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Stability analysis of a spaceframe structure Oberti, Andrea Luca 1969

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STABILITY ANALYSIS OF A SPACEFRAME STRUCTURE by ANDREA LUCA OBERTI "Laurea" ( E l e c t r i c a l Eng.) Polytechnic I n s t i t u t e of Turin, I t a l y , 1965 B.A.Sc. ( C i v i l Eng.) The U n i v e r s i t y of B r i t i s h Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED'SCIENCE i n the Department of CIVIL ENGINEERING We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1969 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission for extensive copying of t h i s thesis for s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Andrea L. Oberti Department of C i v i l Engineering The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada September, 1969 i ABSTRACT Two theories for the s t a b i l i t y a n alysis of a spaceframe structure are presented: the f i r s t uses only L i v e s l e y s t a b i l i t y functions, the second includes i n a d d i t i o n the e f f e c t s due to chord r o t a t i o n and f l e x u r a l end shortening of a member. The c r i t i c a l c ondition i s defined by the load which makes the tangent s t i f f n e s s matrix of the structure become s i n g u l a r . Three methods for obtaining the c r i t i c a l load are presented: determinant p l o t , Southwell p l o t , and l o a d - d e f l e c t i o n curves. The a n a l y s i s i s c a r r i e d out f o r the p l e x i g l a s s model of an a c t u a l c o n i c a l spaceframe, made of glulam timber, and b u i l t f o r the storage of potash. The o v e r a l l c r i t i c a l load for t h i s s tructure i s found to be i n s a t i s -factory agreement with the experimental r e s u l t s obtained i n previous model t e s t s . Some a d d i t i o n a l e f f e c t s , such as geometric imperfections i n the j o i n t coordinates and d i f f e r e n t member e n d - f i x i t y conditions are investigated. The concept of e f f e c t i v e length for a member i s introduced to present the r e s u l t s obtained by varying the height/span r a t i o of the structure. F i n a l l y some design suggestions are given f o r structures of t h i s type. The analyses were made using spaceframe programs based on the s t i f f -ness method, modified to include s t a b i l i t y e f f e c t s . An IBM 360/67 computer was used for the c a l c u l a t i o n s . i i TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENTS i i LIST OF FIGURES i v LIST OF TABLES v i NOTATION v i i ACKNOWLEDGEMENTS i x CHAPTER I INTRODUCTION 1 CHAPTER II GOVERNING EQUATIONS AND SOLUTION PROCEDURE .2.1 Governing Equations 5 2.2 Solution Procedure 8 2.3 Computer Program Outline 10 CHAPTER I I I DESCRIPTION OF THE STRUCTURE AND LOADING SYSTEMS 15 CHAPTER IV RESULTS FOR THE ORIGINAL CONE 4.1 Introduction 25 4.2 Determinant Plot 25 4.3 Southwell Plot 32 4.4 Load-deflection Curves 44 4.5 P a r t i a l Loading 44 4.6 Conclusions about Southwell Plot 51 CHAPTER V RESULTS FOR SPACEFRAMES WITH DIFFERENT HEIGHT/SPAN RATIO 5.1 Conical Shapes 54 i i i TABLE OF CONTENTS (Continued) Page 5.2 S p h e r i c a l Shape 64 CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS 69 APPENDIX 1 DERIVATION OF THE NON-LINEAR RELATIONS GOVERNING THE DEFORMATION OF A MEMBER 73 APPENDIX 2 EXACT SOLUTION FOR TWO-BAR STRUCTURE 82 APPENDIX 3 NOTES ON THE SOUTHWELL PLOT 85 APPENDIX 4 LIST OF REFERENCES 88 i v LIST OF FIGURES Page FIG. 1:1 Geom e t r i c a l l y Non-linear E f f e c t s 2 FIG. 2:1 Member N o t a t i o n 6 FIG. 2:2 Step-by-step S o l u t i o n 10 FIG. 2:3 Check of Computer Program f o r Theory 2 12 FIG. 2:4 Two-Bar S t r u c t u r e 13 FIG. 3:1 The O r i g i n a l Cone 16 FIG. 3:2 Member P r o p e r t i e s 17 FIG. 3:3 Connection D e t a i l s 18 FIG. 3:4 Coding of the S t r u c t u r e 20 FIG. 3:5 ( i ) H a l f Snow Load 19 ( i i ) "Quarter Wave" B u c k l i n g 21 FIG. 3:6 Member End-Forces 21 FIG. 3:7 P e r t u r b a t i o n Force Systems ("Trigger Systems") 23 FIG. 4:1 O r i g i n a l Cone. Determinant P l o t s . F i x i t y C o n d i t i o n (a) 27 FIG. 4:2 O r i g i n a l Cone. Determinant P l o t s . F i x i t y C o n d i t i o n (b) 28 FIG. 4:3 Comparison Determinant P l o t s 30 FIG. 4:4 Southwell P l o t . Exact Geometry 36 FIG. 4:5 Southwell P l o t . Exact Geometry 37 FIG. 4:6 Southwell P l o t . Exact Geometry 38 FIG. 4:7 Southwell P l o t . Random Geometry 39 V LIST OF FIGURES (Continued) Page FIG. 4:8 Southwell P l o t . Random Geometry 40 FIG. 4:9 Southwell P l o t . Random Geometry 41 FIG. 4:10 C r i t i c a l Mode Shape. F i x i t y C o n d i t i o n (a) 42 FIG. 4:11 L o a d - d e f l e c t i o n Curve. F i x i t y C o n d i t i o n (a) 45 FIG; 4:12 L o a d - d e f l e c t i o n Curve. F i x i t y C o n d i t i o n (a) 46 FIG. 4:13 L o a d - d e f l e c t i o n Curve. F i x i t y C o n d i t i o n (b) 47 FIG. 4:14 L o a d - d e f l e c t i o n Curve. F i x i t y C o n d i t i o n (b) 48 FIG. 4:15 Displacements along Middle Ring 49 FIG. 4:16 O r i g i n a l Cone. H a l f Snow Load. Determinant P l o t s 50 FIG. 4:17 Geometric I n s t a b i l i t y (Snap-through) 52 FIG. 5:1 FC1 Cone. 1/2 O r i g i n a l Height. Determinant P l o t 56 FIG. 5:2 FC2 Cone. 1/4 O r i g i n a l Height. Determinant P l o t 57 FIG. 5:3 D i f f e r e n t Behaviour of Adjacent Ribs 62 FIG. 5:4 C o n i c a l Spaceframes. E f f e c t of Var y i n g Height/ Span Ratio 63 FIG. 5:5 S p h e r i c a l Spaceframe 65 FIG. 5:6 S p h e r i c a l Dome SF1. Determinant P l o t 66 FIG. 6:1 E f f e c t of N o n l i n e a r i t i e s on the Margin of Safety 72 V I TABLE 4:1 TABLE 4:2 TABLE 4:3 TABLE 5:1 TABLE 5:2 TABLE 5:3 TABLE 5:4 TABLE 5:5 LIST OF TABLES Determination of S t r u c t u r e C r i t i c a l Load by Determinant P l o t I n f l u e n c e of Geometric Imperfections on A x i a l Loads Determination of S t r u c t u r e C r i t i c a l Load by Southwell P l o t Determination of C r i t i c a l Load by Determinant P l o t f o r D i f f e r e n t Height/Span Ratios O r i g i n a l Cone FCO. a = „ }.c o 3.35 Cone FC1. a. = 1/2 a = — 1 o 6.7 Cone FC2. a 0 = 1/4 a = \ •, 2 o 13.4 Comparison of a C o n i c a l and a S p h e r i c a l Spaceframe Page 26 31 33 55 59 60 61 67 v i i NOTATION x, y, z = member axes; X, Y, Z = s t r u c t u r e axes; {P} = e x t e r n a l load v e c t o r ; {U} = s t r u c t u r e displacement v e c t o r ; [K] = s t r u c t u r e s t i f f n e s s m a t r i x ; [Kj_] = s t r u c t u r e tangent s t i f f n e s s m a t r i x ; {f} = member fo r c e v e c t o r i n member axes; {F} = member f o r c e v e c t o r i n s t r u c t u r e axes; {u} = member displacement v e c t o r i n member axes; [ k ] m = member s t i f f n e s s m a t r i x i n member axes; [k] = member s t i f f n e s s m a t r i x i n s t r u c t u r e axes; [k ] = member geometric s t i f f n e s s m a t r i x i n s t r u c t u r e axes; [k^] = member tangent s t i f f n e s s m a t r i x i n s t r u c t u r e axes; [R] = matrix of d i r e c t i o n cosines of a member; [T] = t r a n s f o r m a t i o n m a t r i x f o r a member; N = a x i a l f o r c e i n a member; X = load f a c t o r ; {P} = e x t e r n a l load v e c t o r at A = 1; o N = l i n e a r a x i a l f o r c e i n a member at X = 1; o A^j = j o i n t displacement component normal to the su r f a c e of the cone; o)^ = j o i n t r o t a t i o n component about an a x i s normal to the sur f a c e of the cone; a = height/span r a t i o ; NOTATION (Continued) L = member j o i n t - t o j o i n t l e n g t h ; A = member c r o s s - s e c t i o n a l area; I = moment of i n e r t i a ; J = t o r s i o n a l constant; r = radius of g y r a t i o n of member c r o s s - s e c t i o n ; E, G = e l a s t i c moduli of m a t e r i a l ; L e = e f f e c t i v e l e n g t h of a member; P = t o t a l a p p l i e d e x t e r n a l load i n v e r t i c a l d i r e c t i o n P = chord r o t a t i o n . i x ACKNOWLEDGEMENTS The author wishes to express h i s thanks to h i s s u p e r v i s o r Dr. R. F. Hooley, and to Drs. D. L. Anderson and N. D. Nathan of the C i v i l Engineering Department, f o r t h e i r encouragement and guidance i n the p e r i o d of residence s t u d i e s and during the p r e p a r a t i o n of t h i s t h e s i s . The f i n a n c i a l support of the N a t i o n a l Research C o u n c i l of Canada, i n the form of a Post-graduate S c h o l a r s h i p , i s g r a t e f u l l y acknowledged. G r a t i t u d e i s a l s o expressed to the U. B. C. Computing Centre f o r the very generous a l l o c a t i o n of computer time and the use of i t s outstanding f a c i l i t i e s . September, 1969 Vancouver, B. C. STABILITY ANALYSIS OF A SPACEFRAME STRUCTURE CHAPTER I INTRODUCTION Various s t r u c t u r a l systems may be used to cover l a r g e areas f o r r e c r e a t i o n a l or i n d u s t r i a l purposes, e.g. systems of t r u s s e s and beams, continuous s h e l l s , or spaceframes. When the geometrical c o n f i g u r a t i o n presents an a x i s of r o t a t i o n a l symmetry, the most w i d e l y used systems are e i t h e r continuous s h e l l s , g e n e r a l l y i n concrete, or framed domes. In c o u n t r i e s where the cost of labour i s h i g h w i t h respect to that o f m a t e r i a l s , the use of spaceframe s t r u c t u r e s may show an economical advantage over the use of continuous s h e l l s ; the former may a l s o be pre-f e r r e d frosm the p o i n t of view of speed of e r e c t i o n and general c o n s t r u c t i o n q u a l i t y , s i n c e f i e l d work i s reduced to a minimum w i t h respect to shop p r e f a b r i c a t i o n . Notable examples of such domes are R. B. F u l l e r ' s geodesic domes, the Astrodome of Houston, Texas, the Schwedler Dome of B e r l i n . In the design of these s t r u c t u r e s i t i s r e l a t i v e l y easy to f i n d the l i n e a r i n t e r n a l f o r c e s and moments due to a s p e c i f i e d e x t e r n a l l o a d , by means of a l i n e a r s t i f f n e s s a n a l y s i s (1). At present, however, q u e s t i o n -able r u l e s are used to f i n d the al l o w a b l e compressive loads i n the members. N o n l i n e a r i t y i n the behaviour of these s t r u c t u r e s under s t a t i c l o a d s , e x c l u d i n g m a t e r i a l n o n l i n e a r i t i e s , a r i s e s from changes i n the ^Numbers i n parentheses r e f e r to references l i s t e d i n Appendix A. geometry of the structure due to i t s deformation: these effects are particularly important in those spaceframes where members meeting at a joint are almost coplanar. The object of this thesis i s to determine the effect of some geometric nonlinearities, namely chord rotation and bowing, as shown i n f i g . I l l , on the evaluation of the elastic c r i t i c a l load for a spaceframe structure, and to give some design suggestions for this type of structure, particularly about the concept of effective length of a member. In addition, an investigation i s made about the applicability of the South-well plot for the evaluation of the c r i t i c a l load of the structure. This w i l l be achieved by analysing an elastic spaceframe having the shape of a conical shell, with 16 straight ribs, 4 main rings, and bracing diagonals, as shown in f i g . 3 t l . A structural model of such a cone has been used in the past to predict the c r i t i c a l load (2). The present work w i l l be carried out analytically using two different structural stiffness Tsatrieesj naai&ly (1) considering the effect of the axial force on the bending stiffness of a member, (2) same as (1), but adding the effects of the change i n axial force due to chord rotation and flexural end-shortening (bowing), as from f i g . 1:1. i n i t i a l position chord rotation bowing FIG. 1:1 - GEOMETRICALLY NONLINEAR EFFECTS 3 Since the r e s u l t s of the two analyses were always very c l o s e , i t was not deemed necessary to analyse s e p a r a t e l y the e f f e c t of chord r o t a t i o n o n l y . I n the s t a b i l i t y a n a l y s i s of framed s t r u c t u r e s the importance of the change i n a member bending s t i f f n e s s , due to the presence of an a x i a l f o r c e , has been shown to be e s s e n t i a l by s e v e r a l i n v e s t i g a t o r s (e.g. L i v e s l e y (3), Merchant & Home (4)). In the f i r s t a n a l y s i s the a x i a l f o r c e i n a member i s determined only by the orthogonal p r o j e c t i o n s of the end-displacements on the i n i t i a l p o s i t i o n of the member, w h i l e the second a n a l y s i s takes i n t o account the a c t u a l d i f f e r e n c e i n l e n g t h between the f i n a l deformed shape and the i n i t i a l c o n f i g u r a t i o n . Thus, w i t h reference to f i g . 1:1, the f i r s t a n a l y s i s would give zero a x i a l f o r c e i n the member, whereas the second would show the presence of a t e n s i l e a x i a l f o r c e . The a d d i t i o n a l e f f e c t s considered i n theory 2 could be s i g -n i f i c a n t f o r s t r u c t u r e s of s h e l l - l i k e c o n f i g u r a t i o n , which do not a l l o w i n e x t e n s i o n a l bending, and have been shown to be of e s s e n t i a l importance i n the p o s t - b u c k l i n g domain (5). Among other methods to p r e d i c t the c r i t i c a l l o a d of a spaceframe dome, there i s the p o s s i b i l i t y of e s t a b l i s h i n g an e q u i v a l e n t homogeneous s h e l l , whose c r i t i c a l load may be obtained from the a v a i l a b l e a n a l y t i c a l and experimental r e s u l t s . This method may be u s e f u l when the number of members i n the spaceframe i s so l a r g e as to make a computer a n a l y s i s i m p o s s i b l e or uneconomical.. The method, however, f a i l s i f the mesh s i z e of the spaceframe i s too coarse, or i f the l a t e r a l s t i f f n e s s of the r i b s i s much sma l l e r than t h e i r s t i f f n e s s normal to the s h e l l s u r f a c e , i n which case b u c k l i n g may occur because of l a t e r a l bending of the members of the spaceframe. The n o n l i n e a r equations governing the deformation of the s t r u c t u r e may be solved by va r i o u s methods (6,7), among which the Newton-Raphson and the step-by-step incremental method have been wid e l y used. I n the present study a modified v e r s i o n of t h i s second method w i l l be used, s i n c e the Newton-Raphson method has been reported to e x h i b i t d i f f i c u l t i e s i n c a l c u l a t i o n s t a b i l i t y and convergence (8). Other advantages of the incremental method are the p o s s i b i l i t y of f o l l o w i n g the p r o g r e s s i v e deformation of the s t r u c t u r e and the f a c t that the determinant of the tangent s t i f f n e s s m a t r i x provides at each step a measure of r e l a t i v e s t a b i l i t y of an e q u i l i b r i u m c o n f i g u r a t i o n . The f o l l o w i n g a d d i t i o n a l c o n d i t i o n s w i l l a l s o be considered i n t h i s i n v e s t i g a t i o n : (a) changing members' end c o n d i t i o n s so as to consider the two cases: i ) P o s s i b i l i t y of b u c k l i n g about both axes of member c r o s s - s e c t i o n admitted, i i ) P o s s i b i l i t y of b u c k l i n g about the weak a x i s ( i . e . " i n plane" b u c k l i n g ) excluded. In the f i r s t case a member provides bending r e s i s t a n c e at a j o i n t w i t h respect to both p r i n c i p a l axes of the c r o s s - s e c t i o n i n the second case no bending s t i f f n e s s e x i s t s f o r r o t a t i o n s about the weak a x i s of the c r o s s ^ s e c t i o n ; (b) a random v a r i a t i o n of j o i n t c o o r d i n a t e s , to simulate p o s s i b l e c o n s t r u c t i o n i m p e r f e c t i o n s ; (c) changes of geometrical c o n f i g u r a t i o n , i . e . c o n i c a l and doubly curved domes of d i f f e r e n t height/span r a t i o . CHAPTER I I GOVERNING EQUATIONS AND SOLUTION PROCEDURE 2.1 Governing Equations As mentioned e a r l i e r , two methods f o r s t a b i l i t y a n a l y s i s are used. The f i r s t makes use of L i v e s l e y s t a b i l i t y f u n c t i o n s , as o u t l i n e d i n (3) and (4), to take i n t o account the v a r i a t i o n s i n the bending s t i f f -ness of a member due to the presence of an a x i a l f o r c e . In t h i s case the s t i f f n e s s m a t r i x of the s t r u c t u r e [K] i s a f u n c t i o n only of the a x i a l loads N_^  i n the members. The governing load-displacement r e l a t i o n s w i l l be: ' {P} = [K(N.)] {U} . . . [1] where {P} i s the e x t e r n a l l o a d v e c t o r , and {U} i s the s t r u c t u r e d i s -placement v e c t o r . Generally a few i t e r a t i o n s , at each load l e v e l , are re q u i r e d to achieve convergence f o r the elements of [ K ] , For s t a b i l i t y a n a l y s i s we can r e w r i t e [1] i n incremental form (assuming that the member a x i a l loads do not change f o r a sm a l l increment o the e x t e r n a l l o a d s ) : {AP} = [K(N )] ' {AU} . . . , 0 [2] Then, by d e f i n i t i o n , at the c r i t i c a l l o a d we have {AP} = {0} f o r some non-zero {AU}, i . e . [K] must become s i n g u l a r , or |K| = o . In other words, at the c r i t i c a l l o a d the e q u i l i b r i u m c o n f i g u r a t i o n of the s t r u c t u r e i s not unique. 6 The second analysis, including the effects due to chord rotation and bowing, is based on the derivation described in references (9) and (10). The assumptions on which this theory rests are: 1. The material is linear elastic. 2. Each member is prismatic and homogeneous. 3. Loads are applied only at the ends of a member. 4. Shear deformations are neglected. 5e Linear strains and squares of the rotations are of the same order of magnitude, and small compared to one. 6. Torsion-flexure coupling and warping restraint are neglected. It can be noted that assumptions 4 and 5 correspond exactly to those made by Von Karman in his "large deflection" theory of plate bending (11). An outline of the derivation of the load-displacement relations for a member, in the two-dimensional case, is shown in Appendix 1. For a general spaceframe member, the member coordinate system, end-forces and end-displacements are shown in fig. 211. The member reference frame does not follow the deformation of the member, but is fixed to its undeformed position. x The complete s et of force-displacement r e l a t i o n s f o r a member can be w r i t t e n , i n m a t r i x form, as: { f } 1 2 x l = ^ 12x12 { u } 1 2 x l " { V l 2 x l ^ where { f } = { F A 1 ' FA2' FA3' M A 1 ' MA2' MA3' F B 1 ' FB2' FB3' ^ 1 ' ^ 2 ' M B 3 } and T { u } = { u A l ' UA2' UA3' UA1» tt)A2' > W B 3 } are the column-vectors of member end-forces and end-displacements; [ k ] m i s the member s t i f f n e s s m a t r i x , i n c l u d i n g L i v e s l e y s t a b i l i t y f u n c t i o n s : {fg} i s the v e c t o r of the n o n l i n e a r geometric terms, a r i s i n g from bowing and chord r o t a t i o n . I n the present a n a l y s i s i t reduces to only two a x i a l terms, s i n c e the sway terms (of the type pF^, where UB2 ~ UA2 p = - ) , which appear i n (10), are i n c l u d e d i n L i v e s l e y s t a b i l i t y functions„ I t should a l s o be noted that f i n i t e r o t a t i o n s do not obey the a d d i t i o n law f o r v e c t o r s , but the e r r o r i s of the order of the square of a r o t a t i o n , which i s n e g l i g i b l e w i t h the assumptions used i n the present d e r i v a t i o n . To o b t a i n the system equations i n s t r u c t u r e , or g l o b a l , co-o r d i n a t e s , l e t [R]3 X3 he the m a t r i x of the d i r e c t i o n cosines f o r the undeformed member d i r e c t i o n s (x,y,z) w i t h respect to the g l o b a l r eference frame (X,Y,Z). Then i t can be shown (1) that the t r a n s f o r m a t i o n matrix [T] f o r a spaceframe member r e s u l t s : 8 [T] 12x12 [R] [0] [0] [0] [0] [R] [0] [0] [0] [0] [R] [0] [0] [0] [0] [R] Then, f o r each member, we have, i n s t r u c t u r e c o o r d i n a t e s : {U} m = [ T ] T {u} {F} = [ T ] T {f} {F G> = [ T ] T {f G> [k] = [ T ] T [ k ] m [T] Adding up the c o n t r i b u t i o n s of a l l the members connected to a j o i n t , the system of j o i n t e q u i l i b r i u m equations i s obtained: [A] where {P} i s the e x t e r n a l l o a d v e c t o r ; [K] i s the s t r u c t u r e s t i f f n e s s m a t r i x , which again i s a f u n c t i o n only of t h e . a x i a l loads i n the members; {U} i s the s t r u c t u r e displacement v e c t o r ; {P }, produced, by {F }, i s e q u i v a l e n t to a loa d v e c t o r , due to G G geometric n o n l i n e a r i t i e s , and i s a f u n c t i o n both of the members' a x i a l loads and end-displacements. {P} = [K] {U} - {Pg} 2.2 S o l u t i o n Procedure To sol v e system [4] of n o n l i n e a r equations, a modified incremental method i s used, whereby the e x t e r n a l loads are a p p l i e d by f i n i t e increments, and an i t e r a t i o n c y c l e by successive s u b s t i t u t i o n s i s c a r r i e d out a f t e r a number of ste p s , that can be v a r i e d at c h o i c e , u s i n g the complete, "exact", set of n o n l i n e a r equations [ 4 ] , To o b t a i n the incremental p a r t of the s o l u t i o n , the expressions f o r the d i f f e r e n t i a l s of the end-actions of a member are needed. F o l l o w i n g reference (10), we can w r i t e , i n member co o r d i n a t e s : {Af} = [ k ] m '{Au} + [ k G ] m {Au} where [kQ]m {Au} = - {Afg}, and [k ] m may be i n t e r p r e t e d as a geometric s t i f f n e s s m a t r i x , which, w i t h our assumptions, happens to be symmetric: [ k r ] m i s given i n Appendix 1. Then we can w r i t e : • {Af} = [ k t ] m {Au} [5] where [ k t ] m = [ k ] m + [ k G ] m i s the member tangent s t i f f n e s s m a t r i x . Assembling the c o n t r i b u t i o n s o f each member we o b t a i n the system equations i n incremental form: ' {AP} = [K t] {AU} [6] where [ K t ] i s now the s t r u c t u r e tangent s t i f f n e s s m a t r i x . Again i n s t a b i l i t y i s reached when equation [6] has a n o n - t r i v i a l s o l u t i o n f o r {AP} = {0}, i . e . when |K | = 0. A second s t a b i l i t y c r i t e r i o n can be e s t a b l i s h e d by the f a i l u r e i n ac h i e v i n g convergence i n the c y c l e of suc c e s s i v e s u b s t i t u t i o n s : f o r t h i s method a s m a l l increment s i z e i s necessary i n the neighborhood of the c r i t i c a l l o a d , i n order to achieve reasonable accuracy. An a d d i t i o n a l method of determining the c r i t i c a l l o a d i s the use of Southwell p l o t s i n connection w i t h s e l e c t e d d i s t u r b i n g f o r c e systems. I t has been shown (12) that i n a plane frame, as long as a di s t u r b a n c e , e i t h e r a loa d or a geometric i m p e r f e c t i o n , e x c i t e s any component of the 10 f i r s t c r i t i c a l mode shape, the corresponding Southwell p l o t w i l l y i e l d the r e l a t i v e c r i t i c a l l o a d . Thus i t was decided to check the a p p l i c a b i l i t y of t h i s method to the present spaceframe s t r u c t u r e . 2.3 Computer Program O u t l i n e A standard spaceframe s t i f f n e s s program, i n c l u d i n g L i v e s l e y s t a b i l i t y f u n c t i o n s , has been modified i n order to b u i l d the tangent s t i f f n e s s m a t r i x of the s t r u c t u r e at each lo a d l e v e l , u s i n g the a x i a l f orces and j o i n t displacements from the previous s t e p , s i n c e [ k ^ ] m , from the assemblage of which [K^J i s made up, i s a f u n c t i o n both of the a x i a l l o a d and the end-displacements of a member. Every few steps a c y c l e of successive s u b s t i t u t i o n s is performed, keeping the e x t e r n a l loads constant, as shown i n f i g . 2:2. d e f l e c t i o n U FIG. 2:2 - STEP-BY-STEP SOLUTION 11 Segment AB corresponds to the incremental step along the tangent to the load-displacement curve, u s i n g equation [ 6 ] , w h i l e BC represents the c o r r e c t i v e i t e r a t i o n c y c l e at constant l o a d . In t h i s c y c l e equation [4] i s used repeatedly: at each i t e r a t i o n the new values of displacements and a x i a l loads are used to b u i l d [K] and {P n}; then the equation can be w r i t t e n as {P} + {Pg} = [K] {U} [7] and solved to o b t a i n a b e t t e r s o l u t i o n f o r the displacement v e c t o r {U}. The step s i z e i s reduced i n geometric p r o g r e s s i o n , as the t o t a l load i s i n c r e a s e d , to al l o w greater accuracy i n the determination of the c r i t i c a l point„ Only a p r o p o r t i o n a l load system i s considered, where at each l e v e l the e x t e r n a l load d i s t r i b u t i o n i s a m u l t i p l e of the i n i t i a l , b a s i c l o a d i n g p a t t e r n , which can be represented by a v e c t o r {P} Q> s o t h a t at each stage we have {P} = X {P} Q [8] where X i s a numerical parameter. Then the c r i t i c a l l o a d w i l l be given by the corresponding value X•• of the load f a c t o r X. F i n a l l y , to check the program, fig„ 2:3 shows an example (a p o r t a l frame) presented i n (10), which was used as the b a s i c reference f o r the present i n v e s t i g a t i o n . I t i s to be noted that the s o l u t i o n of r e f . (10) allowed a 5% convergence t o l e r a n c e , whereas the procedure used by the w r i t e r permitted f u l l convergence, which may e x p l a i n the s l i g h t gap between the two curves. As another simple example of a s t r u c t u r e where geometric non-l i n e a r i t i e s are important, f i g . 214 shows a two-bar s t r u c t u r e w i t h very low r i s e / s p a n r a t i o . With the given c r o s s - s e c t i o n a l p r o p e r t i e s , geometric i n s t a b i l i t y w i l l occur before Euler b u c k l i n g . 12 LOAD - DISPLACEMENT CURVES. 50O0 Displacement A, inches FIG. 2* 3 - C H E C K OF C O M P U T E R P R O G R A M F O R T H E O R Y 2 . •05 10 15 -20 25 30 35 40 45 50 FIG. 2-4- T W O - B A R S T R U C T U R E . 14 The two computer analyses are compared w i t h the exact s o l u t i o n , developed from s t r a i n energy c o n s i d e r a t i o n s : the r e l a t i v e expressions are der i v e d i n Appendix 2. The s t r u c t u r e was analysed as a plane frame w i t h 4 members, as shown below. I t can be seen from f i g . 214 that the r e s u l t s given by theory 2 are very c l o s e to the exact s o l u t i o n ; t h i s i s another check f o r the correc t n e s s of the computer program w r i t t e n f o r theory 2 . 15 CHAPTER I I I DESCRIPTION OF THE STRUCTURE AND LOADING SYSTEMS The f i r s t large structure analysed was the plexiglass model (1:28 l i n e a l r a t i o ) of an actual conical spaceframe, made of glulam timber and b u i l t i n Esterhazy, Saskatchewan, for International Minerals and Chemical Corporation (Canada) Ltd. (2). The geometric and e l a s t i c properties of the structure are shown i n f i g s . 311 and 3T2. The angle 8 i s measured from a v e r t i c a l plane through the axis of a member to~a plane containing the weak p r i n c i p a l axis of the cross-section. The structure consists of 16 r i b s , 64 r i n g members, and 64 diagonals, which develop the function of the web members of a truss: altogether there are 80 j o i n t s and 192 members. Thus the structure i s s t a t i c a l l y determinate i n i t s primary stress system, since we have: 80 j o i n t s y i e l d i n g 240 equations of equilibrium, 192 bars providing 192 unknown a x i a l forces, and 16 x 3 = 48 unknown foundation reaction components. This yields 240 unknowns and 240 equations, and by inspection i t can be seen that there are no redundant members„ To simulate the r i g i d foundation r i n g , each r i b was considered perfectly fixed to the respective foundation j o i n t . In the model the members meeting at a j o i n t were held i n place by two aluminum plates (2), one at the top and one at the bottom, t i g h t l y fastened by b o l t s , as sketched i n f i g . 313. 16 ORIGINAL CONE (Plexiglass Model) Pion View FIG. 3 ' I - THE ORIGINAL CONE 17 Member Size ( i n . x in.) Area A (i n . 2 ) (in!*) (in?*) ( i n . 4 ) e Ribs: 1 I M X 1 .25 .02083 .00130 .00439 0° 2 3/16 X 3/4 .1406 .00659 .000412 .00139 0° 3 3/16 X 11/16 ,'1289 .00508 .000378 .00125 0° 4 3/16 X 9/16 .1055 .00278 .000309 .000976 0° Rings: 1 1/2 X 1/2 .25 .00521 .00521 .00879 34° 2 3/8 X 3/8 .1406 .00165 .00165 .00278 34° 3 1/4 X 1/4 .0625 .000325 .000325 .00055 34° 4 3/16 X 1/2 .0938 .00195 .000275 .000837 34° Diagonals 1 3/16 X 11/16 .1289 .00508 .000378 .00125 2 8 , 2 ° 2 3/16 X 1/2 .0938 .00195 .000275 .000837 2 5 . 2 3 ° 3 3/16 X 3/8 .0703 .000824 o000206 .000566 2 0 . 0 7 ° 4 3/16 X 9/16 .1055 .00278 .000309 .000976 1 1 . 4 1 ° E = 450 k s i G = 187 k s i FIG. 3:2 - MEMBER PROPERTIES 18 FIG. 3:3 - CONNECTION DETAILS In a d d i t i o n , glue and sand were s p r i n k l e d on the edges of the members to in c r e a s e f r i c t i o n between the members and the p l a t e s , so as to t r y and o b t a i n a r i g i d connection w i t h respect to both c r o s s - s e c t i o n a l axes of a member. In the computer a n a l y s i s two f i x i t y c o n d i t i o n s were examined: i n the f i r s t , which w i l l be c a l l e d c o n d i t i o n ( a ) , a l l the members were considered p e r f e c t l y f i x e d to the j o i n t s , except the diagonals which were coded as f i x e d about the strong a x i s , but hinged about the weak a x i s of the c r o s s - s e c t i o n . Because of the way a member s t i f f n e s s m a t r i x i s b u i l t by the computer program, a member coded as hinged at the j o i n t s at i t s ends, w i t h no e x t r a degrees of freedom along i t s l e n g t h , behaves as a simple s t r u t , w i t h no b u c k l i n g considered. Thus the p o s s i b i l i t y of l a t e r a l 19 buckling of the long diagonals, which i n the model were held i n place by s t r i n g s s t r e t c h i n g from r i b to r i b , and i n the act u a l structure by p u r l i n s , was excluded. In the second condition, c a l l e d condition (b), a l l members were considered f i x e d to the j o i n t s f o r bending about the strong a x i s , i . e . i n a d i r e c t i o n normal to the surface of the cone, and hinged about the weak axis, i . e . f o r bending i n the surface of the cone. This second condition, perhaps, represents more c l o s e l y the actual behaviour of the j o i n t s of the model, and c e r t a i n l y allows a better s t a b i l i t y a n alysis of the actual st r u c t u r e , where p u r l i n s and decking provide adequate l a t e r a l bracing to prevent " i n plane" buckling. The following f i g . 3:4 shows the actual coding of the structure f o r the computer a n a l y s i s . Because of symmetry, only h a l f of the structure was analysed: the j o i n t s l y i n g i n the plane of symmetry were permitted only displacements i n that plane, and the members cut by the plane of symmetry were entered with h a l f t h e i r s t i f f n e s s e s . This procedure s t i l l allowed the analysis of p a r t i a l loadings and of buckling modes not possessing r o t a t i o n a l symmetry, as sketched i n f i g . 3:5. symmetric about <fc. I FIG. 3:5 (i) - HALF SNOW LOAD FIG.3-4 ~ C O D I N G OF THE STRUCTURE 21 symmetric about I • up X down FIG. 3 : 5 ( i i ) - "QUARTER WAVE" BUCKLING For each j o i n t the computer a n a l y s i s provides displacements and r o t a t i o n s i n the X, Y, Z d i r e c t i o n s ( s t r u c t u r e axes), and f o r each member the i n t e r n a l forces shown i n f i g . 3 : 6 . FIG. 3 : 6 - MEMBER END-FORCES 22 The b a s i c l o a d i n g c o n d i t i o n examined was a uni f o r m l y d i s t r i b u t e d snow l o a d , which was simulated w i t h concentrated loads at each j o i n t a c cording to the pressure at and the sur f a c e area t r i b u t a r y to the j o i n t . In a l l the f o l l o w i n g graphs the value of P q shown w i l l represent the t o t a l v e r t i c a l load due to such U.D.L. when the load f a c t o r X i s equal to one. The d i f f e r e n t d i s t u r b i n g f o r c e systems used ( " t r i g g e r systems") are shown i n f i g . 3:7. The magnitude of these p e r t u r b a t i o n loads was contained i n a range from 4% to 8% of the given e x t e r n a l l o a d , a c t i n g at the same j o i n t . 23 FIG. 3 7- PERTURBATION F O R C E S Y S T E M S ( TRIGGER S Y S T E M S ) 24 FIG. J 7 (Cont.d) 25 CHAPTER IV RESULTS FOR THE ORIGINAL CONE 4.1 I n t r o d u c t i o n The model t e s t s c a r r i e d out i n the past (2) y i e l d e d a value of 904 ± 80 l b . f o r the t o t a l s t r u c t u r e c r i t i c a l l o a d , determined by Southwell and Lundquist p l o t s , using j o i n t d e f l e c t i o n s normal to the surface of the cone. In the present i n v e s t i g a t i o n , determinant p l o t s , Southwell p l o t s , and l o a d - d e f l e c t i o n curves were used f o r the e v a l u a t i o n of the c r i t i c a l l o a d . 4.2 Determinant P l o t Table 4.1 contains the r e s u l t s from the determinant p l o t s , which are shown i n f i g s . 411 and k°.2 f o r f i x i t y c o n d i t i o n s (a) and (b) r e s p e c t i v e l y . The random v a r i a t i o n of j o i n t c o o r d i n a t e s , s i m u l a t i n g geometric i m p e r f e c t i o n s , was contained i n a range of +.5%. The same "random geometry" was used i n a l l runs thus l a b e l l e d . The t o l e r a n c e shown i n the values of c r i t i c a l l o a d depends e s s e n t i a l l y on the accuracy of the a n a l y s i s i n the neighborhood of the c r i t i c a l l o a d , i . e . on the magnitude of the incremental step and on the number of i t e r a t i o n s performed at each load l e v e l . The values obtained by the determinant graphs are seen to be i n good agreement w i t h the previous experimental determination of the 26 TABLE 4.1. DETERMINATION OF STRUCTURE CRITICAL LOAD BY DETERMINANT PLOT (a) A l l members, except b r a c i n g d i a g o n a l s , p e r f e c t l y f i x e d to the j o i n t s at t h e i r ends. Theory 1 Theory 2 Exact geometry 920 ± 10 l b . 915 ± 10 l b . Random geometry 927 ± 10 l b . 928 ± 10 l b . Theory 1 : S t a b i l i t y f u n c t i o n s o n l y . Theory 2 : S t a b i l i t y f u n c t i o n s plus chord r o t a t i o n and bowing. (b) A l l members f i x e d to the j o i n t s about the s t r o n g a x i s , but hinged about the weak a x i s . Theory 1 Theory 2 Exact geometry 1048 ± 15 l b . 1035 + 15 l b . Random geometry 996 ± 15 l b . 972 ± 15 l b . 27 Load Factor \(Pg=l20lb.) FIG. 4-1- ORIGINAL C O N E . DETERMINANT P L O T S . FIXITY CONDITION (A) 28 7 60 7 80 800 820 8-40 8 60 880 Load Factor \ (P0 = 1201b) FIG. 4' 2-ORIGINAL CONE.DETERMINANT P L O T S . FIXITY CONDITION (B) 29 c r i t i c a l load, especially considering the li k e l y differences between the model and the structure here analysed, owing to material non-linearities (creep of plexiglass), actual joint behaviour, application of the ex-ternal loads, and actual construction imperfections. The higher c r i t i c a l load for f i x i t y condition (b) i s due to the fact that in this case the lateral buckling of individual members about their weak axis i s excluded from the analysis. It i s as i f the members were actually restrained against motion in the plane of the shell , since no degrees of f r e e d o m a r e c o n s i d e r e d a l o n g the length of a member between panel points. It can also be argued that i f member buckling about the weak axis of the cross-section is allowed for, as in condition (a), above a certain load level some members bring negative contributions to the stiffness matrix of the structure. The influence of geometric imperfections and of different pertur-bation force systems on the zeros of the stiffness matrix determinant appears to be minor as shown i n f i g . 4J3. This could be explained by the fact that the structure is st a t i c a l l y determinate i n i t s primary stress system: hence the distribution of the axial loads in the members should not change much for small imperfections, except probably in the neighborhood of the c r i t i c a l load. In effect table 4.2 points out some of the biggest differences in axial loads between the structure with exact geometry and that with random geometry, at about 1/3 of the c r i t i c a l load and near buckling. It can be seen that the differences are magnified as X ->- X , but since b cr* our geometric imperfections are distributed at random, their overall effect on the structure stiffness determinant probably tends to zero. 30 O R I G I N A L CONE.DETERMINANT P L O T S . 8 0 8-20 8 40 8 60 Load Factor \ (Pa= 1201b.) FIG. 4: 3- COMPARISON DETERMINANT P L O T S 31 TABLE 4.2. INFLUENCE OF GEOMETRIC IMPERFECTIONS ON AXIAL LOADS (a) Members perfectly fixed to the joints. (Theory 1 . Trigger system 3). X - 7.70. P = 120 lb. cr o Member load factor X = 2.67 Exact geom. Random geom. No. Exact geom. Random geom. X = 7.52 X = 7.60 3 16.4 lb. 16.2 lb„ 48.9 lb. 52.4 lb. 5 16.4 13.6 48.8 36.6 6 32.3 33.7 90.4 99.5 13 5.65 7.7 14.9 22.6 34 5.46 5.9 15.9 18.8 39 4.1 5.4 10.2 18.3 63 1.4 .77 4.18 2.1 65 1.4 1.87 4.17 6.3 76 2.1 2.75 5.6 10. 85 -1.3 -2. -3.4 -8. Axial loads: +ve compression, -ve tension. (b) Members fixed about strong axis, hinged about weak axis. (Trigger system 9) X - 8.50. P = 120 lb. cr o Member X = 2.67 X = 8.15 No. Exact geom. Random geom0 Exact geom. Random geom. 2 31.9 lb. 32.8 lb. 96.2 lb. 107.6 lb. 3 16.0 15.9 49.6 67.9 6 32.5 34.3 95.6 124.0 8 33.3 32.6 102.4 90.0 15 6.0 5.7 14.8 30.7 23 20.9 20.0 68.2 49.6 34 5.6 6.1 16.8 27.8 42 4.5 3.7 14.5 -11.4 87 -1.5 -.75 -5. 4.2 32 4.3 Southwell P l o t The second method used f o r the e v a l u a t i o n of the c r i t i c a l load was the Southwell p l o t : the r e s u l t s from the graphs are summarized i n t a b l e 4.3. The d e f l e c t i o n components p l o t t e d were displacements and r o t a t i o n s i n a d i r e c t i o n normal to the s u r f a c e of the cone: A., and u„ N N r e s p e c t i v e l y , as sketched below. A few t y p i c a l Southwell p l o t s are shown i n f i g s . 4;4 and f o l l o w i n g . For these d e f l e c t i o n components ( i . e . A^ and to ) i t was observed t h a t i n a l l graphs, the r e s u l t s of which are shown i n t a b l e 4.3, i f the Southwell p l o t was a s t r a i g h t l i n e over a range of d e f l e c t i o n s of amplitude equal to at l e a s t 60% of the value of the l a r g e s t d e f l e c t i o n used, on the s i d e of the l a r g e s t d e f l e c t i o n , then the corresponding c r i t i c a l load was w i t h i n 15% of the value given by the determinant p l o t f o r the same case. This c o n d i t i o n could then be taken as a p r a c t i c a l c r i t e r i o n f o r the use of the Southwell p l o t . A c t u a l l y , i n the m a j o r i t y of the p l o t s using A„ and u>„, i f the above c o n d i t i o n was met, the Southwell and determinant p l o t s gave values i n agreement w i t h i n 10%. 33 TABLE 4.3. DETERMINATION OF STRUCTURE CRITICAL LOAD BY SOUTHWELL PLOT (i) Plotting joint deflections normal to the surface of the cone (A^) A) Members' end f i x i t y condition (a) (fully fixed) Exact geometry Random geometry C r i t i c a l load Joint Trigger C r i t i c a l load Joint Trigger (lb.) System (lb.) System Theory 1 2500 28 9 * 1360 19 1 2560 18 6 1742 19 1 2960 18 3 1712 18 6 3200 8 2 1792 18 10 2080 18 8 1680 18 7 1824 18 3 5240 23 1 Theory 2 2340 28 9 1552 18 3 3680 19 4 1850 18 4 1900 19 4 2256 23 4 Theory 1 : Stability functions only. Theory 2 : Stability functions plus chord rotation and bowing. ^Torsional r i g i d i t y of members put equal to zero (GJ = 0), (cont'd ...) 34 B) Members' f i x i t y c ondition (b) : fi x e d about strong a x i s , hinged about weak ax i s . Exact geometry Random geometry C r i t i c a l load J o i n t Trigger C r i t i c a l load J o i n t Trigger (lb.) System (lb.) System Theory 1 2750 18 3 950 14 6 • 3700 18 7 960 8 6 2160 8 11 980 13 3 1665 23 11 985 28 9 1010 13 6 1010 13 9 1038 18 3 1040 18 7 1086 18 6 1200 19 6 Theory 2 1152 13 9 960 28 9 1166 28 9 1032 13 9 1230 18 5 1090 18 9 1584 39 5 1104 38 9 1648 38 5 Theory 1 : S t a b i l i t y functions only. Theory 2 : S t a b i l i t y functions plus chord r o t a t i o n and bowing. (cont'd ...) ( i i ) P l o t t i n g j o i n t r o t a t i o n s about an a x i s normal to the sur f a c e of the cone (co^) A) Members' f i x i t y c o n d i t i o n (a) Exact geometry Random geometry C r i t i c a l l o a d J o i n t T r i g g e r C r i t i c a l l o a d J o i n t T r i g g e r ( l b . ) System ( l b . ) System Theory 1 890 23 9 1120 9 1 910 28 9 960 24 1 930 34 1 928 13 8 1000 39 1 924 17 8 1048* 9 1 922 18 8 1000* 24 1 936 19 8 1016* 34 1 922 20 8 1040* 39 1 915 23 8 890 14 1 920 28 8 912 18 1 918 18 3 934 18 10 960 13 3 968 13 1 980 18 7 1040 38 1 1070 8 1 Theory 2 916 28 9 880 28 1 1100 9 1 930 18 3 1120 8 1 980 34 1 990 24 1 1000 9 4 1030 8 1 1096 39 1 1168 9 1 Theory 1 : S t a b i l i t y f u n c t i o n s o n l y . Theory 2 : S t a b i l i t y f u n c t i o n s plus chord r o t a t i o n and bowing. * T o r s i o n a l r i g i d i t y of members equal zero (GJ = 0 ) . JOINT No. 18 AN 4*(x/o3) D/'sp fa cement, inches FIG. 4'4-SOUTHWELL PLOT. EXACT GEOMETRY. JOINT No. 18 ^(xio3) 20 5/5 o \ 10 OV c a Fixity Condition (a) Trigger System 8 Theory (7) Po 320 lb. X cr Per I x 5 320 x 2-88 =922lb. 10 20 30 40 Potation, Radians 50 FIG. 4; 5-SOUTHWELL PLOT'. EXACT GEOMETRY. U>N(xto3) 60 38 Joint No. 18 AN FIG. 4 = 6- S O U T H W E L L PLOT. EXACT GEOM. 39 FIG. 4 '7 - SOUTHWELL PLOT. RANDOM GEOM. JOINT No. 13 AN Note • &N + ve down ward 0 40 80 120 160 200 240 Displacement, inches ^ o FIG. 4-8 - SOUTHWELL PLOT. RANDOM GEOMETRY JOINT No. 28 FIG. 4 •• 9 - SOUTHWELL PLOT .RANDOM GEOMETRY. 42 Other d e f l e c t i o n components were i n v e s t i g a t e d , namely the me r i d i o n a l and c i r c u m f e r e n t i a l components, but f o r these the above c r i t e r i o n does not apply, i . e . some Southwell p l o t s , s t r a i g h t over a range of d e f l e c t i o n s as s p e c i f i e d above, y i e l d e d a value of c r i t i c a l l oad more than 30% higher than the c o r r e c t v a l u e . This may be explained by the f a c t that these l a t t e r d e f l e c t i o n s were not components of the lowest c r i t i c a l mode shape. A c t u a l l y , f o r the s t r u c t u r e w i t h exact geometry and members p e r f e c t l y f i x e d to the j o i n t s ( c o n d i t i o n ( a ) ) , i t was p o s s i b l e to f i n d a disturbance f o r c e system, namely system 8, c o n s i s t i n g of two couples about the Y-axis a c t i n g at j o i n t s 18 and 28, e x c i t i n g a mode shape i n which a l l the j o i n t r o t a t i o n s p l o t t e d (see t a b l e 4.3 ( i i ) ) y i e l d e d p r a c t i c a l l y the same value of c r i t i c a l l o a d . The corresponding mode shape i s given below i n f i g . 4:10. V z FIG. 4:10 - CRITICAL MODE SHAPE. FIXITY CONDITION (A) 43 The f a c t that the j o i n t r o t a t i o n s showed a l a r g e r a t e of incre a s e at approximately the same c r i t i c a l l o a d i n d i c a t e s that b u c k l i n g i s a g l o b a l phenomenon, and the s t r u c t u r e as a whole d i s t o r t s . In t h i s case ( i . e . f i x i t y c o n d i t i o n (a)) b u c k l i n g seems to be i n i t i a t e d by bending of the main members about the weak a x i s of the c r o s s - s e c t i o n , which may be c a l l e d " i n plane" b u c k l i n g . This may e x p l a i n why f o r t h i s c o n d i t i o n the normal d e f l e c t i o n i s not a p p l i c a b l e , i n the sense t h a t , i n the range of loads a p p l i e d to the s t r u c t u r e (below the f i r s t c r i t i c a l l o a d ) , i t does not s a t i s f y the p r e v i o u s l y mentioned c r i t e r i o n , i . e . Southwell p l o t s are not s t r a i g h t l i n e s over a wide range. The values of c r i t i c a l l o a d shown i n t a b l e 4:3 f o r these cases were obtained by drawing a s t r a i g h t l i n e i n t e r p o l a t i n g through the l a s t few p o i n t s of the graph, as shown i n f i g . 4:4. There i s the p o s s i b i l i t y t h a t , i f the a n a l y s i s could be c a r r i e d beyond the f i r s t c r i t i c a l l o a d , these graphs would a l s o become s t r a i g h t l i n e s over a wide range, i n d i c a t i n g a higher c r i t i c a l mode. I t should be p o s s i b l e to perform t h i s check by determining the f i r s t few eigenvectors of the s t r u c t u r e s t i f f n e s s m a t r i x ; the process, however, would be lengthy given the s i z e of the m a t r i x to be analysed. In a few cases the t o r s i o n a l r i g i d i t y of the members was put equal to zero, to see whether t h i s a f f e c t e d the graphs, s i n c e the Southwell method was developed from the eigenvalue f o r m u l a t i o n of the beam-column problem (4, 12), where the t o r s i o n a l r i g i d i t y of a member, GJ, does not i n t e r v e n e . This had a c e r t a i n e f f e c t on one p l o t w i t h A.,, no N p r a c t i c a l e f f e c t f o r to . 44 4.4 L o a d - d e f l e c t i o n curves F i g s . 4J11 and f o l l o w i n g show some t y p i c a l l o a d - d e f l e c t i o n curves f o r the two f i x i t y c o n d i t i o n s (a) and ( b ) . These curves are c o n s i s t e n t w i t h the previous r e s u l t s , because i n the f i r s t case ( p e r f e c t l y r i g i d - j o i n t e d spaceframe) the d e f l e c t i o n component appears to be l a r g e l y u n a f f e c t e d by the f i r s t c r i t i c a l l o a d , whereas the r o t a t i o n c l e a r l y i n d i c a t e s the onset of b u c k l i n g . Under c o n d i t i o n (b) instead, becomes a good i n d i c a t i o n of b u c k l i n g , and a change of s i g n f o r the d e f l e c t i o n , probably depending on the t r i g g e r system used, can be noted as the load approaches the c r i t i c a l l e v e l . F i g . 4:15 shows the p a t t e r n of the displacements along the middle r i n g , near b u c k l i n g , f o r f i x i t y c o n d i t i o n (b) and v a r i o u s t r i g g e r systems. I t i s remarkable that the same wavy p a t t e r n , e s s e n t i a l l y due to the p a r t i c u l a r arrangement of the d i a g o n a l s , was a l s o observed i n the experimental t e s t s . I t can a l s o be noted that a s m a l l disturbance system can cause a l a r g e d i s t o r t i o n i n the displacement p a t t e r n . 4.5 P a r t i a l l o a d i n g To conclude the r e s u l t s f o r the o r i g i n a l cone, an i n v e s t i g a t i o n was made of a U. D. L. covering only h a l f of the s t r u c t u r e . F i g . 4:16 shows that the value of i s s l i g h t l y l e s s than f o r the corresponding case of a U. D. L. over the whole s t r u c t u r e , namely JOINT No. Id AN FIG. 4 •• If - LOAD - DEFLECTION CURVE. FIXITY (A). JOINT No. 18 80 40 N I) and (2 Random Geometry Trigger System 3 (T) Stability Functions only •critical load (by Def.) • II- -t- Chord Rototion and Bowing 250 5 0 0 0 ,  7*5°  1 0 0 0 f250 Rotation ioNtRadians 1500 FIG. 4 •• 12 - LOAD-DEFLECTION CURVE. FIXITY (A). JOINT No. 13 4» 8 0. £ 60/-•i o 40 -Random Geometry ^ * Trigger System 9 (7) Stability Functions only (2) //— // + Chord Rototion and Bowing Note • AN+ ve down ward 20 40 60 80 100 Displacement, inches -critibol lood levels 7 T ' ~ 120 -AN (xlO3) FIG. 4* 13-LOAD DEFLECTION CURVE. FIXITY(B). JOINT No. 28 AN 8 0 &&0 •— v. 40 -critical loads ond @ Rondom Geometry Trigger System 9 (7) Stability Functions only (2) //— // + Chord Rototion and Bowing 20 40 60 80 Displacement, inches 100 •AH (xtO3) 00 FIG. 4- 14 - LOAD-DEFLECTION CURVE. FIXITY(B). 49 Fixity (b) Theory I Stability Functions only P0 = 120 lb ( Xc~ - 8-30) w Angular Position ondom Geometrf Trigger System 9 Exact Geometry No Triggers —•Linear Ano lysis No Triggers Exact Geometry Note Positive Down word FIG. 4- 15 - DISPLACEMENT AN ALONG MIDDLE RING. 5 0 Fixity Condition (b) Exact Geometry NTN 2 0 C\j i 1 1 1 1 1 (7) Stability Functions only \ (2) //— // + Chord Rotation and Bowing Load Factor \ (R= 1201b.) FIG 4 • 16- ORIGINAL CONE .HALF SNOW LOAD. DETERMINANT PLOTS 51 X =8.20 vs. 8.62 f o r theory 2 , and c r J ' X =8.31 vs. 8.72 f o r theory 1 cr J In c o n c l u s i o n then,the s t r u c t u r e buckles at about the same load per square foot whether on the whole or h a l f dome. 4.6 Conclusions about Southwell p l o t The l o a d - d e f l e c t i o n curves a l l o w another e x p l a n a t i o n of the d i f f e r e n t r e s u l t s shown i n t a b l e 4.3, depending on the use of or to^: i n f a c t i t i s shown i n Appendix 3 that the Southwell p l o t being a s t r a i g h t l i n e i m p l i e s the corresponding l o a d - d e f l e c t i o n curve being a r e c t a n g u l a r hyperbola having the c r i t i c a l l o a d f o r h o r i z o n t a l asymptote. This i s seen to occur to <A^  f o r f i x i t y c o n d i t i o n ( a ) , and to f o r f i x i t y c o n d i t i o n (b) on l y . This shape of l o a d - d e f l e c t i o n curve ( i . e . r e c t a n g u l a r hyperbola) i s a l s o c o n s i s t e n t w i t h the r e s u l t s obtained (Home and Merchant ( 4 ) , Timoshenko (17)) by usi n g the c r i t i c a l modes f o r the s e r i e s expansion of any d e f l e c t i o n component. Then each term of the s e r i e s i s independently magnified as the e x t e r n a l l o a d approaches the corresponding c r i t i c a l l o a d . By analogy w i t h the l o a d - d e f l e c t i o n curves r e l a t i v e to the E u l e r column problem, i t could a l s o be s a i d that the Southwell p l o t i s w e l l a p p l i c a b l e to i n s t a b i l i t y cases of the b i f u r c a t i o n type. I n s t e a d , from the two-bar s t r u c t u r e example (Appendix 2 ) , i t appears that when geometric i n s t a b i l i t y (snap-through) dominates, the Southwell p l o t would not be a s t r a i g h t l i n e and should not be a p p l i e d (see f i g . 4117). 52 FIG. 4:17 - GEOMETRIC INSTABILITY (SNAP-THROUGH) In f a c t the equations governing geometric i n s t a b i l i t y d i f f e r from those governing E u l e r b u c k l i n g p r e c i s e l y f o r the presence of the a d d i t i o n a l terms due to chord r o t a t i o n and bowing. As long as these a d d i t i o n a l e f f e c t s are n e g l i g i b l e , i t must be p o s s i b l e to use the Southwell p l o t f o r the e v a l u a t i o n of the c r i t i c a l l o a d , provided the p r e v i o u s l y mentioned c r i t e r i o n i s s a t i s f i e d , and a c o r r e c t set of d e f l e c t i o n s and disturbance systems i s used. However, f o r t h i s type of s t r u c t u r e , there may be d i f f i c u l t i e s 53 i n d e t e c t i n g the lowest c r i t i c a l mode and corresponding c r i t i c a l l o a d , as " i n plane" and "out of plane" b u c k l i n g are coupled, because of the curvature of the cone s u r f a c e . 54 CHAPTER V RESULTS FOR SPACEFRAMES WITH DIFFERENT HEIGHT/SPAN RATIO 5.1 C o n i c a l shapes To check the importance of geometric n o n l i n e a r i t i e s , the c r i t i c a l l o a d l e v e l f o r a U. D. L. over the whole s t r u c t u r e was determined f o r two other cones of lower height/span r a t i o but w i t h the same geometric con-f i g u r a t i o n . Only the determinant p l o t was used i n these cases. Let a Q be the height/span r a t i o f o r the o r i g i n a l cone, which w i l l be c a l l e d FCO; the other two s t r u c t u r e s analysed, c a l l e d FC1 and FC2 1 1 r e s p e c t i v e l y , w i l l have height/span r a t i o s = -^a and = Tp3^* Table 5.1 shows the values of c r i t i c a l load f o r the three s t r u c t u r e s , obtained by determinant p l o t s , which are presented i n f i g s . 511 and 512. Only f i x i t y c o n d i t i o n (b) was analysed, s i n c e i t was assumed tha t l a t e r a l b u c k l i n g of the members would be prevented by p u r l i n s and decking. Again i t can be noted t h a t the d i f f e r e n c e s i n the values obtained by t h e o r i e s 1 and 2 are n e g l i g i b l e . However the i n f l u e n c e of non-l i n e a r i t i e s i s a p p r e c i a b l e , s i n c e the o v e r a l l c r i t i c a l l o a d drops f a s t e r than the s i n e of the slope angle, as i t would happen f o r a l i n e a r theory, s i n c e tie s i z e s of the members, hence the maximum load they could c a r r y , have been kept constant throughout. From a dimensional a n a l y s i s p o i n t of view i t can be s a i d that f o r cones having the same geometric c o n f i g u r a t i o n and arrangement of web members, the c r i t i c a l a x i a l l o a d i n a member, N . w i l l be a f u n c t i o n 55 TABLE 5.1. DETERMINATION OF CRITICAL LOAD BY DETERMINANT PLOT FOR DIFFERENT HEIGHT/SPAN RATIOS F i x i t y c o n d i t i o n ( b ) : members f i x e d to the j o i n t s i n the str o n g d i r e c t i o n , hinged i n the weak d i r e c t i o n . Exact geometry: U. D. L 0 over the whole s t r u c t u r e . Theory 1 Theory 2 FCO (a ) o 1048 l b . 1035 l b . FC1 (h 494 l b . • 486 l b . FC2 C 218 l b . 214 l b . 2.21 7.43 1 a 3.35 o Theory Theory 1 2 L i v e s l e y s t a b i l i t y f u n c t i o n s . L i v e s l e y s t a b i l i t y functions plus chord r o t a t i o n and bowing. DETERMINANT PLOT. Fixity condition ( b) Exact geometry No Triggers 3-85 3-90 395 400 4-05 4-/0 4-15 Load Factor \ I P0= 1201b.) Theory (J) • Stability Functions only Theory \2) • "— '/ + Chord Rotation and Bowing FIG. 5-/-FCI CONE 1/2 ORIGINAL HEIGHT 57 DETERMINANT PLOT. No Triggers Fixity con dit ion (b) Exact geometry Load Factor \(Po=l20lb) Theory (7) •' Stability Functions only Theory (2) • // '/ + Chord Rotation and Bowing FIG. 5-2-FC2 CONE -1/4 ORIGINAL HEIGHT 58 only of i t s s t i f f n e s s , i t s l e n g t h and the height/span r a t i o of the cone, a. Thus we can w r i t e : N c r = f[AE, E I , L, a] [*] where L = j o i n t - t o - j o i n t l e n g t h of a member, a = height/span r a t i o of the cone. Having a r e l a t i o n s h i p among 5 v a r i a b l e s , by Buckingham's n-theorem, t h i s can be reduced to a r e l a t i o n s h i p i n v o l v i n g only 3 dimensionless parameters; f o r i n s t a n c e we may choose: c r AEL , _ . . — J 3 I ~ , - g j - , a , o b t a i n i n g N L 2 - ^ f f - - g[~ , a ] [**] L 2 where — i s the slenderness r a t i o of the member, being Ar = 1 . r Now we can de f i n e an e f f e c t i v e l e n g t h of a member, L = kL, e such as 2 n EI N = =• . S u b s t i t u t i n g i n t o [**] we o b t a i n the simple " a e ) 2 r e l a t i o n s h i p L e L — = g' [— , a] , which can be used to present the e f f e c t s of JLi TC a change i n height/span r a t i o f o r s i m i l a r cones. This i s shown n u m e r i c a l l y i n t a b l e s 5.2 to 5.4 and g r a p h i c a l l y i n f i g . 5:4. I t i s to be noted that r i b members behave i n two d i f f e r e n t ways, because of the arrangement of the di a g o n a l s . These would be 59 TABLE 5.2. ORIGINAL CONE FCO. a = -ATF F i x i t y c o n d i t i o n (b) . Exact geometry. Theory 1 Member Number L r N ( l b . ) c r K " L Rib Members: Type 1 2 41.6 105.8 2.46 27 55.5 47.0 2.08 52 60.9 11.8 3.63 77 74.4 4.66 4.28 Type 2 3 41.6 51.5 3.52 28 55.5 34.8 2.41 53 60.9 24.25 2.53 78 74.4 13.4 2.52 Ring Members: 18 93.5 67.9 1.37 43 88.3 58.1 1.17 68 78.5 35.45 1.125 Diagonals: 13 98.4 18.92 1.77 38 114.5 14.61 1.47 63 130.0 4.05 2.13 N = X N , where N = a x i a l f o r c e i n a member at loa d l e v e l c r c r o o X = 1, determined by l i n e a r spaceframe a n a l y s i s . / ~ E I — L = member length ( j o i n t - t o - j o i n t ) ; L = II ^  — — c r 60 TABLE 5.3. CONE FC1. a 1 1 1 2 ao 6.7 F i x i t y condition (b). Exact geometry. Theory 1 . Member Number L r N (lb.) cr L Rib Members: Type 1 2 36.5 85.0 3.12 27 48.6 39.8 2.57 52 53.3 8.6 4.85 77 65.2 4.33 5.05 Type 2 3 36.5 37.0 4.75 28 48.6 27.2 3.1 53 53.3 19.5 3.22 78 65.2 12.3 3.0 Ring Members: 18 93.5 49.95 1.6 43 88.3 55.5 1.2 68 78.5 32.4 1.18 Diagonals: 13 94.0 15.27 2.06 38 107.5 14.7 1.56 63 118.7 4.6 2.19 61 TABLE 5.4. CONE FC2. a 4 o 13.4 F i x i t y c o n d i t i o n (b ) . Exact geometry. Theory 1 . Member Number L r N ( l b . ) c r L Rib Members: Type 1 2 35.15 64.6 3.73 27 46.9 33.72 2.9 52 51.3 8.36 5.11 77 62.6 4.69 5.05 Type 2 3 35.15 26.1 5.85 28 46.9 21.92 3.6 53 51.3 16.11 3.68 78 62.6 11.0 3.3 Ring Members: 18 93.5 27.76 2.14 43 88.3 43.9 1.35 68 78.5 27.7 1.28 Diagonals: 13 92.8 9.85 2.6 38 105.5 11.73 1.79 63 115.6 4.41 2.3 62 s t r e s s l e s s and every r i b (e.g. AB and CD from f i g . 5:3) would behave i d e n t i c a l l y , i f the s t r u c t u r e were analysed as a space t r u s s , because of the r o t a t i o n a l symmetry of the s t r u c t u r e and of the l o a d i n g system (UDL). I f t h i s i s done, i t can be seen that the j o i n t s of a r i b l i k e AB move outward from the sur f a c e of the cone, w h i l e the j o i n t s of adjacent r i b s l i k e CD move inward. I t i s to be noted t h a t , even f o r s m a l l s t r a i n s , as must occur i n the e l a s t i c range, the j o i n t displacements normal to the cone surface are q u i t e l a r g e , because adjacent r i n g members are almost c o l l i n e a r . To r e s t o r e slope c o n t i n u i t y of the r i n g members ''(fSame act-ion), shear f o r c e s , a c t i n g normal to the surface of the cone, are re q u i r e d at the ends of these members. The magnitude of these forces i s about 25% of EFFECT OF VARYING HEIGHT/SPAN RATIO. L ( For L/r shown in Tobies 52 to 5-4) 2 Nu? Xcr No L - member length FIG. 5=4 - CONICAL SPACEFRAMES 64 the corresponding e x t e r n a l j o i n t l o a d . Because of the s m a l l angle between the plane determined by two diagonals meeting at a j o i n t (say No. 22) and the plane tangent to the cone at the same j o i n t , these shear f o r c e s a r i s i n g from frame a c t i o n produce r a t h e r l a r g e compressive loads i n the diagonals of the two lower a r r a y s . These loads c a r r i e d by the d i a g o n a l s , i n t u r n , modify the d i s t r i b u t i o n of a x i a l loads i n two adjacent r i b s l i k e AB and CD. In summary then, the secondary s t r e s s e s set up i n t h i s continuous space t r u s s are much l a r g e r than the secondary s t r e s s e s i n an o r d i n a r y plane t r u s s because members at a j o i n t are almost coplanar. This accounts f o r the two types of r i b members shown i n the f o l l o w i n g t a b l e s and f i g u r e s : type 1 f o r r i b s l i k e CD and type 2 f o r r i b s l i k e AB. The h i g h values of k obtained i n d i c a t e t h a t , under the given c o n d i t i o n s of l o a d i n g and member f i x i t y , b u c k l i n g of the s t r u c t u r e i s a g l o b a l phenomenon, i n v o l v i n g the whole s t r u c t u r e r a t h e r than i n d i v i d u a l members. In some cases, e.g. f o r the upper r i b members of type 1 , the high value of k i s due to overdesign, i . e . the member i s not s t r e s s e d to i t s f u l l c a p a c i t y . This happens because the a c t u a l cone had to support a s u p e r s t r u c t u r e housing mechanical systems f o r the h a n d l i n g of potash ( 2 ) . Since i t has been shown that the e f f e c t of geometric n o n l i n e a r i t i e s i s always minor f o r these c o n i c a l spaceframes, an eigenvalue a n a l y s i s could be c a r r i e d out to detect the a c t u a l c r i t i c a l mode shapes: t h i s could help e x p l a i n the r e s u l t s obtained f o r the e f f e c t i v e l e n g t h s . 5.2 S p h e r i c a l shape To i n v e s t i g a t e the s t r u c t u r a l advantages o f f e r e d by a doubly 65 curved shape w i t h respect to a s i n g l y curved one, an a n a l y s i s was c a r r i e d out f o r a s p h e r i c a l spaceframe dome, having the same height and span as the c o n i c a l spaceframe FC1 ( i . e . a = T^J) as shown i n f i g . 515. FIG. 5:5 - SPHERICAL SPACEFRAME This s p h e r i c a l spaceframe w i l l be c a l l e d FS1. The c r i t i c a l load was determined by determinant p l o t s , shown i n f i g . 516. Again theory 1 and theory 2 y i e l d p r a c t i c a l l y the same val u e . Table 5.5 compares the r e s u l t s obtained by theory 1 f o r the c o n i c a l and s p h e r i c a l dome r e s p e c t i v e l y . Again i t can be noted that k i s g e n e r a l l y g r e a t e r than one, i n d i c a t i n g that b u c k l i n g i n v o l v e s the whole s t r u c t u r e and not i n d i v i d u a l members. For the r i b members the r e d u c t i o n i n the values of k f o r the s p h e r i c a l dome w i t h respect to the c o n i c a l would permit a r e d u c t i o n i n member s i z e s of the order of 50%. 5 • -66 DETERMINANT PLOT. Fixity Condition ( b) Exact Geometry No Triggers U. D. L . (7) Livesley Stobility Functions only (2) Chord Rotation and Bowing 10 O 10 2 10-4 10-6 10 8 I/O Load Factor \(Pa= 1201b.) FIG. 5 : 6 - SPHERICAL DOME SF1 67 TABLE 5.5. COMPARISON OF A CONICAL AND A SPHERICAL SPACEFRAME F i x i t y c o n d i t i o n ( b ) . Exact geometry. U. D. L. C o n i c a l dome FC1 S p h e r i c a l dome FS1 C r i t i c a l l o a d f a c t o r , X ' c r 4.11 10.81 (P = 120 l b . ) o O v e r a l l c r i t i c a l 494 l b . 1300 l b • l o a d ( v e r t i c a l ) Member Number L r N ( l b . ) c r L L r N ( l b . c r ) L " I T Rib Members: . 2 36.5 85.0 3.12 37.2 150.2 2.3 27 48.6 39.8 2.57 48.75 96.0 1.65 53 53.3 19.5 3.22 53.5 51.8 1.98 78 65.2 12.3 3.0 66.5 35.5 1.73 Ring Members: 18 93.5 49.95 1.6 95.4 42.0 1.71 43 88.3 55.5 1.2 91.5 72.6 1.02 68 78.5 32.4 1.18 82.1 61.8 .82 Diagonals: 13 94.0 15.27 2.06 95.2 25.6 1.57 38 107.5 14.7 1.56 109.1 17.15 1.43 63 118.7 4.6 2.19 120.3 7.37 1.71 height/span r a t i o a = 6.7 N = X N : N = a x i a l f o r c e i n a member at X=l determined by l i n e a r c r c r o o ., spacerrame a n a l y s i s EI N cr 68 However, t h i s advantage from a s t r u c t u r a l p o i n t of view would be p a r t i a l l y o f f s e t by f a b r i c a t i o n c o mplications due to the f a c t that the r i b s would no longer be s t r a i g h t throughout. A l s o , p a r t i a l loadings may cause t e n s i o n i n the main members, i n c r e a s i n g the complexity of the connections. F i n a l l y a s p h e r i c a l shape would r e q u i r e a greater q u a n t i t y of decking m a t e r i a l and longer members than the corresponding c o n i c a l shape (but i t would provide a greater volume of covered, a v a i l a b l e space). In c o n c l u s i o n , the p o s s i b i l i t y of usi n g a s p h e r i c a l dome i n s t e a d of a c o n i c a l one seems worthy of d e t a i l e d i n v e s t i g a t i o n . 69 CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS The overall c r i t i c a l load for the original cone determined in this analysis was found to be in satisfactory agreement with the value obtained in the previous model tests. The results of this investigation show also that, for a space-frame of this type and for practical values of height/span ratios, a st a b i l i t y analysis in the elastic range up to the f i r s t c r i t i c a l load can be successfully carried out using only Livesley s t a b i l i t y functions: the additional effects due to chord rotation and bowing are generally negligible. Actually i t should be noted that Livesley's version of st a b i l i t y functions ( 3 , 4) contains sway terms of the type Np, where N is the axial load in a member and p is the chord rotation, as shown in Appendix 1. S t r i c t l y speaking, these terms should be included in the geometric part of the stiffness matrix, as i t is done in ref. (10 ) . This may explain partly the closeness of results obtained by theories 1 and 2 , since Livesley s t a b i l i t y functions were used for theory 1 An investigation of the post-buckling behaviour of this type of spaceframes (statically determinate in their primary stress system) seems to be of l i t t l e practical interest, since the structure does not possess any adequate reserve strength beyond the f i r s t c r i t i c a l load. For other spaceframes, namely those with redundant members, i t may be interesting to follow the deformation of the structure past the lowest c r i t i c a l load: then displacements and rotations may become very 70 l a r g e and the present a n a l y s i s would no longer be a p p l i c a b l e . In such a case, as w e l l as i n the general case of s t r u c t u r e s e x h i b i t i n g t r u l y l a r g e d e f l e c t i o n s w h i l e remaining e l a s t i c , the best way to t a c k l e the problem seems to be to r e s o r t to a f i n i t e element formu-l a t i o n w i t h an incremental step-by-step s o l u t i o n by means of Taylor's expansion theorem, as shown i n references (13) and (14). I f the problem r e q u i r e s i n v e s t i g a t i o n , the l a t e r a l - t o r s i o n a l s t a b i l i t y of an i n d i v i d u a l member can be checked s e p a r a t e l y i n a second stage of the a n a l y s i s , using f o r i n s t a n c e the procedure o u t l i n e d i n references (15) and (16). From the present i n v e s t i g a t i o n i t appears a l s o t h a t , when per-forming a numerical s t a b i l i t y a n a l y s i s , the Southwell p l o t should only be used together w i t h the determinant p l o t , s i n c e i t can be m i s l e a d i n g , e s p e c i a l l y f o r the d i f f i c u l t y , i n a complex s t r u c t u r e , of f i n d i n g a c o r r e c t d i s t u r b i n g f o r c e system and a corresponding set of d e f l e c t i o n s , t r u l y r e p r e s e n t a t i v e of the lowest b u c k l i n g mode. For t h i s purpose, a set of geometric i m p e r f e c t i o n s together w i t h s e v e r a l d i f f e r e n t p e r t u r -b a t i o n f o r c e systems should be used f o r a r e l i a b l e i n v e s t i g a t i o n . I f the Southwell p l o t s thus obtained are s t r a i g h t l i n e s upward from about 40% of the value of the l a r g e s t d e f l e c t i o n used, and y i e l d c o n s i s t e n t values f o r the c r i t i c a l l o a d , then such a value can be r e l i e d upon. In these cases the Southwell plot can give u s e f u l i n f o r m a t i o n about the s e n s i t i v i t y of the s t r u c t u r e to d i f f e r e n t p e r t u r b a t i o n s , and can be r e -garded as an a l t e r n a t e method of f i n d i n g the eigenvector corresponding to the lowest eigenvalue, of the s t r u c t u r e s t i f f n e s s m a t r i x . I t can be noted that the d i f f i c u l t y mentioned above of f i n d i n g 71 a c o r r e c t disturbance system, does not occur i n an a c t u a l model t e s t , s i n c e n a t u r a l i m p e r f e c t i o n s i n the c o n s t r u c t i o n of the model or i n the l o a d i n g system u s u a l l y provide a s u f f i c i e n t p e r t u r b a t i o n f o r the use of the Southwell p l o t . The concept of e f f e c t i v e length of a member w i t h a s s o c i a t e d a l l o w a b l e s t r e s s can be u s e f u l i n the p r e l i m i n a r y design of a s t r u c t u r e of t h i s type. However, s i n c e b u c k l i n g i s g e n e r a l l y not due to f a i l u r e of an i n d i v i d u a l member, but r a t h e r i s a g l o b a l phenomenon i n v o l v i n g the whole s t r u c t u r e , i t cannot be s a f e l y assumed th a t the unsupported length of a member i s the j o i n t - t o - j o i n t l e n g t h , as i t i s o f t e n taken f o r plane t r u s s e s . For i n s t a n c e , f o r the c o n i c a l spaceframes analysed i n the present Le work, the r a t i o j — i s seen to vary approximately from 2 to 4 f o r the r i b members and from 1 to 2 f o r the r i n g members, according to the height/span r a t i o . Thus f o r each given s t r u c t u r e of t h i s type (spaceframe domes) i t i s a d v i s a b l e to c a r r y out a s t a b i l i t y a n a l y s i s w i t h L i v e s l e y s t a b i l i t y f u n c t i o n s . An i n v e s t i g a t i o n should a l s o be made of p a r t i a l loadings and, p o s s i b l y , of the e f f e c t of geometric i m p e r f e c t i o n s , s i n c e they can cause a c o n s i d e r a b l e change i n the a x i a l l o a d d i s t r i b u t i o n i n the members, e s p e c i a l l y c l o s e to the c r i t i c a l load l e v e l . To conclude, w i t h reference to f i g . 6 t l , and quoting from reference ( 4 ) , p. 45, "Since, i n s t r u c t u r e s subject to i n s t a b i l i t y , the margin of s a f e t y as measured by a load f a c t o r may be markedly and dangerously s n a l l e r than the ' f a c t o r of s a f e t y ' measured by a s t r e s s 72 •4 Maximum s t r e s s FIG. 6:1 - EFFECT OF NONLINEARITIES ON THE MARGIN OF SAFETY r a t i o , i t i s important t h a t l o a d f a c t o r s , not s t r e s s f a c t o r s , should be used i n design.". 73 APPENDIX I DERIVATION OF THE NON-LINEAR RELATIONS GOVERNING THE DEFORMATION OF A MEMBER (10) The assumptions made i n t h i s d e r i v a t i o n , as l i s t e d i n Chapter 2, are: 1. The m a t e r i a l i s l i n e a r e l a s t i c . 2. Each member i s p r i s m a t i c and homogeneous. 3. Loads are a p p l i e d only at the ends of a member. 74 4. Shear deformations are neglected. 5. L i n e a r s t r a i n s and squares of the r o t a t i o n s are of the same order of magnitude and s m a l l compared to one. 6. T o r s i o n - f l e x u r e c o u p l i n g and warping r e s t r a i n t are neg l e c t e d . With these assumptions i t can be shown ( r e f . (11), page 466) tha t the E u l e r i a n and Lagrangian d e s c r i p t i o n of the deformation d i f f e r only by higher order i n f i n i t e s i m a l s . Then we can w r i t e , f o r the two-dimensional case: ( e ) ^ = e - 5 K 9u . 1 2 £ = -r— + -7T iii 9x 2 9(0 f i n 9v CO = -r-9x where e = the l o n g i t u d i n a l s t r a i n of the c e n t e r - l i n e of the member, u,v = t r a n s l a t i o n components w i t h respect to x,y d i r e c t i o n s , co = r o t a t i o n of a c r o s s - s e c t i o n about the z - a x i s , K = curvature of the c e n t e r - l i n e of the member. 9u 2 With our assumptions co << 1. Then the a x i a l f o r c e N and bending moment M r e s u l t : ( i n the absence of i n i t i a l s t r a i n s ) N = AE £ . [2] M = EI K and the transverse shear f o r c e V i s : 75 We can now apply the p r i n c i p l e of v i r t u a l displacements, i n the form: 6W. = 6W , where 6W denotes v i r t u a l work. 1 e ' In our case: L / (N6e + M6K )dx = F D 6u„ + F_, 6v„ + M L 6a>D Bx J3 By B a ii o + FAx 6 U A + V V A + M A % ' ' ' M Note t h a t , u s i n g the Lagrangian d e s c r i p t i o n , where a l l v a r i a b l e s are r e f e r r e d to the i n i t i a l s t a t e , we can perform the i n t e g r a t i o n at the L.H.S. of [4] along the undeformed p o s i t i o n of the member. Moreover we can s u b s t i t u t e everywhere t o t a l d e r i v a t i v e s f o r p a r t i a l d e r i v a t i v e s , s i n c e u,v,w, being displacement components of p o i n t s of the c e n t e r - l i n e , are now f u n c t i o n s of x o n l y . Now l e t n(x) and ct(x) be (small) v a r i a t i o n s of u(x) and w(x); then <§e = <5(T~ + TT U ) = — h toa. dx 2 dx We a l s o have: 6K = 6(4~) = 4^  J and l e t t i n g dx dx 3(x) be the v a r i a t i o n of v ( x ) , we have: a(x) = 6a) = 6(4^ ) = 4^  • dx dx S u b s t i t u t i n g i n the L„H.S of equation [4] we o b t a i n ; 76 L L /• (N<Se + M6<)dx = / [ N ( ^ + toa) + M^-] dx dx dx o o i n t e g r a t i n g by p a r t s = Nn + Ma — J Cn -5 Ntoa + a — ) dx dx dx o o . . . [5] We now n o t i c e that n(o) = 611^ , n(L) = 6 u B , a(o) = 6to^ , a(L) = 6co_ , 6 (0) = 6v A , g(L) = 6v„ , and that / a(Nco - ^ ) dx = / I 2 - (Nco - ^ ) dx = g(Nco -dx dx dx dx o o o Thus we get: ( s u b s t i t u t i n g back i n t o [5]) / (NSe + M6K ) dx = N_6VL - N . 6 u . + M ( L ) S<o - M(Q )6co A B B A A B A o , ,.T dM. , , dM,. . . dN , + (Nco - ^ ) B 6 v B - (Nco - ^ ) A 6 v A - / n ^  dx o - / [3 4" (Nco - f ^ ) ] d x dx dx o [6] Comparing [6] w i t h [ 4 ] , f o r a r b i t r a r y v i r t u a l displacements, and by Lagrange's lemma (e.g. r e f . (11), p. 273), we must have: and the boundary conditions: and F = -N Ax A dM \ F = -(Nco)A + ( - ) A - - ( N . ) A - V A \ at x = 0 "Ay M A = - M ( 0 ) F = N Bx B ,dM, FBy = ( N W ) B " fe>B - <Nu>B + VB ^ = M(L) at x=L It should be noted that the term i n v o l v i n g the shear force V i s not present i n the expressions for F, and F„ . I t can be shown that . r Ax Bx th i s i s a consequence of the i n i t i a l assumptions: du _,_ 1 22 . dx 2 K = dx 2 and V = - dM dx or that a> << 1 . I f the f u l l expressions f o r e and K are used, and taking V = -dM ds then a shear force component w i l l appear i n the h o r i z o n t a l components of the end-forces. The above d e r i v a t i o n shows one of the advantages of the v a r i a t i o n a l methods i n obtaining automatically the d i f f e r e n t i a l equations and the boundary conditions for a given problem. We can now follow r e f . ( 1 0 ) i n obtaining: N = const. = F g x [from [7]] N du , 1 2 e = AE = dx 2 " W Hence: AE , s •AE r 2 "Bx = L~ ( UB " V + 2 L f W d x o . . . . [9] 78 The second term at the right-hand s i d e of [9] i n c l u d e s the e f f e c t s of chord r o t a t i o n and bowing. Now [8] becomes: • ~ - - N - ^ = 0 • 2 dx dx 2 2 2 or d M „ d v n , T d v — j ~ N — 2 = > where M = EI — j » dx dx dx which has f o r s o l u t i o n : I - " h <D21 s i n * I  + D22 c o s * !> + D 2 3 I + °24 ' * [ 1 0 ] where: ,2 " F B X L 2 D22 = " K 2 1 W A " K 2 2 W B + K23 P ~ " * K - »*) - (A) D, 21 1-C v B A y V l - G ' 22 D „ = to. + 23 A cj) cb(S-Ccb) . _ cb(cb-S) 21 2(1-C)-Scj) ' 22 2(1-C) -Scb K23 = K 2 1 + K22 and C = cos <j> VB ~ V A c • . P = 1 • S = s i n <j> Note that p represents the "chord r o t a t i o n " of a member. I f F„ i s p o s i t i v e , i . e . the member i s under t e n s i l e a x i a l Bx l o a d , <J) becomes imaginary, and h y p e r b o l i c f u n c t i o n s s u b s t i t u t e the tr i g o n o m e t r i c f u n c t i o n s . 79 Using equation [10] i t i s now p o s s i b l e to perform the i n t e g r a t i o n appearing i n [ 9 ] , namely AE . L 2 • AE r L .dv.2 . 757- J to dx = ^ t - / (-j—) dx 2L Q 2L Q dx ~f {°23 + 2 X I-SD 2 1 + d - C ) D 2 2 ] + \ p 2 . - s i + SC, 2* 2 * Then we can express the end-forces on a member as f u n c t i o n s only of the end-displacements, i . e . 2 1^ = M(L) = EI j = EI j + _ k ^ p ] ^ dx M A-M ( 0 ) [ K 2 1 C O A + K 2 2CO B - K 2 3 p] EI F B y = ^ 2 K 2 3 [~ WA - + 2 p ] + p F B x F = -F Ay By _ AE . . AE " 2, Bx = L" ( U B " V + 2L fQ U d x F = -F Ax Bx > [11] The r e l a t i o n s f o r the three-dimensional case are obtained a s s o c i a t i n g bending i n the (x,y) plane, bending i n the (x,z) plane and unre s t r a i n e d t o r s i o n . These r e l a t i o n s h i p s can be put i n ma t r i x form, 80 i n member c o o r d i n a t e s , as f o l l o w s : {f} = [ k ] m {u} - {f G> [12] where {f} = vector of member end-forces, {u} = v e c t o r of member end-displacements, [ k ] m = member s t i f f n e s s m a t r i x , i n c l u d i n g the s t a b i l i t y f u n c t i o n s , {f } = v e c t o r of g e o m e t r i c a l l y n o n l i n e a r terms - ' i f - ' j ; . 0,0,0,0,0, - f- . A f 0,o,o,o,o}T L 2 2 A = / (co + c o ) d x , 0 y ,m where i f L i v e s l e y ' s form of s t a b i l i t y f u n c t i o n s i s used i n [k] . In f a c t i t should be noted that and p r e v i o u s l y defined correspond e x a c t l y to the s t a b i l i t y f u n c t i o n s s and sc described i n r e f . ( 4 ) , page 52. In incremental form, equation [12] becomes being {Af} = [ k ] m {Au} + [ k j m {Au} = [k ] m {Au} [ k G ] m {Au} = - { A f G } , [ k t ] m - [ k ] m + [ k G ] m ; and where Tk l m  L G J 1 2 x 1 2 t k l G ] 3 x 3 [ k l G ] 3 x 3 L AEp, 0 I I 1 0 1 0 1 T 1 0 0 1 ° " [ k l G ] 3 x 3 0 1 ° T 1 0 A 1 AEp 3 1 1 AEp 2 u 1 L 1 ' L — _ _ h  H-AEp 3 | AEp 3 1 AEp 2P 3 L 1 L 1 L ^ 2 ^ | . AEp' 0 0 0 81 W A w i t h p 3 " L " > " P 2 =-^~L V — V w — w B A B A 82 APPENDIX 2 EXACT SOLUTION FOR TWO-BAR STRUCTURE Because of the symmetry, the problem has one degree of freedom, say u. By the p r i n c i p l e of v i r t u a l work: WSu = 6U , where U i s the s t r a i n energy of the s t r u c t u r e . But now 6U = ~ <5u , hence du W = f [1] du AE 2 For the two b a r s : U = -— (AL) , where Li o AL = L - L o Therefore U = — [2L 2 + u 2 - 2uh - 2L A2 + u 2 - 2uh ] L o o o o 83 Hence I T dU 2AE r , j . u - h . r , W = 7 - = — [u - h - L ZZZZZZZZZZZT" J L 2 J du L o 7^ 5 7. ° A2 + u 2 - 2uh Therefore W = 0 f o r u = 0, h, 2h W = W when 4~ = 0 (tangent s t i f f n e s s equal to zero) c r du dW du = 0 f o r u = u , where c r ' u c r = 1 ± 5 / h / (Lax 2/3 a -1 [3] We see that indeed there are two c r i t i c a l p o i n t s , symmetric w i t h respect to u = h. 84 To find W , we substitute [3] back into [2], obtaining: Wcr = 2AE[l-(f- ) 2 / 3 ] 3 / 2 [ 4 ] o For — = .10, we have a ' u 1000 W - r " « 4 2 3 6 ' - 2 A F T ^ = - 1 9 ° 5 4 APPENDIX 3 NOTES ON THE SOUTHWELL PLOT The Southwell p l o t w i l l be a s t r a i g h t l i n e i f f d(|) , f = a constant = c do Expanding the d i f f e r e n t i a t i o n : i d<7> or — - c = - o P d6 Separating the v a r i a b l e s : d6 4> 86 Integrating: In 6 = In (j - c) + k 1 where k^ is a constant of integration. Therefore In 6 (j - c) = -k x or 6 (-|- - c) = e ^ = c^ ( a constant) [1] which in the plane (P,<5) represents a rectangular hyperbola with a horizontal asymptote for P = P = — , since then J r cr c ' 6 = 1 p " c To see this better, consider the translation of axes: c i 6 = 6 ' - — c P = -P' + -c and substitute back into [1]: C l 1 (6' - - i ) ( — H -c -P« + i - c) = c 1 or (6' - — ) ( ! + cP' - 1) = -C.P' + whence 6'cP' - c,P' = -c.P' + — 1 1 c or 6'P' = —TT = a constant, 87 The above equation i s the usual form f o r a r e c t a n g u l a r hyperbola, i n the plane (P',6'), r e f e r r e d to i t s asymptotes. Note that c was the slope of the Southwell l i n e , and we have thus shown that 1 C = F ~ * c r APPENDIX 4 88 LIST OF REFERENCES J . M. Gere & W. Weaver: A n a l y s i s of Framed S t r u c t u r e s , D. Van Nostrand Co., New York, 1965. B. Madsen: "Unique Design i n Glulam", E. I . C. Engineering J o u r n a l , A p r i l 1962. R. K. L i v e s l e y : M a t r i x Methods of S t r u c t u r a l A n a l y s i s , Pergamon Press L t d . , London, 1964. M. R. Home, W. Merchant: The S t a b i l i t y of Frames , Pergamon Press L t d . , London, 1965. J . H. A r g y r i s : "Continua and D i s c o n t i n u a " , From the Proceedings of the Conference on M a t r i x Methods i n S t r u c t u r a l Mechanics, he l d at Wright-Patterson A i r Force Base, Ohio, 26-28 October, 1965. F. B. Hildebrand: I n t r o d u c t i o n to Numerical A n a l y s i s , McGraw-Hill Book Co., Inc., New York, 1956. R. H. M a l l e t t and L. A. Schmit, J r . : "Nonlinear S t r u c t u r a l A n a l y s i s by Energy Search", ASCE, J o u r n a l of the S t r u c t u r a l D i v i s i o n , June 1967. W. Merchant, D. M. B r o t t o n , M. A. M i l l a r : "A Computer Method f o r the A n a l y s i s of Nonlinear E l a s t i c Plane Frameworks" , Paper No. 10, I n t e r n . Symposium on the Use of E l e c t r o n i c D i g i t a l Computers i n S t r u c t u r a l Engineering, U n i v e r s i t y of Newcastle on Tyne, England, 1966. 89 (9) M. S. Zarghamee and J . M. Shah: " S t a b i l i t y of Spaceframes" ASCE, J o u r n a l of the Engineering Mechanics D i v i s i o n , A p r i l 1968. (10) J . J . Connor, R. D. Logcher, Shing-Ching Chan: "Nonlinear A n a l y s i s of E l a s t i c Framed S t r u c t u r e s " , ASCE, J o u r n a l of  the S t r u c t u r a l D i v i s i o n , June 1968, pp. 1525-1545. (11) Y. C. Fung: Foundations of S o l i d Mechanics , Chapter 16. P r e n t i c e - H a l l , I n c . , Englewood C l i f f s , N. J . , 1965. (12) S. T. Ariaratnam: "The Southwell Method f o r P r e d i c t i n g C r i t i c a l Loads of E l a s t i c S t r u c t u r e s " , Quart. Journ. Mechanics and  A p p l i e d Mathematics , V o l . XIV, P t . 2, 1961. (13) J . T. Oden: "Numerical Formulation of Nonlinear E l a s t i c i t y Problems", ASCE, J o u r n a l of the S t r u c t u r a l D i v i s i o n , June 1967. (14) N. D. Nathan: F i n i t e Element Formulation of G e o m e t r i c a l l y N o n l i n e a r Problems of E l a s t i c i t y , Ph.D. T h e s i s , U n i v e r s i t y of Washington, • S e a t t l e , 1969. (15) R. De V a i l : The L a t e r a l - T o r s i o n a l B u c k l i n g of doubly-symmetric wide-flange SecfciQ.ns y M.A.Sc. T h e s i s , U n i v e r s i t y of B. C , Vancouver, A p r i l 1968. (16) B. A. Z a v i t z : A s t i f f n e s s m a t r i x f o r t w i s t bend b u c k l i n g of narrow r e c t a n g u l a r s e c t i o n s , M.A.Sc. T h e s i s , U n i v e r s i t y of B. C , Vancouver, May 1968. (17) S. P. Timoshenko: Theory of E l a s t i c S t a b i l i t y , 2nd ed., McGraw-H i l l Book Co., Inc., New York, 1961. 

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