - \ l - ' s i ^ 5 S . •V \ l v C 6 S l S S 3 ' S i s* SS , — • S \ v s l S S , • S \ s S S , • S i \ S S , +" 2.4 • SS , • S i s SS 3 • SS_X — 2,1 4 • S I W SS j • SS x . SS t l L ^ + 1 I t • c o s ss* 3 • s s . -ss u u — 9 o • s s t . SS\ . s i s . SS t • S \ s s s s -V &TO • SS,. . S S i . SS*. » S « s S S , ^— I — L _ — • S»s"*" S S 3 • s i s ' 1 , s s 3 +-~l48 • s i s * S S $ - c o s * S S , •V 4 8 • S S t . -=> v s SS , . s i s S S j — 4 8 • S S i . s i s S S , • c o S l « } — <36» • S i s X SS 3 • S i s "SS 3 » s i h S S , •V 1 9 1 • - s i A S S , • S i s S S j . c o s 1 S S j 4- 3 6 0 . S S j • s i s *SSj • s \'v-v s s 3 — 1 X O . S S i . 5>'vv S S ^ . C © s l SS ^  ^  (4.5) i s I t should now be reca l led that the tota l non-l inear member s t i f f ne s s 17. ,4 i u 4. Check of k and k u It is immediately evident that the non-linear stiffness matrix k reduces to the classical linear element stiffness matrix if the ss are taken to be zero - { . E I 2. E X \_ 4- E X u (4.6) Also, the non-linear stiffness matrix k reduces to the classical geometric element stiffness matrix (8) if the ss. are taken to be zero •f = P SS so 2 J -»s € 5 \-e-V-. - J _ NO 2^ L. \S (4.7) The total non-linear stiffness matrix was found to satisfy the criteria of rigid body invariance and it is also obvious that it is symmetric. The non-linear stiffness matrices k and k were used to determine the critical buckling loads and mode shapes for various structural systems. Agreement with the results of other theoretical buckling analyses was obtained. 18. CHAPTER V. FORMULATION OF PROBLEM 1. General Case The sensitivity of a structural system to initial imperfection is largely determined by the post-buckling behaviour of the ideal system and further the rate at which the load capacity of the ideal system changes near the critical load will be important. It is thus evident that of major concern is the initial post-buckling behaviour of the ideal system. The content of this work is concerned with the post-buckling behaviour of elastic structures for which the critical state in the idealized system occurs at a definite point of bifurcation. A stiffness matrix that can be used for the study of general buckling problems, and expressions that can be used to investigate the imper-fection sensitivity of structural systems that exhibit bifurcation buckling, will be developed. A procedure for evaluating the stable equilibrium zones for any structure will be determined and verified by the quantitative study of some examples. It is supposed that the configuration of the ideal structural system can be specified completely by a finite number of generalized co-ordinates r,-and it is further assumed that the structure is subjected to a set of external forces each of which can be specified as the product of a constant and a single variable load parameter P . The total potential energy of the ideal 19. s t ruc tu ra l system i s a s ing le valued function V " that depends on the load parameter P , the generalized co-ordinates V$ and the physical properties of the system. ; Referr ing to equation (3.1), l e t t i n g R be a "mode shape" of external fo rces , and the o r i g i na l vector R be replaced by P B then F ^ C ^ l = F T ( P R ) - P F ^ C O and we have • + z rT * T g ( a t ) (5.D The displacements of the e l a s t i c system can be reduced into "mode forms", the amplitudes of which supply a set of p r i nc ipa l generalized co-ordinates for the s t ructure. This can be accomplished by the use of a l i nea r orthogonal transformation of the form C - $ ^ where: cj i s a vector of p r inc ipa l generalized co-ordinates and <f) i s a transformation matrix such that the determinant ~\t\4 o . I t can be shown that <j> i s the matrix of fundamental mode shapes, wr i t ten as columns. Subst i tut ion fo r r i n equation (5.1) y i e l d s "V = - P cj" 1" 0 T R - P F T ( R ) £ ( ^ a $ o, ) + j £f i T ' (5.2) 20. The n equilibrium equations jj\7 ( P, r { ) ] - O ( 5« 3) may be solved to yield a number of solutions of the type r"; - V; ( each of which specifies an equilibrium path for the ideal system in P - v . space. Suppose one of these paths passes through the region of interest. This path will be the unbuckled equilibrium path and will be referred to as the basic state. From equation (5.2) it can be seen that since «| = Oon the basic state (Figure 3 ) , then V= O on the basic state curve. We explore V in the region about a point A on the basic state in which P = P0 and oj;-=o . By Taylor's theorem the total potential energy at an adjacent point A" is: + 3./ L^PPP lA P* + V 1 , - 1 J 1 J A l ^ j l f c + / U • ' • (5.4) Where the cross product derivatives vanish because of the orthogonality of the mode shapes. Suffix symbols on "V "* indicate partial differentiation ( eq. ~^Jaa Q. = — « — V and all the derivatives of V are evaluated at point A on the basic state where P'Po+p and c\; •= o . However p=o because we are looking for "neighbouring states" at P and thus P = P „ . We note that V ( P o , o ) = 0 by definition and V p \ -~\7<.. 1 = because A is an equilibrium state. 21. GENERALIZED CONFIGURATION SPACE FIGURE 3. 22. We now suppose that point A on the basic state coincides with a point of bifurcation. Realizing that only the diagonal terms appear in the quadratic form of oj. in equation (5.4), then the condition for " V ^ . o . j c|. ^ . to be positive semi-definite is that the determinant |~^ "«\. «\ | — O • This yields If the lowest critical load is identified by P, and if we are restricted to systems for which critical loads are discrete so that P-, i1 P j for i ^ j then, at A , we have " ^ " o j t e j ( = O while " V ^ o, > O for r 7* I Hence when equilibrium equations (5.3) are written there are found to be two types of equations. Partial differentiation of *V(P,^,-) with respect to the critical co-ordinate Q w^ill give equations with no linear term in o, when P = P , , whereas partial differentiation with respect to any non-critical co-ordinate °\r(y* 0 will yield equations containing a linear term in *\r and hence a solution of equations (5.3) would be r = o i ) . Thus to first order we are left with ^_ C P i l;)^ = ^ + I V P I , 1 , 1 a P 1 . ] a ° ( 5 ' 6 ) This equation for the ideal system has solutions c| ^  « o (5.6a) and «!, = - "Vf 1 , % P ( 5 < 6 B ) from which it is obvious that solution (5.6a) corresponds to the basic state and solution (5.6b) represents an adjacent state of equilibrium intersecting the basic state at point A . It is seen that the slope of this initial post-buckling path in P — cj space is eMr* = — 1. "V<^ ^ o ( (5.7) 23. At th i s point i t i s worth noting that the re lat ionsh ip between the change i n c r i t i c a l load and the imperfections of the real system depends on the slope of the i n i t i a l post-buckling curve of the ideal system. 2. Special Case of a Symmetric System I f the buckling mode i s such that there are no odd order terms i n Oj. i n the to ta l potent ia l energy expansion, that i s to say the "V^ a, cj i n equation (5.4) are equal to zero, the system i s termed symmetric. The equi l ibr ium equation of such a system i s wr i t ten as ' ^ M . n . PS. h ^ . l . - * ' . 1,' = O (5.8) considering higher order terms i n the expansion of to ta l potent ia l energy equation (5.4). Again, as before, i t can be seen that the i n i t i a l post-buckling path of the ideal symmetric system i s given by P = -from which the curvature i s determined as d P _ *V c\,o,t^ to,, ( 5 1 0 j I t should be noted here that fo r symmetric systems the re l a t i on between the c r i t i c a l load and the imperfections of the real system depends on the curvature of the i n i t i a l post-buckling path of the ideal system. 24. CHAPTER VI. DERIVATION OF EQUATIONS 1. Total Potent ia l Energy We can rewrite equation (5.2) as Define: P = amplitude of load vector Y j R ^ $ = the usual generalized co-ordinates and forces F = vector of ax ia l compressive forces in members, a r i s i ng from R G = vector of ax ia l forces i n members a r i s i ng from OJ T } the mode shape of the buckled structure £ = vector of member shortenings due to bowing 3 = connect iv i ty matrix such that S •» a \r F i s generally a complicated function of •£ and hence of . G i s a l i n e a r function of <\ . Pjc^ B- w i l l make no contr ibut ion to higher var iat ions of "V . We thus have = F ^ f c S q - 0 G i f d q - ? t f R (6.2) 25. We can determine F f o r each member of a general frame as a function of £ , the member shortenings. I t would appear that i n a l l cases i t w i l l be possible to express F f o r a member i n terms of j u s t one element of £ , s ince i n fact a l l elements of £ are functions of the s ingle quantity ° • However, i t may not be possible to express F i n terms of the -f fo r the same member, since the shortening of some members does not inf luence F . Therefore we or may not be fo r a d i f f e ren t member. Equation (6.2) i s an approximate expression for the potent ia l energy of the external loads. I t may be i n numerical er ror for high order va r i a t i on s , however i t i s much eas ier to evaluate than the exact expression. The trends of the resu l t s using this expression are correct but the numerical parameters are i n e r ro r . assume that F i s determined as a function of f an element of £ which may D i f f e ren t i a t i n g we obtain Vf?" - P F K - G \ £ - p V F V f p p *Lf - ye If G ) V = - V F vr* vr _ F f 26. - P + F } * y f 2: if* 2. i_G >lf i ^ i ^ ] - M (if V yr * 3 y^ f yr* ilT If }f 4 V i i i<^ ^ + * a i r ( yf*^1* ^ i f i f*^  \ ^ / i i i -v- y F v v r i f ii» ><\ + 3 i_F y j : * } V if* i v ^ + 3 ij if iv + F y^r -i if" i ^ ic^ i <\A J L i<^ i<^  i i<^ i«l i ^ i<=t4 J 27. 2. Potent ia l of Loads We now consider a s ing le member in a frame. We can l i m i t the deformations of the member to end ro ta t ions , since any displacements can be reduced to these by a su i tab le l i n ea r transformation. Suppose the mode shape o, , has been normalized on one rotat ion which we c a l l Oj . Then = o>_ Al so , we assume that the member has end rotat ions C, cj and C^cj . ~ From equation (3.2) i t can be seen that the shortening of a member due to bowing i s -f - S \ v > l C , Q +• L. s»v, C t Q \S - I- S V s C , Q s ( v » C ^ Q (6.4) 30 I t i s i n te re s t i ng to note that the shortening i s less when C , and C ^ a r e of the same sign than when they are of opposite s ign. We now obtain the fo l lowing expressions y£ _. Z \- C, C, (j - c o s Ct<| $<\ \S + C t s U C l < | ' C O S C t o| \S " i=. ( C » C.OS C , «\ * s i w C l e | + C ^ s m C , C J • c o s >V •- 2. \~ C* ( c o s S . c , c ^ ) + a L C J " ( c o s a c ^ ) 3o 2- c i C t C O - S C J O J . c o s - C ^ s i s C j O ^ . S » w C ^ ^ ) 28. J 1 * I S \5 1 ' 6 £ \ C ^ 5 W N • c o s ~ C X « S \ v \ C , < ^ - c o -s C x <^  ) ^ = - 8 * ( c o S Z. e^) - 8 \ - C x 4 ( C O S 2 - C ^ c ^ ) " ^ ^ ( * 6 C * C * C t 4 ) ^ i ^ C ^ . Evaluating equations ( 6 . 5 ) , f o r cj =»o , we obtain ! 1 » o • i V I = o y f I = - 2Jr ( 4 C , - c | * C l - C , C J - V 4 r C v ) (6.6) 29. Equations (6.3) can now be evaluated noting that G 1 = o and i . * 1*° ^ G I = O . I f we assume that -f i s the end shortening of a } 1 ^ = o member being considered we obtain ^ so - Eir ( t C * - C , C 4 V I C , 1 ) vs L i s K ^ 1 ' \S .7) 3. S t ra in Energy Recal l from equation (2.9), fo r SS, and SS .we have U = E I — s y s ; ^ v ^ i + $ * * * * * * •V 2 0t 3 "S \ 55", $ » v> SS 3 -*• $ 4 + a 0 + ^ 6 * ^ l s * 5 •v z <p^<j>9 * i ss, • s ; ^ ss a 3 »v» SS 3 • Sivx SS , ss 4 . + 2 ^ ^ , S^ 3 = a e x =-X ^ S ^ 1 " SS, 4- S » S S , • S \ v s SS 3 •+ siv^"*" S S S ^ J 4 E X \ O S l _ ^ \ t s » M ss, + 8 s\*% ss, • s \^ s* i s\ 1 s< , • ss ] (6.8) 30. If now we consider the buckling mode CJ , and note that ss, = c, SS3 = c^cj we obtain 2. EI J" Svvv^C ^ + -svvx c , <*, • C z. *\ + sv^Lc t a l •v- -siwx A c t 2 (6.9) Differentiating we obtain -. E.EX [ - 8 c ,J s\^C,^ • c o s C , a, - C t ( 3 C , + C i ) Sivvc,<| . co* - C , ( c , l + J C , 1 ) c o s c ^ . S ^ C a C j - 8 cAJ s'w-x c i a, . co «& c u 1 ~2 + ^ ^ | ^ . ^ 8 4 c, a s i . c, <^  . c o - S 3 C , cj - 6 4 0 c , S " S v ^ ^ C , ^ • C o s C , ^ -V 48 c: 4* co s 3 c , t | • s i *\ c t ©^  - c, ( "l C,1 + * c t l ) ^ >'wlC, Qj • c o s C j C j ' S i w . C -V \ 44 C^c, 1 s \ v, C , • cos lc,<^ . C G S - \"2. C , (EC,*1 n C ^ ) C O - S C , ^ . c,^ • S i ^ V j . 36 c c t co s l c , CJ . sv^ c z c | * c o S C i c i - \Z. c 2 ( J c ^ + i c ^ ) s''-«rc,^ - s \ * * c ^ • Co-SC + 36C,c J t S\v^C,<^. C O I C ^ • c o ^ C ^ C j -8 c, ( c ^ + SC, 1) co s c, «) «s i ^  3 c t - 2-4 C z (SC,1*--}*:^) si ^ c,^ • - 5 1 ^ 0 ^ - c o - s c , , 31. ^ \ \ 4 C , C t l C O S C T C | - c i. <\ • C o S l C v e j + 4 B C ( C J • c o s * C , C J 4- 3 8 4 C * S V ' H c t cj • c o s i C i < ^ S N 1 C , ^ - 4 - c x c l ( c 1 l ' + c 1 > ) c o « s c l o 1 e C O S C z Oj -V- ( c , 4 + ^ c * C ^  + C t * ) *» *\ ^  c x • S \**x C ^  1 4 E X f c . 4 C o i * C , q - ^ O I X . C , 4 ^ x ^ Z c \ <\ * c o s * c i °[ -+ 6 4 o c , 4 5 > v N ' i C l C J - 4 S C, 1 ( I O C , 1 v i c j " ) c o s ' - c . c j • s i ^ C , C J . s ^  C , c j -V C ^ C j C O S 5 C , C j • C o S C ^ c j + 8 ( e i c , + + I S C ^ C J 1 + c i 4 ) ' S l v > 5 c , ^ . s i s c t c| - ^ ^ » c t c , ( i c , 1 + c i l ) < s * v , 1 ' c i i • C o , s c i c l * C o s C i 1 + W ( C 1 4 H C | 1 C 1 I + C ^ ) -5 i * s X c , C J . - s i s 1 " C f c C J - i 4 ( c 1 l + i c 1 t ) c , 1 c o s ^ c . o , . s i ^ c ^ S \ ^ c Co-S C , • s i s C , C J « C o s C ^ c j - 2-4 C t l ( 3 c j * c ^ ) S ; ^ C , C J • c o ^ C ^ + 8 ( c , 4 + \ 8 c ^ c ^ + y c i ) S » ^  C , cj . s »v,3 c c o s C ^ c j _ 4 8 C l l ( 4 c , l + i o c i l ) « •5 l s C , C J • S » v A C l o J . c o s a C v c j 32. V^8 4 C , 4 c o s 4 C ^ (6.10) Evaluated for a •= o , we obtain O + - c t c t * + i c ^ ] (6.11) Equations (6.11) can be evaluated f o r various ra t io s of C t / c , • Results are shown in Figures 4 and 5. 33. % i*xi SHAPE c o c, 36 L o o 35 U c, o I o - 4-8 Ell c, ^-S, 3*S U - 1 o 4-8 EX c, c, 35 U <s o - 6EI c Si/, o 5 L - ic ex 5^ U C C ' / 2 L EQUATIONS (6.11) EVALUATED FIGURE 4 CHAPTER VII. EXAMPLES 35. 1 EuIer Column The s t ructura l system i s shown in Figure 6. The Euler column i s a symmetric s t ructura l system and the curvature of i t s post-buckl ing curve i s known to be po s i t i v e . It w i l l be shown that the. Euler column i s a symmetric system and equation (5.10) w i l l be employed to determine the curvature of i t s post-buckling curve and ve r i f y that i t i s i n fact po s i t i v e . I t i s ea s i l y seen that F = I and 2JE = o . A l so, the ax ia l force i n the member i s not af fected by shortening due to bowing and hence G = O and = o . mode shape i s known to be The so lut ion for the c r i t i c a l buckling load and P - TT E X and c . = \ From equations (6.7) we determine j - ° - j = o hence - O and the structure i s in fact symmetric. ^ I 4" °1 = ° - L 3 4 - P L = 4 T i ^ E I 3 3V-hence and From equations (6.11) i t i s seen that ^ XI = / 14Q TT* + 144- \ E X V 105 L, / - — L. 3 <\-0 4 8 E X 35 U-36. Buckled shape EULER COLUMN FIGURE 6 37. Now, equation (5.10) gives 3 P - ( \4Q TT + \AA \ E I which i s a pos i t i ve curvature. This confirms the known property of the Euler column: the post-buckling equi l ibr ium curve r e f l e c t s a r i s i n g load and hence the structure i s in sens i t i ve to imperfections. 2, Pitched Frame A pitched frame as shown i n Figure 7 w i l l be studied. The force i n each bar i s P J(S/2.)L « P ( \ S u V"1 where A • -f* . Hence P — ( \ S x \ ^ and d i f f e r e n t i a t i n g we obtain - — f \ + S x Y"'/i» S*- ! W 4- \ A-^-^yJ a ^ h T ^ s i Evaluating for cj = O and s imp l i f y ing y i e l d s F L . = — - L If- I „ L fo r each member. G and i§ are non-zero but are c l e a r l y equal, fo r each member, and of opposite sign and therefore cancel . The so lut ion for the c r i t i c a l buckl ing load and mode shape i s commonly known to be P = 2.TT2" c x s"\v\-0 . _. • ^ ~ and C , = \ j c 2 . From equations (6.7) we determine ^ I •= o a hence ^ "V 1 = O and the s t ructura l system i s symmetric <\ = o = - L-Cj = o 5" S\v-v -O-i 4 - ° - . . I = - ^ 3 5 P L . 38. S PITCHED FRAME FIGURE 7. 39. From equations (6.11) i t i s seen that ^ * ^ ^ o "SS V-For the s t ructure, equation (5.10) y i e ld s - ~ P 4- _L2 ?_ - 8 C X SVVA-Q-} <J »- S - V a ^ 2 - ^ - \ c . n L.1-= (S\v> - e Q E X f - Z.-U-^ ^ IS TT*" - 8 . 1 u x L s + a ^ a - e - 4> i J The above resu l t i s p lo t ted i n Figure 8 as a graph of ^ P V S -O- . from which i t i s seen that the curvature changes from negative to pos i t i ve fo r -9- ^ £4-° a 3. Right-Angle Knee Frame This i s one of the few structures fo r which experimental and theoret ica l resu lts have been obtained and compared. Roorda (5) presented the resu l t s of careful experiments that ind icated qua l i t a t i v e agreement with the general non-l inear theory of Ko i ter (1). Ko i ter (9) obtained a close quant i tat ive comparison between t heo re t i c a l l y and experimentally determined resu l t s fo r the r ight-angle knee frame. He studied the problem in the context of continuum mechanics and, as stated by Roorda (5, p.104) the equations derived are exceedingly cumbersome. The st ructura l system i s shown in Figure 9. The F and F der ivat ives w i l l f i r s t be determined. From equi l ibr ium of the frame i t can be seen that fo r members 1 and 2, F , ° 1 - £«• and F T = £\. respect ive ly . Hence M , - - ° 40. GRAPH OF ^ % VS -e-FIGURE 8. FIGURE 9. 42. 111. - ± u _v_ L. From a buckling analysis based on the derived non-l inear s t i f f ne s s matrix (one element for each member) i t i s found that the c r i t i c a l buckl ing load i s approximately p - \e.t> ELI and the mode shape i s Thus for member 1 and f o r member 2 C , = l.O C , = l.O , C L = - 1.72.5 , c t - -o.s The G and G der ivat ives w i l l now be determined, by using slope def lec t ion equations, to be G and thus i G t - — 3 E.X >1 = 0.S5 EX r e — = 0.55 EI r r ~ G, and G F Cshould be equal fo r overa l l equ i l ib r ium. The fact that they are not i s due to the displacement f i e l d s we are using not being equi l ibr ium f i e l d s . Further subdiv is ion of the members would bring G, nearer to G £ , as the displacement f i e l d approaches the equi l ibr ium f i e l d . I t would be possible to automate th i s procedure using matrix s t ructura l ana lys i s . 43. From equations (6.3) we determine i * -°- I = - 0.64S L i =• 3 . 6 SO E I cj = o L_ I t w i l l be noted that the r ight-angle knee frame i s not a symmetric s t ructura l system. 5 ' ^ Now, equation (5.10) y i e ld s i P s . 2..Q1 E.I = o . i s s P I t i s noted that Ko i ter (9) obtained a slope of O. 1 8 0 5 P . I t w i l l be rea l i zed that the resu l t obtained above depends upon Gr which, in tu rn , r e f l ec t s the shear in the members. Assumed mode shapes are notoriously bad at predict ing shear and end moments of buckled frames, and thus the resu lts obtained may be regarded as an acceptable f i r s t approximation of Ko i t e r ' s value. I t i s probable that subdiv is ion of the members into smaller elements would give a better quant i tat ive r e su l t . 44. CHAPTER VI I I. CONCLUSION A general non-l inear s t i f f ne s s matrix was developed to study the post-buckl ing s t a b i l i t y of s t ructura l systems. It was tested and found to s a t i s f y a l l the required c r i t e r i a . A procedure was developed for determining the stable equi l ibr ium zones f o r any s t ructure. With some addit ional work i t would be poss ible to completely automate the procedure. Examples were invest igated to show correctness of trend in re su l t s . Su f f i c i en t work was not done to show convergence to the few ava i lab le solut ions and the accuracy pursued in the study of examples was only that necessary to ind icate trends. The examples were of structures that exh ib i t a de f i n i t e point of b i f u r ca t i on . The results suggested the correctness of the method. With the procedure developed i t would be possible to introduce imperfections in geometry and load d i r e c t l y ; however, more work i s necessary to include these e f f e c t s . 45. BIBLIOGRAPHY Ko i ter , W.T., "On the s t a b i l i t y of E l a s t i c Equ i l i b r i um" , D i s se r ta t i on , Polytechnic I n s t i t u t e , De l f t , 1945. Ko i te r , W.T., " E l a s t i c S t a b i l i t y and Post-Buckling Behaviour", Non-Linear  Problems, Edited by R.E. Langer, Univers i ty of Wisconsin Press, Madison, Wis., 1963. Budiansky, B. and Hutchinson, J.W., "Dynamic Buckling of Imperfection-Sens i t ive St ructures " , Proceedings of the XI International Congress  of Applied Mechanics, Edited by H. Gor t le r . Ju l iu s Spr inger-Verlag, Be r l i n 1964, pp. 636-651. Hutchinson, J.W. and Amazigo, J . C , " Imperfect ion-Sens i t iv i ty of Eccen t r i ca l l y S t i f fened Cy l i nd r i c a l S h e l l s " , Journal of the American  I n s t i tu te of Aeronautics and Astronaut ics, Vo l . 5, No. 3, pp. 392-401, March 1967. Roorda, J . , " S t a b i l i t y of Structures with Small Imperfections", Journal of the Engineering Mechanics D i v i s i on , Proceedings of the A.S.C.E., Vol . 91, No. EMI, February 1965. B r i t vec , S.J. and Ch i l ve r , A.H., " E l a s t i c Buckling of R i g id l y - Jo in ted Braced Frames", Journal of the Engineering Mechanics D i v i s i o n , Proceedings of the A.S.C.E., Vo l . 89, No. EM6, December 1963. Langhaar, H.L., Energy Methods i n Applied Mechanics, John Wiley & Sons Inc., New York 1962. Mart in, H . C , "On the Derivat ion of S t i f fness Matrices for the Analysis of Large Def lect ion and S t a b i l i t y Problems", Conference on Matrix  Methods in Structural Mechanics, Wright Patterson A i r Force Base, 1965. Ko i te r , W.T., "Post-Buckl ing Analysis of a Simple Two-Bar Frame", Report No. 312 of the Laboratory of Engineering Mechanics, Department of Mechanical Engineering, Technological Un i ve r s i t y , De l f t . 

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