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Ultimate load analysis of fixed arches 1985

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ULTIMATE LOAD ANALYSIS OF FIXED ARCHES by Andrew John M i l l B.A.Sc, The University 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 i n THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept this 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 thesis i n p a r t i a l f u l f i l l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may 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 publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C i v i l Engineering The University of B r i t i s h Columbia 1956 Main Mall • Vancouver, Canada V6T 1Y3 Date October, 1985 ABSTRACT The advent of Limit States Design has created the necessity for a better understanding of how structures behave when loaded beyond f i r s t l o c a l yielding and up to collapse. Because the problem of determining the ultimate load capacity of structures i s complicated by geometric and material non-linearity, a closed form solution for anything but the simplest of structure i s not p r a c t i c a l . With t h i s as motivation, the ultimate capacity of fixed arches i s examined i n this thesis. The results are presented i n the form of dimensionless collapse 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 plotted as a function of slenderness. The ultimate capacity of a structure i s often determined by P l a s t i c Collapse analysis or E l a s t i c Buckling. P l a s t i c Collapse i s attained when su 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 analysis has been proven v a l i d for moment r e s i s t i n g frames subjected to large amounts of bending and whose second order effects are minimal. E l a s t i c buckling i s defined when a second order structure s t i f f n e s s matrix becomes singular or negative d e f i n i t e . Pure e l a s t i c buckling correctly predicts the ultimate load i f a l l components of the structure 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 clear that a reasonable ultimate load analysis must include both p l a s t i c hinge formation and second order effects i n order to evaluate a l l ranges of arch slenderness. - i i - A computer program available at the University of B r i t i s h Columbia accomplishes the task of combining second order analysis with p l a s t i c hinge formation. This ultimate load anaysis program, cal l e d "ULA", i s in 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 ultimate. A second order analysis i s continually performed on the structure. 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 altered to r e f l e c t this hinge formation, and a new structure 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 loss 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 different load cases were considered i n 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 i s a function 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 location or pattern of the loading to produce a minimum dimensionless load. Because of the multitude of parameters governing arch action i t was not possible 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 variations was given. Being three times redundant, a fixed arch p l a s t i c 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 s t i f f n e s s due to - i i i - the formation of fewer hinges than required for 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 fixed arches. F i n a l l y , the collapse curves were applied to three existing 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. - i v - TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i i i LIST OF FIGURES i x LIST OF SYMBOLS x i ACKNOWLEDGEMENTS x i i i CHAPTER 1 - INTRODUCTION 1 1.1 Basic Design Philosophies 1 1.2 Reserve Capacity 2 1.3 Application to Arches 2 1.4 Computer Program Theory and Underlying Assumptions 4 1.4.1 El a s t o - P l a s t i c 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 Analysis 8 1.4.4 Moment Ax i a l Interaction 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 Interactive Graphic Display 13 CHAPTER 2 - AN ECCENTRICALLY LOADED COLUMN 16 2.1 Governing Parameters 16 2.2 Comparison of An a l y t i c a l Equation with Correct Analysis and Experimental Results for a Pa r t i c u l a r Cross Section . 20 CHAPTER 3 - PRESENTATION AND DISCUSSION OF STANDARD ARCH BEHAVIOUR CURVES 25 3.1 Nonlinear Arch Behaviour 25 - v - TABLE OF CONTENTS (Continued) Page 3.1.1 Computer Model 25 3.1.2 Governing Parameters 27 3.1.3 The Standard Arch 28 3.1.4 Loading for Minimum Strength 30 3.1.5 Point Loading for Minimum Strength 30 3.1.6 Unbalanced Uniform Loading for Minimum Strength .. 40 3.2 Discussion of Hinge Formation Curves and Collapse Envelopes 41 3.2.1 Collapse Envelopes 41 3.2.2 Effect of L/r on Type of Collapse 42 3.2.3 Effect of Dead Load on Type of Collapse 42 3.2.4 E l a s t i c Buckling and the Limiting Slenderness Ratio 44 3.2.5 C r i t i c a l Loading Pattern, x/L Results 44 CHAPTER 4 - ANALYTICAL BOUNDS 49 4.1 A n a l y t i c a l Bounds for Low L/r 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 Interaction 55 4.1.2.1 P l a s t i c Collapse, Low L/r, Including A x i a l Interaction, 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 Interaction, U.D.L. Case 57 4.2 A n a l y t i c a l Bounds for High L/r 60 4.2.1 High L/r; Fu l l , Uniform Live Load E l a s t i c Buckling 60 4.2.2 High L/r; Point Load, One Hinge Buckling 61 - v i - TABLE OF CONTENTS (Continued) Page 4.2.3 A n a l y t i a l Solution for the Theoretical Slenderness Limit 62 4.3 Comparison of A n a l y t i c a l Bounds With Collapse Envelopes 62 CHAPTER 5 - VARIATION OF STANDARD PARAMETERS 66 5.1 Variation of E/o 66 y 5.2 Variation of f/L 67 5.3 Variation of y/r 77 5.4 Variation of Z/S 77 CHAPTER 6 - CONCLUSION 81 6.1 Hinge Locations and Formation Sequence 81 6.2 Typical Load Deflection Behaviour 82 6.3 Application of Load and Performance Factors 85 6.4 Application to Existing 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 - v i i - LIST OF TABLES Page Table I Hinge Formation Sequence, Point Loading 82 I I Hinge Formation Sequence, UDL Loading 82 - v i i i - 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 .... 7 4 Second Order El a s t o - P l a s t i c Response 9 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 Failure 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 Ecce n t r i c a l l y Loaded Column 16 10 Moment-Axial-Curvature Relation 22 11 Cooling Residual Stress Pattern Assumed by Galambos and Ketter 22 12 Effect of E l a s t o - P l a s t i c Assumption on Column Capacity 23 14 Effect of El a s t o - P l a s t i c Assumption on a Typical Load- Response 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 - i x - LIST OF FIGURES (Continued) Page Figure 25 Variation of Dimensionless Load Parameter with Load Location 39 27 Stubby Arches, No Second Order Amplification 46 28 Haunch Moments i n Slender Arches 47 29 Variations of (L/r) 43 trans 30 Free Body Diagrams of Point Loaded Arch 50 33 Four Hinge P l a s t i c Collapse Under Unbalanced U.D.L. Loading 53 34 F.B.D. of Right Side 54 35 Discrepancies Between An a l y t i c a l Solutions and Collapse Curves, Point Loading 64 36 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 Analysis 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 y, Point Loading, a=0.20 .... 69 39 S e n s i t i v i t y Analysis of E/o y, Uniform Loading, a=0.10 .. 70 40 S e n s i t i v i t y Analysis of E/o y, 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 Analysis 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 Analysis 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 Analysis of Z/S, Uniform Loading 79 47 Load-Response of a Typical Standard Arch 84 48 Application of Collapse Curves to La Conner and Capilano Bridges 91 49 Application of Collapse Curves to the Arvida Bridge .. 92 - x - LIST OF SYMBOLS A Cross sectional area a^ Intercept of facet i of a y i e l d surface with the p axis Intercept of facet i of a y i e l d surface with the m axis c Distance from the centroid of a symmetrical cross-section to the outer f i b r e E Young's Modulus e Ecce n t r i c i t y of applied load f Rise of an arch F^ Load cases F D,F 0,F p,F Load Vectors h^ Height to loca t i o n i on an arch I Moment of i n e r t i a of a cross-section K Stiffness matrix k. Ef f e c t i v e length factor L Span of an arch M Bending moment M P l a s t i c moment P M Y i e l d moment y m |M|M ' 1 P P Point load Point load of which event i occurs P A x i a l force required to cause f u l l cross sections y i e l d i n g p with no moment present p P/P P r Radius of gyration of a cross section - x i - LIST OF SYMBOLS (Continued) S E l a s t i c section 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. Uniformly distributed dead load d Unbalanced U.D.L. at which event i occurs w Ultimate U.D.L. u x Parameter in d i c a t i n g loading pattern y The distance from the center of gravity of a symmetrical cross section to the centre of gravity of either the upper or lower half Z P l a s t i c section modulus a Dead load r a t i o ot̂  Load factors applied to load cases F^ X Load l e v e l <J> Curvature (J)i m/b^ + p/a i 0 Yield stress - x i i - ACKNOWLEDGEMENT The author wishes to thank. Dr. Roy F. Hooley for his 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 efforts i n aiding Canadian research i s acknowledged. F i n a l l y the author would l i k e to extend his thanks to the UBC Computing Center for their provision of such excellent services. - x i i i - 1. CHAPTER 1 INTRODUCTION 1.1 Basic Design Philosophies The basic philosophy of st r u c t u r a l design has seen many changes. Allowable stress design has been very common and i s s t i l l used today i n many applications. In allowable stress 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 stress i s normally the y i e l d stress divided by some factor of safety, i . e . ; STRESSES DUE TO D.L. + L.L. < o /N (1.1) y where D.L. = dead load, L.L. = l i v e load, o = y i e l d stress, y N = factor of safety. Eq. (1.1) implies that both the dead load and l i v e load are subject to the same factor of safety. S t a t i s t i c a l studies of loads and materials have been used to develop a contemporary design philosophy. The object of this new design method, call e d Limit States Design, i s to ensure that the pr o b a b i l i t y of reaching a given l i m i t state, such as collapse or unserviceability i s below an acceptable value. To accomplish t h i s , the dead and l i v e loads must each have their own factors of safety, N, and N 2 simply because the dead load i s better defined than the l i v e load. The basic Limit State Design philosophy can now be summarized as follows: Nj D.L. + N 2 L.L. < fl) R 2. (1.2) where D.L. + N 2 L.L. ™ Effect of applied loads R = the resistance of a member, connection or structure, and o> = the capacity reduction factor accounting for material v a r i a t i o n : A s l i g h t change i n nomenclature accompanies the new design method such that Nĵ  and N 2 are now referred to as load factors. 1.2 Reserve Capacity I t i s common today to use e l a s t i c analysis to f i n d the response of the structure to the factored loads. If R i s taken as f i r s t y i e l d , there exists additional capacity beyond that load l e v e l . This w i l l be referred to as reserve capacity. Unless a structure i s exceedingly slender and f a i l s due to e l a s t i c buckling prior to reaching f i r s t y i e l d , the reserve capacity i s at least 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 structure f a i l s after the formation of one p l a s t i c hinge, therefore a more redundant structure would generally possess a higher reserve capacity. 1.3 Application to Arches With the preceeding discussion of Limit States Design and reserve capacity as motivation, this thesis w i l l examine the ultimate load, or collapse l i m i t state of fixed arched r i b s . This w i l l ultimately lead to 3. a better understanding of the reserve capacity of fixed arches as well as the factors on which i t depends. The key to the success of t h i s work i s a r e l i a b l e analysis technique which must include a l l prevalent types of behaviour. Conventionally, p l a s t i c analysis i s used i n determining collapse loads for 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 evalua- t i o n of the ultimate capacity 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 unsymmet- r i c a l l i v e loads, the ultimate strength may be governed by p l a s t i c collapse, e l a s t i c buckling, or by some intermediate form of i n s t a b i l i t y with less p l a s t i c hinges than required for a collapse mechanism. The reserve capacity of an arch i s therefore governed by non-linear behaviour. This non-linearity arises from p l a s t i c hinging and P-A second order e f f e c t s . A computer analysis combining both these factors i s outlined i n Section 1.2. In an age of increasing a c c e s s i b i l i t y to 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 to present his results 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 similar conventional techniques? Or, i s i t the researcher's r e s p o n s i b i l i t y to present the result of hours of computer analysis so as to inform and enlighten the reader and to give the reader conceptual ideas and guidelines, 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 to use the results for his or her own part i c u l a r and specialized 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 this work i s "ULA" (Ultimate Load A n a l y s i s ) 1 . I t i s a plane frame s t i f f n e s s program which combines second order analysis with p l a s t i c hinge formation. ULA i s an interactive program which allows the user to monitor the structure and to place p l a s t i c hinges when necessary as the load i s increased to ultimate. One of the requirements of l i m i t states design i s that the structure not f a i l when subjected to each of a number of load vector F Q where F 0 = a1¥1'+ a 2 F 2 + O3F3 + ... . = F D + F ( 1 . 3 ) FQ i s then a l i n e a r combination of load cases Fj_ augmented by the appropriate load factor ot£. In analyzing f o r ultimate load, the response of the structure at any load l e v e l X to the force vector F must be determined where F = F D + X F ( 1 . 4 ) The o r i g i n a l load vector FQ i s the sum of vectors and F. In performing the analysis to determine ultimate load, only F i s augmented by load vector X. This makes i t possible to maintain a constant dead load factor, 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 basic methods of e l a s t o - p l a s t i c analysis. The f i r s t 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 for different mechanisms and mechanism 5. combinations. The second method i s by load increments whereby the structure 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 at which point i t remains at M with free r e l a t i v e P P rotati o n of adjacent members. 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. This continues u n t i l a collapse mechanism i s obtained. The second method i s preferred because i t lends i t s e l f to computer simulation and i t makes the i n c l u s i o n of second order effects 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 increasing X for a single bay frame. I t i s important to note that each of the structures No. 0 through 4 are different and each i s v a l i d only for a s p e c i f i c range of X. Each structure has a di 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 actually place a hinge i n the structure at the appropriate load l e v e l , an additional slave j o i n t i s created at the hinge location which has the same tr 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 rotation. The load vector F i s then augmented by ±M^ between each master and slave pair, so the new load vector Is now F = F_ + XF + F , where F contains r ' D p' p only ± M p * This hinge placement i s depicted i n Fig. 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 single bay frame of F i g . 1 forms the polygonal shape i n Fig. 3 indicating the loss of s t i f f n e s s i n the structure as each hinge forms. In the method described above for hinge placement, a hinge can be placed at any load l e v e l . Because each of the structures of F i g . 1 i s unique, the response at load l e v e l X , for B example, can be obtained by a f i r s t order analysis of structure #2 from zero load l e v e l to X,, along a secant OB, and not along the facets of the a polygon. IOX 5X no plastic hinges Structure_#0 0 < X< X, IOX 5X A two plastic hinges 7777 Structure # 2 x2 < x < x3 IOX 5X 1 \ one plastic hinge M, M p three plastic a > M P i > M p , n 9 e s Structure^* I X, < X< X 2 5X1 1IOX 5X1 M P M p four plastic ^ M p ^ h i n g e s ^ M p Structure # 3 X 3 <X <"X4 Structure #4 Mechanism Fig. 1. E l a s t o - P l a s t i c Hinge Formation. <fcnox h e r e Joint L Tension Face before hinge placed Joint L Joint L+ I (Master) (Slove) after hinge ploced Fig. 2. Hinge Placement. 7. £VF=\F 0 only Mechanism max load = XFn A 0 Structure No.l 0 I I Response Fig. 3. F i r s t Order Elasto-Plastic Response, 1.4.2 Second Order Analysis Second order analysis requires the structure to be e l a s t i c and to be i n equilibrium 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. Details 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 including Gere and Weaver2. The s t a b i l i t y func- tions depend on the a x i a l forces, and the axi a l forces depend on the 8. deflected shape. I t Is therefore necessary to Iterate towards a solution several times at each load l e v e l . This Is nicely handled by the Inter- active format of the program because the analyst can view the determinant of the structure s t i f f n e s s matrix and use that as a c r i t e r i a for converg- ence. Normally, only a small number of cycles, 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 - sion of the second order e f f e c t s . Of course a few more cycles 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 ultimate load theories, p l a s t i c collapse 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 structure i s either especially 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 actual ultimate 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 analysis. I t i s apparent that i n order to establish the maximum load capacity, and hence an idea of the probability of reaching the ultimate l i m i t state, a combination of the two theories i s needed for many p r a c t i c a l structures. 1.4.3 Second Order E l a s t o - P l a s t i c Analysis An incremental approach i s a common method for combining second order and el a s t o - p l a s t i c analysis. The incremental forces and deflec- tions due to a small increment, dX, i n load l e v e l , X^ are obtained using a tangent s t i f f n e s s matrix. 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 to determine the necessity of placing a p l a s t i c hinge. The t o t a l response i s then the sum of a l l the incremental responses. However, errors arise because the tangent s t i f f n e s s matrix i s approximate, hinges may not be placed at M/M • 1, and round o f f occurs P due to a multitude of steps. These cumulative errors can be minimized by using a small dX. This, however, becomes more expensive and does not assure convergence. This incremental approach i6 not adopted for t h i s work. A simpler ultimate load analysis system i s used which should require less computing time and certainly avoids any cumulative errors. The system adopted i s a simple combination of second order analysis, and hinge placement. A second order elasto - p l a s t i c response curve shown In F i g . 4 i s s i m i l a r to 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 difference being the presence of arc segments ^ ' Linear onalysis Structure 0 . . Mechanism >- A 0 Structure No. I 0 I 2 h— 3-H F i g . 4. Second Order Ela s t o - P l a s t i c Response. 10. between hinge formations instead of l i n e a r facets. To determine the response at a certain load l e v e l , f i r s t order (linear) analysis i s simply replaced by second order analysis, for each of the four structures. To determine the response of the structure at load l e v e l Xg, for example, i t i s not necesasry to methodically increment the load l e v e l and follow the arced segments from 0 to B. As long as the location of the hinge i s known, i n this case at the base of the frame, then a l l that i s required i s a second order analysis with the structure #2 loaded with XgFQ and the p l a s t i c moments shown. The t o t a l response from zero load l e v e l to Xg i s along secant OB. Unlike the incremental approach, any errors due to placing a hinge when M/Mp ± 1 i s a l o c a l error and not cumulative, so that the response at higher load levels w i l l not be affected. 1.4.4 Moment A x i a l Interaction 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 for the cross-section being analyzed. The y i e l d surface can be defined by a series of straight l i n e s , and i s described to the program by the i n t e r - sections of the facets. By including only symmetrical sections, and hence the absolute value of the bending moment, only the top half 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 ariation 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 large 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. The quantity $ i i s defined as follows: $± = m/b± + p/a± 11. C A N 3 - S I 6 . I - M 7 8 1-0.19,0.95) OUTSIDE Y I E L D S U R F A C E , i.O^ ANY <£j> I (0.19,0.95) -1.0 C O M P R E S S I O N F i g . 5. Y i e l d Surface P 1.0 T E N S I O N , - £ - = p where m = IM|/M , p • P/P = P/Ao , and a J and bJ are the intercepts of P p y i i each facet with the p and m axis respectively. Now the convenience of an interactive format becomes apparent. At each load l e v e l , once the second order convergence i s obtained, a plot of the structure appears on the screen with 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^ for 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 th 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 at which the next hinge should form. This i s accomplished by extrapolating l i n e a r l y from two known points inside the yi 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 Fig. 5 i s straight and that p i s line a r with A. 12. 1.4.5 Moment Curvature A perfect e l a s t o - p l a s t i c behaviour i s assumed for the analysis of fixed arches. Fig. 6 shows an idealized moment curvature relationship. A loss of bending 6 t i f f n e s s i n any section 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, M̂ , as the cross-section becomes f u l l y p l a s t i c . This, as well as the effect of residual stresses are neglected i n t h i s work. Chapter 2 w i l l examine the consequences of these assumptions. MOMENT CURVATURE F i g . 6. Idealized E l a s t o - P l a s t i c Behaviour. The effect of neglecting s t i f f n e s s loss and residual stresses i s examined i n Chapter 2. 1.4.6 C r i t e r i a for Reaching Ultimate Load Ultimate load i s defined here as.the load l e v e l at 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 singular. This i s accomplished by monitoring the determinant of K. A zero determinate implies a singular and unsolvable matrix. A nega- t 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 equilibrium solution i s then possible, i t corresponds to unstable equilibrium and w i l l not be permitted. F i g . 7 shows a beam-column bent i n double curvature due to equal and opposite end e c c e n t r i c i t i e s . The load-deflection curve shows diagram- matically the collapse c r i t e r i a discussed above. The Choleski method used for the solution of the s t i f f n e s s equations i s only coded for real numbers. Because of t h i s , the routine stops when |K| • 0 and signals an unstable structure. P Fig. 7. Failure C r i t e r i a Applied to a Beam-Column i n Double Curvature. 1.4.7 Interactive Graphic Display Interactive graphic display helps the user i n making necessary decisions such as the number of P-delta convergence cycles, hinge place- ment, and selection 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 available on command at any given load l e v e l . Other displays are available which give the analyst enlightened appreciation of how the structure i s behaving. The f i r s t of these i s a display of the y i e l d 14 . surface, as shown In F i g . 5. Superimposed on the y i e l d surface Is a trace of the m and p coordinates for each member end from load l e v e l to load l e v e l . 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 display of the reserve capacity of each member. A self-explanatory example of this display i s shown i n F i g . 8. X = 0.80 I 1.0 F i g . 8. 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 forces, or bending moments. It i s also apparent where the next hinges should form, as the reserve capacities of the locations are approaching zero. Other features such as s t r a i n hardening and hinge closure are also available. 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 particular 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 . It i s this program that w i l l be used investigate the non-linear and ultimate behaviour of f i x e d arches. CHAPTER 2 16. 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 . They are both compression members subject to bending. 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 sta r t with the discussion 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 intention of this chapter to develop an a n a l y t i c a l solution for an e c c e n t r i c a l l y loaded column based on the 6ame assumptions to be used for the ultimate load of arches as outlined i n Section 1.4. This a n a l y t i c a l solution w i l l then be compared to an ex i s t i n g more exact solution and experimental r e s u l t s . An indication 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 ecce n t r i c a l l y loaded column chosen for comparison. Fig. 9. An Eccentrically Loaded Column. 17. As outlined i n Chapter 1, el a s t o - p l a s t i c material behaviour w i l l be assumed neglecting s t r a i n hardening and residual stresses. The elasto- 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 st i f f n e s s between M and M produces a P y P 6 t i f f e r structure and a non-conservative r e s u l t . The ultimate column capacity P^ i s a function of the following s i x parameters; P = f {e, L, EI, AE, P , M } (2.1) u P P where EI = linear e l a s t i c bending s t i f f n e s s AE = li 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 = Ao P y = maximum possible a x i a l load with no moment present and M = Zo P y = maximum possible 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 to 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 ratios are needed to describe the system as follows: 18. The awkward parameters of Eq. (2.2) are chosen because they simplify into the more fam i l i a r ratios shown below; where and so that and P /P = f{e/y, L/r, E/o , y/r} (2.3) u p y E = Young's modulus, A = cross-section (area) I = moment of i n e r t i a r = /I/A - /EI/AE = radius of gyration, y = the distance from the centre of gravity of the symmetrical section to the centre of gravity of either the upper or lower h a l f , o = y i e l d stress, AE/P = AE/Ao = E/o , p y y L L •EI/AE R 6 = e/y M /P P P M _ E = y / r P /EI/AE P The maximum moment of the eccentrically loaded column of Fig. 9 occurs at the midspan. According to Timoshenko3, 19. M - P(e + A) « Pe sec(kL/2) (2.4) max ' \ • y / where k 2 - P/EI 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. To develop an a n a l y t i c a l solution for the column capacity, we need an int e r a c t i o n equation. Facet 1 of Fig. 5 w i l l be used as i t i s v a l i d for |M/Mp| < 0.95. Eq. (2.5) describes this interaction; 0.85 M/M + P/P -=1 (2.5) P P where M • a Ay = P y P y P and P « o A P y Therefore: or where Eq. (2.6) becomes M = Pe sec ^ = 1.18 M (1 - P/P ) max 2 p p k£ (1 - P/P ) - 0.85 (P/P ) (e/y) sec (2.6) p p I kfc _ JP_ £ _ iP_ i_ m / p _ ^y_ £_ 2 EI 2 AE 2r P E 2r P 20. P /P - — — (2.7) u p 1 + 0.85 e/y s e c ( / | - ^ j r P Eq. (2.7) i s an a n a l y t i c a l expression for column capacity under eccentric loading based on the same assumptions that w i l l be used to analyse the ultimate capacity of arches. 2.2 Comparison of A n a l y t i c a l Equation with Correct Analysis and Experimental Results for a P a r t i c u l a r Cross-Section Galambos and Ketter 4* present dimensionless curves for 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 difference between the derivation 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 . Galambos and Ketter use a correct r e l a t i o n l i k e curve B of Fig. 6. In this thesis, 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, similar to curve A of F i g . 6. The method used by Galambos and Keffer i s based on numerically integrating values on a s p e c i f i c M-<|> curve and i t e r a t i n g towards a correct deflected shape. I n s t a b i l i t y arises when the i t e r a t i o n s do not converge. Because this 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 for every different cross-section, a closed form solution i s not available. I t i s now possible to compare the results of Eq. 2.7 with Galambos and Ketter for 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 for t h i s beam i s shown i n F i g . 10, based on an assumed residual stress pattern shown i n Fig. 11. Of course, the 21. reduction of M due to the presence of a x i a l i n the derivation of Eq. 2.7 P i s handled by incorporating the y i e l d surface of F i g . 5. Fig . 12 i s a dimensionless plot of an ultimate load parameter F^/?^ versus slenderness L/r against an ecce n t r i c i t y parameter e c / r 2 . The quantity c Is measured from the centroid of the symmetric cross section to the outer f i b r e . The results according to the assumptions of this t hesis, l a b e l l e d "ULA" are cl e a r l y non-conservative compared to the more a n a l y t i c a l l y correct results of Galambos and Ketter. The discrepancy i s Indicated by a shaded region and i s as much as ten percent. Experimental results have also been included i n the plot of Fi g . 12 and appear to be bounded by the two a n a l y t i c a l solutions. I t was necessary to make a sl i g h t modification to Eq. 2.7 i n order to plot 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 arithmetic to transform from one to the other once the cross-section properties are known. In t h i s case, e c / r 2 = c y / r 2 (e/y) = 0.85 e/y. Also, E/ay = 30,000/33 = 909. Therefore, Eq. 2.7 becomes: P P . JL = -° + i (2.8) p y 1 + 0.73 ec/r 2 sec (/P7P~ L/60.3) It i s Eq. 2.8 that i s actually plotted on Fig. 12 and labelled "ULA". An e c c e n t r i c a l l y loaded column i s a determinate structure which f a i l s after 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 to extrapolate the results and make some judgement on the effect of i d e a l i z i n g behaviour as elasto-plastic on 22. Fig. 11. Cooling Residual Stress Pattern Assumed by Galambos & Ketter. 23. Fi g . 12. Effect of Ela s t o - P l a s t i c Assumption on Column Capacity. the ultimate strength of fixed arches. A fixed arch i s three times redundant. Most p l a s t i c hinges formed p r i o r to collapse would already be i n the p l a s t i c region where the moment-curvature behaviour (Fig. 10) lev e l s out to: constant reduced only by the presence of a x i a l forces. Any errors during the formation of p l a s t i c hinges p r i o r to the l a s t hinge are l o c a l errors, not cumulative, and do not effect 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. This i s demonstrated q u a l i t a t i v e l y i n F i g . 14. I t i s therefore proposed that the 24. L O A D A L E V E L R E S P O N S E Fig. 14. Effect of Ela s t o - P l a s t i c Assumption on a Typical Load-Response Curve. effect 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 fix 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. It i s worth pointing out at 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 further eliminate any small non-conservatism. The basic case of an eccen 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 fixed arches. 25. CHAPTER 3 PRESENTATION AND DISCUSSION OF STANDARD ARCH BEHAVIOUR CURVES 3.1 Nonlinear Arch Behaviour I t i s the object of th i s chapter to present the nonlinear behaviour of fixed arches. Because of the multitude of parameters governing arch action I t w i l l not be possible to 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 to various parameter variations w i l l be investigated. 3.1.1 Computer Model Since ULA considers only straight 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 difference between this and a continuous curve would be less than 1%. If the r e a l arch r e a l l y has twenty straight segments then of course the error i n this model i s zero. I f , on the other hand, the real arch has say, four segments, then the error may be too large for 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 for dead load; a shape somewhere between a parabola and a catinary. The 21 nodes were placed on a parabola for this study together with 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 analysis and no attempt was made to factor i t out. Arches 26. constructed so that dead load moment due to r i b shortening i s minimized w i l l then have smaller moments than calculated with this model. In summary then, the model consisted of a twenty sided polygon with the nodes l y i n g on a parabola. n I X 1 1 1 1 1 I 1 1 1 1 1 1 N * 1 I I 1 1 1 1 1 1 1 W: M i l l L O A D I L O A D 2 J O I N T I Fig. 16. Arch Loading. J O I N T 21 Two load cases were considered to act on the model as shown i n F i g . 16. Load one consists of the dead load plus w a point load P located x d 1 from the l e f t end. Load two consists of a dead load w, plus a l i v e w. on d l a loaded length of x-Xj^. The distributed loads ŵ  and ŵ  were modelled as point loads at the polygon nodes i n order to eliminate l o c a l bending on the straight segments. The l i v e load ŵ  or ?^ was gradually increased In ULA with ŵ  held constant. The subscript i i s used to denote the load at which s p e c i f i c events occured as follows: 27. w or P Load at which y i e l d stress f i r s t occurred at some point on the e e r i b Wĵ  or Pj Load at which f i r s t hinge formed w2 or P 2 Load at which second hinge formed w3 or P 3 Load at which t h i r d hinge formed ŵ  or P^ Load at which fourth hinge formed w or P i s the ultimate load which may be any of the above loads as u 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 w1 or Pj 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 ŵ  and P^ i s the ultimate. Each of ŵ  and P^ was minimized by varying x (and x^). In general It was found that x^ was zero and x for minimum load varied with i . 3.1.2 Governing Parameters The load ŵ  or P^ i s a function of nine parameters as follows: w± (or P i) = f[ L , f, x, wd, EI, AE, P ?, Mp, My] where L = span f = r i s e w, = dead load d EI = bending s t i f f n e s s AE = a x i a l s t i f f n e s s 28. P » Ao m p l a s t i c a x i a l load with no moment action p y M = A o = za = p l a s t i c moment with no a x i a l load action p y y y M = So = moment at which y i e l d occurs with no a x i a l y y Since these ten parameters l i n k only the two dimensions of force and length, the Buckingham II theorem shows that only eight dimensionless parameters govern the system. The following eight are chosen for convenience: w.L2 P.L . _ w,L2 _ i _ ( o r JL.) = f r i i i k i £ I _ d _ M M LL' L' r' r' s' o' 8fP -P P y P The parameter Z/S w i l l only effect the f i r s t y i e l d condition and not hinge formation. The parameter w^L 2/8fP^ i s chosen to represent, 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 It i s c l e a r l y impractical to evaluate numerically the dimensionless load of Eq. (3.1) as a function of seven independent parameters. I t i s p r a c t i c a l though to 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 analysis to show their 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 indication 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 analysis for cases remote from this 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 follows: 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 i6 d e f i n i t e l y applicable to steel and close to concrete. Behaviour of other materials w i l l come from the s e n s i t i v i t y analysis. 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 analysis. A s o l i d rectangular section 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 factor Z/S varies 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 closer to a st 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 2 P iL L w dL 2 ~M~" ( ° r M"""' = f [ P SIP"] P P P for the standard arch. A study of existing arches shows that a = wX 2/8fP d p ranges from hear zero to approximately 0.2. I t was decided to produce curves of V 2 P i L L - | — (or = f(p for a - 0, 0.1, 0.2 P P 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 plotted i n Figs. 17 through 24. It should be noted that the parameter L/r involves the span length and not the c l a s s i c " e f fective" length kL. For a fixed arch, the effe c t i v e slenderness i s given by kL/r = 0.37 L/r. 3.1.4 Loading for Minimum Strength Influence l i n e s have been invaluable i n the li n e a r analysis of arches to determine the loading for maximum moment, thrust, stress, etc. They are of l i t t l e use though with nonlinear behaviour because superposition i s not applicable. For the case at hand i t i s necessary t numerically vary x^/L and x/L to produce a minimum dimensionless load. This method was necessary for a l l ŵ  and P since x/L and Xj/L depend upon i . 3.1.5 Point Loading for 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 function of x/L fo the standard arch with a given value of L/r. I t i s apparent from this behaviour that a single minimum exists for the ultimate load and a l l hinges formed after the f i r s t hinge. However, two l o c a l minima exist fo the f i r s t y i e l d and f i r s t hinge curves. These two minima arise because the f i r s t hinge may form at two different locations on the arch, each location corresponding to a diff e r e n t value of x/L. However, once the ARCH COLLAPSE ENVELOPE - PT. LOAD , -rf-=- = 0 22 2 0 18 16 14 It Mp 12 10 8 6 4 PLASTIC.NO INTERACTION FOUR HINGES THREE HINGES TWO HINGES ONE HINGE FIRST YIELD SYMBOL O • */ L 0.15 0.25 0. 30 STANDARD ARCH E/<Ty= 750 f/L =0.15 i/S = 1.15 y/r =0.95 100 2 0 0 5 0 0 6 0 0 3 0 0 4 0 0 L/r F i g . 17. Hinge Formation Curves and Collapse Envelope, Point Loading a - 0.0. 7 0 0 w L ARCH COLLAPSE ENVELOPE - PT. LOAD , -^-=0.10 0 100 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 L/r Fig. 19. Hinge Formation Curves and Collapse Envelope, Point Loading a - 0.20. Fig. 20. Hinge Formation Curves and Collapse Envelope, Uniform Loading a = 0.0. ARCH COLLAPSE ENVELOPE - U.D.L., ^ = - = 0.10 8f Pr ELASTIC BUCKLING k s 0.37 x / L = I .0 300 500 600 400 L/r Fig. 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 f / L =0.15 z/S = 1.15 y/r = 0.95 ELASTIC BUCKLING x/L= 1.0 , k s 0 .377 200 300 500 600 400 L/r Fig. 22. Hinge Formation Curves and Collapse Envelope, Uniform Loading a » 0.20. Fig. 23. Fixed Arch Collapse Envelopes, Point Loading. ARCH COLLAPSE CURVES - U.D.L. Fig. 24. Fixed Arch Collapse Envelopes, Uniform Loading. 0 0.10 0.20 0.30 0.40 x_ L F i g . 25. Variation of Dimensionless Load Parameter with Load Location. f i r s t hinge forms, there exists only one possible remaining location for each of the subsequent hinges, therefore, t h e i r behaviour yields a single minimum. The f i r s t hinge forms either at the l e f t haunch or at the location of the point load. If 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 location of the point load. If the f i r s t hinge forms at the location 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 right haunch and the fourth hinge forms at, or near the right 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 either one may govern depending on the slenderness. 3.1.6 Unbalanced Uniform Loading for Minimum Strength Two dimensionless parameters, Xj/L and x/L, are required to describe the location of the unbalanced uniform load. During preliminary analysis, i t quickly became evident that the value of x ^ L required to load for minimum strength was zero. This means that a uniformly di s t r i b u t e d load s t a r t i n g at the l e f t haunch and extending part way along the span w i l l minimize w^L2/M^. This loading was used for a l l ŵ  so that only x/L needed variation to produce a minimum. The behaviour of the load parameter w.L2/M as a function of x/L i s I P similar to that of the point load of Fig. 25. Two l o c a l minima exist for the f i r s t y i e l d and f i r s t hinge conditions, and one unique minimum exists for each of the subsequent hinges. As before, once the f i r s t hinge forms, the location 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 thi r d hinge forms near the right quarter point, and the fourth 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 to the formation of any of these hinges. As previously mentioned, the arch was discretized into twenty members. This means that the values of x/L for minimum strength for either the point loaded arch, or the uniformly loaded arch, could be 41. incorrect by as much as ±2 1/2%. This would res u l t i n only negligible errors i n the minimum load parameters. 3.2 Discussion of Hinge Formation Curves and Collapse Envelopes The standard arch behaviour curves of Figs. 17 to 24 are the funda- mental results of this thesis. The curves are bounded by a n a l y t i c a l solutions 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 discussion w i l l include a summary of arch behaviour by regions on the plots. 3.2.1 Collapse Envelopes The curve which defines the ultimate load as a function of L/r i s actually an envelope of the hinge formation curves, Figs. 17 to 23. Once the hinge formation curves are plotted, and the collapse enve- lope generated, the results can be summarized on a separate graph showing the collapse envelopes only. Two such plots are required; one for the point loaded arch, Fig. 23, and one for the uniformly loaded arch, F i g . 24. These graphs of P L/M or w L2/M versus L/r each show three u p u p collapse envelopes corresponding to a = 0.0, 0.10, and 0.20. As expected, there are f i v e different types of collapse; e l a s t i c buckling, and one, two, three, or four hinge collapse. The governing collapse mechanism for 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 collapse curves of Figs. 23 and 24 corresponding to the different mechanisms of collapse. 42. 3.2.2 Effect of L/r on Type of Collapse I t Is of no surprise by now that less hinges are required for collapse with increasing slenderness. On any given collapse 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 cusp i s actually the end of a hinge formation curve. 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 this point, the formation of a t h i r d hinge i s not possible because loss 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 th i r d hinge has a chance to form. The e f f e c t of L/r can be summarized by contrasting 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 collapse mechanism as dictated 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 effects are minimial. The opposite occurs at high L/r where second order effects are prevalent and f a i l u r e i s instigated by the loss 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 Effect of Dead Load on Type of Collapse Having discussed the effect of slenderness on the type of collapse, i t remains to discuss how and why the dead load r a t i o a influences the mode of collapse. The values of slenderness marking the t r a n s i t i o n between two different 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 ŵ . Any increase i n dead load would increase the dead load thrust and hence increase any second order effects. It i s therefore correct to conclude that the collapse curves corresponding to higher values of a are A 3 . influenced more by second order ef f e c t . Therefore, i t i s not surprising that and decrease with increasing a. The dead load also effects the value of (L/r) marking the loca- trans t i o n of a cusp. There must exist some values of L/r for which a lower value of o would permit an additional hinge to form due to 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 to show q u a l i t a t i v e l y the range of L/r for which two different 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 for trans higher values of a. L/r F i g . 29. Variation of (L/r) trans 44. 3.2.4 E l a s t i c Buckling and the Limiting Slenderness Ratio Examining Figs. 23 and 24, i t i s apparent that the collapse envelopes cross the horizontal axis where the l i v e load i s zero. At this point, 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. Theoretically, t h i s i s the maximum possible slenderness for a given dead load r a t i o a, and i s referred to as the slenderness l i m i t . Figs. 23 and 24 also show that the behaviour of arches just p r i o r to reaching t h i s slenderness l i m i t i6 different 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, Fig. 23, does e l a s t i c buckling govern the ultimate 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 entire span of the standard arch. The uniform load w required to cause e l a s t i c buckling was smaller than the half 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 close 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 impractical 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 fixed arch w i l l r arely, i f ever, govern design. 3.2.5 C r i t i c a l Loading Pattern, x/L Results Indicated on a l l the hinge formation curves Is the value of x/L which minimized the dimensionless load. These are shown by the use of symbols plotted s l i g h t l y above the actual data points for c l a r i t y . The results for the ultimate load for each loading condition are reasonably 45. consistent. In general, the non-linear behaviour dictates that loading 55 to 60 percent of the span governs for the ultimate capacity of a uniformly loaded arch, and placing the point load at x/L = 0.25 or 0.30 governs for 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 either load parameter i s plotted as a function of x/L only, two l o c a l minima a r i s e , each corresponding to different f i r s t hinge locations. However, i t remains to explain why one l o c a l minima governs for low L/r, and the other for higher L/r. Under uniform loading, the f i r s t hinge (and f i r s t yield) 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, to x/L • 0.6, corresponding to a hinge forming at the right haunch. To explain this phenomena, i t i s necessary to define a moment due to r i b shortening, M̂ ,̂ and a second order amplification factor (j). Two separate cases w i l l be examined, a stubby arch with L/r approaching zero and a very slender arch with 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 less 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 difference. Simple superposition says that the l e f t haunch moment with x/L m 0.4 and the right haunch moment with x/L = 0.6 are equal, however this excludes the effect of r i b shortening. I t i s important to note that M acts to increase the l e f t haunch moment, but v rs decrease the right haunch moment. This explains why the t o t a l moment at joi n t 1 with x/L = 0.4 i s the largest, 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 (M w+ M R S)I.O x/L=0.60 JOINT 21 (0.75MW- MRS)I.O l 1 l lw . 1 1 1 1 3 * . JOINT I (0.75M w + MRS)I.O (M w - MRS)I.O F i g . 27. Stubby Arches, No Second Order Amplification, $ = 1.0. form there at low L/r. Fi 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 respectively. The moment due to r i b shortening becomes i n s i g n i f i c a n t at large L/r because the r a t i o M /M varies rs w inversely with L/r. For large L/r, the second order effect now over- shadows any effe c t of r i b shortening. The maximum j o i n t 1 moment i s <J>, M . The maximum jo 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 +0)<£( x/L=0.60 JOINT I (4 (0.75Mw + 0)aS2 JOINT 21 (0.75Mw-0)<£ I I I w • I I I Fig* 28. Haunch Moments i n Slender Arches. 60% loaded than for a load over only 40% of the span. Therefore, $ 2 i s greater than $ 1 and the moment at j o i n t 21 with x/L = 0.6 i s the largest. For slender arches, the f i r s t hinge w i l l form at the right haunch, joi n t 21, with the span 60% loaded. A s i m i l a r phenomenon arises 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 with x/L - 0.30 at the location of the point load. Thus, the reason for this i s similar to the explanation given for a uniform loading and w i l l not be repeated. 48. In t h i s chapter, the main results of t h i s thesis were presented i n the form of hinge formation curves and collapse envelopes, Figs. 17 through 22. Conventional a n a l y t i c a l solutions for ultimate load are plotted on these figures as a n a l y t i c a l bounds to the results generated. I t remains to derive these bounds and to discuss any discrepancies between the collapse curves and the a n a l y t i c a l solutions. The following chapter w i l l accomplish t h i s . 49. CHAPTER 4 ANALYTICAL BOUNDS The fixed arch collapse curves were presented and discussed i n Chapter 3. A n a l y t i c a l bounds were also plotted to served as reference. It i s the aim of this 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 analysis and to compare these to the collapse 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 for a four hinge p l a s t i c collapse mechanism. The a n a l y t i c a l solution i s therefore based on conventional p l a s t i c analysis with no second order e f f e c t s . At high L/r, the point loaded arch i s bounded by one hinge collapse, 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 for Low L/r Two solutions w i l l be derived for each of the two loading cases. The f i r s t solution w i l l neglect the effect of any reduction of M due to P the presence of a x i a l force, and the second solution w i l l include this a x i a l reduction of M . In both cases, no second order magnification i s P considered. 4.1.1 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 . The point load i s placed at the l e f t quarter point. This i s a reasonable assumption and i s confirmed by the results of Chapter 3 which indicated 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 1 1 K V L w d 4 L/4 1 I I INd V R " w d X 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 ri g h t . The locations A, B and D of three of the four hinges are known to be at the haunches and at the point load. However, the location C of the fourth hinge must be estab- lished and i s represented by the unknown variable x^. The following equilibrium equations apply to the three free body diagrams of F i g . 30: F.B.D. #1, IV = 0 gives P + w L, u d (4.1) F.B.D. #1, IM = 0 gives 2M + V L - P L/4 - ( W jL)(L/2) = 0, p R u d (4.2) F.B.D. #2, ZM B = 0 gives (4.3) and F.B.D. #3, ZM = 0 gives c w*-2M + V Dx = Hh + w, p R c d 2 (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 solution to t h i s problem involves the s i x unknowns, H, V, , V„, x , P L R c u and h and only fi v e equations. The fi v e equations 4.1 to 4.5 relate the si x unknowns. Elimination of H, V , V and h gives Li R IT- = ^ + 3(x /L) A(x /L)'> <4'6> p c c ' It i s important to note that the dead load has no effect on the result for four hinge p l a s t i c collapse i f a x i a l reduction of and second order effects of neglected. This arises because i n the a n a l y t i c a l solution 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 effects and reduction of M . P It remains to determine the location of the fourth hinge by minimiz- ing P^L/Mp i n Eq. (4.6) with respect to * C/L. This can be accomplished by maximizing D where D = 3(x c/L) - 4 ( x c / L ) 2 . D i f f e r e n t i a t i n g and setting dD/dx equal to zero gives x c/L = .318 for minimum collapse load. This minimum collapse load i s then P L ^ - = 22 j = 22.22 (4.7) P This means that the ultimate point load parameter i s constant i f a x i a l reduction of M and second order effects are neglected. Eq. (4.7) P i s plotted on Figs. 17, 18 and 19 as a straight l i n e l a b e l l e d , " P l a s t i c , No Interaction". This result i s grossly non-conservative because a x i a l i n t e r a c t i o n to reduce M i s prevalent at low L/r, and second order P effects are not ne g l i g i b l e , especially at intermediate and high L/r. A similar a n a l y t i c a l solution for low L/r and neglecting reduced p l a s t i c moment must now be derived for 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 locations are known. Fi g . 33 i s a free body diagram of a parabolic arch loaded by an unbalanced uniform load. For the purposes of th i s analysis, the dead load i s not considered because, as we have just seen, i t i s of no consequence I f a x i a l i n teraction and second order effects are neglected. The method of solution i s exactly analogous to the point load case. Moment and force equilibrium arch yields expressions for V and V . As we l l , moment equilibrium of a free body diagram from A to B w i l l result In an expression for horizontal thrust H, just as f o r the point load case. These three reactions are as follows: 2/2 M wx L and H - f - |wxb(i - |r-) + 2M £ - 2M -?r2} h. 1 2L p L p 2 1 (4.8) r 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 free body from C to D i n Fig. 34 w i l l y i e l d the f i n a l equation as 2M^ + V Rc - H(h c). Substituting for the known reactions H and V gives the following: K 2M (4.9) i n order to simplify, l e t h = _c / f 2 j . i f -4 p c 2 + 4 j c »b _ 4 l 2 b 2 + 4 l b r-^r>2 Eq. 4.9 now i s a function of the hinge locations b/L and c/L as shown below. w L' u M 2(c/L + y b/L - y - 1) a 2c ab ... a N — - - Y — (1 ) 2L : 2L Ybf. 2L 2 (4.10) P H F i g . 34. F.B.D. of Right Side. 55. It now remains to minimize the ultimate load parameter with respect to hinge locations and loaded length. To accomplish t h i s , a simple computer program was written which evaluted wuL2/M^ for various combinations of b/L, c/L and x/L, to determine the minimum. The results were as follows; b/L = 0.30, c/L = 0.30, x/L • 0.50 and w L^ u M - 93.33 (4.11) Eq. 4.9 i s the result of a four hinge p l a s t i c collapse analysis neglecting a x i a l i n teraction and second order e f f e c t s . I t i s plotted as a horizontal l i n e i n Figs. 20, 21 and 22 and i s evidently grossly non-conservative. I t i s worth noting that for both loading cases, the ultimate load ratios are independent of f/L, E/o^ and y/r and a. 4.1.2 Low L/r, Including A x i a l Interaction 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 solution to obtain a more reasonable bound at low L/r. To make the arch behaviour amendable to a closed form solution, 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 entire span of the arch i s assumed constant and equal to 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 int 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 yi e l d surface of Fig. 5. 56. This i s the same y i e l d surface used for the non-linear analysis 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 Interaction, i n Point Load Case Now, for the point load case, Eq. (4.7) must be rewritten as PuL/M - 22.22 (4.12) where M i s the reduced p l a s t i c moment due to a x i a l P. For the same reason, Eq. 4.3 i s sim p l i f i e d and rewritten as P L w,L2 - H " J T + 8f f p L O M U , „ ZM / / I O N 4 T + a P p T ( 4' 1 3 ) The y i e l d surface i s represented by the following two equations: P/P + 0.85 M/M = 1.0 for M/M < 0.95 (4.14) P P P and 0.26 P/P + M/M = 1.0 for M/M > 0.95 (4.15) P P P Combining Eqs. 4.12 and 4.13 with 4.14 and then 4.15 gives lit (L/r)(1.0-ci) _ f o r g / M > 0 > 9 5 ( A < 1 6 ) M p 0.16 + .0385 L/r 57. and V (L/r)(1.0-a) f o r - / M < 0 < 9 5 ( 4 > 1 7 ) M p 0.0416 {|70 + .045(L/r) Substituting the standard arch values of y/r = 0.95 and f/L = 0.15 into 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 easily shown that Eq. 4.18 governs for L/r < 111 and Eq. 4.19 governs for L/r > 111. These two equations are plotted on Figs. 17, 18 and 19 and labelled as " P l a s t i c , B i l i n e a r Interaction". As expected t h i s curve Is vastly different from the " P l a s t i c , No Interaction" curve for low L/r. This i s because the hinges do not form at a moment M̂ , 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 ?UL/M as L/r approaches zero i s 22.22. 4.1.2.2 P l a s t i c Collapse, Low L/r, Including A x i a l Interaction, U.D.L. Case Having derived expressions for four hinge p l a s t i c collapse includin a x i a l interaction for a point loaded arch, i t remains to repeat this derivation for an arch loaded by unbalanced U.D.L. Eq. 4.11 must be rewritten as follows: w L2/M = 93.33 u 58. (4.20) Substituting x/L - 0.5, b/L - 0.3, c/L « 0.3 into Eq. (4.8) and adding the dead load thrust, the expression for a x i a l force i n the arch becomes: 8.036xl0" 2 w L c M P = H = I L - - 5 _ p _ + a P f/L 3 L(f/L) (4.21) Combining Eqs. (4.20) and (4.21) with each of the interaction Eqs. (4.14) and (4.15) gives and w L< u M w L' M (L/r)(1.0-ct) 0.0625 + 9.1071xl0 _ 3(L/r) (L/r)(1.0-a) 0.01625 + 1.07l4xl0 _ 2(L/r) for M/M < 0.95 (4.22) P for M/M > 0.95 (4.23) P Substituting the standard values of f/L = 0.15 and y/r = 0.15 results i n the following two equations: w Ld u (L/r)(1.0-a) M p 0.396 + 9.11xl0" 3(L/r) and w L z (L/r)(1.0-ct) M p ' 0.103 + 1.071xl0 - 2(L/r) for M/M < 0.95 (4.24) P for M/M > 0.95 (4.25) P Equating (4.24) and (4.25) indicates that for L/r < 182 Eq. (4.24) w i l l govern, and for L/r > 182 Eq. (4.25) w i l l govern. This result i s plotted 59. on Figs. 20, 21 and 22 and la b e l l e d , " P l a s t i c , B i l i n e a r Interaction". 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 collapse. At very low L/r, the governing f a i l u r e mechanism could be f u l l cross-section 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 solution for th 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 as follows: P y w L 2 w .L 2 F s » = -*- + -tr w L 2 U + a P = P 8f " *p p Substituting M /P r = 0.95 yields: P P w L 2 U (.95r) = 1.0 -a 8fM P Simplifying and substituting f/L = 0.15 results i n the following: w L 2  u M P 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 for L/r < 43. Eq. (4.26) i s plotted on Figs. 20, 21 and 22 and labelled "Axial Y i e l d " . 60. 4.2 A n a l y t i c a l Bounds for High L/r As the slenderness, L/r, approaches the theoretical 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 for each of the slender collapses mentioned. A x i a l reduction of M i s not s i g n i f i c a n t at high P L/r and i s therefore not included In this a n a l y t i c a l derivation. 4.2.1 High L/r, F u l l Uniform Live Load E l a s t i c Buckling An expression for the e l a s t i c buckling load parameter as a function of L/r can be derived by equating f u l l l i v e load and dead load thrust to the Euler 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 horizontal thrust so that V l + V l w?EI 8 f 8 f = ( k L ) 2 or _ , W u L 2 ir 2EAr 2 P 8 f ( k L ) 2 where kL i s the effective length of a fixed arch. Including the ide n t i t y y/r = M /P r and simplifying gives P P w L 2 u 8f/L r i r 2 E 1 y i \ - — = —.— I a L/r J (4.27) M y/r v „ 2 oy L/r 1 y/r KYz ay The a n a l y t i c a l solution requires a value of the effective length factor k. This was obtained by examining the resu l t of a standard arch ULA analysis at L/r = 700 for 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 res u l t . Substituting standard arch values of f/L = 0.15, E/oy = 750 and y/r = 0.95 as w e l l as k = 0.377 gives a f i n a l r e s u l t : 61. w L 2 u M P 1-26 a (L/r) (4.28) Eq. (4.28) describes the uniform ultimate load parameter for e l a s t i c buckling as a function of the dead load r a t i o a and slenderness L/r. This i s plotted on Figs. 20, 21 and 22 under the label " E l a s t i c Buckling, k = 0.377". 4.2.2 High L/r, Point Load, One Hinge Buckling An expression for 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 line a r f i r s t order moment PL/17 with P second order magnification. f K - H / H > ° M p ( 4 ' 2 9 ) cr * PL/17 was determined by evaluating the maximum moment from linear f i r s t order s t i f f n e s s analysis. For large L/r, v i r t u a l l y a l l the thrust comes from the dead load. w dL 2 Therefore, i t i s assumed that H = ac. = a P . Now, Eq. (4.29) becomes: * 8f p M p ~ 17 ( otP } 1 - — 2 - ( k L ) 2 1 T 2EI Setting P = P^ and rearranging: p L ° M J L . m 1 7 ( 1 - 2 - J L (iSk)2) M *• 2 E r ' P Substituting k = 0.37 and E/oy = 750 yields the f i n a l result: 62 . P L 17(1 - 1.920xlCT5 a ( L / r ) 2 ) (4.30) P Eq. (4.30) i s an an a l y t i c a l solution for the point load r a t i o required for one hinge i n s t a b i l i t y as a function of the dead load r a t i o a and the slenderness L/r. I t i s plotted as "One Hinge A n a l y t i c a l , k = 0.377" on Figs. 17, 18 and 19. 4.2.3 A n a l y t i c a l Solution for the Theoretical Slenderness Limit An expression for the theoretical slenderness l i m i t can be obtained by solving either Eq. (4.27) or Eq. (4.29) for L/r when w uL 2/M p or P,L/Mr o are zero. If P L/M = 17(1 - % -2- ( — ) 2 ) = 0 u p T T z E r ' J u then ( L / r ) o . I /iFjL (4.31) where ( L / r ) 0 i s the slenderness l i m i t . Substituting K = 0.377, E/o^ 750 and a = 0.1 then 0.2 indicates that ( L / r ) Q = 720 for a = 0.1 and ( L / r ) n = 510 for a = 0.2 4.3 Comparison of A n a l y t i c a l Bounds With Collapse Envelopes There ex i s t s a discrepancy between the a n a l y t i c a l bounds derived i n this chapter and the collapse curves generated by non-linear ultimate load analysis. These ar i s e due to the inadequacies of the conventional a n a l y t i c a l solutions. The graphical explanations for the discrepancies 63. are presented by slenderness regions i n Figs. 35 and 36. The plots of a =0.1 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 Fig. 35. Discrepancies Between Analytical 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 8 0 0 Fig. 36. Discrepancies Between Analytical Solutions and Collapse Curves, U.D.L. Loading. CHAPTER 5 66. VARIATION OF STANDARD PARAMETERS The non-linear behaviour of standard fixed arches are summarized by hinge formation curves and collapse envelopes i n Chapter 3. To make th 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 indicative of a t y p i c a l steel box girder or wide flange arch. I t i s the purpose of t h i s chapter to vary these four standard parameters and examine the effect on the non-linear performance of fix e d arches. This should f a c i l i t a t e the extrapolation of the results of this thesis to include actual arches whose parameters w i l l c e r t a i n l y deviate from the standard values. In the following sections only one parameter at a time i s altered; a l l others are kept at the standard value. 5.1 Variation of E/o y The dimensionless parameter E/o^ i s a material property and not a function of arch- geometry or cross-section. I t ranges t y p i c a l l y from approximately 375 or 400 for aluminum to about 900 for reinforced concrete. Eqs. 4.26 and 4.29 serve as a n a l y t i c a l bounds for behaviour at long L/r for uniform loading and point loading respectively. The second order reduction terms are [1 - (a/ir 2)(o /E(kL/r)] for point loading and it 2 E 1 [— — -j-yY - aL/r] for uniform loading. I t i s apparent from these terms that second order effects are proportional to E/o . Thus, a reduction of 67. E/o from the standard value of 750 w i l l increase any second order y e f f e c t s and therefore decrease the capacity of the arch. This has been confirmed by computer analysis 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 for E/o^, i t i s obvious that the effect of reducing E/o^ becomes less pronounced with decreasing L/r. This i s because any second order effects are propor- t i o n a l to ( L / r ) 2 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 ultimate 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 collapse 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 i s proportional to the square root of E/o^. This supports the reduction of the slender- ness l i m i t due to the halving of E/o^ indicated by Figs. 37 through 40. 5.2 Variation 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 to 0.30 for bridge arch ribs i n s t e e l , concrete or aluminum. The standard arch has an assumed value of 0.15. At low L/r, four hinge p l a s t i c collapse i s described by Eqs. 4.16 and 4.17 for point loading and Eqs. 4.22 and 4.23 for unbalanced uniform loading. The quantity f/L appears i n both these a n a l y t i c a l solutions. It i s evident from these equations that Increasing only f/L results i n an increase i n the ultimate load r a t i o s P L/M or w L2/M . Also, the effect u p u p of varying f/L diminishes with increasing 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 STANDARD PARAMETER E/tr y P ARCH COLLAPSE ENVELOPE - PT. LOAD , 37^" = 0 - 2 0 VARIATION OF STANDARD PARAMETER E/cry p 6 0 0 A R C H C O L L A P S E E N V E L O P E - U . D . L . , ^ - ^ - = 0.10 VARIATION OF STANDARD PARAMETER E/cr y 8 f P P PLASTIC.NO INTERACTION 0 100 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 L/r Fig. 39. Se n s i t i v i t y Analysis of E/o , Uniform Loading, o » 0.10 100 r - ARCH COLLAPSE ENVELOPE - U.D.L., ^ - ^ - = 0.20 VARIATION OF STANDARD PARAMETER E/crw 8 f P l P L A S T I C , N O I N T E R A C T I O N F O U R H I N G E S \ T H R E E H I N G E S . T W O H I N G E S \ O N E H I N G E F I R S T Y I E L D S T A N D A R D A R C H • E / i r y * 750 f / L « O . I S l / S • 1.19 y/r »0.93 \ \ \ - E L A S T I C B U C K L I N G * / L = 1.0 , k = 0 . 3 7 7 E / o y = 37S 6 0 0 Fig. 40. Sen 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 to u p 0.25 i s only 9%. This behaviour i s confirmed by ULA computer analysis for 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 Figs. 41 through 44. I t i s interesting to note that at high L/r, an increase i n f/L actually causes a small decrease i n the ultimate load parameters. At f i r s t , t h i s may appear as an anomaly when compared with Eq. 4.27 describing the a n a l y t i c a l bounds for uniform loading at high L/r because wuL2/Mp appears to be l i n e a r l y proportional to f/L. However, the value of K i s assumed In the derivation of Eq. 4.27 as a f r a c t i o n of the span length L when i n fact i t i s more correctly interpreted as a fraction of the arc length 1. Eq. 4.27 can be rewritten i n the form of Eq. 5.1 using kL as the effective length w L 2 M 8(f/L)(L/r) , I T 2 E = i o (— rr- o) (5.1) p (y/r) ( k L / r ) 2 °y Realizing that an increase i n f/L causes the effective length kL to increase due to a larger arc length i t i s evident that increasing f/L at large slenderness can act to reduce the ultimate load parameter. A s i m i l a r argument holds true for the point loading case. As a f i n a l comment before leaving the discussion of variation of f/L, a p r a c t i c a l note i s now made. The r a t i o f/L was changed from 0.15 to 0.25 i n the computer analysis by increasing the r i s e f by that r a t i o 5/3. For a v a l i d comparison, a l l other dimensionless r a t i o s must be unchanged. This meant that for the dead load r a t i o a •* w,L2/8fP , the d p dead load ŵ  has to be increased by 5/3 to maintain a = 0.1 or 0.2. This ARCH COLLAPSE ENVELOPE - PT. LOAD , -^-=0.10 VARIATION OF STANDARD P A R A M E T E R f / L p L/r Fig. 41. Sen 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 p Fig. 42. Sensi t i v i t y Analysis of f/L, Point Loading, ct= 0.20. 2 ARCH COLLAPSE ENVELOPE - U.D.L., ^ - ^ - = 0.10 VARIATION OF STANDARD PARAMETER f / L 8 f P P PLASTIC.NO INTERACTION Fig. 43. Sens 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 100r- VARIATION OF STANDARD PARAMETER f / L 8 f P P P L A S T I C . N O I N T E R A C T I O N 90 - 80 - Fig. 44. Sensi t i v i t y Analysis of f/L, Uniform Loading, a = 0.20. 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 6 o l i d 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 in 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 wJL2/Mp by 4.1 percent, and P^L/M^ by 0.8 percent. ULA computer analysis confirms the insignificance of the variation of the quantity y/r. It 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 ratio of the plastic section modulus to the elastic section modulus, Z/S is also a ratio of the plastic moment of a cross-section to i t s yield moment, M /M and i s often referred to as a shape factor. It 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 , -^-=0.10 VARIATION OF STANDARD PARAMETER z/S 0 t P P L/r Fig. 45. Sensi t i v i t y Analysis of Z/S, Point Loading, a » 0.10. 2 ARCH COLLAPSE ENVELOPE - U.D.L., ^ - ^ - = 0.10 VARIATION OF STANDARD 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 Fig. A6. Sen s i t i v i t y Analysis of Z/S, Uniform Loading, a - 0.10. 80. corresponding to the approximate shape factor f o r box and wide flange section. As pointed out i n Chapter 3, the shape factor effects only the f i r s t y i e l d condition and not hinge formation or ultimate load. In the l i m i t - ing 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 factor, 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 increasing Z/S would decrease P L/M or w L2/M . This i s e p e p easily confirmed by second order e l a s t i c computer analysis, the results of which are superimposed on standard curves f o r a = 0.10 i n Figs. 45 and 46. 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 ratios E/o^, f/L, Z/S and y/r, an indication of the s e n s i t i v i t y of the load parameters to these ratios was obtained. I t i s concluded that the standard arch hinge formation curves and collapse envelopes are reasonable for any values of y/r and Z/S and for values of f/L i n the range from 0.10 to 0.30. However, as shown cl e a r l y i n Figs. 37 through 40, the standard arch curves are s i g n i f i - cantly sensitive to v a r i a t i o n i n the material parameter E/o . This 3 y cannot be overlooked when applying the arch behaviour curves. CHAPTER 6 81. CONCLUSION 6.1 Hinge Locations and Formation Sequence By considering both second order effects and member p l a s t i c i t y the behaviour of standard fixed arches loaded to ultimate has been summarized using hinge formation curves and collapse envelopes. A v a r i a t i o n of parameters which defined the standard arch was carried out to examine the s e n s i t i v i t y of the response to these parameters. Throughout this work, I t became clear 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 buckling e l a s t i c a l l y . The location of the p l a s t i c hinges and the sequence of formation have yet to be discussed completely. These results are summarized for the di f f e r e n t collapse mechanisms i n Tables 1 and 2 for point loading and uniform load- ing respectively. Each row i n the tables describes a different collapse mechanism. The numbers i n the body of the table indicate which hinge, i f any, formed at a certain location on the arch. For example, three hinge i n s t a b i l i t y under uniform loading occurs with the f i r s t hinge forming at the right 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 right quarter point. The loading, defined by x/L i s for minimum ultimate strength and i s indicated i n the collapse envelopes of Figs. 17 to 22. 82. TABLE I. Hinge Formation Sequence, Point Loading Point Load Left Haunch Near Left 1/4 Point Near Right 1/4 Point Right Haunch 4 Hinge Collapse 2 1 4 3 3 Hinge Collapse 2 1 3 2 Hinge Collapse (2)* 1 (2)* 1 Hinge Collapse 1 *Second hinge may form at either haunch, depending on L/r and a. TABLE I I . Hinge Formation Sequence, UDL Loading UDL Load Left Haunch Near Left 1/4 Point Near Right 1/4 Point Right Haunch 4 Hinge Collapse 1 4 3 2 3 Hinge Collapse 2 3 1 2 Hinge Collapse 2 1 1 Hinge Collapse 1 6.2 Typical Load Deflection Behaviour It i s common to monitor the behaviour or response of a structure due to increasing load l e v e l to compare experimental results with a n a l y t i c a l work. Unfortunately, no experimental results are available, 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 load- response basis, i n the belief that the second order e l a s t o - p l a s t i c analysis used i n t h i s work, closely models actual behaviour. Figs. 3 and 4 of Chapter 1 contrast f i r s t and second order elasto- p l a s t i c response of a hypothetical single bay frame. Such a comparison w i l l now be applied to a t y p i c a l fixed arch. The l i v e load applied i s uniformly distributed over six-tenths of the span and the dead load i s of course applied to the entire span. The response i s the maximum arch defle c t i o n . The arch chosen to evaluate load-deflection 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 ind i c a t i v e of slender arched ribs of highway bridges. Several load deflection curves are plotted on F i g . 47 for the above mentioned arch. These generated curves contrast the second order elasto- p l a s t i c "ULA" response with f i r s t order elasto p l a s t i c behaviour, with and without moment a x i a l interaction. Because an assumed dead load was Included i n the analysis, the load deflection curves do not st a r t at the o r i g i n . The deflection corresponding to u)L2/Mp = 0.0 i s the dead load defle c t i o n . 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 neglect- ing second order effects i n determining a collapse mechanism. This i s best summarized by noticing 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 analysis, the arch has actually either formed, or i s very near, a three hinge collapse mechanism. Any discrepancies between the di f f e r e n t load deflection curves would be even more pronounced i f the dead load parameter a were greater than 0 1 2 3 4 5 6 7 8 9 DIMENSIONLESS TOTAL DEFLECTION 1000 S/L 00 Fig. 47. Load-Response of a Typical Standard Arch. 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 Application of Load and Performance Factors There has been l i t t l e discussion thus far on the application of load factors and performance factors as dictated by Limit States Design. The collapse curves and hinge formation curves have a l l been based on a computer analysis. 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 material properties, are appropriately factored. Thus, before entering the curves, a l l factors must f i r s t be applied when calculating the required dimensionless parameters, then the l i v e load P^ and ŵ  obtained from the curves are factored loads. This ensures complete f l e x i b i l i t y because any factors may be used. For example the dead load w must be interpreted as aw where a i s the dead load factor and ŵ g i s the specified load. S i m i l a r l y , the p l a s t i c moment M indicated as part of several dimensionless r a t i o s must actually P be calculated as 4>ZOy. Of course, i t w i l l almost certainly be necessary to interpolate between curves with d i f f e r e n t a ra t i o s to obtain meaningful values of P^ and ŵ . The following section w i l l deal with application of the dimensional analysis to existing arches where load and performance factors must be applied. 6.4 Application to Ex i s t i n g Arches A very common use of the arch as a st r u c t u r a l form i s for highway bridge r i b s . A span which i s too long for a truss, and yet not long enough to warrant a suspension or cable stayed structure, i s commonly 86. bridged by two or more arch r i b s . If the foundation conditions are stable enough, a fixed 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 to support l i v e loads greater than the o r i g i n a l design loads. Most existing 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 stress approach. Thus properly evaluating an existing bridge, as well as designing a new bridge by Limit 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 structure 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 factored loads cause a response i n t h i s region, then the structure may be deemed safe from a strength point of view. The t y p i c a l hinge formation curves and collapse envelopes of Figs. 21 to 27 w i l l now be applied to the fixed arches of three existing 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 Arvida Bridge i n Arvida, Quebec. The arched r i b s of these bridges are made of structural s t e e l , reinforced concrete and aluminum respectively. As a result of a l l three arches having long spans, the designs were governed by lane loading as opposed to truck loading. The arch collapse curves for unbalanced uniform loading w i l l be used. A point load was required i n addition to the uniform lane loading for the La Conner and Capilano bridges. The analysis i n t h i s work did not Include t h i s additional point load, however both loaded lengths are quite long and any error due to the omission of the point load should not be serious. The o r i g i n a l design loads are used along with a Limit States Design dead load factor of 1.3 and performance factor of o) = 0.90 applied to 87. reduce P and M . For each bridge r i b the value of wL2/M i s plotted on P P P the appropriate collapse curves. The value of w i s the unfactored uniform design load per r i b . This includes an impact factor and sidewalk pedestrian loading. A reduction i n gross area due to any r i v e t or bolt holes was considered i n calculating M̂ . When calculating r, f u l l cross- sectional area was used. The load case examined here does not include such things as temperature, wind or earthquake and i s therefore by no means a complete analysis, however a very good conceptual idea of the load factor required to cause f i r s t y i e l d and the load factor required to cause collapse i s indicated. In the analysis used for this work, a constant cross-section 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 cross-section i s commonly used to increase the moment resistance at the haunch where f i r s t y i e l d normally occurs. This results i n a var i a t i o n i n M and r. Thus, the key dimensionless ratios wL2/M P P and L/r w i l l not have one single value each, but a range of values. The re s u l t i n g plots on Figs. 48 and 49 w i l l therefore consist of a service load l e v e l region as opposed to a single point for each bridge examined. 6.4.1 The La Conner Bridge The La Conner Bridge, also known as the Swinomish Chanel Bridge, i s located at La Conner, Washington. This fixed steel box arch spans 167.6 metres (550 f e e t ) . I t was designed by H.R. Powell and Associates of Seattle, 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 = 0.16 wL2/M 7.37 to 8.89 P and L/r = 185 to 191. The corresponding service 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 collapse curves for both o = 0.10 and o = 0.20 are shown on Fig. 48 so that an interpolation between the two curves can be made by the reader. The actual 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. I t includes two reinforced concrete arch ribs 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 Ltd. for 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 describing 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 Is symmetrically reinforced r e s u l t i n g i n as much compression s t e e l as tension steel 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 noticeable v a r i a t i o n i n cross-section i s apparent i n a reinforced concrete arch than a metal arch, thus the service load region plots larger. 89. Data from the design drawings give: f/L = 0.168 E/o - 900 y a = 0.14 L/r = 115 to 181 and wL2/M = 14.6 to 20.2 P The design load region for the Capilano Canyon bridge i s plotted on Fig. 48. The curves shown i n F i g . 48 are conservative when applied to the concrete Capilano arch because they correspond to E/o^ = 750, when i n fact E/o = 900 for concrete. y 6.4.3 The Arvida Bridge The f i r s t aluminum highway bridge on the American continent was b u i l t i n Arvida, Quebec, i n 1950. This, the Arvida Bridge, has a main span which i s a fixed arch 88.4 meter (290 f t . ) center to center of skewbacks, spanning the Saguenay River. The following dimensionless parameters were calculated 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 L/r = 151 to 156 a = 0.11 y/r =0.93 and wL2/M = 8.40 to 9.84 . 90. The available 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 incorrect, however the resulting non-conservatism i s not serious at the low L/r corresponding to the Arvida arch. 6.4.4 Further Research I t would be interesting to compare the theoretical solutions presented herein with an experimental study on model arches. As w e l l , the results herein are centered around a moment a x i a l interaction curve for a material such as s t e e l . Some investigation should be made using the somewhat unique interaction curve for reinforced concrete. F i n a l l y , the current results could possibly be si m p l i f i e d into 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 APPLICATION TO EXISTING ARCHES PLASTIC.NO INTERACTION 8 f P„ ELASTIC BUCKLING k =0.37 x/L =1.0 800 Fig. 48. Application of Collapse Curves to La Conner and Capilano Bridges. 100 90 Fig. 49. Application of Collapse Curves to the Arvida Bridge. 93. REFERENCES Hooley, R.F. and Mulcahy, F.X., 1982. Nonlinear Analysis by Interative 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 . 1st ed. McGraw-Hill, New York, pp. 36-38. Galambos, T.V. and Ketter, R.L., 1961. Columns Under Combined Bending and Thrust. American Society of C i v i l Engineers, Transactions, Vol. 126(1). Plmenoff, C.J., 1949. The Arvida Bridge. E.I.C. Journal, Vol. 32, No. 4.

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