ULTIMATE LOAD ANALYSIS OF FIXED ARCHES by Andrew John M i l l B.A.Sc, The U n i v e r s i t y of B r i t i s h Columbia, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1985 ©Andrew John M i l l , 1985 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l l m e n t of the requirements f o r 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. agree that permission for extensive purposes may I further copying of t h i s t h e s i s f o r s c h o l a r l y be granted by the Head of my Department or by h i s or her rep- resentatives. 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 g a i n s h a l l not be allowed without my w r i t t e n permission. 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 1956 Main M a l l • Vancouver, Canada V6T 1Y3 Date October, 1985 ABSTRACT The advent of L i m i t States Design has created the necessity for a better understanding of how s t r u c t u r e s behave when loaded beyond f i r s t l o c a l y i e l d i n g and up to c o l l a p s e . Because the problem of determining the u l t i m a t e load capacity of s t r u c t u r e s i s complicated by geometric and material n o n - l i n e a r i t y , a closed form s o l u t i o n for anything but the simplest of s t r u c t u r e i s not p r a c t i c a l . With t h i s as motivation, ultimate capacity of f i x e d arches i s examined i n t h i s t h e s i s . r e s u l t s are presented i n the form of dimensionless the The c o l l a p s e curves. The form of these curves i s analogous to column capacity curves i n that an ultimate load parameter w i l l be p l o t t e d as a f u n c t i o n of slenderness. The ultimate capacity of a structure i s often determined by P l a s t i c Collapse a n a l y s i s or E l a s t i c Buckling. P l a s t i c Collapse i s a t t a i n e d when s u f f i c i e n t p l a s t i c hinges form i n a structure to create a mechanism. This a n a l y s i s has been proven v a l i d f o r moment r e s i s t i n g frames subjected to large amounts of bending and whose second order e f f e c t s are minimal. E l a s t i c buckling i s defined when a second order s t r u c t u r e s t i f f n e s s matrix becomes s i n g u l a r or negative d e f i n i t e . correctly Pure e l a s t i c buckling p r e d i c t s the u l t i m a t e load i f a l l components of the s t r u c t u r e remain e l a s t i c . This may occur i n slender structures loaded to produce large a x i a l forces and small amounts of bending. Because arches are subject to a considerable amount of both a x i a l and bending, i t i s c l e a r that a reasonable ultimate load a n a l y s i s must include both p l a s t i c hinge formation and second order e f f e c t s i n order to evaluate a l l ranges of arch slenderness. - i i - A computer program a v a i l a b l e at the U n i v e r s i t y of B r i t i s h Columbia accomplishes the task of combining second order a n a l y s i s with p l a s t i c hinge formation. This u l t i m a t e load anaysis program, c a l l e d "ULA", i s i n t e r a c t i v e , allowing the user to monitor the behaviour of the structure as the load l e v e l i s increased to u l t i m a t e . A second order a n a l y s i s i s c o n t i n u a l l y performed on the s t r u c t u r e . Whenever the load l e v e l i s s u f f i c i e n t to cause the formation of a p l a s t i c hinge, the s t i f f n e s s matrix and load vector are a l t e r e d to r e f l e c t t h i s hinge formation, and a new s t r u c t u r e i s created. I n s t a b i l i t y occurs when a s u f f i c i e n t l o s s of s t i f f n e s s brought on by the formation of hinges causes the determinant of the s t i f f n e s s matrix to become zero or negative. Two d i f f e r e n t load cases were considered i n t h i s work. point load and a uniformly d i s t r i b u t e d l o a d . These are a Both load cases included a dead load d i s t r i b u t e d over the e n t i r e span of the arch. The load, e i t h e r point load or uniform l o a d , at which c o l l a p s e occurs i s a f u n c t i o n of several independent parameters. I t i s convenient to use the Buckingham n Theorem to reduce the number of parameters which govern the behaviour of the system. For both load cases, i t was necessary to numerically vary the l o c a t i o n or p a t t e r n of the loading to produce a minimum dimensionless load. Because of the multitude of parameters governing arch a c t i o n i t was not p o s s i b l e to describe a l l arches. Instead, the dimensionless behaviour of a standard arch was examined and the s e n s i t i v i t y of t h i s standard to various parameter v a r i a t i o n s was given. Being three times redundant, a f i x e d arch p l a s t i c c o l l a p s e mechanism r e q u i r e s four hinges. This indeed was the case at low L/r. However, at intermediate and high values of slenderness, the loss of s t i f f n e s s due to - iii - the formation of fewer hinges than required f o r a p l a s t i c mehanism was s u f f i c i e n t to cause i n s t a b i l i t y . As w e l l , i t was determined that pure e l a s t i c buckling r a r e l y , i f ever, governs the design of f i x e d arches. F i n a l l y , the collapse curves were applied to three e x i s t i n g arch bridges; one aluminum arch, one concrete arch, and one s t e e l arch. The ultimate capacity tended to be between three and f i v e times the service l e v e l l i v e loads. - iv - TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES viii LIST OF FIGURES ix LIST OF SYMBOLS xi ACKNOWLEDGEMENTS xiii CHAPTER 1 - INTRODUCTION 1 1.1 Basic Design Philosophies 1 1.2 Reserve Capacity 2 1.3 A p p l i c a t i o n t o Arches 2 1.4 Computer Program Theory and Underlying Assumptions 4 1.4.1 Elasto-Plastic Analysis 4 1.4.2 Second Order Analysis 7 1.4.3 Second Order E l a s t o - P l a s t i c A n a l y s i s 8 1.4.4 Moment A x i a l I n t e r a c t i o n 10 1.4.5 Moment Curvature 12 1.4.6 C r i t e r i a for Reaching Ultimate Load 12 1.4.7 I n t e r a c t i v e Graphic Display 13 CHAPTER 2 - AN ECCENTRICALLY LOADED COLUMN 16 2.1 Governing Parameters 16 2.2 Comparison of A n a l y t i c a l Equation with Correct Analysis and Experimental Results f o r a P a r t i c u l a r Cross Section . 20 CHAPTER 3 - PRESENTATION AND DISCUSSION OF STANDARD ARCH BEHAVIOUR CURVES 3.1 Nonlinear Arch Behaviour 25 25 - v - TABLE OF CONTENTS (Continued) Page 3.2 3.1.1 Computer Model 25 3.1.2 Governing Parameters 27 3.1.3 The Standard Arch 28 3.1.4 Loading f o r Minimum Strength 30 3.1.5 Point Loading f o r Minimum Strength 30 3.1.6 Unbalanced Uniform Loading f o r Minimum Strength .. 40 Discussion of Hinge Formation Curves and Collapse Envelopes 41 3.2.1 Collapse Envelopes 41 3.2.2 E f f e c t of L/r on Type of Collapse 42 3.2.3 E f f e c t of Dead Load on Type of Collapse 42 3.2.4 E l a s t i c Buckling and the L i m i t i n g Slenderness Ratio C r i t i c a l Loading P a t t e r n , x/L Results 44 44 3.2.5 CHAPTER 4 - ANALYTICAL BOUNDS 49 4.1 A n a l y t i c a l Bounds f o r Low L/r 4.2 49 4.1.1 Low L / r ; Neglecting A x i a l Reduction of 49 4.1.2 Low L / r ; Including A x i a l I n t e r a c t i o n 55 4.1.2.1 P l a s t i c Collapse, Low L / r , Including A x i a l I n t e r a c t i o n , Point Load Case 56 4.1.2.2 P l a s t i c Collapse, Low L / r , Including A x i a l I n t e r a c t i o n , U.D.L. Case 57 A n a l y t i c a l Bounds f o r High L/r 60 4.2.1 High L / r ; F u l l , Uniform Live Load E l a s t i c Buckling 60 4.2.2 High L / r ; P o i n t Load, One Hinge Buckling 61 - vi - TABLE OF CONTENTS (Continued) Page 4.2.3 4.3 A n a l y t i a l Solution f o r the Theoretical Slenderness L i m i t Comparison of A n a l y t i c a l Bounds With Collapse Envelopes CHAPTER 5 - VARIATION OF STANDARD PARAMETERS 62 62 66 5.1 V a r i a t i o n of E/o y 66 5.2 V a r i a t i o n of f/L 67 5.3 V a r i a t i o n of y/r 77 5.4 V a r i a t i o n of Z/S 77 CHAPTER 6 - CONCLUSION 81 6.1 Hinge Locations and Formation Sequence 81 6.2 Typical Load D e f l e c t i o n Behaviour 82 6.3 A p p l i c a t i o n of Load and Performance Factors 85 6.4 A p p l i c a t i o n to E x i s t i n g Arches 85 6.4.1 The La Conner Bridge 87 6.4.2 The Capilano Canyon Bridge 88 6.4.3 The Arvida Bridge 89 6.4.4 Further Research 90 REFERENCES 93 - vii- LIST OF TABLES Page Table I Hinge Formation Sequence, Point Loading 82 II Hinge Formation Sequence, UDL Loading 82 - viii - LIST OF FIGURES Page Figure 1 E l a s t o - P l a s t i c Hinge Formation 6 2 Hinge Placement 6 3 F i r s t Order E l a s t o - P l a s t i c Response 4 Second Order E l a s t o - P l a s t i c Response 5 Y i e l d Surface 11 6 Idealized E l a s t o - P l a s t i c Behaviour 12 7 F a i l u r e C r i t e r i a Applied to a Beam-Column i n Double Curvature 13 8 Member Reserve Capacity 14 9 An E c c e n t r i c a l l y Loaded Column 16 10 Moment-Axial-Curvature R e l a t i o n 22 11 Cooling Residual Stress Pattern Assumed by Galambos and K e t t e r 22 12 E f f e c t of E l a s t o - P l a s t i c Assumption on Column Capacity 23 14 E f f e c t of E l a s t o - P l a s t i c Assumption on a T y p i c a l LoadResponse Curve 24 16 Arch Loading 26 17 Hinge Formation Curves and Collapse Envelope, Point Loading a=0.0 31 18 Hinge Formation Curves and Collapse Envelope, Point Loading cx=0.10 32 19 Hinge Formation Curves and Collapse Envelope, Point Loading a=0.20 33 20 Hinge Formation Curves and Collapse Envelope, Point Loading a=0.0 34 21 Hinge Formation Curves and Collapse Envelope, Point Loading a=0.10 35 22 Hinge Formation Curves and Collapse Envelope, Point Loading a=0.20 36 23 Fixed Arch Collapse Envelopes, Point Loading 37 24 Fixed Arch Collapse Envelopes, Uniform Loading 38 - ix - .... 7 9 LIST OF FIGURES (Continued) Page Figure 25 V a r i a t i o n of Dimensionless Load Parameter w i t h Load Location 39 27 Stubby Arches, No Second Order A m p l i f i c a t i o n 46 28 Haunch Moments i n Slender Arches 47 29 V a r i a t i o n s of ( L / r ) trans Free Body Diagrams of P o i n t Loaded Arch Four Hinge P l a s t i c Collapse Under Unbalanced U.D.L. Loading 43 34 F.B.D. of Right Side 54 35 Discrepancies Between A n a l y t i c a l Solutions and Collapse Curves, P o i n t Loading 64 30 33 36 50 53 Discrepancies Between A n a l y t i c a l Solutions and Collapse Curves, U.D.L. Loading 65 37 S e n s i t i v i t y A n a l y s i s of E/o^, Point Loading, ct=0.10 .... 68 38 S e n s i t i v i t y Analysis of E/o , Point Loading, a=0.20 .... 69 39 S e n s i t i v i t y Analysis of E/o , Uniform Loading, a=0.10 .. 70 40 S e n s i t i v i t y Analysis of E/o , Uniform Loading, ct=0.20 .. 71 41 S e n s i t i v i t y Analysis of f/L, Point Loading, a=0.10 73 42 S e n s i t i v i t y A n a l y s i s of f/L, Point Loading, a=0.20 74 43 S e n s i t i v i t y Analysis of f/L, Uniform Loading, a=0.10 ... 75 44 S e n s i t i v i t y A n a l y s i s of f/L, Uniform Loading, a=0.20 ... 76 45 S e n s i t i v i t y Analysis of Z/S, Point Loading, ct=0.10 78 46 S e n s i t i v i t y A n a l y s i s of Z/S, Uniform Loading 79 47 Load-Response of a T y p i c a l Standard Arch 84 48 A p p l i c a t i o n of Collapse Curves to La Conner and Capilano Bridges 91 A p p l i c a t i o n of Collapse Curves t o the A r v i d a Bridge .. 92 49 y y y - x - LIST OF SYMBOLS A Cross s e c t i o n a l area a^ Intercept of facet i of a y i e l d surface with the p a x i s Intercept of f a c e t i of a y i e l d surface with the m a x i s c Distance from the c e n t r o i d of a symmetrical c r o s s - s e c t i o n to the outer f i b r e E Young's Modulus e E c c e n t r i c i t y of applied load f Rise of an arch F^ Load cases F ,F ,F ,F Load Vectors h^ Height t o l o c a t i o n i on an arch I Moment of i n e r t i a of a c r o s s - s e c t i o n K S t i f f n e s s matrix k. E f f e c t i v e length f a c t o r L Span of an arch M Bending moment D M P 0 p P l a s t i c moment M Y i e l d moment y m |M|M 1 P ' P Point load Point load of which event i occurs P p A x i a l force required to cause f u l l cross sections y i e l d i n g with no moment present p P/P r P Radius of g y r a t i o n of a cross s e c t i o n - xi - LIST OF SYMBOLS (Continued) S E l a s t i c s e c t i o n modulus V ,V V e r t i c a l reactions w Uniformly d i s t r i b u t e d load (U.D.L.) w. d Uniformly d i s t r i b u t e d dead load Unbalanced U.D.L. a t which event i occurs w u Ultimate U.D.L. x Parameter i n d i c a t i n g loading pattern y The distance from the center of g r a v i t y of a symmetrical cross s e c t i o n t o the centre of g r a v i t y of e i t h e r the upper or lower h a l f Z P l a s t i c s e c t i o n modulus a Dead load r a t i o ot^ Load f a c t o r s a p p l i e d to load cases F^ X Load l e v e l <J> Curvature (J) m/b^ + p / a 0 Yield stress i i - x i i- ACKNOWLEDGEMENT The author wishes to thank. Dr. Roy F. Hooley f o r h i s invaluable i n s p i r a t i o n and assistance towards the completion of t h i s work. This research might never have been started without the generous f i n a n c i a l support of the Natural Sciences and Engineering Research Council. Their e f f o r t s i n aiding Canadian research i s acknowledged. F i n a l l y the author would l i k e to extend h i s thanks t o the UBC Computing Center f o r t h e i r p r o v i s i o n of such excellent s e r v i c e s . - xiii - 1. CHAPTER 1 INTRODUCTION 1.1 Basic Design Philosophies The basic philosophy of s t r u c t u r a l design has seen many changes. Allowable s t r e s s design has been very common and i s s t i l l used today i n many a p p l i c a t i o n s . In allowable s t r e s s design, dead and l i v e loads are applied to a structure such that nowhere i n the structure does any stress exceed allowable. The allowable s t r e s s i s normally the y i e l d s t r e s s divided by some f a c t o r of safety, i . e . ; STRESSES DUE TO D.L. + L.L. < o /N y where (1.1) D.L. = dead load, L.L. = l i v e load, o y N = yield stress, = f a c t o r of s a f e t y . Eq. (1.1) i m p l i e s that both the dead load and l i v e load are subject t o the same f a c t o r of s a f e t y . S t a t i s t i c a l studies of loads and materials have been used t o develop a contemporary design philosophy. The object of t h i s new design method, c a l l e d L i m i t States Design, i s t o ensure that the p r o b a b i l i t y of reaching a given l i m i t s t a t e , such as collapse or u n s e r v i c e a b i l i t y i s below an acceptable value. To accomplish t h i s , the dead and l i v e loads must each have t h e i r own f a c t o r s of s a f e t y , N, and N i s b e t t e r defined than the l i v e load. philosophy 2 simply because the dead load The b a s i c L i m i t State Design can now be summarized as follows: 2. Nj D.L. where D.L. + N 2 + N 2 (1.2) L.L. < fl) R L.L. ™ E f f e c t of applied loads R = the r e s i s t a n c e of a member, connection or s t r u c t u r e , and o> = the capacity reduction f a c t o r accounting for m a t e r i a l variation: A s l i g h t change i n nomenclature accompanies the new design method such that Nj^ and N 1.2 2 are now referred to as load f a c t o r s . Reserve Capacity I t i s common today to use e l a s t i c a n a l y s i s to f i n d the response of the structure to the factored loads. I f R i s taken as f i r s t y i e l d , there e x i s t s a d d i t i o n a l capacity beyond that load l e v e l . to as reserve This w i l l be r e f e r r e d capacity. Unless a s t r u c t u r e i s exceedingly slender and f a i l s due to e l a s t i c buckling p r i o r to reaching f i r s t y i e l d , the reserve capacity i s at l e a s t the increase i n load required to form the f i r s t p l a s t i c hinge, and at most, the increase i n load required to obtain a p l a s t i c collapse mechanism. A determinate s t r u c t u r e f a i l s a f t e r the formation of one p l a s t i c hinge, therefore a more redundant structure would generally possess a higher reserve 1.3 capacity. A p p l i c a t i o n to Arches With the preceeding d i s c u s s i o n of L i m i t States Design and reserve capacity as motivation, t h i s thesis w i l l examine the ultimate load, or c o l l a p s e l i m i t s t a t e of f i x e d arched r i b s . This w i l l u l t i m a t e l y lead to 3. a b e t t e r understanding of the reserve capacity of f i x e d arches as w e l l as the f a c t o r s on which i t depends. The key t o the success of t h i s work i s a r e l i a b l e a n a l y s i s technique which must include a l l prevalent types of behaviour. Conventionally, p l a s t i c a n a l y s i s i s used i n determining c o l l a p s e loads f o r moment r e s i s t ing frames and continuous beams. E l a s t i c buckling i s used i n the evaluat i o n of the u l t i m a t e c a p a c i t y of slender columns. Considering that an arch i s b a s i c a l l y a compression member subject to bending by unsymmetr i c a l l i v e loads, the u l t i m a t e strength may be governed by p l a s t i c c o l l a p s e , e l a s t i c b u c k l i n g , or by some intermediate form of i n s t a b i l i t y with l e s s p l a s t i c hinges than required f o r a c o l l a p s e mechanism. The reserve capacity of an arch i s therefore governed by non-linear behaviour. This n o n - l i n e a r i t y a r i s e s from p l a s t i c hinging and P-A second order e f f e c t s . A computer a n a l y s i s combining both these f a c t o r s i s o u t l i n e d i n S e c t i o n 1.2. I n an age of i n c r e a s i n g a c c e s s i b i l i t y t o computer hardware and software, a d i f f i c u l t question faces the researcher. Is i t a researcher's r e s p o n s i b i l i t y t o present h i s r e s u l t s i n the form of design or a n a l y t i c a l equations based on curve f i t t i n g or s i m i l a r conventional techniques? Or, i s i t the researcher's r e s p o n s i b i l i t y t o present the r e s u l t of hours of computer a n a l y s i s so as to inform and enlighten the reader and t o give the reader conceptual ideas and g u i d e l i n e s , assuming that the reader has the computer f a c i l i t i e s to duplicate some part of the researcher's work and t o use the r e s u l t s f o r h i s or her own p a r t i c u l a r and s p e c i a l i z e d purpose? The l a t t e r approach has been chosen here. 4. 1.4 Computer Program Theory and Underlying Assumptions The computer program used i n t h i s work i s "ULA" Analysis) . 1 (Ultimate Load I t i s a plane frame s t i f f n e s s program which combines second order a n a l y s i s with p l a s t i c hinge formation. ULA i s an i n t e r a c t i v e program which allows the user to monitor the s t r u c t u r e and to place p l a s t i c hinges when necessary as the load i s increased to u l t i m a t e . One of the requirements of l i m i t states design i s that the s t r u c t u r e not f a i l when subjected to each of a number of load vector F F 0 = a ¥ '+ a F 1 = F 2 1 D 2 + Q O3F3 + ... . + F (1.3) FQ i s then a l i n e a r combination of load cases Fj_ augmented by appropriate load f a c t o r ot£. where the In analyzing f o r u l t i m a t e load, the response of the s t r u c t u r e at any load l e v e l X to the force vector F must be determined where F = F D + X F The o r i g i n a l load vector FQ i s the sum of vectors (1.4) and F. In performing the a n a l y s i s to determine ultimate load, only F i s augmented by load vector X. This makes i t p o s s i b l e to maintain a constant dead load f a c t o r , for example, and increase only the l i v e load u n t i l collapse. 1.4.1 E l a s t o - P l a s t i c Analysis There are two b a s i c methods of e l a s t o - p l a s t i c a n a l y s i s . The first i s an energy method whereby the. external energy created by the loading i s equated to the i n t e r n a l energy f o r d i f f e r e n t mechanisms and mechanism 5. combinations. The second method i s by load increments whereby the s t r u c t u r e i s analyzed as l i n e a r e l a s t i c u n t i l a member moment reaches the p l a s t i c moment M P at which point i t remains at M r o t a t i o n of adjacent members. P with f r e e r e l a t i v e The load l e v e l i s then increased and the structure i s analyzed l i n e a r l y u n t i l another hinge i s to form. continues u n t i l a c o l l a p s e mechanism i s obtained. This The second method i s preferred because i t lends i t s e l f to computer s i m u l a t i o n and i t makes the i n c l u s i o n of second order e f f e c t s p r a c t i c a l . F i g . 1 shows a t y p i c a l hinge formation sequence with i n c r e a s i n g X f o r a s i n g l e bay frame. I t i s important to note that each of the structures No. 0 through 4 are d i f f e r e n t and each i s v a l i d only f o r a s p e c i f i c range of X. Each s t r u c t u r e has a d i f f e r e n t s t i f f n e s s matrix K and each w i l l be analyzed under the loads shown. To a c t u a l l y place a hinge i n the s t r u c t u r e at the appropriate load l e v e l , an a d d i t i o n a l slave j o i n t i s created at the hinge l o c a t i o n which has the same t r a n s l a t i o n as the master j o i n t , but d i f f e r e n t r o t a t i o n . The load vector F i s then augmented by ±M^ between each master and slave p a i r , so the new load vector Is now F = F_ + XF + F , where F contains ' D p' p r only ± M p* This hinge placement i s depicted i n F i g . 2. F u l l d e t a i l s are given i n reference 1. The l i n e a r e l a s t o - p l a s t i c response of the s i n g l e bay frame of F i g . 1 forms the polygonal shape i n F i g . 3 i n d i c a t i n g the l o s s of s t i f f n e s s i n the s t r u c t u r e as each hinge forms. In the method described above f o r hinge placement, a hinge can be placed at any load l e v e l . Because each of the s t r u c t u r e s of F i g . 1 i s unique, the response at load l e v e l X , f o r B example, can be obtained by a f i r s t order a n a l y s i s of s t r u c t u r e #2 from zero load l e v e l to X,, along a secant OB, and not along the f a c e t s of the a polygon. 5X IOX 5X IOX \ 1 one plastic hinge no plastic hinges M, Structure^* I X, < X< X Structure_#0 0 < X< X, IOX 2 5X1 5X M three plastic A p two plastic hinges a> Pi> p M M , n 9 e s 1IOX M P 5X1 M p four plastic ^Mp^hinges^Mp 7777 Structure # 3 X < X <"X Structure # 2 x <x < x 2 3 3 F i g . 1. E l a s t o - P l a s t i c Hinge Formation. <fcnox Joint L 4 Structure # 4 Mechanism h e r e Tension Face before hinge placed Joint L Joint L+ I (Master) (Slove) after hinge ploced F i g . 2. Hinge Placement. 7. £VF=\F Mechanism max load = XF A only 0 n Response 0 Structure No.l 0 I I F i g . 3. F i r s t Order E l a s t o - P l a s t i c Response, 1.4.2 Second Order A n a l y s i s Second order a n a l y s i s requires the s t r u c t u r e t o be e l a s t i c and to be i n e q u i l i b r i u m i n the deformed shape. The l a t t e r i s achieved by using s t a b i l i t y functions i n the member matrix. D e t a i l s of these s t a b i l i t y functions w i l l not be discussed here as they are standard and presented by many other authors i n c l u d i n g Gere and Weaver . 2 The s t a b i l i t y func- tions depend on the a x i a l forces, and the a x i a l forces depend on the 8. deflected shape. I t I s therefore necessary to I t e r a t e towards a s o l u t i o n several times at each load l e v e l . This I s n i c e l y handled by the I n t e r - a c t i v e format of the program because the analyst can view the determinant of the s t r u c t u r e s t i f f n e s s matrix and use that as a c r i t e r i a f o r convergence. Normally, only a small number of c y c l e s , perhaps two, i s required for convergence as the a x i a l force changes only s l i g h t l y with the i n c l u s i o n of the second order e f f e c t s . Of course a few more c y c l e s are required when more hinges are placed due to the increased f l e x i b i l i t y and load l e v e l . The two previously mentioned u l t i m a t e load t h e o r i e s , p l a s t i c c o l l a p s e and e l a s t i c i n s t a b i l i t y , would each give a collapse load. However, unless a p a r t i c u l a r s t r u c t u r e i s e i t h e r e s p e c i a l l y stubby to collapse p l a s t i c a l l y , or slender to buckle e l a s t i c a l l y , then the a c t u a l u l t i m a t e load behaviour i s somewhere between these two extremes, and the value of the ultimate load i s lower than that obtained by p l a s t i c or second order a n a l y s i s . I t i s apparent that i n order t o e s t a b l i s h the maximum load capacity, and hence an idea of the p r o b a b i l i t y of reaching the u l t i m a t e l i m i t s t a t e , a combination of the two theories i s needed f o r many p r a c t i c a l s t r u c t u r e s . 1.4.3 Second Order E l a s t o - P l a s t i c A n a l y s i s An incremental approach i s a common method f o r combining second order and e l a s t o - p l a s t i c a n a l y s i s . The incremental forces and d e f l e c - t i o n s due to a small increment, dX, i n load l e v e l , X^ are obtained a tangent s t i f f n e s s matrix. using At each load l e v e l , the r a t i o of moment to p l a s t i c moment, M/M , i s checked t o determine the necessity of p l a c i n g a p l a s t i c hinge. responses. The t o t a l response i s then the sum of a l l the incremental However, e r r o r s a r i s e because the tangent s t i f f n e s s matrix i s approximate, hinges may not be placed a t M/M P due t o a multitude of steps. • 1, and round o f f occurs These cumulative errors can be minimized by using a small dX. T h i s , however, becomes more expensive and does not assure convergence. work. This incremental approach i 6 not adopted for t h i s A simpler u l t i m a t e load a n a l y s i s system i s used which should require less computing time and c e r t a i n l y avoids any cumulative e r r o r s . The system adopted i s a simple combination of second order a n a l y s i s , and hinge placement. A second order e l a s t o - p l a s t i c response curve shown In F i g . 4 i s s i m i l a r t o the f i r s t order e l a s t o - p l a s t i c response curve shown i n F i g . 3; the d i f f e r e n c e being the presence of arc segments ^ ' Linear onalysis Structure 0 A 0 Structure No. I 0 F i g . 4. I 2 . . Mechanism >- h— 3-H Second Order E l a s t o - P l a s t i c Response. 10. between hinge formations instead of l i n e a r f a c e t s . To determine the response at a c e r t a i n load l e v e l , f i r s t order ( l i n e a r ) a n a l y s i s i s simply replaced by second order a n a l y s i s , f o r each of the four s t r u c t u r e s . To determine the response of the s t r u c t u r e at load l e v e l Xg, f o r example, i t i s not necesasry to methodically increment the load l e v e l and f o l l o w the arced segments from 0 to B. As long as the l o c a t i o n of the hinge i s known, i n t h i s case at the base of the frame, then a l l that i s required i s a second order a n a l y s i s w i t h the s t r u c t u r e #2 loaded w i t h XgF Q and the p l a s t i c moments shown. l e v e l to Xg i s along secant OB. The t o t a l response from zero load Unlike the incremental approach, any e r r o r s due to p l a c i n g a hinge when M/Mp ± 1 i s a l o c a l e r r o r and not cumulative, so that the response at higher load l e v e l s w i l l not be affected. 1.4.4 Moment A x i a l I n t e r a c t i o n Consideration must now be given to the reduction of the p l a s t i c moment due to the presence of an a x i a l load P i n the member. To do t h i s , the analyst must f i r s t decide on an appropriate y i e l d surface f o r the cross-section being analyzed. The y i e l d surface can be defined by a s e r i e s of s t r a i g h t l i n e s , and i s described to the program by the i n t e r sections of the f a c e t s . By i n c l u d i n g only symmetrical s e c t i o n s , and hence the absolute value of the bending moment, only the top h a l f of a y i e l d surface need be considered. The y i e l d surface used throughout t h i s work i s shown i n F i g . 5. It i s a s l i g h t v a r i a t i o n on CAN3-S16.1-M84. A p l a s t i c hinge forms when moments and a x i a l s becomes l a r g e enough to reach the y i e l d surface. A parameter d>, i s defined for each facet i such that when the maximum d>. = i l 1, the y i e l d surface has been reached and a hinge should be placed. quantity $ i i s defined as f o l l o w s : $ ± = m/b ± + p/a ± The 11. OUTSIDE CAN 3 - SI6.I-M78 YIELD i.O^ ANY 1-0.19,0.95) -1.0 <£j> I (0.19,0.95) P TENSION, -£-=p COMPRESSION Fig. SURFACE, 1.0 5. Y i e l d Surface where m = IM|/M , p • P/P = P/Ao , and a and b are the i n t e r c e p t s of P p y i i J J each facet with the p and m axis r e s p e c t i v e l y . Now the convenience of an i n t e r a c t i v e format becomes apparent. At each load l e v e l , once the second order convergence i s obtained, a plot of the s t r u c t u r e appears on the screen w i t h a l i s t of the f i v e maximum a)j values, where o)j i s the maximum of a l l 4^ f o r member end j . At a glance, the analyst can t e l l how close the structure i s to forming a p l a s t i c hinge, and where t h i s hinge w i l l form. To f a c i l i t a t e the analyst's cho.ice of load l e v e l , the program estimates the load l e v e l a t which the next hinge should form. This i s accomplished by e x t r a p o l a t i n g l i n e a r l y from two known points i n s i d e the y i e l d surface to the y i e l d surface i t s e l f . The basic assumptions here are that l i n e 1-2-H i n F i g . 5 i s s t r a i g h t and that p i s l i n e a r with A. 12. 1.4.5 Moment Curvature A p e r f e c t e l a s t o - p l a s t i c behaviour i s assumed f o r the a n a l y s i s of f i x e d arches. A F i g . 6 shows an i d e a l i z e d moment curvature r e l a t i o n s h i p . l o s s of bending 6 t i f f n e s s i n any s e c t i o n occurs from the f i r s t y i e l d moment, M^, to the p l a s t i c moment, M^, as the c r o s s - s e c t i o n becomes f u l l y plastic. T h i s , as w e l l as the e f f e c t of r e s i d u a l stresses are neglected i n t h i s work. Chapter 2 w i l l examine the consequences of these assumptions. MOMENT CURVATURE Fig. 6. Idealized E l a s t o - P l a s t i c Behaviour. The e f f e c t of neglecting s t i f f n e s s loss and r e s i d u a l stresses i s examined i n Chapter 2. 1.4.6 C r i t e r i a f o r Reaching Ultimate Load Ultimate load i s defined here as.the load l e v e l a t which the second order e l a s t o - p l a s t i c s t i f f n e s s matrix K assembled i n the ULA program becomes s i n g u l a r . This i s accomplished by monitoring the determinant of K. A zero determinate implies a singular and unsolvable matrix. A negat i v e d e f i n i t e s t i f f n e s s matrix occurs when the determinant i s negative, 13. and although an e q u i l i b r i u m s o l u t i o n i s then p o s s i b l e , i t corresponds to unstable e q u i l i b r i u m and w i l l not be permitted. Fig. 7 shows a beam-column bent i n double curvature due t o equal and opposite end e c c e n t r i c i t i e s . The l o a d - d e f l e c t i o n curve shows diagram- m a t i c a l l y the c o l l a p s e c r i t e r i a discussed above. The Choleski method used f o r the s o l u t i o n of the s t i f f n e s s equations i s only coded f o r r e a l numbers. Because of t h i s , the routine stops when |K| • 0 and s i g n a l s an unstable s t r u c t u r e . P Fig. 1.4.7 7. F a i l u r e C r i t e r i a Applied to a Beam-Column i n Double Curvature. I n t e r a c t i v e Graphic Display I n t e r a c t i v e graphic d i s p l a y helps the user i n making necessary decisions such as the number of P-delta convergence c y c l e s , hinge placement, and s e l e c t i o n of the next load l e v e l . Of course, the standard displays such as member bending moments, a x i a l s , shears, and deflected shapejare a v a i l a b l e on command a t any given load l e v e l . are a v a i l a b l e which give the analyst enlightened s t r u c t u r e i s behaving. Other displays appreciation of how the The f i r s t of these i s a d i s p l a y of the y i e l d 14. surface, as shown I n F i g . 5. Superimposed on the y i e l d surface I s a trace of the m and p coordinates load l e v e l . for each member end from load l e v e l t o This gives the analyst a quick and easy way of determining whether groups of members are behaving as bending members, compression or tension members, or some combination. The f i n a l feature i s a d i s p l a y of the reserve capacity of each member. A self-explanatory example of t h i s d i s p l a y i s shown i n F i g . 8. X = 0.80 F i g . 8. I 1.0 Member Reserve Capacity. The analyst can now determine at a glance how much of each cross-section i s being used up by a x i a l f o r c e s , or bending moments. I t i s a l s o apparent where the next hinges should form, as the reserve c a p a c i t i e s of the l o c a t i o n s are approaching zero. Other features such as s t r a i n hardening and hinge closure are also a v a i l a b l e . The program ULA, with i t s i n t e r a c t i v e graphic format, gives the analyst a complete and quickly understood appreciation of how a p a r t i c u l a r structure i s behaving with 15. increasing load l e v e l , and where i t may need redesign or where materi i s not being used e f f i c i e n t l y . I t i s t h i s program that w i l l be used i n v e s t i g a t e the non-linear and ultimate behaviour of f i x e d arches. 16. CHAPTER 2 AN ECCENTRICALLY LOADED COLUMN 2.1 Governing Parameters An arch and a column posess many s i m i l a r i t i e s . compression members subject to bending. They are both A column bends when loaded e c c e n t r i c a l l y , and an arch bends when loaded unsymmetrically. Rather than s t a r t with the d i s c u s s i o n of arches, the s i m i l a r , more f a m i l i a r , and simpler problem of e c c e n t r i c a l l y loaded columns w i l l be considered. It i s the i n t e n t i o n of t h i s chapter to develop an a n a l y t i c a l s o l u t i o n f o r an e c c e n t r i c a l l y loaded column based on the 6ame assumptions to be used f o r the ultimate load of arches as outlined i n Section 1.4. This a n a l y t i c a l s o l u t i o n w i l l then be compared to an e x i s t i n g more exact s o l u t i o n experimental r e s u l t s . An i n d i c a t i o n of error due to the o r i g i n a l assumptions w i l l be shown. F i g . 9 shows the e c c e n t r i c a l l y loaded column chosen for comparison. F i g . 9. and An E c c e n t r i c a l l y Loaded Column. 17. As o u t l i n e d i n Chapter 1, e l a s t o - p l a s t i c m a t e r i a l behaviour w i l l be assumed neglecting s t r a i n hardening and r e s i d u a l s t r e s s e s . The e l a s t o - p l a s t i c assumption e s s e n t i a l l y means that moment curvature remain l i n e a r up to M . Neglecting the loss of s t i f f n e s s between M and M produces a P y P structure and a non-conservative r e s u l t . 6tiffer The u l t i m a t e column capacity P^ i s a f u n c t i o n of the f o l l o w i n g s i x parameters; P where u = f {e, L, E I , AE, P , M } P P (2.1) EI = l i n e a r e l a s t i c bending s t i f f n e s s AE = l i n e a r e l a s t i c a x i a l s t i f f n e s s P P = Ao y = maximum possible a x i a l load with no moment present and M P = Zo y = maximum p o s s i b l e bending moment with no a x i a l present. With several independent parameters, i t i s convenient to use the Buckingham II Theorem t o reduce the number of parameters which govern the behaviour of the system. With seven parameters i n Eq. (2.1) dependent on the two dimensions of force and length, only f i v e dimensionless r a t i o s are needed to describe the system as follows: 18. The awkward parameters of Eq. (2.2) are chosen because they s i m p l i f y i n t o the more f a m i l i a r r a t i o s shown below; P /P = f{e/y, L / r , E/o , y/r} u p y (2.3) E = Young's modulus, where A = c r o s s - s e c t i o n (area) I = moment of i n e r t i a r = /I/A - /EI/AE = radius of g y r a t i o n , y = the distance from the centre of g r a v i t y of the symmetrical s e c t i o n to the centre of g r a v i t y of e i t h e r the upper or lower h a l f , o and = yield stress, so that AE/P = AE/Ao p y L L •EI/AE 6 M /P P P = E/o , y R = e/y and M _E P P = /r y /EI/AE The maximum moment of the e c c e n t r i c a l l y loaded column of F i g . 9 occurs a t the midspan. According t o Timoshenko , 3 19. M max - P(e + A)' « Pe sec(kL/2) \ • y k - P/EI (2.4) / where 2 The column i s determinate and w i l l therefore f a i l once the hinge forms at the midspan. The p l a s t i c moment must be reduced i n the presence of an a x i a l load according to the y i e l d surface of F i g . 5. analytical solution equation. 0.95. f o r the column capacity, we need an p interaction; 0.85 M/M M and P interaction Facet 1 of F i g . 5 w i l l be used as i t i s v a l i d f o r |M/M| < Eq. (2.5) describes t h i s where To develop an P • a Ay = P y y P P « o A y P + P/P P -=1 (2.5) Therefore: M max = Pe sec ^ 2 = 1.18 M p (1 - P/P p ) or (1 - P/P ) - 0.85 p where Eq. (2.6) becomes kfc _ JP_ £ _ i P _ i_ 2 EI 2 AE 2r k£ (P/P ) (e/y) sec p I m / p _ ^y_ P E P £_ 2r (2.6) 20. P /P u p —— 1 + 0.85 (2.7) e/y s e c ( / | - ^ P jr Eq. (2.7) i s an a n a l y t i c a l expression for column capacity under e c c e n t r i c loading based on the same assumptions that w i l l be used to analyse the ultimate capacity of 2.2 arches. Comparison of A n a l y t i c a l Equation with Correct A n a l y s i s and Experimental Results f o r a P a r t i c u l a r Cross-Section Galambos and Ketter * present dimensionless 4 curves f o r the ultimate strength of a t y p i c a l I-beam under a x i a l load with equal end e c c e n t r i c i t i e s causing bending i n the strong d i r e c t i o n . The fundamental d i f f e r e n c e between the d e r i v a t i o n of Eq. 2.7 and the Galambos and Ketter approach i s the assumed moment versus curvature r e l a t i o n . Ketter use a correct r e l a t i o n l i k e curve B of F i g . 6. Galambos and In t h i s t h e s i s , the moment curvature r e l a t i o n i s s i m p l i f i e d by i d e a l i z i n g e l a s t o - p l a s t i c behaviour, s i m i l a r to curve A of F i g . 6. The method used by Galambos and K e f f e r i s based on numerically i n t e g r a t i n g values on a s p e c i f i c M-<|> c o r r e c t d e f l e c t e d shape. converge. curve and i t e r a t i n g towards a I n s t a b i l i t y a r i s e s when the i t e r a t i o n s do not Because t h i s method r e l i e s on a known moment-axial-curvature r e l a t i o n , which i s unique f o r every d i f f e r e n t c r o s s - s e c t i o n , a closed form s o l u t i o n i s not a v a i l a b l e . I t i s now p o s s i b l e to compare the r e s u l t s of Eq. 2.7 with Galambos and Ketter f o r a s p e c i f i c I-beam, namely a 315.7. The required moment-axial-curvature r e l a t i o n f o r t h i s beam i s shown i n F i g . 10, based on an assumed r e s i d u a l stress pattern shown i n F i g . 11. Of course, the 21. reduction of M due t o the presence of a x i a l i n the d e r i v a t i o n of Eq. 2.7 P i s handled by i n c o r p o r a t i n g the y i e l d surface of F i g . 5. F i g . 12 i s a dimensionless plot of an ultimate load parameter F^/?^ versus slenderness L / r against an e c c e n t r i c i t y parameter e c / r . 2 The quantity c I s measured from the c e n t r o i d of the symmetric cross s e c t i o n to the outer f i b r e . The r e s u l t s according to the assumptions of t h i s t h e s i s , l a b e l l e d "ULA" are c l e a r l y non-conservative compared t o the more a n a l y t i c a l l y correct r e s u l t s of Galambos and K e t t e r . The discrepancy i s Indicated by a shaded region and i s as much as t e n percent. Experimental r e s u l t s have also been included i n the plot of F i g . 12 and appear to be bounded by the two a n a l y t i c a l s o l u t i o n s . I t was necessary to make a s l i g h t m o d i f i c a t i o n to Eq. 2.7 i n order to p l o t the ULA curve. The dimensionless parameter chosen by Galambos and Ketter to r e f l e c t e c c e n t r i c i t y was e c / r . 2 This d i f f e r s from the r a t i o e/y used i n Eq. 2.7 and i t i s a simple matter of a r i t h m e t i c to transform known. from one to the other once the cross-section properties are In t h i s case, e c / r 2 = cy/r 2 (e/y) = 0.85 e/y. A l s o , E/ay = 30,000/33 = 909. Therefore, Eq. 2.7 becomes: P JL p = P . -° + y i sec (/P7P~ L/60.3) 1 + 0.73 e c / r 2 (2.8) I t i s Eq. 2.8 that i s a c t u a l l y plotted on F i g . 12 and l a b e l l e d "ULA". An e c c e n t r i c a l l y loaded column i s a determinate s t r u c t u r e which f a i l s a f t e r the formation of one p l a s t i c hinge. The purpose of the comparison presented i n F i g . 12 was t o extrapolate the r e s u l t s and make some judgement on the e f f e c t of i d e a l i z i n g behaviour as e l a s t o - p l a s t i c on 22. F i g . 11. Cooling Residual Stress Pattern Assumed by Galambos & K e t t e r . 23. F i g . 12. E f f e c t of E l a s t o - P l a s t i c Assumption on Column Capacity. the ultimate strength of f i x e d arches. redundant. A f i x e d arch i s three times Most p l a s t i c hinges formed p r i o r to c o l l a p s e would already be i n the p l a s t i c region where the moment-curvature behaviour ( F i g . 10) l e v e l s out to: constant reduced only by the presence of a x i a l forces. Any e r r o r s during the formation of p l a s t i c hinges p r i o r t o the l a s t hinge are l o c a l e r r o r s , not cumulative, and do not e f f e c t the f i n a l r e s u l t . Any non-conservatism should only occur i n the l a s t hinge formed. demonstrated q u a l i t a t i v e l y i n F i g . 14. This i s I t i s therefore proposed that the 24. LOAD A LEVEL RESPONSE F i g . 14. E f f e c t of E l a s t o - P l a s t i c Assumption on a T y p i c a l Load-Response Curve. e f f e c t of i d e a l i z i n g behaviour as e l a s t i c - p l a s t i c i s not as s i g n i f i c a n t i n the case of f i x e d arched r i b s as i t i s i n the case of a beam-column and would therefore be appreciably less than ten percent. I t i s worth p o i n t i n g out a t t h i s time that the b e n e f i c i a l e f f e c t of s t r a i n hardening i s not considered here, and might serve to f u r t h e r eliminate any small non-conservatism. The basic case of an e c c e n t r i c a l l y loaded column w i l l now be expanded to the study of the ultimate strength of f i x e d arches. 25. CHAPTER 3 PRESENTATION AND DISCUSSION OF STANDARD ARCH BEHAVIOUR CURVES 3.1 Nonlinear Arch Behaviour I t i s the object of t h i s chapter t o present the nonlinear behaviour of f i x e d arches. Because of the multitude of parameters governing arch a c t i o n I t w i l l not be p o s s i b l e t o describe a l l arches. Instead, the dimensionless behaviour of a standard arch w i l l be given. In Chapter 5 the s e n s i t i v i t y of t h i s standard t o various parameter v a r i a t i o n s w i l l be investigated. 3.1.1 Computer Model Since ULA considers only s t r a i g h t members between nodes, the r i b w i l l be a polygon. This polygon was chosen to be twenty segments connecting twenty-one nodes because experience has shown that the d i f f e r e n c e between t h i s and a continuous curve would be less than 1%. I f the r e a l arch r e a l l y has twenty s t r a i g h t segments then of course the error i n t h i s model i s zero. I f , on the other hand, the r e a l arch has say, four segments, then the e r r o r may be too l a r g e f o r p r a c t i c a l applications. Most arches are designed so that the dead load produces no moment except, perhaps, from r i b shortening. The shape i s then the moment diagram f o r dead load; a shape somewhere between a parabola and a catinary. The 21 nodes were placed on a parabola for t h i s study together w i t h 19 equal point loads so as to produce no moment under dead load except for r i b shortening. Rib shortening i s automatically included i n a s t i f f n e s s a n a l y s i s and no attempt was made to f a c t o r i t out. Arches 26. constructed so that dead load moment due t o r i b shortening i s minimized w i l l then have smaller moments than calculated with t h i s model. I n summary then, the model consisted of a twenty sided polygon with the nodes l y i n g on a parabola. I n I X 1 1 1 1 1 1 1 1 1 1 1 N LOAD I LOAD 2 * 1 I I 1 1 1 1 1 1 1 W: M i l l J O I N T 21 JOINT I F i g . 16. Arch Loading. Two load cases were considered to act on the model as shown i n F i g . 16. Load one c o n s i s t s of the dead load plus w a point load P located x d 1 from the l e f t end. Load two c o n s i s t s of a dead load w, plus a l i v e w. on d a loaded length of x-Xj^. l The d i s t r i b u t e d loads w^ and w^ were modelled as point loads at the polygon nodes i n order to eliminate l o c a l bending on the s t r a i g h t segments. The l i v e load w^ or ?^ was gradually increased I n ULA w i t h w^ held constant. The s u b s c r i p t i i s used t o denote the load a t which s p e c i f i c events occured as f o l l o w s : 27. w e or P Load at which y i e l d s t r e s s f i r s t occurred at some point on the e rib Wj^ or P j Load at which f i r s t hinge formed w w 2 or P 2 Load a t which second hinge formed 3 or P 3 Load at which t h i r d hinge formed w^ or P^ Load a t which fourth hinge formed w i s the ultimate load which may be any of the above loads as u or P u w i l l be explained l a t e r . Since the arch i s three times redundant, up to four hinge w i l l form before f a i l u r e occurs. For very slender arches, the system may buckle as soon as the f i r s t hinge forms so that w 1 or P j i s the ultimate load. For stocky arches, a l l four hinges w i l l form before f a i l u r e occurs as a mechanism so that w^ and P^ i s the ultimate. Each of w^ and P^ was minimized by varying x (and x ^ ) . In general I t was found that x^ was zero and x for minimum load varied with i . 3.1.2 Governing Parameters The load w^ or P^ i s a f u n c t i o n of nine parameters as f o l l o w s : w where ± (or P ) = f [ L , f , x, w , i L = span f = rise d w, = d dead load EI = bending s t i f f n e s s AE = axial stiffness E I , AE, P , M , ? p M] y 28. P p » Aoy M m p l a s t i c a x i a l load with no moment a c t i o n = A o = za = p l a s t i c moment with no a x i a l load a c t i o n y y y M = So = moment at which y i e l d occurs with no a x i a l y y p Since these ten parameters l i n k only the two dimensions of force and length, the Buckingham II theorem shows that only eight parameters govern the system. dimensionless The f o l l o w i n g eight are chosen f o r convenience: w.L _i_ M P P.L 2 ( o r JL.) M P . _ = ri ii k i f L w,L £ I 2 _d_ L ' L' r ' r ' s' o' 8fP y P The parameter Z/S w i l l only e f f e c t the f i r s t y i e l d c o n d i t i o n and not hinge formation. The parameter w^L /8fP^ i s chosen t o represent, 2 approximately, the f r a c t i o n of a x i a l capacity P^ used up by dead load thrust. 3.1.3 The Standard Arch I t i s c l e a r l y i m p r a c t i c a l t o evaluate numerically the dimensionless load of Eq. (3.1) as a function of seven independent parameters. It is p r a c t i c a l though t o define a standard, or average, or p r a c t i c a l arch by assigning s p e c i f i c values to these seven parameters and then to run a s e n s i t i v i t y a n a l y s i s to show t h e i r r e l a t i v e importance. Such a system w i l l give the' s p e c i f i c behaviour i n a p r a c t i c a l region and an i n d i c a t i o n of what might happen some distance from that region. In general though i t w i l l be necessary to run a f u l l a n a l y s i s for cases remote from t h i s standard arch. 29. With the above i n mind, four parameters were given s p e c i f i c values to define the standard arch as f o l l o w s : E/o - 30,000 k s i / 40 k s i = 210,000 MPa / 280 MPa = 750 y f/L - 0.15 y/r • 0.95 Z/S = 1.15 An E/Oy of 750 i 6 d e f i n i t e l y a p p l i c a b l e to s t e e l and close to concrete. Behaviour o f other m a t e r i a l s w i l l come from the s e n s i t i v i t y a n a l y s i s . An f/L of 0.15 ha6 been used for many bridges but higher structures w i l l be covered i n the s e n s i t i v i t y a n a l y s i s . A s o l i d rectangular s e c t i o n has y/r = 0.866 while two flanges with no web has y/r = 1.00. The chosen y/r = 0.95 i s then a reasonable value. The shape f a c t o r Z/S v a r i e s from 1.5 for a s o l i d rectangle to 1.00 for two flanges with no web. The chosen value of 1.15 i s then c l o s e r to a s t e e l box or wide flange. With x/L chosen so as to minimize the dimensionless load t h i s leaves w L P L ~M~" ° M"""' P P 2 ± for the standard arch. ( r w L P SIP"] P 2 i = f [ L d A study of e x i s t i n g arches shows that a = wX /8fP d p 2 ranges from hear zero to approximately 0.2. I t was decided to produce curves of V -|— P 2 P i (or L L = P f(p f o r a - 0, 0.1, 0.2 30. to give the behaviour of the standard arch. Numerous runs on the Amdahl V8 of the UBC computing centre then defined the functions of Eq. (3.1) which are shown p l o t t e d i n F i g s . 17 through 24. It should be noted that the parameter L/r i n v o l v e s the span length and not the c l a s s i c " e f f e c t i v e " length kL. For a f i x e d arch, the e f f e c t i v e slenderness i s given by kL/r = 0.37 L / r . 3.1.4 Loading f o r Minimum Strength Influence l i n e s have been i n v a l u a b l e i n the l i n e a r a n a l y s i s of arches to determine the loading for maximum moment, t h r u s t , s t r e s s , e t c . They are of l i t t l e use though w i t h nonlinear behaviour because superposition i s not a p p l i c a b l e . For the case at hand i t i s necessary t numerically vary x^/L and x/L t o produce a minimum dimensionless load. This method was necessary for a l l w^ and P since x/L and Xj/L depend upon i . 3.1.5 Point Loading f o r Minimum Strength To minimize P^L/M^ i t i s only necessary to vary the one parameter x/L. F i g . 25 shows a t y p i c a l v a r i a t i o n of P^L/M^as a f u n c t i o n of x/L f o the standard arch with a given value of L / r . I t i s apparent from t h i s behaviour that a s i n g l e minimum e x i s t s f o r the u l t i m a t e load and a l l hinges formed a f t e r the f i r s t hinge. However, two l o c a l minima e x i s t fo the f i r s t y i e l d and f i r s t hinge curves. These two minima a r i s e because the f i r s t hinge may form at two d i f f e r e n t l o c a t i o n s on the arch, each l o c a t i o n corresponding t o a d i f f e r e n t value of x/L. However, once the ARCH COLLAPSE ENVELOPE - PT. LOAD , -rf-=- = 0 22 FOUR HINGES THREE HINGES TWO HINGES ONE HINGE FIRST YIELD P L A S T I C . N O INTERACTION 20 18 STANDARD ARCH E/<T = 750 y f/L =0.15 i/S = 1.15 y/r =0.95 16 14 ItMp 12 10 SYMBOL 8 O • 6 */ L 0.15 0.25 0. 30 4 100 200 300 400 500 600 L/r F i g . 17. Hinge Formation Curves and Collapse Envelope, Point Loading a - 0.0. 700 w L ARCH COLLAPSE ENVELOPE - PT. LOAD , - ^ - = 0 . 1 0 0 F i g . 19. 100 200 300 L/r 400 500 Hinge Formation Curves and Collapse Envelope, Point Loading a - 0.20. 600 F i g . 20. Hinge Formation Curves and Collapse Envelope, Uniform Loading a = 0.0. ARCH COLLAPSE ENVELOPE - U.D.L., ^ = - = 0.10 8f P r E L A S T I C BUCKLING k s 0.37 x/L= I .0 300 400 L/r 500 600 F i g . 21. Hinge Formation Curves and Collapse Envelope, Uniform Loading a = 0.10. 700 800 ARCH COLLAPSE ENVELOPE - U.D.L., ^ - ^ - = 0 20 8fP„ STANDARD ARCH E/try = 750 ELASTIC f/L =0.15 z/S = 1.15 y/r = 0.95 BUCKLING x / L = 1.0 , k s 0 . 3 7 7 200 300 400 L/r 500 600 F i g . 22. Hinge Formation Curves and Collapse Envelope, Uniform Loading a » 0.20. F i g . 23. Fixed Arch Collapse Envelopes, Point Loading. ARCH COLLAPSE CURVES - U.D.L. F i g . 24. Fixed Arch Collapse Envelopes, Uniform Loading. 0 0.10 0.20 0.30 0.40 x_ L Fig. 25. V a r i a t i o n of Dimensionless Load Parameter with Load Location. f i r s t hinge forms, there e x i s t s only one possible remaining l o c a t i o n f o r each of the subsequent hinges, therefore, t h e i r behaviour y i e l d s a s i n g l e minimum. The f i r s t hinge forms e i t h e r at the l e f t haunch or a t the l o c a t i o n of the point load. I f the f i r s t hinge forms at the l e f t haunch, then the second hinge w i l l always form at the l o c a t i o n of the point load. I f the f i r s t hinge forms at the l o c a t i o n of the point load, then the second hinge w i l l form at the l e f t haunch. The t h i r d hinge forms at the r i g h t haunch and the fourth hinge forms a t , or near the r i g h t quarter point. 40. This assumes of course that s t a b i l i t y permits the formation of a l l the hinges. I t was necessary to examine both l o c a l minima on the f i r s t y i e l d and f i r s t hinge curves because e i t h e r one may govern depending on the slenderness. 3.1.6 Unbalanced Uniform Loading f o r Minimum Strength Two dimensionless parameters, Xj/L and x/L, are required t o describe the l o c a t i o n of the unbalanced uniform load. During preliminary a n a l y s i s , i t q u i c k l y became evident that the value of x ^ L required to load f o r minimum strength was zero. This means that a uniformly d i s t r i b u t e d load s t a r t i n g a t the l e f t haunch and extending part way along the span w i l l minimize w^L /M^. 2 This loading was used f o r a l l w^ so that only x/L needed v a r i a t i o n to produce a minimum. The behaviour of the load parameter w.L /M as a f u n c t i o n of x/L i s I P 2 s i m i l a r to that of the point load of F i g . 25. Two l o c a l minima e x i s t f o r the f i r s t y i e l d and f i r s t hinge c o n d i t i o n s , and one unique minimum e x i s t s for each of the subsequent hinges. As before, once the f i r s t hinge forms, the l o c a t i o n of each of the subsequent hinges i s uniquely defined. The f i r s t hinge forms at one of the haunches, depending on L / r . The second hinge w i l l then always form at the opposite haunch. The t h i r d hinge forms near the r i g h t quarter p o i n t , and the f o u r t h hinge forms near the l e f t quarter point, assuming i n s t a b i l i t y has not already occurred p r i o r t o the formation of any of these hinges. As previously mentioned, the arch was d i s c r e t i z e d i n t o twenty members. This means that the values of x/L f o r minimum strength f o r e i t h e r the point loaded arch, or the uniformly loaded arch, could be 41. i n c o r r e c t by as much as ±2 1/2%. This would r e s u l t i n only n e g l i g i b l e e r r o r s i n the minimum load parameters. 3.2 D i s c u s s i o n of Hinge Formation Curves and Collapse Envelopes The standard arch behaviour curves of F i g s . 17 t o 24 are the funda- mental r e s u l t s of t h i s t h e s i s . The curves are bounded by a n a l y t i c a l s o l u t i o n s which w i l l be derived i n Chapter 4. I t i s the purpose of t h i s segment of the work to discuss the collapse envelopes and the hinge formation curves themselves. The d i s c u s s i o n w i l l include a summary of arch behaviour by regions on the p l o t s . 3.2.1 Collapse Envelopes The curve which defines the u l t i m a t e load as a f u n c t i o n of L/r i s a c t u a l l y an envelope of the hinge formation curves, F i g s . 17 to 23. Once the hinge formation curves are p l o t t e d , and the c o l l a p s e envelope generated, the r e s u l t s can be summarized on a separate graph showing the c o l l a p s e envelopes only. Two such p l o t s are required; one f o r the point loaded arch, F i g . 23, and one f o r the uniformly loaded arch, F i g . 24. These graphs of P L/M or w L /M versus L/r each show three u p u p 2 collapse envelopes corresponding to a = 0.0, 0.10, and 0.20. As expected, there are f i v e d i f f e r e n t types of c o l l a p s e ; e l a s t i c buckling, and one, two, three, or four hinge c o l l a p s e . The governing c o l l a p s e mechanism f o r the standard arches examined i s dependent on the slenderness L / r , and the dead load r a t i o a. This gives r i s e to regions on the arch c o l l a p s e curves of F i g s . 23 and 24 corresponding to the d i f f e r e n t mechanisms of c o l l a p s e . 42. 3.2.2 E f f e c t of L/r on Type of Collapse I t Is of no s u r p r i s e by now that l e s s hinges are required f o r c o l l a p s e w i t h i n c r e a s i n g slenderness. On any given c o l l a p s e envelope, the value of L/r which marks the t r a n s i t i o n from one type of f a i l u r e to another i s c l e a r l y v i s i b l e by a cusp i n the curve. the end of a hinge formation curve. The cusp i s a c t u a l l y For example, the t r a n s i t i o n between three hinge f a i l u r e and two hinge f a i l u r e i s the end of the t h i r d hinge formation curve. For any value of slenderness beyond t h i s point, the formation of a t h i r d hinge i s not p o s s i b l e because l o s s of s t i f f n e s s causes i n s t a b i l i t y to occur before the t h i r d hinge has a chance to form. The e f f e c t of L/r can be summarized by c o n t r a s t i n g the f a i l u r e modes at low L/r and high L / r . A four hinge p l a s t i c c o l l a p s e mechanism as d i c t a t e d by c l a s s i c a l p l a s t i c theory occurs only at low L/r where second order e f f e c t s are m i n i m i a l . The opposite occurs at high L/r where second order e f f e c t s are prevalent and f a i l u r e i s i n s t i g a t e d by the l o s s of s t i f f n e s s due to the formation of the f i r s t hinge or complete e l a s t i c buckling. 3.2.3 E f f e c t of Dead Load on Type of Collapse Having discussed the e f f e c t of slenderness on the type of c o l l a p s e , i t remains to discuss how and why the dead load r a t i o a influences the mode of c o l l a p s e . The values of slenderness marking the t r a n s i t i o n between two d i f f e r e n t collapse mechanisms w i l l be termed (L/r) v trans The r a t i o a i s the only parameter which contains the dead load w^. Any increase i n dead load would increase the dead load thrust and hence increase any second order e f f e c t s . I t i s therefore c o r r e c t to conclude that the c o l l a p s e curves corresponding to higher values of a are A3. influenced more by second order e f f e c t . that and Therefore, i t i s not s u r p r i s i n g decrease with increasing a. The dead load a l s o e f f e c t s the value of ( L / r ) marking the locatrans t i o n of a cusp. There must e x i s t some values of L/r f o r which a lower value of o would permit an a d d i t i o n a l hinge to form due t o a lessening of the second order e f f e c t . A t y p i c a l segment of two superimposed collapse envelopes i s shown i n F i g . 29 t o show q u a l i t a t i v e l y the range of L/r f o r which two d i f f e r e n t types of collapse are prevalent. Because the range of L/r described by F i g . 29 must e x i s t , ( L / r ) must be lower f o r trans higher values of a. L/r F i g . 29. V a r i a t i o n of ( L / r ) trans 44. 3.2.4 E l a s t i c Buckling and the L i m i t i n g Slenderness Ratio Examining F i g s . 23 and 24, i t i s apparent that the c o l l a p s e envelopes cross the h o r i z o n t a l a x i s where the l i v e load i s zero. At t h i s p o i n t , the dead load alone i s s u f f i c i e n t to cause e l a s t i c buckling. T h e o r e t i c a l l y , t h i s i s the maximum p o s s i b l e slenderness load r a t i o a, and i s referred to as the slenderness f o r a given dead limit. F i g s . 23 24 a l s o show that the behaviour of arches j u s t p r i o r to reaching slenderness and this l i m i t i 6 d i f f e r e n t for the point loaded arch than for the uniformly loaded arch and so each w i l l be discussed separately. Nowhere on the point load collapse curve, F i g . 23, does e l a s t i c buckling govern the u l t i m a t e load except i n the l i m i t as P approaches zero where the dead load alone causes e l a s t i c buckling. Under uniform loading, the region of e l a s t i c buckling i s very small. In t h i s region x/L = 1.0, which means the l i v e load was applied over the e n t i r e span of the standard arch. was The uniform load w required to cause e l a s t i c buckling smaller than the h a l f span load required to form the f i r s t hinge. This e l a s t i c buckling region i s so c l o s e to the t h e o r e t i c a l slenderness l i m i t , where the l i v e load to dead load r a t i o becomes zero that i t i s i m p r a c t i c a l and l i k e l y impossible to a t t a i n . In summary, in-plane e l a s t i c buckling of a f i x e d arch w i l l r a r e l y , i f ever, govern design. 3.2.5 C r i t i c a l Loading P a t t e r n , x/L Results Indicated on a l l the hinge formation curves I s the value of x/L which minimized the dimensionless load. These are shown by the use of symbols p l o t t e d s l i g h t l y above the a c t u a l data points f o r c l a r i t y . The r e s u l t s for the ultimate load for each loading c o n d i t i o n are reasonably 45. consistent. I n general, the non-linear behaviour d i c t a t e s that loading 55 t o 60 percent of the span governs f o r the ultimate capacity of a uniformly loaded arch, and p l a c i n g the point load a t x/L = 0.25 or 0.30 governs f o r a point loaded arch. There are two d i s t i n c t values of x/L governing f i r s t y i e l d and the formation of the f i r s t hinge. This was expected because, as previously explained, when e i t h e r l o a d parameter i s p l o t t e d as a f u n c t i o n of x/L only, two l o c a l minima a r i s e , each corresponding to d i f f e r e n t f i r s t hinge locations. However, i t remains t o e x p l a i n why one l o c a l minima governs for low L / r , and the other f o r higher L / r . Under uniform loading, the f i r s t hinge (and f i r s t y i e l d ) curves show a d e f i n i t e t r a n s i t i o n from x/L = 0.4, corresponding to a hinge forming at the l e f t haunch, t o x/L • 0.6, corresponding t o a hinge forming a t the r i g h t haunch. To explain t h i s phenomena, i t i s necessary to define a moment due t o r i b shortening, M^^, and a second order a m p l i f i c a t i o n f a c t o r (j). Two separate cases w i l l be examined, a stubby arch with L / r approaching zero and a very slender arch w i t h high L / r . F i g . 27 shows the approximate haunch moments In a stubby arch loaded with 40% and then 60% of f u l l l i v e load. The maximum haunch moment caused by the unbalanced uniform l i v e load w alone i s given the symbol M^. The oppo- s i t e haunch moment i s l e s s than M and i s a r b i t r a r i l y taken as 0.75 M to w w emphasize the d i f f e r e n c e . Simple superposition says that the l e f t haunch moment w i t h x/L m 0.4 and the r i g h t haunch moment w i t h x/L = 0.6 are equal, however t h i s excludes the e f f e c t of r i b shortening. I t i s important t o note that M acts t o increase the l e f t haunch moment, but rs v decrease the r i g h t haunch moment. This e x p l a i n s why the t o t a l moment at j o i n t 1 with x/L = 0.4 i s the l a r g e s t , thus allowing the f i r s t hinge to 46. x/L = 0.40 1. I 1 l» 1 I 1 I 1 I 1 l«d JOINT I JOINT 21 (M + M )I.O w (0.75M - M )I.O RS W RS x/L=0.60 l lw 1 . 1 1 1 1 l 3*. JOINT I (0.75M + M )I.O w RS ( M - M )I.O w RS F i g . 27. Stubby Arches, No Second Order A m p l i f i c a t i o n , $ = 1.0. form there at low L / r . F i g . 28 shows the haunch moments of two slender arches loaded by 40% and 60% of f u l l l i v e load r e s p e c t i v e l y . The moment due to r i b shortening becomes i n s i g n i f i c a n t a t l a r g e L/r because the r a t i o M /M v a r i e s rs w i n v e r s e l y with L / r . For large L / r , the second order e f f e c t now over- shadows any e f f e c t o f r i b shortening. The maximum j o i n t 1 moment i s <J>, M . The maximum j o i n t 21 moment i s d>,M . The second order magnification w 'w 47. x/L= 0.40 i ) i i i i r JOINT I (M w JOINT 21 +0)<£ (0.75M -0)<£ ( w III • x/L=0.60 w III JOINT I (4 (0.75M + 0)aS w 2 F i g * 28. Haunch Moments i n Slender Arches. 60% loaded than for a load over only 40% of the span. greater than $ 1 Therefore, $ i s 2 and the moment a t j o i n t 21 w i t h x/L = 0.6 i s the l a r g e s t . For slender arches, the f i r s t hinge w i l l form at the r i g h t haunch, j o i n t 21, w i t h the span 60% loaded. A s i m i l a r phenomenon a r i s e s when an arch i s loaded by a point load. Lower L/r implies that x/L - 0.15 and the f i r s t hinge forms at the l e f t haunch. At higher L / r , the f i r s t hinge forms w i t h x/L - 0.30 at the l o c a t i o n of the point load. Thus, the reason f o r t h i s i s s i m i l a r to the explanation given f o r a uniform loading and w i l l not be repeated. 48. In t h i s chapter, the main r e s u l t s of t h i s t h e s i s were presented i n the form of hinge formation curves and collapse envelopes, F i g s . 17 through 22. Conventional a n a l y t i c a l s o l u t i o n s f o r u l t i m a t e load are p l o t t e d on these f i g u r e s as a n a l y t i c a l bounds to the r e s u l t s generated. I t remains t o derive these bounds and to discuss any discrepancies between the collapse curves and the a n a l y t i c a l s o l u t i o n s . chapter w i l l accomplish t h i s . The f o l l o w i n g 49. CHAPTER 4 ANALYTICAL BOUNDS The f i x e d arch collapse curves were presented and discussed i n Chapter 3. A n a l y t i c a l bounds were a l s o p l o t t e d t o served as reference. I t i s the aim of t h i s chapter to derive these a n a l y t i c a l solutions based on t r a d i t i o n a l a n a l y s i s and to compare these to the c o l l a p s e envelopes. The a n a l y t i c a l solutions serve as bounds at low L/r and high L / r . At low L/r, the ultimate load approaches that f o r a four hinge p l a s t i c collapse mechanism. The a n a l y t i c a l s o l u t i o n i s therefore based on conventional p l a s t i c a n a l y s i s w i t h no second order e f f e c t s . At high L / r , the point loaded arch i s bounded by one hinge c o l l a p s e , and the uniformly loaded arch by e l a s t i c buckling. 4.1 A n a l y t i c a l Bounds f o r Low L/r Two s o l u t i o n s w i l l be derived f o r each of the two loading cases. The f i r s t s o l u t i o n w i l l neglect the e f f e c t of any reduction of M due to P the presence of a x i a l force, and the second s o l u t i o n w i l l include t h i s a x i a l reduction of M . P considered. 4.1.1 In both cases, no second order magnification i s Low L/r; Neglecting A x i a l Reduction of The point loaded arch w i l l be examined f i r s t . placed a t the l e f t quarter point. The point load i s This i s a reasonable assumption and i s confirmed by the r e s u l t s of Chapter 3 which i n d i c a t e d that x/L = 0.25 at low L/r. F i g . 30 shows three free body diagrams. One diagram of a para- b o l i c arch under dead load and a point l i v e load, the second of the l e f t 50. 1 I I 1 I 1 1 K V V L w d R " w d INd X 4 L/4 F.B.D. $2 - Left Quarter F.B.D. 03 - Right Side F i g . 30. Free Body Diagrams of Point Loaded Arch, quarter of the arch, and the t h i r d of the r i g h t . The l o c a t i o n s A, B and D of three of the four hinges are known to be at the haunches and at the p o i n t load. However, the l o c a t i o n C of the f o u r t h hinge must be estab- l i s h e d and i s represented by the unknown v a r i a b l e x^. The f o l l o w i n g e q u i l i b r i u m equations apply to the three free body diagrams of F i g . 30: F.B.D. #1, IV = 0 gives P u + w L, d (4.1) F.B.D. #1, IM = 0 gives 2M + V L - P L/4 - ( L ) ( L / 2 ) = 0, p R u d Wj F.B.D. #2, ZM B (4.2) = 0 gives (4.3) and F.B.D. #3, ZM c = 0 gives 2M w*+ V x = Hh + w, p R c d2 D (4.4) A f i f t h equation can be obtained from the geometry of the parabolic arch h = -4(^)2 + 4 ( ^ ) (4.5) 52. The s o l u t i o n t o t h i s problem involves the s i x unknowns, H, V, , V„, x , P L R c u and h and only f i v e equations. s i x unknowns. The f i v e equations 4.1 to 4.5 r e l a t e the E l i m i n a t i o n of H, V , V ITp = ^ + and h gives R Li 3(x /L) A(x c /L)'> c ' <'> 4 6 I t i s important to note that the dead load has no e f f e c t on the r e s u l t f o r four hinge p l a s t i c c o l l a p s e i f a x i a l reduction of e f f e c t s of neglected. and second order This a r i s e s because i n the a n a l y t i c a l s o l u t i o n the dead load only causes a x i a l forces and no bending, and a x i a l forces contribute only to second order e f f e c t s and reduction of M . P I t remains to determine the l o c a t i o n of the fourth hinge by minimizing P^L/Mp i n Eq. (4.6) with respect t o * / L . This can be accomplished by maximizing D where D = 3 ( x / L ) - 4 ( x / L ) . D i f f e r e n t i a t i n g and C 2 c c s e t t i n g dD/dx equal t o zero gives x / L = .318 f o r minimum c o l l a p s e load. c This minimum collapse load i s then PL ^P = 22 j = 22.22 (4.7) This means that the ultimate point load parameter i s constant i f a x i a l reduction of M and second order e f f e c t s are neglected. P Eq. (4.7) i s p l o t t e d on F i g s . 17, 18 and 19 as a s t r a i g h t l i n e l a b e l l e d , " P l a s t i c , No I n t e r a c t i o n " . This r e s u l t i s grossly non-conservative because a x i a l i n t e r a c t i o n t o reduce M i s prevalent a t low L/r, and second order P e f f e c t s are not n e g l i g i b l e , e s p e c i a l l y a t intermediate and high L / r . A s i m i l a r a n a l y t i c a l s o l u t i o n f o r low L/r and neglecting reduced p l a s t i c moment must now be derived f o r the uniformly loaded arch. This 53. loading case i s s l i g h t l y more complicated because only two of the four hinge l o c a t i o n s are known. F i g . 33 i s a free body diagram of a parabolic arch loaded by an unbalanced uniform load. a n a l y s i s , the dead load i s not considered For the purposes of t h i s because, as we have just seen, i t i s of no consequence I f a x i a l i n t e r a c t i o n and second order e f f e c t s are neglected. The method of s o l u t i o n i s exactly analogous t o the point load case. Moment and force e q u i l i b r i u m arch y i e l d s expressions f o r V and V . As w e l l , moment e q u i l i b r i u m of a free body diagram from A to B w i l l r e s u l t In an expression f o r h o r i z o n t a l thrust H, j u s t as f o r the point load case. These three reactions are as f o l l o w s : wx /2 L 2 M and - f - |wxb(i - |r-) + 2M £ - 2M h. 2L p L p H -?r } 2 2 1 r [ (4.8) 1 X b c L F i g . 33. Four Hinge P l a s t i c Collapse Under Unbalanced U.D.L. Loading. 54. Now. moment equilibrium about C of the f r e e body from C t o D i n F i g . 34 w i l l y i e l d the f i n a l equation as 2M^ + V c - H ( h ) . R the known reactions H and V c Substituting for gives the f o l l o w i n g : K 2M (4.9) i n order to s i m p l i f y , l e t = h _c »b / p c2 + i4 j c -4 f _ l 4 2 2 b j . f 2 + lb r-^r>2 4 Eq. 4.9 now i s a function of the hinge l o c a t i o n s b/L and c/L as shown below. w L' u M 2(c/L + y b/L - y - 1) ac ab ... — - - Y — (1 2L 2 : a 2L N ) Ybf. 2L 2 P H F i g . 34. F.B.D. of Right Side. (4.10) 55. I t now remains t o minimize the u l t i m a t e load parameter with respect to hinge l o c a t i o n s and loaded length. To accomplish t h i s , a simple computer program was w r i t t e n which evaluted w L /M^ f o r various 2 u combinations of b/L, c/L and x/L, to determine the minimum. The r e s u l t s were as f o l l o w s ; b/L = 0.30, c/L = 0.30, x/L • 0.50 and w L^ u - 93.33 M (4.11) Eq. 4.9 i s the r e s u l t of a four hinge p l a s t i c collapse a n a l y s i s n e g l e c t i n g a x i a l i n t e r a c t i o n and second order e f f e c t s . I t i s p l o t t e d as a h o r i z o n t a l l i n e i n F i g s . 20, 21 and 22 and i s evidently grossly non-conservative. I t i s worth noting that f o r both loading cases, the ultimate load r a t i o s are independent of f/L, E/o^ and y/r and a. 4.1.2 Low L / r , Including A x i a l I n t e r a c t i o n Neglecting a x i a l reduction of at low L/r i s a serious omission. This w i l l now be included i n the a n a l y t i c a l s o l u t i o n to obtain a more reasonable bound a t low L/r. To make the arch behaviour amendable t o a closed form s o l u t i o n , two assumptions are now made. F i r s t , the d i s t r i b u t i o n of a x i a l force over the e n t i r e span of the arch i s assumed constant and equal t o the thrust H. This i s a f a i r assumption for arches whose r i s e to span r a t i o , f / L , i s not abnormally high. Second, the i n t e r a c t i o n between a x i a l and bending i s assumed b i l i n e a r as shown by the y i e l d surface of F i g . 5. 56. This i s the same y i e l d surface used f o r the non-linear a n a l y s i s i n ULA so the comparisons should be v a l i d . 4.1.2.1 P l a s t i c Collapse, Low L / r , Including A x i a l I n t e r a c t i o n , i n Point Load Case Now, f o r the point load case, Eq. (4.7) must be r e w r i t t e n as P L/M - 22.22 (4.12) u where M i s the reduced p l a s t i c moment due t o a x i a l P. For the same reason, Eq. 4.3 i s s i m p l i f i e d and r e w r i t t e n as P L " H J T p w,L - 2 8f + f OM L U , 4 T „ + a P ZM / / I O N p T (4 ' 13) The y i e l d surface i s represented by the f o l l o w i n g two equations: P/P + 0.85 M/M = 1.0 P P for M/M 0.26 P/P for M/M P < 0.95 (4.14) > 0.95 (4.15) and P + M/M P = 1.0 P Combining Eqs. 4.12 and 4.13 with 4.14 and then 4.15 gives lit M p (L/r)(1.0-ci) 0.16 _ + .0385 L/r f o r g / M > 0 > 9 5 ( A < 1 6 ) 57. and V M (L/r)(1.0-a) p 0.0416 {|70 + f o r - / M < 0 < 9 5 ( 4 > 1 7 ) .045(L/r) S u b s t i t u t i n g the standard arch values of y/r = 0.95 and f/L = 0.15 i n t o the above equations gives: and ^ - ..iS'i'&S'U) ^ 5 / M > 0 . 9 5 (4.19, P By equating Eqs. 4.18 and 4.19 i t i s e a s i l y shown that Eq. 4.18 governs f o r L/r < 111 and Eq. 4.19 governs f o r L/r > 111. These two equations are p l o t t e d on F i g s . 17, 18 and 19 and l a b e l l e d as " P l a s t i c , B i l i n e a r Interaction". As expected t h i s curve I s v a s t l y d i f f e r e n t from the " P l a s t i c , No I n t e r a c t i o n " curve f o r low L / r . This i s because the hinges do not form at a moment M^, they form at M, and M « as L/r approache zero. The l i m i t of P L/M as L/r approaches zero i s zero, however the u p l i m i t of ? L/M as L/r approaches zero i s 22.22. U 4.1.2.2 P l a s t i c C o l l a p s e , Low L / r , I n c l u d i n g A x i a l I n t e r a c t i o n , U.D.L. Case Having derived expressions f o r four hinge p l a s t i c c o l l a p s e i n c l u d i n a x i a l i n t e r a c t i o n f o r a point loaded arch, i t remains to repeat t h i s d e r i v a t i o n f o r an arch loaded by unbalanced U.D.L. Eq. 4.11 must be r e w r i t t e n as f o l l o w s : 58. w L /M u = 93.33 2 (4.20) S u b s t i t u t i n g x/L - 0.5, b/L - 0.3, c/L « 0.3 i n t o Eq. (4.8) and adding the dead load t h r u s t , the expression f o r a x i a l f o r c e i n the arch becomes: 8.036xl0" w L M I L - - 5 _ p _ f/L 3 L(f/L) 2 P = H = c + a (4.21) P Combining Eqs. (4.20) and (4.21) with each of the i n t e r a c t i o n Eqs. (4.14) and (4.15) gives w L< u M (L/r)(1.0-ct) 0.0625 + for M/M for M/M 9.1071xl0 (L/r) _3 P < 0.95 (4.22) > 0.95 (4.23) and w L' M (L/r)(1.0-a) 0.01625 + 1.07l4xl0 (L/r) _2 P S u b s t i t u t i n g the standard values of f/L = 0.15 and y/r = 0.15 r e s u l t s i n the f o l l o w i n g two equations: w L u Mp d (L/r)(1.0-a) for M/M for M/M 0.396 + 9.11xl0" (L/r) 3 P < 0.95 (4.24) > 0.95 (4.25) and w L Mp z (L/r)(1.0-ct) ' 0.103 + 1.071xl0 (L/r) -2 P Equating (4.24) and (4.25) i n d i c a t e s that f o r L/r < 182 Eq. (4.24) w i l l govern, and f o r L/r > 182 Eq. (4.25) w i l l govern. This r e s u l t i s p l o t t e d 59. on F i g s . 20, 21 and 22 and l a b e l l e d , " P l a s t i c , B i l i n e a r I n t e r a c t i o n " . Unlike the point load case, the arch behaviour at low L/r under U.D.L. i s not e n t i r e l y governed by four hinge c o l l a p s e . the governing At very low L/r, f a i l u r e mechanism could be f u l l c r o s s - s e c t i o n a x i a l y i e l d - ing under f u l l span l i v e load (x/L = 1.0). An a n a l y t i c a l s o l u t i o n f o r t h i s behaviour i s obtained simply by equating the thrust caused by f u l l dead and l i v e laod to f u l l a x i a l y i e l d P • Ao P w L F as f o l l o w s : y w .L 2 2 » = -*- -tr s + w L U 8f 2 + a P " *p = P p S u b s t i t u t i n g M /P r = 0.95 y i e l d s : P P w L 2 U 8fM (.95r) = 1.0 -a P S i m p l i f y i n g and s u b s t i t u t i n g f/L = 0.15 r e s u l t s i n the f o l l o w i n g : w L u M P 2 1.263(1.0-a)(L/r) (4.26) Eq. (4.26) describes the ultimate f u l l span uniform l i v e load required to cause a x i a l y i e l d i n g of a standard arch. Equating Eq. (4.26) with four hinge p l a s t i c collapse Eq. (4.24) shows that f u l l l i v e load a x i a l y i e l d i n g only governs f o r L/r < 43. 21 and 22 and l a b e l l e d " A x i a l Y i e l d " . Eq. (4.26) i s p l o t t e d on F i g s . 20, 60. 4.2 A n a l y t i c a l Bounds f o r High L / r As the slenderness, L / r , approaches the t h e o r e t i c a l slenderness l i m i t , the arch under unbalanced U.D.L. collapses by e l a s t i c buckling, whereas the point loaded arch buckles a f t e r the formation of the f i r s t hinge. A n a l y t i c a l solutions w i l now be derived f o r each of the slender c o l l a p s e s mentioned. i s not s i g n i f i c a n t a t high P L/r and i s therefore not included In t h i s a n a l y t i c a l d e r i v a t i o n . 4.2.1 A x i a l reduction of M High L / r , F u l l Uniform L i v e Load E l a s t i c Buckling An expression f o r the e l a s t i c buckling load parameter as a f u n c t i o n of L/r can be derived by equating f u l l l i v e load and dead load thrust to the E u l e r buckling load. Again, i t i s assumed that the a x i a l force i n the arch i s constant and equal to the h o r i z o n t a l thrust so that V l + V l 8 f w?EI 8 f = (kL) 2 or _ , u W P ir EAr L 2 2 (kL) 8 f 2 2 where kL i s the e f f e c t i v e length of a f i x e d arch. Including the i d e n t i t y y/r = M /P r and s i m p l i f y i n g gives P P w L u -— M 2 8f/L i r E 1 —y/r . — Iz ay y/r „ 2 oy L/r 2 r = K Y v yi \ a L/r J (4.27) 1 The a n a l y t i c a l s o l u t i o n requires a value of the e f f e c t i v e length f a c t o r k. This was obtained by examining the r e s u l t of a standard arch ULA a n a l y s i s at L/r = 700 f o r a = 0.10 where the governing ultimate load behaviour was e l a s t i c buckling under f u l l l i v e load. A value of k = 0.377 was chosen such that Eq. 4.27 would agree with the ULA r e s u l t . S u b s t i t u t i n g standard arch values of f/L = 0.15, E/oy = 750 and y/r = 61. 0.95 as w e l l as k = 0.377 gives a f i n a l r e s u l t : w L u M P 2 1-26 a (L/r) (4.28) Eq. (4.28) describes the uniform ultimate load parameter for e l a s t i c b u c k l i n g as a f u n c t i o n of the dead load r a t i o a and slenderness L/r. This i s p l o t t e d on F i g s . 20, 21 and 22 under the l a b e l " E l a s t i c Buckling, k = 0.377". 4.2.2 High L / r , Point Load, One Hinge Buckling An expression f o r one hinge i n s t a b i l i t y i s derived by equating the p l a s t i c moment M to the approximate l i n e a r f i r s t order moment PL/17 with P second order m a g n i f i c a t i o n . f K - H / H cr > ° PL/17 M p* ( 4 was determined by evaluating the maximum moment from l i n e a r ' 2 9 ) first order s t i f f n e s s a n a l y s i s . For l a r g e L / r , v i r t u a l l y a l l the t h r u s t comes from the dead load. w L Therefore, i t i s assumed that H = . = a P . Now, Eq. (4.29) becomes: * 8f p 2 d ac M p ~ 17 ( otP } 1 - — 2 - (kL) 2 1T EI 2 S e t t i n g P = P^ and rearranging: p ° L JL. M P m 17(1- 2 - J L *• 2 E M r (iSk)2) ' S u b s t i t u t i n g k = 0.37 and E/oy = 750 y i e l d s the f i n a l r e s u l t : 62. P L 17(1 - 1.920xlCT a ( L / r ) ) 5 (4.30) 2 P Eq. (4.30) i s an a n a l y t i c a l s o l u t i o n f o r the point load r a t i o required f o r one hinge i n s t a b i l i t y as a f u n c t i o n of the dead load r a t i o a and the slenderness L / r . I t i s p l o t t e d as "One Hinge A n a l y t i c a l , k = 0.377" on F i g s . 17, 18 and 19. 4.2.3 A n a l y t i c a l S o l u t i o n f o r the T h e o r e t i c a l Slenderness L i m i t An expression f o r the t h e o r e t i c a l slenderness l i m i t can be obtained by s o l v i n g e i t h e r Eq. (4.27) or Eq. (4.29) f o r L/r when w L /M or P,L/M u o are zero. I f P L/M = 17(1 - % -2- ( — ) ) = 0 u p T T E r ' 2 u p r 2 J z then ( where ( L / r ) 0 L / r i s the slenderness ) . I o /iFjL limit. (4.31) S u b s t i t u t i n g K = 0.377, E/o^ 750 and a = 0.1 then 0.2 i n d i c a t e s that (L/r) Q = 720 for a = 0.1 (L/r) n = 510 for a = 0.2 and 4.3 Comparison of A n a l y t i c a l Bounds With Collapse Envelopes There e x i s t s a discrepancy between the a n a l y t i c a l bounds derived i n t h i s chapter and the collapse curves generated by non-linear ultimate load a n a l y s i s . These a r i s e due t o the inadequacies of the conventional a n a l y t i c a l s o l u t i o n s . The graphical explanations f o r the discrepancies 63. are presented by slenderness regions i n F i g s . 35 and 36. =0.1 The p l o t s of a were a r b i t r a r i l y chosen here, however the explanation holds for a l l three dead load r a t i o s examined. ARCH COLLAPSE ENVELOPE - PT. LOAD , = 0.10 F i g . 35. Discrepancies Between A n a l y t i c a l Solutions and Collapse Curves, Point Loading. A R C H C O L L A P S E E N V E L O P E - U.D.L., ^ - L 8fP„ = 0.10 ELASTIC BUCKLING k » 0.37 x/L= I .0 800 F i g . 36. Discrepancies Between A n a l y t i c a l Solutions and Collapse Curves, U.D.L. Loading. 66. CHAPTER 5 VARIATION OF STANDARD PARAMETERS The non-linear behaviour of standard f i x e d arches are summarized by hinge formation curves and c o l l a p s e envelopes i n Chapter 3. To make t h i s examination of arch behaviour possible i t was necessary to define a standard arch by assuming that E/o^ = 750, f/L = 0.15, Z/s = 1.15 and y/r = 0.95. These standard values are i n d i c a t i v e of a t y p i c a l s t e e l box g i r d e r or wide flange arch. I t i s the purpose of t h i s chapter t o vary these four standard parameters and examine the e f f e c t on the non-linear performance of f i x e d arches. This should f a c i l i t a t e the e x t r a p o l a t i o n of the r e s u l t s of t h i s thesis to include a c t u a l arches whose parameters w i l l c e r t a i n l y deviate from the standard values. In the f o l l o w i n g sections only one parameter at a time i s a l t e r e d ; a l l others are kept at the standard value. 5.1 V a r i a t i o n of E/o y The dimensionless parameter E/o^ i s a m a t e r i a l property and not a f u n c t i o n of arch- geometry or c r o s s - s e c t i o n . I t ranges t y p i c a l l y from approximately 375 or 400 f o r aluminum t o about 900 f o r r e i n f o r c e d concrete. Eqs. 4.26 and 4.29 serve as a n a l y t i c a l bounds f o r behaviour at long L/r f o r uniform loading and point loading r e s p e c t i v e l y . The second order reduction terms are [1 - ( a / i r ) ( o /E(kL/r)] f o r point loading and 2 it E 2 1 [— — -j-yY - aL/r] f o r uniform l o a d i n g . I t i s apparent from these terms that second order e f f e c t s are p r o p o r t i o n a l t o E/o . Thus, a reduction of 67. E/o y from the standard value of 750 w i l l increase any second order e f f e c t s and therefore decrease the capacity of the arch. This has been confirmed by computer a n a l y s i s and i s presented i n F i g . 37 through 40 for E/a - 375. y Upon examination of these v a r i a t i o n of parameter curves f o r E/o^, i t i s obvious that the e f f e c t of reducing E/o^ becomes l e s s pronounced with decreasing L / r . t i o n a l to ( L / r ) This i s because any second order e f f e c t s are propor2 and therefore die out at low L/r. Extending t h i s argument to the l i m i t i n g case as L/r approaches zero, i t i s evident that' E/Oy has no e f f e c t on the u l t i m a t e load parameter. This l i m i t i n g case i s governed by a four hinge p l a s t i c c o l l a p s e mechanism according to Eqs. 4.16 and 4.17 which do not contain the parameter E/cJy' The l i m i t i n g slenderness l i m i t defined by Eq. 4.37 to the square root of E/o^. i s proportional This supports the reduction of the slender- ness l i m i t due to the halving of E/o^ i n d i c a t e d by F i g s . 37 through 5.2 V a r i a t i o n of f/L The r i s e to span r a t i o , f/L, i s commonly i n the range of 0.10 0.30 40. f o r bridge arch r i b s i n s t e e l , concrete or aluminum. arch has an assumed value of The standard 0.15. At low L / r , four hinge p l a s t i c c o l l a p s e i s described by Eqs. and 4.17 loading. to for point loading and Eqs. 4.22 and 4.23 4.16 for unbalanced uniform The quantity f/L appears i n both these a n a l y t i c a l s o l u t i o n s . I t i s evident from these equations that Increasing only f/L r e s u l t s i n an increase i n the u l t i m a t e load r a t i o s P L/M or w L /M . u p u p 2 A l s o , the e f f e c t of varying f/L diminishes with i n c r e a s i n g L/r and i s almost non-existant i n the intermediate range of L / r . For example, at a r e l a t i v e l y low value ARCH COLLAPSE ENVELOPE - PT. LOAD , - ^ - - = 0.10 VARIATION OF S T A N D A R D P A R A M E T E R E/tr y P ARCH COLLAPSE ENVELOPE - PT. LOAD , 37^" VARIATION OF STANDARD PARAMETER E/cr y = 0 - 2 0 p 600 ARCH COLLAPSE ENVELOPE - U.D.L., VARIATION OF S T A N D A R D P A R A M E T E R E/cr ^ - ^ - = 0.10 P 8 y f P PLASTIC.NO INTERACTION 0 100 200 300 400 500 600 700 L/r F i g . 39. S e n s i t i v i t y Analysis of E/o , Uniform Loading, o » 0.10 ARCH COLLAPSE ENVELOPE - U.D.L., ^ - ^ - = 0.20 100 r PLASTIC,NO VARIATION OF S T A N D A R D P A R A M E T E R E/cr 8 f P w l INTERACTION STANDARD E/ir FOUR HINGES \ THREE HINGES. TWO HINGES \ ONE HINGE FIRST YIELD -ELASTIC \ \ \ y * 750 f/L «O.IS l/S • 1.19 y/r »0.93 BUCKLING * / L = 1.0 , k = 0 . 3 7 7 E/oy ARCH = 37S 600 Fig. 40. S e n s i t i v i t y Analysis of E/o , Uniform Loading, a = 0.20 • 72. of L/r of 100, the Increase i n P L/M due to a change i n f/L from 0.15 u p 0.25 i s only 9%. to This behaviour i s confirmed by ULA computer a n a l y s i s f o r point loading and uniform loading with an f/L value of 0.25. The r e s u l t s are superimposed on standard arch behaviour curves In F i g s . 41 through 44. I t i s i n t e r e s t i n g to note that at high L/r, an increase i n f/L a c t u a l l y causes a small decrease i n the ultimate load parameters. f i r s t , t h i s may appear as an anomaly when compared w i t h Eq. At 4.27 d e s c r i b i n g the a n a l y t i c a l bounds for uniform loading at high L/r because w L /Mp appears to be l i n e a r l y p r o p o r t i o n a l to f/L. 2 u of K i s assumed In the d e r i v a t i o n of Eq. 4.27 However, the value as a f r a c t i o n of the span length L when i n f a c t i t i s more c o r r e c t l y i n t e r p r e t e d as a f r a c t i o n of the arc length 1. Eq. 4.27 can be r e w r i t t e n i n the form of Eq. 5.1 using kL as the e f f e c t i v e length w L Mp 2 = 8 ( f / L ) ( L / r ) i, (y/r) IT (— E 2 (kL/r) 2 rr°y o o) (5.1) R e a l i z i n g that an increase i n f/L causes the e f f e c t i v e length kL to increase due to a l a r g e r arc length i t i s evident that i n c r e a s i n g f/L at large slenderness can act to reduce the ultimate load parameter. A s i m i l a r argument holds true f o r the point loading case. As a f i n a l comment before leaving the d i s c u s s i o n of v a r i a t i o n of f/L, a p r a c t i c a l note i s now made. to 0.25 5/3. The r a t i o f/L was changed from 0.15 i n the computer a n a l y s i s by increasing the r i s e f by that r a t i o For a v a l i d comparison, a l l other dimensionless unchanged. r a t i o s must be This meant that for the dead load r a t i o a •* w,L /8fP , the d p dead load w^ has to be increased by 5/3 to maintain a = 0.1 or 0.2. This 2 ARCH COLLAPSE ENVELOPE - PT. LOAD , - ^ - = 0 . 1 0 V A R I A T I O N OF S T A N D A R D P A R A M E T E R f/L p L/r Fig. 41. S e n s i t i v i t y Analysis of f/L, Point Loading, a » 0.10. ARCH COLLAPSE ENVELOPE - PT. LOAD , - ^ - = 0.20 VARIATION OF STANDARD PARAMETER f / L F i g . 42. S e n s i t i v i t y Analysis of f/L, Point Loading, ct= 0.20. p 2 ARCH COLLAPSE ENVELOPE - U.D.L., ^ - ^ - = 0.10 VARIATION OF STANDARD P A R A M E T E R f/L 8 f P P PLASTIC.NO INTERACTION F i g . 43. S e n s i t i v i t y Analysis of f/L, Uniform Loading, a = 0.10. ARCH COLLAPSE ENVELOPE - U.D.L., ^ - ^ - = 0 . 2 0 VARIATION OF STANDARD PARAMETER f / L P 100r- 8 f PLASTIC.NO INTERACTION 90 80 - F i g . 44. S e n s i t i v i t y Analysis of f/L, Uniform Loading, a = 0.20. P 77. should be i n mind when examining the small effect of varying f/L i n Figs. 41 to 44. 5.3 Variation of y/r The ratio y/r i s a cross section property ranging from 0.866 for a 6olid rectangular section to 1.0 for an idealized section with a l l of i t s material concentrated at two flanges. Again, i n Eqs. 4.16 and 4.17 for point loaded plastic collapse and Eqs. 4.22 and 4.23 for U.D.L. plastic collapse the quantity y/r i s apparent. An increase i n y/r w i l l cause a decrease i n the ultimate load parameters. The sensitivity of the load ratios to any change i n y/r i s the same as for f/L, however, the range of y/r i s very limited whereas f/L may vary considerably. As an example, the analytical bound equations at a value of L/r of 100, varying y/r through i t s entire feasible range from 0.866 to 1.0 only changes wJL /Mp by 4.1 percent, and P^L/M^ by 0.8 2 percent. ULA computer analysis confirms the insignificance of the variation of the quantity y/r. I t i s therefore reasonable to conclude that the standard arch non-linear behaviour curves are practical for a l l values of y/r. No additional plots are needed. 5.4 Variation of Z/S The r a t i o of the plastic section modulus to the e l a s t i c section modulus, Z/S i s also a ratio of the plastic moment of a cross-section to i t s y i e l d moment, M /M and i s often referred to as a shape factor. I t p y i s a cross-sectional property and varies from 1.0 for an idealized section with a l l i t s material at two flanges to 1.50 for a solid rectangular section. The value assumed for the standard arch i s 1.15 ARCH COLLAPSE ENVELOPE - PT. LOAD , - ^ - = 0 . 1 0 VARIATION OF STANDARD PARAMETER z/S 0 t P P L/r Fig. 45. S e n s i t i v i t y Analysis of Z/S, Point Loading, a » 0.10. 2 ARCH COLLAPSE ENVELOPE - U.D.L., ^ - ^ - = 0.10 V A R I A T I O N OF S T A N D A R D P A R A M E T E R z/S 8 f P l PLASTIC.NO INTERACTION HINGES 1INGES HINGES HINGE ELASTIC BUCKLING k » 0.37 x/L =1.0 800 F i g . A6. S e n s i t i v i t y Analysis of Z/S, Uniform Loading, a - 0.10. 80. corresponding to the approximate shape f a c t o r f o r box and wide flange section. As pointed out i n Chapter 3, the shape f a c t o r e f f e c t s only the f i r s t y i e l d c o n d i t i o n and not hinge formation or ultimate load. ing In the l i m i t - case of Z/S = 1.0, the f i r s t y i e l d and f i r s t hinge curves would coincide. For any other values of the shape f a c t o r , the f i r s t y i e l d curve must l i e below the f i r s t hinge curve. I t i s therefore simple to conclude that i n c r e a s i n g Z/S would decrease P L/M or w L /M . e p e p 2 This i s e a s i l y confirmed by second order e l a s t i c computer a n a l y s i s , the r e s u l t s of which are superimposed on standard curves f o r a = 0.10 46. i n F i g s . 45 and The s h i f t i n the f i r s t y i e l d curve i n the low and intermediate ranges are very nearly the r a t i o of the change i n Z/S. By varying the four standard r a t i o s E/o^, f/L, Z/S and y/r, an i n d i c a t i o n of the s e n s i t i v i t y of the load parameters to these r a t i o s obtained. was I t i s concluded that the standard arch hinge formation curves and c o l l a p s e envelopes are reasonable for any values of y/r and Z/S for values of f/L i n the range from 0.10 t o 0.30. and However, as shown c l e a r l y i n F i g s . 37 through 40, the standard arch curves are s i g n i f i cantly s e n s i t i v e to v a r i a t i o n i n the m a t e r i a l parameter E/o . y cannot be overlooked when applying the arch behaviour curves. 3 This 81. CHAPTER 6 CONCLUSION 6.1 Hinge Locations and Formation Sequence By considering both second order e f f e c t s and member p l a s t i c i t y the behaviour of standard f i x e d arches loaded to ultimate has been summarized using hinge formation curves and c o l l a p s e envelopes. A v a r i a t i o n of parameters which defined the standard arch was c a r r i e d out to examine the s e n s i t i v i t y of the response to these parameters. Throughout t h i s work, I t became c l e a r that the collapse mechanism depends on slenderness and ranges from one hinge i n s t a b i l i t y at high L/r to a four hinge p l a s t i c collapse mechanism at low L / r , with a few extremely slender uniformly loaded arches b u c k l i n g e l a s t i c a l l y . The l o c a t i o n of the p l a s t i c hinges and the sequence of formation have yet to be discussed completely. These r e s u l t s are summarized f o r the d i f f e r e n t collapse mechanisms i n Tables 1 and 2 f o r point loading and uniform loading r e s p e c t i v e l y . Each row i n the tables describes a d i f f e r e n t c o l l a p s e mechanism. The numbers i n the body of the table i n d i c a t e which hinge, i f any, formed at a c e r t a i n l o c a t i o n on the arch. For example, three hinge i n s t a b i l i t y under uniform loading occurs w i t h the f i r s t hinge forming at the r i g h t haunch, the second hinge forming at the l e f t haunch, and the t h i r d and f i n a l hinge forming near the r i g h t quarter point. The loading, defined by x/L i s for minimum ultimate strength and i s indicated i n the c o l l a p s e envelopes of F i g s . 17 to 22. 82. TABLE I . Hinge Formation Sequence, Point Loading Point Load Left Haunch 4 Hinge Collapse 2 1 3 Hinge Collapse 2 1 3 2 Hinge Collapse (2)* 1 (2)* 1 Hinge Collapse Near L e f t 1/4 Point Near Right 1/4 Point Right Haunch 4 3 1 *Second hinge may form at e i t h e r haunch, depending on L/r and a. TABLE I I . Hinge Formation Sequence, UDL Loading UDL Load Left Haunch 4 Hinge Collapse 1 3 Hinge Collapse 2 2 Hinge Collapse 2 Near L e f t 1/4 Point 4 1 Hinge Collapse 6.2 Near Right 1/4 Point Right Haunch 3 2 3 1 1 1 T y p i c a l Load D e f l e c t i o n Behaviour I t i s common to monitor the behaviour or response of a s t r u c t u r e due to i n c r e a s i n g load l e v e l to compare experimental r e s u l t s with a n a l y t i c a l work. Unfortunately, no experimental r e s u l t s are a v a i l a b l e , therefore 83. d i f f e r e n t common a n a l y t i c a l techniques w i l l be compared on a loadresponse b a s i s , i n the b e l i e f that the second order e l a s t o - p l a s t i c a n a l y s i s used i n t h i s work, c l o s e l y models a c t u a l behaviour. F i g s . 3 and 4 of Chapter 1 contrast f i r s t and second order e l a s t o p l a s t i c response of a h y p o t h e t i c a l s i n g l e bay frame. w i l l now be applied to a t y p i c a l f i x e d arch. Such a comparison The l i v e load applied i s uniformly d i s t r i b u t e d over s i x - t e n t h s of the span and the dead load i s of course applied to the e n t i r e span. deflection. The response i s the maximum arch The arch chosen t o evaluate l o a d - d e f l e c t i o n i s a standard arch as previously defined with slenderness L / r «= 222 and dead load r a t i o a • 0.10. These parameters were chosen as they are i n d i c a t i v e of slender arched r i b s of highway bridges. Several load d e f l e c t i o n curves are p l o t t e d on F i g . 47 f o r the above mentioned arch. These generated curves contrast the second order e l a s t o - p l a s t i c "ULA" response with f i r s t order e l a s t o p l a s t i c behaviour, with and without moment a x i a l i n t e r a c t i o n . Because an assumed dead load was Included i n the a n a l y s i s , the load d e f l e c t i o n curves do not s t a r t at the origin. The d e f l e c t i o n corresponding to u)L /Mp = 0.0 i s the dead load 2 deflection. Several observations can be made from these load-response p l o t s , the most obvious being the s i g n i f i c a n t non-conservatism a r i s i n g from neglecting second order e f f e c t s i n determining a c o l l a p s e mechanism. This i s best summarized by n o t i c i n g that at the load l e v e l when the f i r s t hinge would form according to a f i r s t order a n a l y s i s , the arch has a c t u a l l y e i t h e r formed, or i s very near, a three hinge collapse mechanism. Any discrepancies between the d i f f e r e n t load d e f l e c t i o n curves would be even more pronounced i f the dead load parameter a were greater than 0 1 2 3 4 5 6 7 DIMENSIONLESS T O T A L DEFLECTION 1000 S/L F i g . 47. Load-Response of a Typical Standard Arch. 8 9 00 85. 0.10 because that would increase any second order e f f e c t s . This i s indeed the case for many long span arches with a between 0.10 and 0.20. 6.3 A p p l i c a t i o n of Load and Performance Factors There has been l i t t l e d i s c u s s i o n thus f a r on the a p p l i c a t i o n of load f a c t o r s and performance factors as d i c t a t e d by L i m i t States Design. The c o l l a p s e curves and hinge formation curves have a l l been based on a computer a n a l y s i s . I t must be assumed that a l l the parameters r e l a t i n g to the curves, be they loads or m a t e r i a l p r o p e r t i e s , are appropriately factored. Thus, before entering the curves, a l l factors must f i r s t be a p p l i e d when c a l c u l a t i n g the required dimensionless parameters, then the l i v e load P^ and w^ obtained from the curves are factored loads. This ensures complete f l e x i b i l i t y because any f a c t o r s may be used. For example the dead load w must be i n t e r p r e t e d as a w dead load f a c t o r and w^g i s the s p e c i f i e d load. moment M P where a i s the S i m i l a r l y , the p l a s t i c i n d i c a t e d as part of several dimensionless r a t i o s must a c t u a l l y be c a l c u l a t e d as 4>ZOy. Of course, i t w i l l almost c e r t a i n l y be necessary to i n t e r p o l a t e between curves with d i f f e r e n t a r a t i o s to obtain meaningful values of P^ and w^. The f o l l o w i n g s e c t i o n w i l l deal with a p p l i c a t i o n of the dimensional a n a l y s i s to e x i s t i n g arches where load and performance factors must be applied. 6.4 A p p l i c a t i o n to E x i s t i n g Arches A very common use of the arch as a s t r u c t u r a l form i s f o r highway bridge r i b s . A span which i s too long for a t r u s s , and yet not long enough t o warrant a suspension or cable stayed s t r u c t u r e , i s commonly 86. bridged by two o r more arch r i b s . I f the foundation conditions are stable enough, a f i x e d arch can be constructed. Throughout the l i f e of a bridge, i t w i l l l i k e l y be required t o support l i v e loads greater than the o r i g i n a l design loads. Most e x i s t i n g arched bridges were designed e l a s t i c a l l y , and l i k e l y by means of an allowable s t r e s s approach. Thus properly evaluating an e x i s t i n g bridge, as w e l l as designing a new bridge by L i m i t States Design both require a knowledge of behaviour beyond the c r i t e r i o n of f i r s t y i e l d . I f a s t r u c t u r e has s i g n i f i c a n t reserve capa- c i t y beyond f i r s t y i e l d and f a c t o r e d loads cause a response i n t h i s region, then the s t r u c t u r e may be deemed safe from a strength point of view. The t y p i c a l hinge formation curves and c o l l a p s e envelopes of F i g s . 21 to 27 w i l l now be applied to the f i x e d arches of three e x i s t i n g bridges. These bridges are the La Conner Highway Bridge i n Washington State, the Capilano Canyon Highway Bridge i n Vancouver, B r i t i s h Columbia, and the A r v i d a Bridge i n A r v i d a , Quebec. The arched r i b s of these bridges are made of s t r u c t u r a l s t e e l , r e i n f o r c e d concrete and aluminum respectively. As a r e s u l t of a l l three arches having long spans, the designs were governed by lane loading as opposed to truck loading. curves f o r unbalanced uniform loading w i l l be used. The arch c o l l a p s e A point load was required i n a d d i t i o n to the uniform lane loading f o r the La Conner and Capilano bridges. The a n a l y s i s i n t h i s work d i d not Include t h i s a d d i t i o n a l point load, however both loaded lengths are quite long and any e r r o r due t o the omission of the point load should not be s e r i o u s . The o r i g i n a l design loads are used along with a L i m i t States Design dead load f a c t o r of 1.3 and performance f a c t o r of o) = 0.90 applied to 87. reduce P and M . For each bridge r i b the value of wL /M i s p l o t t e d on P P P 2 the appropriate c o l l a p s e curves. The value of w i s the unfactored uniform design load per r i b . This includes an impact f a c t o r and sidewalk pedestrian l o a d i n g . A reduction i n gross area due t o any r i v e t o r b o l t holes was considered i n c a l c u l a t i n g M^. s e c t i o n a l area was used. When c a l c u l a t i n g r , f u l l cross- The load case examined here does not include such things as temperature, wind o r earthquake and i s therefore by no means a complete a n a l y s i s , however a very good conceptual idea of the load f a c t o r required t o cause f i r s t y i e l d and the load f a c t o r required to cause c o l l a p s e i s i n d i c a t e d . I n the a n a l y s i s used for t h i s work, a constant c r o s s - s e c t i o n was assumed. The r e a l i t y , however, i s that a small v a r i a t i o n i n c r o s s - s e c t i o n i s commonly used to increase the moment r e s i s t a n c e a t the haunch where f i r s t y i e l d normally occurs. in a variation i n M and r . This r e s u l t s Thus, the key dimensionless r a t i o s wL /M 2 P P and L/r w i l l not have one s i n g l e value each, but a range of values. The r e s u l t i n g p l o t s on F i g s . 48 and 49 w i l l therefore c o n s i s t of a s e r v i c e load l e v e l region as opposed to a s i n g l e point for each bridge examined. 6.4.1 The La Conner Bridge The La Conner Bridge, a l s o known as the Swinomish Chanel Bridge, i s located at La Conner, Washington. metres (550 f e e t ) . This f i x e d s t e e l box arch spans 167.6 I t was designed by H.R. Powell and Associates of S e a t t l e , Washington i n 1955. Data from the design drawings give: f/L = 0.167 E/o = 600 y Z/S - 1.18 y/r = 0.95 88. = a wL /M 0.16 7.37 2 P to 8.89 and L/r = 185 to 191. The corresponding s e r v i c e load l e v e l region i s plotted as a square on F i g . 48. The f i r s t y i e l d and c o l l a p s e curves f o r both o = 0.10 and o = 0.20 are shown on F i g . 48 so that an i n t e r p o l a t i o n between the two curves can be made by the reader. The a c t u a l cruves f o r the La Conner Arch would plot s l i g h t l y below the standard arch curves due to the d i s crepancy between the standard value of E/a = 750 and the La Conner value y of 600. 6.4.2 The Capilano Canyon Bridge The Capilano Canyon Bridge i s part of the Trans-Canada Highway. It includes two r e i n f o r c e d concrete arch r i b s which span 103.4 m (339.4 f t . ) across the Capilano Canyon supporting a four lane concrete deck. The bridge was designed by Choukalos Woodburn Hooley and McKenzie L t d . f o r the B.C. Department of Highways i n 1956. Although t h i s research was o r i g i n a l l y geared towards metal arches, reasonable estimates can be made of the important parmeters d e s c r i b i n g the arch such as slenderness and a p l a s t i c moment. As i s common to a l l concrete arches, the Capilano arch I s symmetrically r e i n f o r c e d r e s u l t i n g i n as much compression s t e e l as tension s t e e l for bending. This implies s i g n i f i c a n t d u c t i l i t y and c a p a b i l i t y of hinge formation. A much more n o t i c e a b l e v a r i a t i o n i n c r o s s - s e c t i o n i s apparent i n a r e i n f o r c e d concrete arch than a metal arch, thus the s e r v i c e load region p l o t s l a r g e r . 89. Data from the design drawings give: and f/L = 0.168 E/o y - 900 a = 0.14 L/r = 115 t o 181 wL /M = 14.6 t o 20.2 2 P The design load region for the Capilano Canyon bridge i s plotted on F i g . 48. The curves shown i n F i g . 48 are conservative when a p p l i e d to the concrete Capilano arch because they correspond to E/o^ = 750, when i n f a c t E/o = 900 f o r concrete. y 6.4.3 The A r v i d a Bridge The f i r s t aluminum highway bridge on the American continent was b u i l t i n A r v i d a , Quebec, i n 1950. This, the Arvida Bridge, has a main span which i s a f i x e d arch 88.4 meter (290 f t . ) center to center of skewbacks, spanning the Saguenay R i v e r . The following dimensionless parameters were c a l c u l a t e d from information i n an a r t i c l e by C.J. Pimenoff: 5 f/L - 0.16 Z/S = 1.12 E/a - 210 y and L/r = a = 0.11 y/r =0.93 wL /M = 2 151 t o 156 8.40 to 9.84 . 90. The a v a i l a b l e hinge formation curves f o r E/o^ = 375 and 0.10 are applied to the Arvida arches i n F i g . 49. Again, the value of E/o^ i s i n c o r r e c t , however the r e s u l t i n g non-conservatism i s not serious at the low L/r corresponding t o the Arvida arch. 6.4.4 Further Research I t would be i n t e r e s t i n g t o compare the t h e o r e t i c a l s o l u t i o n s presented herein with an experimental study on model arches. As w e l l , the r e s u l t s h e r e i n are centered around a moment a x i a l i n t e r a c t i o n curve f o r a m a t e r i a l such as s t e e l . Some i n v e s t i g a t i o n should be made using the somewhat unique i n t e r a c t i o n curve f o r r e i n f o r c e d concrete. F i n a l l y , the current r e s u l t s could possibly be s i m p l i f i e d i n t o a design system more r e a l i s t i c than that used today. ARCH COLLAPSE ENVELOPE - U.D.L., ^ - = 0.10 8 f P„ APPLICATION TO EXISTING ARCHES PLASTIC.NO INTERACTION ELASTIC BUCKLING k =0.37 x/L =1.0 800 F i g . 48. Application of Collapse Curves to La Conner and Capilano Bridges. 100 90 F i g . 49. Application of Collapse Curves to the Arvida Bridge. 93. REFERENCES Hooley, R.F. and Mulcahy, F.X., 1982. Nonlinear Analysis by I n t e r a t i v e Graphics. Canadian Society f o r C i v i l Engineering, Annual Conference. Gere, J.M. and Weaver, W., 1965. Analysis of Framed Structures. Van Nostrand Reinhold Co., New York, pp. 428-431. Timoshenko, S., 1936. Theory of E l a s t i c S t a b i l i t y . McGraw-Hill, New York, pp. 36-38. 1st ed. Galambos, T.V. and K e t t e r , R.L., 1961. Columns Under Combined Bending and Thrust. American Society of C i v i l Engineers, Transactions, V o l . 126(1). Plmenoff, C.J., 1949. No. 4. The Arvida Bridge. E.I.C. J o u r n a l , V o l . 32,
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Ultimate load analysis of fixed arches Mill, Andrew John 1985
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Title | Ultimate load analysis of fixed arches |
Creator |
Mill, Andrew John |
Publisher | University of British Columbia |
Date Issued | 1985 |
Description | The advent of Limit States Design has created the necessity for a better understanding of how structures behave when loaded beyond first local yielding and up to collapse. Because the problem of determining the ultimate load capacity of structures is complicated by geometric and material non-linearity, a closed form solution for anything but the simplest of structure is not practical. With this as motivation, the ultimate capacity of fixed arches is examined in this thesis. The results are presented in the form of dimensionless collapse curves. The form of these curves is analogous to column capacity curves in that an ultimate load parameter will be plotted as a function of slenderness. The ultimate capacity of a structure is often determined by Plastic Collapse analysis or Elastic Buckling. Plastic Collapse is attained when sufficient plastic hinges form in a structure to create a mechanism. This analysis has been proven valid for moment resisting frames subjected to large amounts of bending and whose second order effects are minimal. Elastic buckling is defined when a second order structure stiffness matrix becomes singular or negative definite. Pure elastic buckling correctly predicts the ultimate load if all components of the structure remain elastic. This may occur in slender structures loaded to produce large axial forces and small amounts of bending. Because arches are subject to a considerable amount of both axial and bending, it is clear that a reasonable ultimate load analysis must include both plastic hinge formation and second order effects in order to evaluate all ranges of arch slenderness. A computer program available at the University of British Columbia accomplishes the task of combining second order analysis with plastic hinge formation. This ultimate load analysis program, called "ULA", is interactive, allowing the user to monitor the behaviour of the structure as the load level is increased to ultimate. A second order analysis is continually performed on the structure. Whenever the load level is sufficient to cause the formation of a plastic hinge, the stiffness matrix and load vector are altered to reflect this hinge formation, and a new structure is created. Instability occurs when a sufficient loss of stiffness brought on by the formation of hinges causes the determinant of the stiffness matrix to become zero or negative. Two different load cases were considered in this work. These are a point load and a uniformly distributed load. Both load cases included a dead load distributed over the entire span of the arch. The load, either point load or uniform load, at which collapse occurs is a function of several independent parameters. It is convenient to use the Buckingham π Theorem to reduce the number of parameters which govern the behaviour of the system. For both load cases, it was necessary to numerically vary the location or pattern of the loading to produce a minimum dimensionless load. Because of the multitude of parameters governing arch action it was not possible to describe all arches. Instead, the dimensionless behaviour of a standard arch was examined and the sensitivity of this standard to various parameter variations was given. Being three times redundant, a fixed arch plastic collapse mechanism requires four hinges. This indeed was the case at low L/r. However, at intermediate and high values of slenderness, the loss of stiffness due to the formation of fewer hinges than required for a plastic mechanism was sufficient to cause instability. As well, it was determined that pure elastic buckling rarely, if ever, governs the design of fixed arches. Finally, the collapse curves were applied to three existing arch bridges; one aluminum arch, one concrete arch, and one steel arch. The ultimate capacity tended to be between three and five times the service level live loads. |
Subject |
Arches |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-05-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0062587 |
URI | http://hdl.handle.net/2429/25121 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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