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Ultimate load analysis of fixed arches Mill, Andrew John 1985

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

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