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UBC Theses and Dissertations

Slender concrete column analysis using direct methods Oldridge, Dennis Philip 1981

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SLENDER CONCRETE COLUMN ANALYSIS USING DIRECT METHODS by DENNIS PHILIP OLDRIDGE •A.Sc. (1975), The University of B r i t i s h Columbia A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE DEPARTMENT OF CIVIL ENGINEERING We accept t h i s t h e s i s as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA jrs June, 1981 In p r e s e n t i n g 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 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 o f B r i t i s h Columbia, I agree t h a t 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 r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s 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 p e r m i s s i o n . Department o f 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 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 D a t e J u n e 2 6 ' 1 9 8 1 /"7Q \ Abstract Current design formulas for slender concrete columns are inaccurate when f a i l u r e i s due to i n s t a b i l i t y . Three d i r e c t methods, the energy method, c o l l o c a t i o n and f i n i t e differences, were applied to the problem i n an attempt to obtain a design method suitable for code use. Of the three methods, the c o l l o c a t i o n method appeared to give the best r e s u l t s since the equations obtained from energy and f i n i t e differences were much too complex for the required degree of accuracy. The methods require the adoption of a suitable moment-curvature model, and i t was found that a str a i g h t l i n e and two parabola model could adequately represent the re a l moment-curvature r e l a t i o n . - i i -TABLE OF CONTENTS Page Abstract - i -Table of Contents - i i -L i s t of Tables - i v -L i s t of Figures -v-Nomenclature - v i i -Acknowledgements -x-Chapter 1: Introduction 1 1.1 The Slender Column Problem 1 1.2 ACI and CEB Approximate Methods 2 1.2.1 ACI Method 2 1.2.2 The CEB-FIP Methods 6 1.3 Scope 8 Chapter 2: The Energy Method 9 2.1 Introduction 9 2.2 The Deflected Shape 9 2.3 The Moment-curvature Relationship 10 2.4 The Pote n t i a l Energy Functional 11 2.5 Determining the Shape Parameter - Mm 12 Chapter 3: Solutions Using the Energy Method 14 3.1 Linear Moment-curvature Relation 14 3.2 Parabolic Moment-curvature Relation 16 3.3 Moment-curvature Relation Defined by Two Parabolas 20 3.4 B i - l i n e a r Moment-curvature Relation 29 - i i i -Page Chapter 4: S i m p l i f i c a t i o n of the Energy Solution for the 30 Two Parabola Method. 4.1 Extremely Slender Columns: Mm < My 30 4.2 R e l a t i v e l y Short Columns: My < Me 32 4.3 Slender Columns: Me < My < Mm. 33 4.3.1 Simplify 0 Terms 33 4.3.2 Reduce Form of Equation From Cubic to Quadratic 34 4.3.3 Simplify Expression f o r Work Due to Flexural 35 Shortening - I 4.3.4 Simplify Expression f o r Work Due to Flexural 35 Shortening - II 4.3.5 Reduce Equation to One Unknown and Non- 36 Dimensionalize Chapter 5: Moment-curvature Models and the S e n s i t i v i t y of 40 Solutions to Model Variations 5.1 Introduction 40 5.2 S e n s i t i v i t y of the Two Parabola Moment-curvature Model 40 5.3 The Straight Line and Two Parabola Model 43 Chapter 6: Other Methods 46 6.1 Introduction 46 6.2 The Method of Coll o c a t i o n 46 6.3 The F i n i t e Difference Method 57 Chapter 7: Conclusions 59 7.1 Summary 59 7.2 Recommendations 60 Bibliography 61 - i v -LIST OF TABLES P a ? e Table 3.1 Comparison of Moment Magnification Equations 16 3.2 Comparison of Column Deflection Curve (CDC) Method and 19 Energy Method for the Parabolic Moment-curvature Relation. 3.3 Comparison of Real and Modelled Column Results Using the 23 Program of Nathan. 3.4 Comparison of Nathan's 'Exact' (CDC) Method and the 25 Energy Method f or the Two Parabola Model with Me>My. 3.5 Comparison of Nathan's 'Exact' (CDC) Method and the 28 Energy Method for the Two Parabola Model with Me<My<Mm. 4.1 Comparison of Solutions Obtained Using the Derived Cubic 31 and the Si m p l i f i e d Cubic for Mm<My with the CDC Method. 4.2 Comparison of Solutions Obtained Using the Derived Cubic 32 and the S i m p l i f i e d Cubic f o r My<Me with the CDC Method. 5.1 Comparison of Results Obtained Using the Real Moment- 44 curvature Relation with Results Obtained Using a Straight Line and Two Parabola Model. 6.1 Comparison of the Moment magnification Equation Obtained 49 Using Co l l o c a t i o n with the Exact Secant Formula. 6.2 Comparison of Solutions Obtained Using Collocation with 54 Solutions Obtained Using Nathan's Program for Both the Real and Modelled Moment-curvature Curves - Tee Section, P/P„ = 0.1. o 6.3 Comparison of Solutions Obtained Using Co l l o c a t i o n with 55 Solutions Obtained Using Nathan's Program for Both the Real and Modelled Moment-curvature Curves - Tee Section, P/P = 0.5. o 6.4 Comparison of Solutions Obtained Using Collocation with 56 Solutions Obtained Using Nathan's Program for Both the Real and Modelled Moment-curvature Curves - Square Section, P/PQ = 0.5. - V -LIST OF FIGURES Figure Page l a E c c e n t r i c a l l y Loaded Slender Column. 63 lb Moment i n E c c e n t r i c a l l y Loaded Slender Column. 63 2 Load Moment Interaction Diagram Indicating Material F a i l u r e . 63 3 Typical Moment Curvature Relationships. 64 4 Slender Column Interaction Curves f o r Square Prestressed 65 Concrete Column (Ref. 12). 5 Slender Column Interaction Curves for Prestresses Double 66 Tee Section (Ref. 12). 6 Typical Moment Diagram for a Pin-ended Column with Equal 67 End Moments. 7 Real Moment Diagram for a Square Prestressed Column with 68 P / P q = 0.05 and a Slenderness Ratio of 25 Compared to a Parabolic Moment Diagram. 8 Real Moment Diagram for a Square Prestressed Column with 69 P/PQ = 0.05 and a Slenderness Ratio of 150 Compared to a Parbolic Moment Diagram. 9 Real Moment Diagram for a Square Prestressed Column with 70 P / P q = 0.7 and a Slenderness Ratio of 75 Compared to a Parabolic Moment Diagram. 10 Typical Moment-curvature Relationships for a Prestressed 71 Concrete Column Subjected to Di f f e r e n t A x i a l Loads. 11a Straight Line Moment-curvature Relationship. 72 l i b Parabolic Moment-curvature Relationship. 72 12a B i - l i n e a r Moment-curvature Relationship. 73 12b Two Parabola Moment-curvature Relationship. 73 13 Work Due to External Load on an E c c e n t r i c a l l y Loaded Pin- 74 ended Column. 14 Typical Energy Solution for the Maximum Mid-height Moment 75 as Determined f o r a Parabolic Moment-curvature Relationship for a Column Loaded at Constant E c c e n t r i c i t y . 15 Real Moment-curvature Curves Compared to Single Parabola 76 Moment-curvature Relationship. 16 Prestressed Tee Section; 400 PSI Prestress. 77 - v i -Page 17a Real Moment-curvature Curve and Two Parabola Moment- 78 curvature Model for Prestressed Tee Section (Fig. 16); P/P = 0.1. o 17b Real Moment-curvature Curve and Two Parabola Moment- 79 curvature Model for Prestressed Tee Section (Fig. 16); P/P = 0.5. o 18 Parabolic Moment Diagram with Y i e l d Point. 80 19a-k Beta Functions for Two Parabola Energy Solution. 81-86 20a-k Beta Functions for Non-dimensionalized Two Parabola Energy 87-92 Solution. 21a-d Lower Order Polynomials F i t t e d to Beta Function Curves. 93-94 22 Square Prestressed Section; 400 PSI Prestress. 95 23a-c S e n s i t i v i t y - Two Parabola Model for Tee Section (Fig. 16), 96-98 P/P^ = 0.1. o 24a-c S e n s i t i v i t y - Two Parabola Model for Tee Section ( F i g . 16), 99-101 P/PQ = 0.5. 25a-c S e n s i t i v i t y - Two Parabola Model for Square Section ( F i g . 102-104 22), P/PQ = 0.1. 26a-c S e n s i t i v i t y - Two Parabola Model for Square Section ( F i g . 105-107 22), P/P = 0.5. o 27 Single Straight Line and Two Parabola Moment-curvature 108 Model. 28a-c S e n s i t i v i t y - One Straight Line and Two Parabola Model for 109-111 Square Section (Fig. 22), P/PQ = 0.5. 29 S e n s i t i v i t y - One Straight Line and Two Parabola Model for 112 Tee Section ( F i g . 16), P / P q = 0.5, L/r = 150. 30a S e n s i t i v i t y - One Straight Line and Two Parabola Model f o r 113 Square Section ( F i g . 22), P/PQ = 0.5, L/r = 75. 30b S e n s i t i v i t y - One Straight Line and Two Parabola Model for 114 Square Section ( F i g . 22), P / P q = 0.5, L/r = 125. 30c S e n s i t i v i t y - One Straight Line and Two Parabola Model for 115 Tee Section (Fig. 16), P/PQ = 0.1, L/r = 150. - v i i -NOMENCLATURE A Shape parameter for moment-curvature model B Shape parameter for moment-curvature model. C Shape parameter for moment-curvature model. Cm ACI factor relating the actual moment diagram to an equivalent uniform moment diagram. dy/dx First derivative of y with respect to x; equal to the slope of the deflected shape. d 2y/dx 2 Second derivative of y with respect to x. 9II/8Mm First variation of the potential energy with respect to the unknown parameter Mm. 82II/9Mm2 Second variation of the potential energy with respect to the unknown parameter Mm. Ec Modulus of elasticity of concrete. EI Flexural stiffness of column. Es Modulus of elas t i c i t y of reinforcement. Ig Moment of inertia of gross concrete section about the centroidal axis, neglecting reinforcement. Ise Moment of inertia of reinforcement about the section centroidal axis. L Column Length. M Moment at a given location on a column. 24B M^ Term used to simplify collocation formula; u)pjj2. 24C Mc Term used to simplify collocation formula; -5^ 2• Md Term used to simplify energy solutions; Me-My. Me End moment for a slender column loaded eccentrically with equal end eccentricities. Mm Maximum mid-height moment for a slender column loaded eccentrically with equal end eccentricities. Mpa Term used to simplify energy solutions; -^ 2 - v i i i -Mpb Term used to sim p l i f y energy solutions; Mu Ultimate moment of a section as governed by material f a i l u r e . My Y i e l d moment for the two parabola moment-curvature model. Mya F i r s t y i e l d moment for the st r a i g h t l i n e and two parabola moment-curvature model. Myb Second y i e l d moment for the st r a i g h t l i n e and two parabola moment-curvature model. M^ Smaller factored column end moment from an e l a s t i c frame analysis; p o s i t i v e i f column i s i n single curvature and negative i f column i s i n double curvature. M2 Larger factored end moment from an e l a s t i c frame a n a l y s i s . P Factored column a x i a l load. Per Column a x i a l load which causes the column to become unstable. Po Ultimate column load i f the load i s applied at the p l a s t i c centroid of the column. r Radius of gyration of column cross-section. u Increase i n i n t e r n a l s t r a i n energy as column assumes the deflected p o s i t i o n . W Work performed by the external loads as the column assumes the deflected p o s i t i o n . W^  External work r e s u l t i n g from f l e x u r a l shortening. W2 External work r e s u l t i n g from ro t a t i o n of column ends. x Distance along the column length measured from an o r i g i n at one end of the column. y Column d e f l e c t i o n at a distance x along the column. ymax Maximum mid-height d e f l e c t i o n . 3 Term used to define yielded portion of the column with the two parabola moment-curvature model. 3d ACI factor which accounts for creep; equal to the r a t i o of the absolute value of the maximum factored dead load moment to the maximum factored t o t a l load moment. - i x -Y Non-dimensional term for two parabola model energy solution; Non-dimensional term f o r two parabola model energy so l u t i o n ; A Me PL 2 e Non-dimensional term for two parabola model energy solution; ^. II Total p o t e n t i a l energy of a system. <f> Curvature of a column section <j>m Maximum mid-height curvature. <|>u Ultimate curvature of a section as governed by material f a i l u r e . <|>y Y i e l d curvature for the two parabola moment-curvature model. <|>ya F i r s t y i e l d curvature for the st r a i g h t l i n e and two parabola moment-curvature model. <(>yb Second y i e l d curvature for the s t r a i g h t l i n e and two parabola moment-curvature model. - x -ACKNOWLEDGEMENTS I would l i k e to express my appreciation to my advisor Dr. N.D. Nathan f o r h i s guidance and assistance i n the preparation of t h i s t h e s i s . I would also l i k e to thank Dr. D.L. Anderson who, during a course of study on plate a n a l y s i s , introduced me to the methods of c o l l o c a t i o n and f i n i t e d i f ferences. Without f i n a n c i a l assistance from the National Research Council and the Precast Concrete I n s t i t u t e , t h i s thesis would not have been poss i b l e . I would also l i k e to express my thanks to K e l l y Lamb f o r her assistance i n typing and preparing t h i s t h e s i s . F i n a l l y t h i s undertaking would not have been possible without the patience and understanding of my wife Susan. 1. 1. INTRODUCTION 1 . 1 The S l e n d e r Column Problem A s l e n d e r column i s one i n which the s l e n d e r n e s s o f the column has an e f f e c t upon t h e u l t i m a t e column l o a d . The a x i a l l o a d c r e a t e s a d d i t i o n a l b e n d i n g moment due t o l a t e r a l d e f l e c t i o n o f the column. The s l e n d e r c o n c r e t e column problem has been ad d r e s s e d i n v a r i o u s ways by d i f f e r e n t codes and, i n g e n e r a l , d e s i g n recommendations f o l l o w one o f two p r a c t i c e s , depending on t h e degree o f s l e n d e r n e s s . For v e r y s l e n d e r columns a second o r d e r a n a l y s i s which can take i n t o account the n o n - l i n e a r i t i e s o f the problem i s r e q u i r e d . Such an a n a l y s i s i s complex and r e q u i r e s t h e use o f a comprehensive computer program. Two t y p e s o f n o n - l i n e a r b e h a v i o u r c o m p l i c a t e t h e a n a l y s i s ; geometric n o n - l i n e a r i t y due t o the e f f e c t s o f l a t e r a l d e f l e c t i o n on the bending moments and m a t e r i a l n o n - l i n e a r i t i e s due t o t h e n o n - l i n e a r n a t u r e o f the s t r e s s - s t r a i n r e l a t i o n s h i p s o f the m a t e r i a l s used i n r e i n f o r c e d c o n c r e t e c o n s t r u c t i o n . MacGregor (1) has o u t l i n e d t h e r e q u i r e m e n t s f o r " e x a c t " second-order frame a n a l y s e s . Even f o r i n d i v i d u a l columns s e p a r a t e d out from the e f f e c t s o f moment r e d i s t r i b u t i o n w i t h i n the frame, the a n a l y s i s i s v e r y c o m p l i c a t e d . Most computer programs i n use assume c o n s t a n t d e f o r m a t i o n p r o p e r t i e s over " f i n i t e elements" o f the column. One such program i s d e s c r i b e d by C h a n d i v a n i R. and Nathan (2) and Nathan ( 3 ) • A l t h o u g h both the ACI and t h e CEB codes encourage the use o f such programs f o r a l l s i t u a t i o n s , both codes p r o v i d e a l t e r n a t i v e approximate methods f o r the d e s i g n o f l e s s s l e n d e r columns. 2. 1.2 ACI and CEB Approximate Methods  1.2.1 ACI Method MacGregor et a l (4) outlined the basis for s e l e c t i o n of the approximate design method recommended by the ACI and also presented the r a t i o n a l e behind the method; commonly referred to as the moment magnification method. The analysis i s based on a pinned-pinned column with equal single-curvature end moments, but MacGregor et a l also gave the basis for the ACI adjustment f a c t o r C which accounts for unequal end m moments. The expression used i n the ACI code i s : M l C = 0.6 + 0.4 — > 0.4 (1.1) m M„ 2 This equation was adopted from the American I n s t i t u t e of Steel Construction and i s applicable when the moments are determined using an e l a s t i c frame analysis and where the moment diagram i s a s t r a i g h t l i n e and sidesway i s prevented. Other formulas are discussed i n the Guide to S t a b i l i t y Design C r i t e r i a f o r Metal Structures (5). Nathan (6) suggests that for prestressed concrete columns the ACI formula should be replaced by: M C = 0.7 + 0.3 — . (1.2) M 2 The e f f e c t s of various end conditions are taken care of by the e f f e c t i v e length factor k. For sway frames and p a r t i a l l y braced frames, the e f f e c t i v e length factor method has been shown by MacGregor and Hage (7) to be quite inaccurate. This i s l a r g e l y due to the e f f e c t of beam moments on the f i r s t order a n a l y s i s . For these frames, i n p a r t i c u l a r , a strong recommendation can be made for the use of a comprehensive second order a n a l y s i s . However, since the f a c t o r and the e f f e c t i v e length factor 3. are a v a i l a b l e to deal with various end conditions and loading conditions, at l e a s t i n an approximate manner, the discussion i n t h i s thesis w i l l be li m i t e d to the pin-ended case with equal end moments. A pin-ended slender column loaded e c c e n t r i c a l l y i s shown i n F i g . l a . The end moment i s Pe and the mid-height moment or maximum moment i s P(e+6) as shown i n F i g . l b . The two types of f a i l u r e that can occur are shown i n Fi g . 2. Material f a i l u r e i s indicated where the ends of the column follow path OA^ as the load i s applied and the mid-height moment follows path OB Fa i l u r e occurs when the mid-height moment reaches point B on the 1. J-"short column" i n t e r a c t i o n curve. The maximum end moment that can be applied i s at the point which corresponds to a point on the long column i n t e r a c t i o n curve. S t a b i l i t y f a i l u r e i s indicated where the ends of the column follow path 0A^ and the mid-height moment follows path OB C I n s t a b i l i t y occurs when the mid-height moment reaches point B and 2 ^. ^ the corresponding point on the long column i n t e r a c t i o n curve i s D 2. The ACI moment magnification method amplifies the equal end moments as follows: Me M m = l - ( P / P ) ( 1 ' 3 ) cr The moment magnification fa c t o r ; 1 1-P/P cr (1.4) i s a s i m p l i f i c a t i o n of the secant formula f o r e l a s t i c beam columns which Timoshenko (8) proposed. Timoshenko and Gere (9) i s one of but many texts on e l a s t i c s t a b i l i t y i n which the exact secant formula i s derived. Timoshenko and Gere state that the error between the exact secant formula and the s i m p l i f i e d formula i s less than 2% for values of P/P l e s s than cr 0.6. This i s inc o r r e c t , however, as shown i n Table 5.1 of Park and Paulay 4. (10) the error at P/P of 0.6 i s approximately 13%. A column i s deemed c acceptable by the moment magnification method i f the magnified moment f a l l s within the short column i n t e r a c t i o n curve. There are, however, two problems with t h i s method. F i r s t the EI used to determine the c r i t i c a l load may vary, along the length of the column. This v a r i a t i o n i s due to the non-linear material properties and the v a r i a -t i o n of moment along the column. F i g . 3 shows t y p i c a l moment curvature curves for a prestressed concrete column with two l e v e l s of loading. The moment-curvature points corresponding to the end moments and maximum mid-height moments f o r several slenderness r a t i o s are delineated on the curves. These were obtained with the a i d of the program of Nathan (3). It i s worthwhile to note some of the c h a r a c t e r i s t i c s of these curves. The column with an a x i a l load that i s 0.05 of the ultimate column a x i a l load Po, exhibits a high degree of d u c t i l i t y and i t can be seen that i f the slenderness r a t i o i s not too large, the enti r e column length w i l l f a l l on the r e l a t i v e l y s t r a i g h t portion of the curve i n which y i e l d i n g i s taking place. This would in d i c a t e , that for these columns, a l i n e a r approximation f or EI may not be out of place. For higher slenderness r a t i o s , the slope of the moment curvature curve, and thus EI, vary considerably over the length of the column. The column with an a x i a l load of 0.7 Po exhibits very l i t t l e d u c t i l i t y . For t h i s column the magnitude of the slenderness r a t i o i s l i m i t e d since the c r i t i c a l length of the column must not be exceeded. For lower slenderness r a t i o s , i t can be seen that a portion of the column f a l l s on the i n i t i a l l i n e a r e l a s t i c portion of the moment curvature curve. Near the centre of the column, however, the slope of the moment curvature curve can vary considerably. 5. The ACI gives two approximate formulas for determining EI: (Edg/5) + Es ise  E I - 1 + 3d ( 1 ' 5 ) or conservatively EcIg/2.5 E I = l + g d ( 1 ' 6 ) where (1+Bd) i s a factor which accounts f o r creep e f f e c t s . The EI thus determined i s constant along the length of the column and does not r e f l e c t the fa c t that columns with d i f f e r e n t slenderness r a t i o s l i e on d i f f e r e n t portions of the moment curvature curves and could thus possibly have a d i f f e r e n t approximation f o r EI. MacGregor et a l (4) compare these formulas with a t h e o r e t i c a l EI derived from load-moment-curvature diagrams. A more accurate formula i s given by MacGregor et a l (11), however, t h e i r recommendations were not adopted i n ACI 318-77. Nathan (6) proposed the use of the following f o r -mula for prestressed columns: EI = Eclg/X (1.7) where X = 15-25 (P/Po) > 2.5 (1.8) The ACI moment magnification method assumes that the f a i l u r e mecha-nism i s due to material f a i l u r e . However, as previously pointed out, there i s a po t e n t i a l for a s t a b i l i t y type f a i l u r e . This i s the second . s i g n i f i c a n t problem with the ACI method. Usually s t a b i l i t y f a i l u r e s are li m i t e d to very slender columns with a low load l e v e l r e l a t i v e to Po and with a large e c c e n t r i c i t y . However, Nathan (12), has shown that with prestressed columns the f a i l u r e can be due to i n s t a b i l i t y at r e l a t i v e l y low l e v e l s of slenderness r a t i o L/r. This i s e s s e n t i a l l y due to the low st e e l r a t i o encountered i n these columns but the e f f e c t may be enhanced by the cross-sectional shape c h a r a c t e r i s t i c of the columns. Two figures from 6. Ref. (12), F i g . 4 and F i g . 5 i l l u s t r a t e t h i s e f f e c t . F i g . 4 shows i n t e r -action curves for a 24 inch square column with 200 p s i prestress. The regions of i n s t a b i l i t y and material f a i l u r e are depicted d i f f e r e n t l y and also, i f the i n s t a b i l i t y occurs within 90% of the material f a i l u r e , t h i s i s i n dicated. These curves were produced using Nathan's program. Also shown are the ACI moment magnification predictions for t h i s column f o r three slenderness r a t i o s , L/r = 25, L/r = 75 and L/r = 150. There i s considerable discrepancy between the ACI method and the more exact second order analysis. At the higher load l e v e l s the discrepancy i s on the con-servative side, but at low load l e v e l s , the deviation i s unconservative. The i n t e r a c t i o n curves f o r a prestressed double tee are shown i n F i g . 5. This tee section i s the 8DT12 standard section described i n the PCI Design Handbook (13). The f i g u r e shows that, even f o r a slenderness r a t i o as low a 25, for most a x i a l loads the f a i l u r e i s from i n s t a b i l i t y . Also, the discrepancy between the ACI method and the more exact analysis i s greater. 1.2.2 The CEB-FIP Methods The CEB-FIP (14) present three "approximate" methods; the model column method, the s i m p l i f i e d method based on the equilibrium state, and the approximate method for c a l c u l a t i n g the supplementary moment. The model column method i s based on a set of design aids derived from an "exact" a n a l y s i s . The code l i m i t s the use of t h i s method to slenderness r a t i o s less than 140. Their usefulness i s also r e s t r i c t e d by the design aids, which cover a l i m i t e d number of cross-sections and l i m i t e d range of material properties. Prestressed tee sections for instance could not be designed using the aids presented. Within t h e i r range of usefulness, however, they should be reasonably accurate, since they were developed from a more general a n a l y s i s . 7. The second method i s more general i n that i t can be applied to sections of any cross-section; i t i s a graphical procedure based on p l o t s of e c c e n t r i c i t y vs. curvature. It i s possible, however, to apply the method using more f a m i l i a r p l o t s of moment vs. curvature. The maximum mid-height d e f l e c t i o n i s assumed to be: <J>mL2 ymax = J L ^ - (1.9) This corresponds to the assumption of a sinusoidal deflected shape with the approximation that IT2 = 10. The method requires a reasonably accurate p l o t of the moment-curvature diagram f o r the column section at the load l e v e l under consideration. A sloping s t r a i g h t l i n e corresponding to the equation; Pd>L2 M = Me + | Q (1.10) i s superimposed on the moment-curvature p l o t . The st r a i g h t l i n e w i l l normally intercept the moment-curvature r e l a t i o n at two points corresponding to equilibrium p o s i t i o n s . The f i r s t point (closest to the orig i n ) i s a stable p o s i t i o n and the second point corresponds to an unstable equilibrium p o s i t i o n . If the sloping s t r a i g h t l i n e of equation 1.10 i s moved up u n t i l i t i s tangent to the moment-curvature r e l a t i o n , the maximum end moment can be obtained from the intercept with the moment axis, and the corresponding maximum mid-height moment i s obtained from the tangent point. If the l i n e does not intercept the moment-curvature r e l a t i o n at any point, then the column i s unstable at the end moment thus defined. The method gives quite reasonable r e s u l t s , but i t has two substantial drawbacks. F i r s t , the moment-curvature r e l a t i o n must be obtained reasonably accurately. Secondly, the procedure i s a graphical one not r e a d i l y adaptable to code use. The t h i r d method consists of evaluating a supplementary moment using 8. an approximate expression based on t e s t s and theory. This method i s comparable to the ACI moment magnification method using the more conserva-t i v e formula for EI, and as such would be subject to the same l i m i t a t i o n s . The method i s not included i n the model code and i t i s suggested as being well suited for preliminary design, since i t i s independent of the amount of reinforcement. 1.3 Scope The use of slender reinforced and prestressed concrete column sections has become prevalent i n recent years. In p a r t i c u l a r they are commonly used for load bearing precast wall s t a t i o n s . I t has been shown that the f a i l u r e load of very slender reinforced concrete columns, prestressed columns and e s p e c i a l l y prestressed tee sections cannot be determined very accurately using the approximate methods set f o r t h i n the ACI and CEB codes. It i s desirable then, to develop an approximate method which would provide solutions for these s i t u a t i o n s . This t h e s i s explores the p o s s i b i l i t y of using a formulation of the Pot e n t i a l Energy Method to obtain an expression which w i l l give the maximum moment and w i l l p r e d i c t the c r i t i c a l load for a slender concrete column. A solution i s obtained with the moment-curvature r e l a t i o n modelled by two parabolas and the accuracy of the r e s u l t s i s demonstrated using Nathan's program. The s e n s i t i v i t y of the solution with respect to v a r i a t i o n s i n the shape of the two parabola model i s examined and compared to the s e n s i t i v i t y of other possible models. Two other methods are also considered; the Method of C o l l o c a t i o n , and the F i n i t e Difference Method. 9. 2. THE ENERGY METHOD 2.1 Introduction With the approximate methods currently i n use, the most s i g n i f i c a n t errors occur when the f a i l u r e i s due to i n s t a b i l i t y . It i s therefore desirable that the r a t i o n a l e behind a new approximate method be such that the method w i l l be capable of p r e d i c t i n g the buckling type of f a i l u r e . I t i s well known that an energy approach can be used to obtain an approximate upper bound for the c r i t i c a l load for a slender column; see for instance Timoshenko & Gere (9). The presence of material n o n - l i n e a r i t i e s i n the problem under consideration complicates the solution of the problem, but should not a f f e c t the a b i l i t y of the method to y i e l d a s o l u t i o n . The procedure of the method i s as follows: 1. Assume a deflected shape for the column i n terms of one or more para-meters which r e f l e c t the desired unknown quantity. 2. E s t a b l i s h a moment-curvature r e l a t i o n s h i p f o r the column. 3. Set up the p o t e n t i a l energy functional and reduce i t to a function by i n s e r t i n g the assumed deflected shape. 4. Determine the shape parameter(s) and hence the moment magnification by s e t t i n g the f i r s t v a r i a t i o n of the p o t e n t i a l energy function equal to zero. The i n s t a b i l i t y point i s established by simultaneously s e t t i n g the second v a r i a t i o n of the p o t e n t i a l energy equal to zero, a l t e r n a t i v e l y , i t may be recognized by the fact that, above some value of the load, no r e a l equilibrium configuration occurs. 2.2 The Deflected Shape The assumed deflected shape must s a t i s f y the forced boundary condi-tio n s , that i s the boundary conditions pertaining to d e f l e c t i o n s and 10. slopes. In the present work, the approximating function i s constructed by assuming a pl a u s i b l e bending moment-variation, deducing therefrom the curvature, and int e g r a t i n g the curvature twice to obtain a shape function. The correct boundary conditions can be enforced i n t h i s procedure, and i t i s possible then to express the assumed shape d i r e c t l y i n terms of the unknown maximum mid-height moment, Mm. For the pin-ended case with equal end moments, the moment diagram w i l l be as shown i n F i g . 6. If a parabolic shape i s assumed for the second order moment, the moment i n the column can be expressed as follows: M = Me + 4 (Mm-Me) [ ( 7 ) - ( 7 ) 2 ] (2 . 1 ) This implies a fourth order equation for the assumed deflected shape. A comparison between the assumed parabolic second order moment diagram and moment diagrams constructed using Nathan's program for three slender columns are shown i n Fig s . 7, 8, and 9. The figures indicate that the adoption of a parabolic shape for the second order moments i s a reasonable assumption. 2.3 The Moment-Curvature Relationship A set of moment-curvature re l a t i o n s h i p s for a prestressed concrete column i s shown i n F i g . 10. These were produced using the program of Nathan. I t can be seen that the shape of the curves i s highly dependent upon the r a t i o P/PQ. Since the p o t e n t i a l energy functional i s set up using the assumed moment-curvature r e l a t i o n s h i p , the accuracy of the sol u t i o n w i l l be dependent upon how c l o s e l y the assumed moment-curvature r e l a t i o n s h i p follows the r e a l one. Also the solution obtained w i l l be an upper bound to the hypothetical problem with the assumed r e l a t i o n s h i p , but i s no longer an upper bound to the r e a l problem. 11 . In order to evaluate the proposed method, a s t r a i g h t l i n e moment-curvature r e l a t i o n s h i p and a single parabolic moment-curvature r e l a t i o n s h i p as shown i n F i g . 11a and l i b were assumed. The single s t r a i g h t l i n e moment-curvature r e l a t i o n s h i p corresponds to a l i n e a r e l a s t i c column. Models composed of two s t r a i g h t l i n e s and two parabolas, as shown i n F i g . 12a and 12b were also t r i e d i n order to achieve a close approximation of the r e a l moment-curvature r e l a t i o n s h i p s . The a b i l i t y of these and other p l a u s i b l e moment-curvature r e l a t i o n s h i p s to accurately predict the c r i t i c a l load for r e a l columns i s discussed i n greater d e t a i l i n Chapter 5. 2.4 The P o t e n t i a l Energy Functional The t o t a l p o t e n t i a l energy, II, can be expressed as U-W, where U i s the i n t e r n a l s t r a i n energy stored throughout the volume of the material, and W i s the work done by the surface t r a c t i o n s as the column moves into the deflected p o s i t i o n . The i n t e r n a l s t r a i n energy can be expressed i n functional form as: U = / [M.$]^ dx (2.2) L <t> where [M.$] i s the area under the moment-curvature diagram up to the curvature <|>. The external work performed by the surface t r a c t i o n s becomes the work done by the load P and the end moment Me as the column moves into the deflected p o s i t i o n . Thus, as shown i n F i g . 13, the external work can be expressed by two terms; the work due to the f l e x u r a l shortening of the column and the work r e s u l t i n g from the r o t a t i o n of the end moments as the ends of the column rotate. Neither the i n t e r n a l nor the external work due to a x i a l shortening enter into the problem because they do not contribute to the f l e x u r a l f a i l u r e mechanism under consideration; they can be seen as the datum from which energy i s measured. The work due to f l e x u r a l 12. shortening can be expressed as / ( f ) 2 * (2 -3 ) L and the work r e s u l t i n g from the rot a t i o n of the end moments i s ; (2.4) W = -2Me ^ 2 dx |x=0 where the negative sign i s required to maintain a consistent sign conven-t i o n . The work due to the external t r a c t i o n s i s then dependent upon the slope of the column, and the i n t e r n a l s t r a i n energy i s a function of the curvature. The p o t e n t i a l energy functional can then be expressed as: n = /[M-tl+dx - ^ J(?)2dx + 2Me ^ ' L J 2 dxy dx x=0 ( 2 ' 5 ) L L This functional can then be reduced to the form of a function by i n s e r t i n g the assumed moment diagram shape and the assumed moment-curvature r e l a t i o n . 2.5 Determining the Shape Parameter - Mm The assumed "shape" i s a one parameter function of Mm. Therefore obtaining the f i r s t v a r i a t i o n of the p o t e n t i a l energy function s i m p l i f i e s to taking the f i r s t p a r t i a l derivative with respect to the unknown parameter Mm. 311 Setting -j ^ j - = 0 (2.6) an expression for Mm i s obtained, the solution of which y i e l d s the maximum moment(s) which s a t i s f y the equilibrium conditions of the problem as defined. These equilibrium states may be stable or unstable. The c r i t i -c a l point - the maximum load for which stable positions e x i s t - may be recognized by the "blowing up" of the maximum moment thus obtained or i t may have to be determined by se t t i n g the second v a r i a t i o n of the po t e n t i a l 13. energy functional equal to zero; 32n 9 ^ 2 " ° ( 2 ' 7 ) The so l u t i o n for Mm which s a t i s f i e s t h i s equation may then be substituted back into the equation for the equilibrium c r i t e r i o n and the c r i t i c a l load can be determined from the r e s u l t i n g expression. 14. 3. SOLUTIONS USING THE ENERGY METHOD 3.1 Linear Moment-Curvature Relation The l i n e a r moment-curvature r e l a t i o n corresponds to the l i n e a r e l a s t i c problem which has the solut i o n : Per = ^  (3.1) Let t i n g the slope of the moment-curvature diagram ( F i g . 11a) be A, and sub s t i t u t i n g i n the assumed shape for the moment diagram, r e s u l t s i n the following expression for the curvature: 4, = «!Z = I = I { M e + 4(Mm-Me) [ £ H 7 ) 2 ] > (3.2) , o A A L VL y VL y J dx^ which upon in t e g r a t i o n , with s a t i s f a c t i o n of the zero slope boundary conditions at the mid-height, y i e l d s an expression for the slope: 1 - i «*[(!) - |i+ <-*>[»(f)2 - ! ( ! ) 3 - in <3-3> The area under the moment diagram becomes: r M.$ M2 = ^ { M e 2 + 8(Mm-Me)[(^-(^) 2] + 16(Mm-Me) 2 [ ( ^ ) 2 - 2 ^ ) 3 + (2S)f] } ( 3 . 4 ) The s t r a i n energy i s obtained by su b s t i t u t i n g the above expression into equation 2.2 and performing the necessary int e g r a t i o n to y i e l d : U = ^ [-^Me2 + -|Me(Mm-Me) + -j-^ (Mm-Me) 2 ] (3.5) Substitution of 3.3 into equation 2.3 leads to the following expression f o r the work due to f l e x u r a l shortening: 3 W1 = ^ - [ " ^ M e 2 + -^Me(Mm-Me) + ^ -^(Mm-Me) 2 ] (3.6) A2 Evaluating 3.3 at x = 0, and sub s t i t u t i n g into equation 2.4 y i e l d s the external work due to the rotation of the end moments: 15. W2 = A ^ M e 2 + 3 M e ( M ™ ' ^ J ] ( 3 . 7 ) The above expression ( 3 .7 ) has the same multiples as ( 3 . 5 ) so that can be d i r e c t l y subtracted from V: U-W2 = ^[""^Me2 + ^ r(Mm-Me)2] ( 3 . 8 ) A l l of the energy solutions behave s i m i l a r l y so that the t o t a l p o t e n t i a l energy, II, consists of the re s i d u a l s t r a i n energy remaining a f t e r the work due to rot a t i o n of the ends i s subtracted, minus the work due to f l e x u r a l shortening. The t o t a l p o t e n t i a l energy i s then expressed as follows: II = -[-^Me 2 + 4^(Mm-Me)2l - ^ ^[-^Me244rMe(Mm-Me)4-^7(Mm-Me) 21 ( 3 . 9 ) A L 2 15 J o 2 4 1 5 6 3 0 A Setting the f i r s t v a r i a t i o n of the p o t e n t i a l energy with respect to the unknown parameter Mm equal to zero y i e l d s : 9n L r 8 , . . . . . , i P L 3 r 1 . . . . 17 ^ = ^ [^(Mm-Me) ] " — [ l T M e + "7r7"(Mm-Me) ] = 0 (3.10) 3Mm A L15 J ? L15 315 J A^ Solving for Mm r e s u l t s i n : r r8 4PL2-,,f8 17PL 2 >n„ , , . . . ^ = K l ? + TTEK^ls ~ 3 T 5 A " ^ M e ( 3 ' 1 1 } This expression "blows up" when the denominator i s equal to zero which implies: P = M M (3.12) C r 17L 2 The slope of the moment-curvature diagram A i s EI for small d e f l e c t i o n beam theory, and the accuracy of t h i s equation can be determined by comparing 168/17 to n 2 . The error i s approximately 0.13% which i s n e g l i g i b l e . Equation 3.12 can also be obtained by se t t i n g the second v a r i a t i o n of the po t e n t i a l energy equal to zero. It i s also desirable to determine the accuracy of the moment magnification factor predicted by t h i s method. Substituting A/L = (17 Pcr)/168 back into 3.11 and si m p l i f y i n g r e s u l t s i n a moment magnification 16. fa c t o r : 1+4P/17 P cr 1-P/P (3.13) cr A comparison of the above formula with the ACI approximate formula and the exact secant formula i s given i n Table 3.1 for a range of P/P . The cr close agreement between the moment magnification factor determined using Table 3.1 Comparison of Moment Magnification Equations Equation P/P cr 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 < A C I ) ' cr 1+4P/17 P r 1-P/P cr cr (Energy Soln) sec(•£/_£-) (Secant 1.111 1.137 1.137 1.250 1.309 1.310 1.429 1.529 1.533 1.667 1.824 1.832 2.000 2.235 2.252 2.500 2.853 2.884 5.000 5.941 6.058 ' cr formula) the energy method and the exact secant formula i s an i n d i c a t i o n that the assumed deflected shape i s acceptable, at le a s t for the e l a s t i c case thus solved. 3.2 Parabolic Moment-Curvature Relation To t e s t the method for the non-linear problem, a moment-curvature r e l a t i o n c o n s i s t i n g of a single parabola with d<j)/dM=0 at the o r i g i n ( F i g . l i b ) was assumed. F i g . 10 indicates that such a curve would represent the r e a l r e l a t i o n s h i p more accurately than does a str a i g h t l i n e . The equation for the curvature becomes: 1 7 . M 2 • = — (3.14) where A i s a constant which defines the end point of the curve; A=Mu2/<|>u. Applying the assumed deflected shape to equation 3.14 and solving for the po t e n t i a l energy function as i n the l i n e a r case r e s u l t s i n the following expression f o r the t o t a l p o t e n t i a l energy: o 8 o 32 , O T P L 3 r l u N = I"L7 M E + "TTMe(Mm-Me) + -—-r(Mm-Me) _ _ 2 ^ - _ M e H 2 o 52 o o 88 o + —Me^Mm-Me) + 777-Me z ( Mm-Me) z + —-Me(Mm-Me)3 15 105 945 + 5°975 (Mm-Me)1*] (3.15) Setting the f i r s t v a r i a t i o n of the po t e n t i a l energy equal to zero r e s u l t s i n a cubic equation which can be expressed either i n terms of the addi t i o n a l or second order moment (Mm-Me): ,o r1815 1485 A ? (Mm-Me)d + l^J-Me - p^ 2) (Mm-Me) zr2145 , 3465 AMe^, % 3465 o , , . , - x + ^ i i l ^ 6 " 262 ^ ( M n - M e ) + ^ M e 3 = o (3.16) or i n terms of Mm: o ,243Me 1485 A x , ,2475 AMe 87 Me2^ m + (~5iT - U T T Z 2 ^ + C 2 6 2 P L 2 + 1iT->ta + (495 A Me2 + | y e 3 = q { 3 i i ? ) 262 PL 2 The so l u t i o n of a cubic equation of the form y J + p y i + qy + r = 0 i s achieved by f i r s t reducing i t to the form x 3 + a x + b = 0 b y su b s t i t u t i n g for y the value, x - p/3 with a = q - p 2/3 (3.18) and b = 2p 3/27 - pq/3 + r (3.19) 18. This equation i s then solved using a trigonometric s u b s t i t u t i o n : l e t t i n g x = m cos 6 implies; 1 2 m = 2(-a/3) (3.20) 3b and i t follows that: cos(36) = — (3.21) am Any s o l u t i o n 9^ which s a t i s f i e s equation 3.21 w i l l also imply solutions 6^ +2TT/3 and 6^ +4IT/3. For load l e v e l s below the c r i t i c a l load, only one of the three roots of equation 3.17 thus obtained i s the r e a l maximum midheight moment. This was found to correspond to the solu t i o n of the reduced cubic of 9^+4n/3 where 8^  i s the smallest angle that s a t i s f i e s equation 3.21. For the c r i t i c a l load, the reduced cubic equation has three r e a l roots, two of which are equal. These two roots correspond to the maximum mid-height moment. If the load exceeds the c r i t i c a l load at the given e c c e n t r i c i t y , these two roots become imaginary. This then defines the "blowing up" c r i t e r i o n for determining the c r t i c a l load. The c r i t i c a l load can also be determined by s e t t i n g the second v a r i a t i o n of the p o t e n t i a l energy with respect to Mm equal to zero. The r e s u l t i n g equation: o r81 990 A > r825 AMe 29 . o , „ „„. Mm + — — Me - — 2)Mm + f—— 7 + — „ Me^J = 0 (3.22) l-262 131 P L Z J ^262 P i / 524 ; i s a quadratic equation with two r e a l roots. Only one of these roots, however, defines an equilibrium p o s i t i o n and thus must also s a t i s f y equation 3.17 when that equation i s about to "blow up". This i s depicted g r a p h i c a l l y i n the example of F i g . 14. When the load i s increased at constant e c c e n t r i c i t y the maximum mid-height moment as predicted by the correct root of equation 3.17 increases u n t i l the i n s t a b i l i t y point i s reached. At the same load, equation 3.22 gives a solution for Mm which 19. represents the passage from s t a b i l i t y to i n s t a b i l i t y ; i t corresponds to an equilibrium p o s i t i o n only at the c r i t i c a l load where the maximum mid-height moment i t predicts i s the same as that predicted by equation 3.17. The accuracy of the solu t i o n was checked using a modified version of the program of Nathan i n which the moment-curvature r e l a t i o n was alte r e d to coincide with the parabola assumed for the energy s o l u t i o n . T y p i c a l r e s u l t s comparing the two methods are shown i n Table 3.2. The procedure used to obtain t h i s comparison was to f i r s t run the program of Nathan for Table 3.2 Comparison of Column Deflection Curve (CDC) Method and Energy Method for the Parabolic Moment-Curvature Relation. C r i t i c a l Load Maximum Mid-Height (kips) Moment (kip-feet) CDC Method 163.0 193. Energy Method 161.11 185.4 a r e a l column with a s p e c i f i e d load and length. The program was then modified to use a single parabola moment-curvature r e l a t i o n with d<f>/dM = 0 at the o r i g i n , terminating at the ultimate moment and curvature predicted by the f i r s t run. The r e s u l t i n g maximum mid-height moment at i n s t a b i l i t y f o r the s p e c i f i e d load i s indicated i n Table 3.2. The program also gives the maximum end moment for the s p e c i f i e d load from which the e c c e n t r i c i t y at i n s t a b i l i t y can be determined. A small computer program was written to solve equation 3.17 and the solutions were c a r r i e d out for increasing load l e v e l s with the e c c e n t r i c i t y held constant at the value determined from the CDC method. The c r i t i c a l load was revealed by noting at which load l e v e l two of the roots became imaginary. If the ultimate column load i s 20. governed by material f a i l u r e rather than i n s t a b i l i t y f a i l u r e , the maximum mid-height moment that causes the solution of equation 3.17 to "blow up" w i l l be greater than the ultimate moment which i s defined by the end point of the moment curvature diagram. The comparison of Table 3.2 indicates that the c r i t i c a l load and the maximum mid-height moment could be determined with s a t i s f a c t o r y accuracy using the energy approach, i f the moment-curvature r e l a t i o n s h i p were parabolic. The single parabola, however, cannot accurately represent the r e a l moment-curvature r e l a t i o n s h i p s as indicated i n F i g . 15. It i s therefore necessary to develop curves which can more accurately depict the true moment-curvature r e l a t i o n . 3.3 Moment-Curvature Relation Defined by Two Parabolas B i - l i n e a r or t r i - l i n e a r i d e a l i z e d moment-curvature curves have been used i n reinforced concrete design (10), but, these are more suitable f o r sections reinforced with mild s t e e l bars with a well defined y i e l d plateau. Sections reinforced with high strength prestressing strands tend to have a more rounded curve due to the lack of a well defined y i e l d plateau. The presence of a s i g n i f i c a n t a x i a l force also subtracts from the l i n e a r i t y of the moment-curvature diagram because a large portion of the concrete may be i n the non-linear region of i t s s t r e s s - s t r a i n curve. A r e l a t i o n described by two parabolas was then chosen to represent the moment-curvature, since the procedure worked well for the single parabola model and i t was f e l t that the r e a l problem could be better represented by such a curve. A two parabola moment-curvature diagram i s shown i n F i g . 12b. The i n i t i a l parabola has d<|>/dM = 0 at My, <|>y. Although a better representation of the r e a l moment-curvature r e l a t i o n could have been 21. obtained by enforcing continuity of slope at My, <j>y, the equations were greatly s i m p l i f i e d by having a zero i n i t i a l slope for the second parabola. In order to v e r i f y that the r e a l moment-curvature r e l a t i o n could be modelled with s a t i s f a c t o r y accuracy by means of the two parabolas, i t was again necessary to use an al t e r e d version of Nathan's program. The procedure used was as follows: 1. Nathan's program was used to determine the moment-curvature r e l a t i o n , the maximum permissable end moments, the corresponding maximum mid-height moments f o r a given column crossection with varying lengths, and subjected to a p a r t i c u l a r a x i a l load. This was repeated f o r several d i f f e r e n t a x i a l loads. Various column sections were treated in t h i s manner, only one of which i s reprted on here (see F i g . 16) as an example. 2. The program was then a l t e r e d to use a two parabola model f o r the moment-curvature r e l a t i o n which would c l o s e l y match the r e a l moment-curvature diagram for the p a r t i c u l a r load. To achieve t h i s , the r e a l ultimate moment and ultimate curvature were used i n the model, and a y i e l d moment, My, and y i e l d curvature, <|>y, were selected such that the model c l o s e l y resembled the r e a l moment-curvature diagram. The r e s u l t i n g maximum end moments were compared to the maximum end moments for the r e a l column. The p o s i t i o n of My and $y was then systematically varied and the program re-run u n t i l the best r e s u l t s for maximum end moment and mid-height moment were achieved. The r e s u l t s f o r the prestressed column tee section of F i g . 16 are shown i n Table 3.3. The table compares the maximum possible end moments and corresponding maximum mid-height moments for the r e a l column with those obtained using two parabolas with zero i n i t i a l slopes and using two parabolas which have continuous tangent at My, <|>y. Two d i f f e r e n t load l e v e l s are compared, with slenderness r a t i o s varying from 25 to 150 i n steps of 25. The r e a l moment-curvature curves and the two parabola models with zero i n i t i a l slopes which gave the best r e s u l t s are shown i n F i g . 17a and b. Table 3.3 i l l u s t r a t e s several f a c t s . F i r s t , there i s v i r t u a l l y no difference between the model using the two parabolas with zero slopes and the model with the two parabolas tangent at My, <|>y. Therefore, i n developing the energy method for a two parabola model, the use of a simpler model with zero i n i t i a l slopes i s j u s t i f i e d . Secondly, the maximum mid-height moments predicted by such models are not very accurate. It i s s u f f i c i e n t i n design p r a c t i c e , however, to have determined the maximum end moment with reasonable accuracy. For low load l e v e l s the maximum end moments obtained with the models are reasonably close to those obtained for the r e a l column. For higher load l e v e l s (P/P = 0.5), i t was o not possible to obtain reasonable solutions using the models throughout the e n t i r e range of slenderness r a t i o s . For the example presented, slenderness r a t i o s up to 100 could be adequately represented. Attempts to reduce the maximum end moments of the columns with slenderness r a t i o s of 125 and 150 to bring them i n l i n e with the maximum end moments for the r e a l column resulted i n considerable loss of accuracy f or the smaller slenderness r a t i o s . The reason for t h i s i s , that for these slender columns, the moment-curvatures for the enti r e column l i e on the f i r s t parabola of the modelled column, and on the r e a l column, the moment curvature r e l a t i o n i s very close to being l i n e a r over the same moment-curvature range. In chapter 5 i t w i l l be shown that more complex models can be developed which w i l l more accurately model the moment-curvature r e l a t i o n s h i p s . The two parabola model can, however, adequately represent Table 3.3 Comparison of Real and Modelled Column Results Using the Program of Nathan Moment-Curvature P/P Q Column Length Maximum End Maximum Mid-Height Relation (inches) Moment ( f t ) Moment (kip-feet) 189 170.3 204. 378 141.3 158. 567 128.2 149. Real 756 115.8 145. 945 103.4 141. 1134 90.9 136. 189 160.8 171. Two parabolas with 378 149.3 160. d<j>/dM=0 at My, 567 138.2 158. <()y f o r second 0.1 756 124.0 158. parabola. 945 107.8 156. 1134 89.5 156. 189 160.5 171. Two parabolas with 378 149.0 160. slopes tangent at 567 137.9 158. My, <j>y. 756 123.9 156. 945 107.0 156. 1134 89.4 156. 189 421.3 480. 378 321.8 435. Real 567 224.9 390. 756 130.2 345. 945 39.3 305. 1134 0.0 189 380.3 445. Two parabolas with 378 303.7 385. d<|>/dM=0 at My,<j>y 567 221.4 375. for second para- 0.5 756 134.7 300. bola. 945 86.2 190. 1134 59.8 135. 189 377.5 445. Two parbolas with 378 299.6 380. slopes tangent at 567 219.1 370. My, <(>y. 756 134.7 300. 945 86.2 190. 1134 59.8 135. 24.: c e r t a i n classes of column problems, and the complexity of the solution equations obtained using such a model i s i n d i c a t i v e of those which would be obtained using more complex models. Therefore an energy sol u t i o n was developed for the two parabola model with both parabolas having zero i n i t i a l slopes. The assumed shape used for the moment diagram was the same as was used for previous energy solutions. For the two parabola model three d i f f e r e n t cases must be considered; very slender columns with Mm < My, intermediate columns with Me < My < Mm and r e l a t i v e l y short columns with Me > My. For extremely slender columns the entire length of the column w i l l have moments and corresponding curvatures which l i e only on the f i r s t parabola which begins at the o r i g i n . The solution i n t h i s case, where Mm i s less than My, i s i d e n t i c a l to that given by equations 3.16 or 3.17 except that the parameter A i s defined by; A = My2/c|>y. For r e l a t i v e l y short columns the moments and curvatures w i l l f a l l e n t i r e l y within the domain of the second parabola which begins at My, <|>y. In t h i s case where Me i s greater than My, s e t t i n g the f i r s t v a r i a t i o n of the p o t e n t i a l energy equal to zero r e s u l t s i n the following cubic equation: , o 1-1815, , 1485 B -i o r2145, x ? (Mm-Me) 6 + |_5^-(Me-My) - 1 3 1 P L 2 J (Mm-Me)z + [-jr^-(Me-My) z ^ 1485 B My2 3465 B , n , 3465 + 1048 A " i6T^ 2 ( M e- M y ) ] ( M m " M e ) + 2M6 B [(Me-My) 3 + — My2(Me-My)] = 0 - (3.23) where A = My2/<j>y and B = (Mu-My) 2 / ( <|>u-<|>y). This equation can be solved i n a manner s i m i l a r to the single parabola solution previously given. Table 3.4 compares r e s u l t s obtained using equation 3.23 with r e s u l t s obtained using a modified version of Nathan's program. The column section i s that of F i g . 16 with P/P = 0 and a slenderness r a t i o of 25. Table 3.4 Comparison of Nathan's 'Exact' (CDC) Method and the Energy Method for the Two Parabola Model with Me > My. C r i t i c a l Load Maximum Mid-Height (kips) Moment (kip-feet) CDC Method 163.0 171. Energy Method 162.80 171.4 The sol u t i o n f o r columns which have moments and corresponding curva-tures on both parabolas i s long and tedious. A moment diagram with the assumed parabolic moment d i s t r i b u t i o n i s shown i n F i g . 18. The point at which the moment reaches y i e l d i s given by: L _ L (^ My-Mel 2 2 (Mm-Me) Defining the yielded length each side of the mid-height as 3L implies that: 1 . (My-Me) 3 = T / I - ./ „ 3.25) 2 (Mm-Me) from which: Mm-Me = M y M e (3.26) 1-4 e 2 This then shows that g i s a function of Mm. The contribution of the s t r a i n energy and work due to f l e x u r a l short-ening to the p o t e n t i a l energy functional must be obtained by integra t i n g over the yielded portions and the unyielded portions of the column separ-ately and then summing the r e s u l t s . For the f l e x u r a l shortening term the slope dy/dx of the inner or yielded portion of column i s determined from the zero slope boundary condition at the column mid-height. The 26. using a modified version of Nathan's program. The column section i s that of F i g . 16 with P/P = 0 and a slenderness r a t i o of 25. o The s o l u t i o n for columns which have moments and corresponding curvatures on both parabolas i s long and tedious. A moment diagram with the assumed parabolic moment d i s t r i b u t i o n i s shown i n F i g . 18. Table 3.4 Comparison of Nathan's 'Exact' (CDC) Method and the Energy Method for the Two Parabola Model with Me > My. C r i t i c a l Load Maximum Mid-Height (kips) Moment (kip-feet) CDC Method 163.0 171. Energy Method 162.80 171.4 The point at which the moment reaches y i e l d i s given by: L _ L (MjcMBl 2 2 (Mm-Me) Defining the yielded length each side of the mid-height as 3L implies that: 1 (My^Mel p 2 (Mm-Me) from which: Mm-Me = M y ~ M e (3.26) 1-4 e 2 This then shows that 3 i s a function of Mm. The contribution of the s t r a i n energy and work due to f l e x u r a l shortening to the p o t e n t i a l energy functional must be obtained by inte g r a t i n g over the yielded portions of the column separately and then summing the r e s u l t s . For the f l e x u r a l shortening term the slope dy/dx of the inner or yielded portion of column i s determined from the zero slope boundary condition at the column mid-height. The expression thus obtained i s used to evaluate the slopes at the y i e l d points which can then be used as the boundary condition for the unyielded or outer portions of the column. In p r a c t i c e i t i s necessary only to integrate over h a l f the column and double the r e s u l t . The l i m i t s of integration thus introduce 3 into the p o t e n t i a l energy function. In taking the f i r s t v a r i a t i o n of the po t e n t i a l energy i t i s necessary to note that: 9 3 _ = 1-4 3 2 3Mm 8 3(Mm-Me) ' The expression that r e s u l t s from s e t t i n g the f i r s t v a r i a t i o n of the po t e n t i a l energy equal to zero contains high order powers of 3 and i s no longer a simple cubic. I t i s possible to put i t i n t o the form a cubic, however, r e s u l t i n g i n the following equation: r PL 3 f4192_ 11 14 3 32 k 272 5 _ 2432 6 _1600 7 L" ~ T ^51975 _ 30 3 " 45 3 + 3 3 " 25 3 45 3 + 21 3 A . 512 8 Mi?. R9 2048 1 0 1536 n ^ _ PL3. (44 3 32 5 5 P ~ 27 P 25 11 AB 45 25 P 6 .1024 7 2048 8 54784 9 8192 1 0 17408 x l ] 3 + ~21~ 3 + 15 3 " 315 3 75 3 105 * J 2048 45 _ III (1048 8 _ 8192 9 _ 81|2 1 0 3 g | 8 , , } } 3 + { _ ^ o ^ 45 189 225 693 ; J L A z [ i i , _ £ 8 3 64 , _ 416 5 _ 1024 6 2176 7 JI024 3 8 L315 5 15 P 3 P 15 P 15 P 21 P 15 P _ 1024 9 ) _ P L i ,20 3 _ 1£ 5 _ 2048 6 5312 7 2048 8 9 P } AB ^9 45 45 105 45 P - ^ 3 9 ) + 1 - 4 3 + 1 6 3 3 - 3 5 + ¥ 3 ? ) + kGr 33 35 A v35 5 7 B 3 - ^ 3 ^ + |H 37)](Mm-Me)2 + [_ ^ ! M e 2 (IM . i 3 + IP- 33 + 3 , 15 35 , 9 315 3 9 3 A*-96 o 5 128 o G 192 o7>, PL 3 r4 .3 32 o 5 64 L - - 3 5 - — 3 6 + — 3 7 ) - — Me(- 3 3 - - 3 5 + - 3 7 ) + - Me 28. - 43 + f 3 3 - f 35)](Mm-Me) - f 3 5) = o (3.28) A s o l u t i o n to equation 3.28 was obtained using a s i m i l a r program to that used to solve the cubic r e s u l t i n g from the single parabola model. Since 3 must be between zero and one-half, i t was only necessary to have the program solve the cubic with 3 varying betwen zero and one-half i n steps of 0.01. The maximum mid-height moments thus determined were sub-s t i t u t e d into equation 3.25 and the r e s u l t i n g 3 was compared to the assumed value. Agreement between the assumed value for 3 and the value ca l c u l a t e d from equation 3.25 determined the correct value of 3 and thus the correct value for Mm the maximum mid-height moment. The c r i t i c a l load was found by increasing the load at a constant e c c e n t r i c i t y u n t i l a point was reached at which for any value of the assumed 3, the 3 calculated by equation 3.25 would not agree with the assumed 3« The r e s u l t i n g maximum end moments and maximum mid-height moments were again compared to r e s u l t s obtained with a version of Nathan's program, modified so that the CDC's were obtained using the two parabola model. Close agreement of the two methods was obtained as i s shown by the t y p i c a l example presented i n Table 3.5. Table 3.5 Comparison of the CDC Method and the Energy Method f o r the Two Parabola Moment-curvature Relation C r i t i c a l Load (kips) Maximum Mid-height Moment (kip-feet) CDC Method 163.0 158. Energy Method 162.98 158.2 29. Equation 3.28 i s obviously much too combersome to be of any use as a design formula and even the solu t i o n of the cubics f o r the cases where Mm < My and My < Me requires considerable e f f o r t , although the l a t t e r are not unduly d i f f i c u l t i f a programmable c a l c u l a t o r i s used. Attempts at the s i m p l i f i c a t i o n of these equations with the intent of developing a design formula are discussed i n the following chapter. 3.4 B i - l i n e a r Moment-curvature Relation The length of equation 3.28 prompted an attempt at the solu t i o n of a b i - l i n e a r model. It was, however, determined that the length of the equa-t i o n f o r the b i - l i n e a r model which corresponded to equation 3.28 was approximately the same as that of equation 3.28 and the accuracy of the maximum mid-height amounts obtained was not s a t i s f a c t o r y when compared to the r e a l column. Since the b i - l i n e a r curve did not have any advantage over the two parabola model, the e f f o r t s at s i m p l i f i c a t i o n were di r e c t e d at reducing equations 3.16, 3.23 and 3.28 to a managable form. 30. 4. SIMPLIFICATION OF THE ENERGY SOLUTION FOR THE TWO PARABOLA MODEL 4.1 Extremely Slender Columns: Mm < My The procedure f o r solving the cubic equations 3.16 or 3.17, which govern the case where the maximum mid-height moment i s less than the y i e l d moment can be s i m p l i f i e d somewhat by determing a and b of equations 3.18 and 3.19 i n terms of the known and unknown parameters. Some rounding o f f w i l l also lead to a simpler approximate s o l u t i o n . It i s convenient to express a and b as u = a/3 and v = b/2. When the appropriate terms of equation 3.16 are substituted into equations 3.18 and 3.19 and the r e s u l t rounded o f f , the following expressions are obtained for u and v. u = (Me2 + 140 Mpa Me - 460 Mpa2)/32 (4.1) v = (Me3 + 100 Mpa Me2 + 9600 Mpa2 Me - 21000 Mpa3)/392 (4.2) where Mpa = A/PL 2 = My2/<|>yPL2. If u 3 + v 2 i s greater than zero, the column i s unstable and i f u 3 + v 2 = 0, the column i s i n neutral equilibrium. The maximum moment i s determined from the expression; 4TT Mm = m cos (9 + — ) + (170 Mpa - 7 Me)/45 (4.3) where m = 2/-u and 9 = [cos - 1(2v/um)]/3 There i s a constant value for Me/Mpa which w i l l cause equations 4.1 and 4.2 to s a t i s f y the neutral equilibrium c r i t e r i o n ; u 3+v 2=0. Through t r i a l and error s u b s t i t u t i o n t h i s value was determined to be 2.41. Substituting 2.41 Mpa for Me into equation 4.1 and 4.2 indicates that the f i r s t terms i n each equation are r e l a t i v e l y small and may be ignored. The r e s u l t i n g equations are: u = (140 Mpa Me - 460 Mpa2)/32 (4.4) and v = (100 Mpa Me2 + 9600 Mpa 2 Me - 21000 Mpa3)/392 (4.5) The constant value for Me/Mpa which s a t i s f i e s the neutral equilibrium c r i t e r i o n using equations 4.4 and 4.5 i s 2.424, i n d i c a t i n g that the difference i n the c r i t i c a l load predicted by equations 4.4 and 4.5 w i l l be only 0.6% d i f f e r e n t than would be predicted using equations 4.1 and 4.2. I t i s possible to solve (Per e)/Mpa = 2.42 f o r the c r i t i c a l load, however, i t should be noted that for higher load l e v e l s , the two parabola model did not give good r e s u l t s as indicated by Table 3.3. Table 4.1 compares the r e s u l t s using Nathan's program modified so that the two parabola moment-curvature r e l a t i o n was employed to calculate the column d e f l e c t i o n curves with the solutions obtained using the derived cubic expression of equation 3.16 and the s i m p l i f i e d solution of equations 4.3 - 4.5. The column section i s that of F i g . 16 with P/P =0.1 and a slenderness r a t i o of 200. o The y i e l d moment and y i e l d curvature were the same as was used for Table 3.3, 155 Kip-feet and 0.000035 i n - 1 r e s p e c t i v e l y . Table 4.1 Comparison of Solutions Obtained Using the Derived Cubic (Eqn. 3.16) and the S i m p l i f i e d Cubic (Eqns. 4.3-4.5) f o r Mm < My with the CDC Method. C r i t i c a l Load Maximum Mid-height (kips) Moment (Kip-feet) CDC Method 163.0 119. Derived Cubic 163.11 117.3 S i m p l i f i e d Cubic 163.14 117.2 The r e s u l t s confirm that there i s no loss of accuracy using the s i m p l i f i e d solution and further s i m p l i f i c a t i o n s may be pos s i b l e . An attempt was also made to replace the arc-cosine and cosine with the f i r s t few terms of the appropriate series expansion. For the arc-cosine term t h i s appeared to give reasonable r e s u l t s , but the angle for 32. the cosine term can be greater than one, r e s u l t i n g i n a slow convergence. With the advent of the pocket c a l c u l a t o r , expressions containing t r i g n o -metric functions should present no d i f f i c u l t y to a designer and i t was therefore decided to leave the solution i n the s i m p l i f i e d form as presen-ted. 4.2 R e l a t i v e l y Short Columns My < Me The cubic equation 3.23 i s the derived expression f o r the s i t u a t i o n where the y i e l d moment My i s less than or equal to the end moment Me. This equation can be treated i n a manner s i m i l a r to the previous treatment of equation 3.16 to y i e l d : u = (140 Md Mpb - 460 Mpb2 + 30 C My2)/32 (4.6) v = (100 Md 2 Mpb + 9600 Md Mpb2 - 21000 Mpb3 - 320 C My 2 Md + 2100 C My 2 Mpb)/392 (4.7) where Md = Me-My, Mpb = B/PL 2 and C = B/A. Again the column i s unstable i f u 3+v 2 i s greater than zero. The expression for the maximum moment becomes: 4TT Mm = m cos (6 + — ) + (170 Mpb - 7Me + 52My)/45 (4.8) where m = 2/-u and 6 = [cos - 1(2v/um)]/3 as before. Table 4.2 presents the r e s u l t s corresponding to the column section presented i n Table 3.4 but with a length of 10 fe e t . With t h i s length, the s i m p l i f i e d s o l u t i o n to the cubic i s given by equations 4.6, 4.7, and 4.8. Table 4.2 Comparison of Solutions Obtained Using the Derived Cubic (Eqn. 3.23) and the Si m p l i f i e d Cubic f o r Me < My with the CDC Method. C r i t i c a l Load Maximum Mid-height (kips) Moment (Kip-feet) CDC Method 163.0 171. Derived Cubic 162.80 171.4 Sim p l i f i e d Cubic 161.57 171.1 .33. For the very long column case, Mm < My, and the short column case, My < Me, the s i m p l i f i e d solutions of the cubic expressions derived using the p o t e n t i a l energy method are of such accuracy that the errors i n solution are l i k e l y to be n e g l i g i b l e compared to the errors i n modelling the moment-curvature r e l a t i o n s h i p . 4.3 Slender Columns with Me < My < Mm The solution of equation 3.28 governs for a l l columns that cannot be solved using the s i m p l i f i e d expressions given i n sections 4.1 and 4.2. Equation 3.28 i s not a cubic because 3 i s a function of (Mm-Me) and therefore the s i m p l i f i c a t i o n cannot proceed i n the same manner as i t d i d f o r the other cases. Various attempts were made to s i m p l i f y equation 3.28. These can be summaried as follows: 1. Simplify 3 terms 2. Reduce form of equation from cubic to quadratic 3. Simplify expression for work due to f l e x u r a l shortening I 4. Simplify expression for work due to f l e x u r a l shortening II 5. Reduce equation to one unknown and non-dimensionalize. A summary of the r e s u l t s of these s i m p l i f i c a t i o n attempts follows. 4.3.1 Simplify 3 Terms Each of the terms of equation 3.28 includes a function of 3; f o r example the l a s t term i n the equation has the function: 2_ _ 1 3 + ± 3 _ § 5 15 2 P 3 P 5 P These functions of 3 were plotted and simpler functions of 3 which approximated the actual 3 functions were obtained by f i t t i n g curves to the p l o t s . The r e s u l t s are shown i n F i g . 19a-k. 34. These simpler expressions f o r the 3 functions reduced the length of equation 3 .28 to y i e l d : r 4192 PL 3 „ o „q PL 3 n o 536576 PL 3 „ 7 , [- (l - 6 3 + 1 2 3 2 - 8 0 3 ) - — ( 3 2 - 2 3 3 ) - TTq^i ( 3 7 ) J(Mm-Me) 351975 A2 B 2 r 88 PL 3 „ 9 „o 25 PL 3 c 32 L o o , + I- Me (l - 6 3 + 1 2 3 2 - 8 3 3 ) - „, , n Me ( 3 3 - 4 3 b ) + T — ( l - 6 3+12 3 z - 8 3 d ] 315 A2 + f | i | (3 3)](Mm-Me) 2 + [ - ^ p ^ M e 2 (1-63+123 2 -8 3 3 ) - | f - ^ M e 2 ( 3 3 ) A2 16 MeL o T 2 PL 3 o 9 + — (1-43+43 2 ) J (Mm-Me) - — Me3 (1 -43+43 2 ) = 0 (4 .9 ) 15 A J 15 , 9 Ab-solutions using t h i s equation resulted i n loss of accuracy i n the order of 10%. However, equation 4 .9 i s not s i g n i f i c a n t l y easier to solve than equation 3 .28 i n that an i t e r a t i v e type computer program i s necessary to obtain a so l u t i o n . 4 . 3 . 2 Reduce Form of Equation From Cubic to Quadratic The p o s s i b i l i t y that the terms containing (Mm-Me)3 i n equation 4 . 9 might be i n s i g n i f i c a n t when compared to those containing (Mm-Me) was examined with a view to reducing the cubic form of the equation to a quadratic. If Mm-Me i s of the same order as the end moment Me, then the f i r s t term 4192/51975 i s approximately 30% of the corresponding term from the (Mm-Me)2 expression; 8 8 / 3 1 5 . If Mm-Me was su b s t a n t i a l l y smaller than the end moment then t h i s s i m p l i f i c a t i o n would be j u s t i f i e d . However, f o r slender columns, the second order moment Mm-Me may be as large or larger than the end moment at the c r i t i c a l load, and in fact a t r i a l quadratic form of the equation which eliminated the cubic terms i n (Mm-Me) and 35. increased the square terms i n (Mm-Me) to compensate did not give good r e s u l t s . The fa c t that equation 3.28 or the s i m p l i f i e d form 4.9 are not ac t u a l l y cubic expressions because of the dependence of 3 upon (Mm-Me) may have influenced the r e s u l t as well. 4.3.3 Simplify Expression f o r Work Due to Fle x u r a l Shortening I Eight of the eleven terms of equation 3.28 are from the energy term which corresponds to the work done by the a x i a l load when the column shortens as i t bends ( F i g . 13). Equation 3.28 could therefore be s i g n i f i c a n t l y reduced i f a simpler expression could be obtained which would adequately represent the f l e x u r a l shortening terms. One approach, which was t r i e d , was to evaluate the terms i n equation 3.28 which originate from the f l e x u r a l shortening work, W^ , at 3 equal to zero and 3 equal to one-half. The difference between the r e s u l t i n g expressions was then found and i t was postulated that the f l e x u r a l shortening terms could be represented by the value with 3 equal to zero plus some function of 3 times the difference. The r e s u l t i n g expression became lengthy and included higher powers of 3 so that no advantage would be obtained from t h i s approach. 4.3.4 Simplify Expression f o r Work Due to Fle x u r a l Shortening II The deflected shapes of a column section for several d i f f e r e n t lengths were compared to a parabola and found to be reasonably close. It was therefore postulated that a simpler expression for the f l e x u r a l shortening terms could be obtained by assuming a parabolic deflected shape. The r e s u l t i n g external work was found to be approximately 5% greater for the parabolic deflected shape than that obtained with the parabolic moment diagram assumption. The following expression was obtained for Wn : 36. L P , 8 (ymax) 2 W l = -2 / fe)2dx - 3 L ( 4 ' 1 0 ) where ymax i s the maximum mid-height d e f l e c t i o n . Substituting for ymax, however, and putting the resultant expression i n terms of one unknown, 3, resulted i n an expression that was at least as complicated as the o r i g i n a l expression derived using the parabolic moment diagram of F i g . 18 . 4 . 3 . 5 Reduce Equation to One Unknown and Non-dimensionalize Equation 3 .28 can be put i n terms of one unknown, 3, by su b s t i t u t i n g into the equation the expression for Mm-Me given by equation 3 . 2 6 . It i s necessary to multiply through by ( 1 - 4 3 2 ) 3 to eliminate the denominators thus formed. In order to non-dimensionalize the r e s u l t i n g expression, i t i s necessary to also multiply through by A 2/PL 3. The following non-dimensional terms are defined: A e = - ( 4 .12 ) B (4 .13 ) MePL2 Y = — ( 4 .14 ) ' Me When these expressions are substituted i n , the following equation i s obtained: [ - f (B) - ef (0) - e 2f 3 ( 3 ) ] (Y -D 3 + [-f4< 3) - ef 5< 3) + fifg(B) + 6ef 7(0) ] (Y -D 2 + [-f 8(B) " ef 9(B) + <5f 1 Q( 3) ]( Y"D - f = 0 < 4 ' 1 5) where f (3), f (3) etc are functions of 3 given as follows: f ,o, 4192 11 J 4 3 32 ^ _ 272 5 _ 2432 6 ^600 7 512 f l ( 3 ) " 51975 " 30 3 " 45 5 + 3 B 25 B 45 3 + 21 3 + 5 37. 4480 27 59 _ 1048 1 0 + 25 1536 11 i l l 44 o 32 r — 0 3 + — 0 b 45 P 25 P 2048 fi 1024 n l 204* 45 P 21 P 15 2048 o f l 38 54784 315 l9 -8192 75 a o 17408 105 i l l 2048 45 8 _ 8192 9 _ 8192 189 9 _ O J - ^ fllO j . 3 2 7 6 8 g l l 225 693 88 315 6 5 152 2 315 P 16 „o 64 „, 448 — 0^  + — 0 4 -3 P 3 P 15 05 -768 f- 22528 n l 8 b + &' 5 P 105 P 1024 „„ 33280 „ q + 0B - 8 9 -3 63 4096 „ i n 4096 „•>•> 3IO + 3 I I 15 P 9 P 20 3 d -416 45 3 5 -2048 g 6 + 16384 ft7 + 2048 ft8 _ 45 315 8' + 0C I i i 2 09 _ 8192 p l Q 21 45 9216 o l l + - 3 5 - 0 1 1 32 „ 128 „ o „o 512 „ c 6656 n 7 1024 „ q - - 40 - — 8 2 + 32 03 _ _ + _ _ 3 7 . _ _ g9 32 3 = ^ 03 - ~ 3 896 5 8704 7 2048 9 15 P 105 P- 35 P 104 _ 4 315 3 832 „o 116 O o 5024 n U 2624 „ s 896 n f i 68224 0 2 + — 0 3 + T77~ 3 4 - -TT- 3 5 - — 3 6 + 315 315 45 315 2560 18432 „„ „q 2048 „ i n 3072 „ n + 0° - 0 9 - 0 l ° + — — 0H 9 35 9 P 7 4 „ o 64 „ c — 0d - 0 b " P 5 P 1408 35 g 7 _ l£g4 3 9 + ig|4 P L L 21 35 16 „ 128 n 7 128 256 „ u 2432 5 4096 7 1024 8 15 3 + 15 3 ~ 5 3 _ 2_ 3 £ 2 22 3 32 4 208 5 _ 128 6 576 7 _ 2432 8 - 15 " 2 " 5 3 + 3 3 + 5 3 5 3 15 3 + 5 3 15 3 512 38. These functions of 3 were then p l o t t e d as shown i n F i g . 20a-k. A l l of the p l o t s have the same absolute maximum (or minimum ) and are scaled to s u i t . In t h i s manner i t can be seen that several of the functions are, with the a p p l i c a t i o n of a scale factor, nearly i d e n t i c a l . The eleven functions can be reduced to four functions with appropriate scale f a c t o r s . The four groups are: 1) f (3), f.(0)f f,(B)f f (3), f i n ( 3 ) , f„(3)i 1 4 6 8 10 11 2) f 2 ( 3 ) , f 5 ( S ) , f g ( 3 ) ; 3) f 3 ( 3 ) ; 4) f ( B ) . These curves cannot be adequately represented over t h e i r e n t i r e length by either a cubic or quadratic. I t i s possible, however, to approximate portions of the functions with a quadratic or lower order polynomial i n 3« If the curves are segmented for 0 < 3 < 0.2, 0.2 < 3 < 0.4 and 0.4 < 3 < 0.5, reasonable lower order functions can be obtained. Figures 21a-d show representative curves for the four functions with possible approximations fo r the three segments. In t h i s manner i t would be possible to obtain quadratic equations i n 3 which are r e a d i l y solved to y i e l d an approximate value f o r 3 and hence the maximum mid-height moment. There are, however, several disadvantages with the methods as presented. I t i s e a s i l y recognized from the known parameters of a given problem when the end moment, Me, i s greater than the y i e l d moment, My, and equations 4.6 - 4.8 apply. However, i t i s not easy to determine i f the maximum mid-height moment Mm, i s less than My, so that equations 4.3 - 4.5 apply, or i f Mm i s i n the range where a quadratic equation, as determined i n t h i s section, would apply. Four p o s s i b i l i t i e s would then e x i s t : 39. 1) Mm < My 2) 0 < 3 < 0.2 3) 0.2 < 3 < 0.4 4) 0.4 < 3 < 0.5 and i t can only be determined by t r i a l and error which of the four p o s s i b i l i t i e s govern. I t could then be necessary, to solve a l l four i n order to obtain the s o l u t i o n . The quadratic equations obtained by f i t t i n g curves as indicated i n Figs. 21a-d, would also be lengthy and are not very su i t a b l e for a code formula. F i n a l l y , the accuracy of the method i s very dependent upon how well the two parabola moment-curvature curve can model the r e a l curve. 40. 5. MOMENT-CURVATURE MODELS AND THE SENSITIVITY  OF SOLUTIONS TO MODEL VARIATIONS 5.1 Introduction The two parabola model, for which solutions were obtained using the energy method, did not adequately represent a l l columns. I t was shown i n Table 3.3 that the solutions obtained for very slender columns with large a x i a l loads were considerably i n error . For those columns which can be represented by the two parabola moment-curvature model, the y i e l d moment and y i e l d curvature are not e a s i l y determined when considering the r e a l column. It i s then desirable to know how the solutions vary with changes i n the y i e l d moment and curvature. If the solutions are very se n s i t i v e to va r i a t i o n s i n the y i e l d moments and curvatures, then i t becomes much more d i f f i c u l t to achieve good r e s u l t s with any method that uses an approximate moment-curvature r e l a t i o n . For those columns which cannot be represented adequately by the two parabola model, i t i s necessary to use a model which provides a better representation of the r e a l moment-curvature r e l a t i o n . The s e n s i t i v i t y of the two parabola model to v a r i a t i o n s i n the y i e l d moment and curvature i s examined i n the following section. A t h i r d section examines the accuracy and s e n s i t i v i t y of a more su i t a b l e model f o r the moment-curvature r e l a t i o n . 5.2 S e n s i t i v i t y of the Two Parabola Moment-curvature Model In order to evaluate the s e n s i t i v i t y of the two parabola moment-curvature model, i t i s necessary to e s t a b l i s h the c r i t e r i a to be used i n performing the a n a l y s i s . The best possible model w i l l r e s u l t i n some error between the solutions obtained for the modelled column and those obtained f or the r e a l column. In order that the o v e r a l l accuracy remain 41. reasonable, v a r i a t i o n s which r e s u l t only from the model should be kept to within 10% of the best f i t model. The method used to test the s e n s i t i v i t y of the model to changes i n the y i e l d moment and y i e l d curvature was as follows: 1. The two-parabola model which gave the best f i t between the end moments for the r e a l column and the modelled column was obtained using the program of Nathan, following the procedure, outlined i n Section 3.4. 2. For the columns considered, moments were calculated which were 10% larger and 10% smaller than the maximum end moments thus found. 3. The y i e l d moments and y i e l d curvatures were then a l t e r e d i n turn and the program re-run u n t i l y i e l d moments and y i e l d curvatures were found which caused the end moments to agree with the increased or decreased moments as calculated i n step 2. In some cases the y i e l d moment or y i e l d curvature was c u r t a i l e d by the natural boundaries of the moment-curvature r e l a t i o n ; that i s zero moment or curvature and the ultimate moment and curvature. The procedure was c a r r i e d out for two column sections, the Tee section of F i g . 16 and the rectangular prestressed section shown i n F i g . 22. Two load l e v e l s were considered for each column; P/P =0.5 and three slender-o ness r a t i o s were examined for each load l e v e l ; L/r = 25, L/r = 75, L/r = 150. The comparison was based e n t i r e l y on the maximum end moments and i n a design o f f i c e , t h i s would be a l l that would be necessary to s a t i s f y the required safety c r i t e r i a . For the c o l l o c a t i o n method presented i n the following chapter, however, the maximum mid-height moments should also be reasonably accurately determined by the model. Obviously the closer the 42. model moment-curvature curve i s to the r e a l moment-curvature curve the better the p r e d i c t i o n w i l l be for the maximum mid-height moment. The r e s u l t s of the s e n s i t i v i t y tests are shown i n Figs. 23a-c, 24a-c, 25a-c and 26a-c. The figures show the two parabola models which, resulted i n maximum end moments close s t to those obtained for the r e a l column. A rectangular shaded area bounds the region where the movement of either the y i e l d moment or curvature r e s u l t s i n a 10% change i n the maximum end moments predicted by the program. Where material f a i l u r e governs, the program indicated that i t was immaterial which path the moment-curvature r e l a t i o n followed, so long as i t ended at the correct values which caused material f a i l u r e . No attempt was made to determine the shape of the in t e r a c t i o n surface for v a r i a t i o n s of y i e l d moment and y i e l d curvature together. The data presented, however, give a good 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 models. The figures show that the s e n s i t i v i t y of the two parabola model to vari a t i o n s i n y i e l d moment and y i e l d curvature increases as the slender-ness increases and also as load l e v e l increases. This indicates that i t would be very d i f f i c u l t to obtain the correct two parabola model for the moment-curvature r e l a t i o n for very slender columns even f o r the cases with low load l e v e l s where a good two parabola model does e x i s t . I t i s there-fore necessary to examine the s u i t a b i l i t y of other models taking the following into consideration: 1. A moment-curvature model must be found which i s sui t a b l e f o r use with the energy method or one of the other methods described i n Chapter 6, and which w i l l s a t i s f a c t o r i l y model a l l column behaviour regardless of slenderness and load l e v e l . 2. The parameters which describe the model must be e a s i l y determined from the r e a l column properties and must not be overly s e n s i t i v e to v a r i a t i o n s i n these parameters. The primary reason that the two parabola model cannot give accurate r e s u l t s f o r very slender columns with high load l e v e l s i s that the r e a l moment-curvature r e l a t i o n i s very l i n e a r i n the region where most of the columns moments and curvatures l i e , that i s on the f i r s t parabola. Therefore i t i s postulated that a model c o n s i s t i n g of a l i n e a r portion followed by two parabolas would give better r e s u l t s . 5.3 The Straight Line and Two Parabola Model The computer program of Nathan was al t e r e d to use a stra i g h t l i n e and two parabola model for the moment-curvature r e l a t i o n s h i p s . With t h i s model there are four parameters which can be varied; Mya, <|>ya, Myb and <)>yb, see F i g . 27. These parameters were varied systematically u n t i l the solutions obtained for the maximum end moments of the column section of figur e 16 were i n close s t agreement with those obtained for the r e a l column. The r e s u l t i n g maximum end moments and maximum mid-height moments for the modelled column are compared with the r e a l r e s u l t s for load l e v e l s of P/P = 0.1 and P/P = 0.5 i n Table 5.1. It can be seen that a close o o agreement f o r the end moments can be achieved, even f o r the heavily loaded, very slender column. The maximum mid-height moments do not compare as c l o s e l y as the maximum end moments, but the error i s not unduly large. , S e n s i t i v i t y analyses were then performed f o r t h i s model. Results of the analyses using the str a i g h t l i n e and two parabolas to model the moment-curvature r e l a t i o n of the square section of F i g . 22 are presented i n F i g . 28a-c. These r e s u l t s are for a load l e v e l of P/P =0.5 and three o 44. slenderness r a t i o s are considered L/r = 25, L/r = 75, and L/r = 125. A simi l a r analysis for the tee section of F i g . 16 with a load l e v e l Table 5.1 Comparison of Results Obtained Using the Real Moment-curvature Relation with Results Obtained Using a Straight Line and Two Parbola Model Moment-curvature P/P Q Column Length Maximum End Maximum Mid-height Relation (inches) Moment(k-ft) Moment (kip-feet) 189 170.3 204. 378 141.3 158. Real 567 128.2 149. 756 115.8 145. 945 103.4 141. 1134 90.9 136. n i One s t r a i g h t l i n e U . X 189 168.2 182. and two parabolas 378 144.9 174. with d<f>/dM = 0 567 123.3 149. at Mya, <|>ya and 756 112.1 134. Myb, <}>yb for each 945 103.3 127. parabola. 1134 94.6 123. 189 421.3 480. 378 321.8 435. Real 567 224.9 390. 756 130.2 345. 945 39.3 305. 1134 0.0 n u • D One st r a i g h t l i n e 189 422.8 475. and two parabolas 378 309.9 425. with d(J>/dM = 0 567 222.9 365. at Mya, <|>ya and 756 134.5 345. Myb, <}>yb for each 945 40.9 330. parabola. 1134 0.0 — — — of P/PQ = 0.1 and a slenderness r a t i o of 150 i s presented i n F i g . 29. In each of these figures the parameters Mya, <(>ya, Myb and <f>yb were varied separately and the l i m i t which resulted i n a 10% discrepancy i s again delineated by the shaded rectangular areas. The r e s u l t s again show that the s e n s i t i v i t y of the model increases with load l e v e l and slenderness, 45. p a r t i c u l a r l y at Mya, <j>ya. For the rectangular column with a load l e v e l of P / P q =0.5 and a slenderness r a t i o of 125, the r e s u l t s obtained using the model are very s e n s i t i v e to changes i n Mya and <j>ya. I f , however, the moments and curvatures are varied such that they f a l l on the st r a i g h t l i n e passing through the Mya, <|>ya of the o r i g i n a l best f i t model, considerably more l a t i t u t d e i s permitted within the 10% v a r i a t i o n c r i t e r i o n . This i s demonstrated by F i g . 30a-c which shows the r e s u l t s of varying Mya and <j>ya i n the above described manner, maintaining the 10% error l i m i t a t i o n . I t i s therefore very important that the slope of the s t r a i g h t l i n e portion be correct i f the column behaviour i s to be accurately modelled using the strai g h t l i n e and two parabola model. The obtaining of the parameters Mya, <f>ya, etc., from the r e a l column properties i s l e f t to further study, however, a few comments are i n order. F i r s t , the slope of the straight l i n e portion varies with P / P . This i s o because as the load l e v e l increases, the concrete i n the column becomes stressed to a l e v e l where the concrete stress s t r a i n curve i s no longer l i n e a r . It may be possible to account for t h i s by modifying Ec by a factor which i s dependent on P / P . Secondly, the ultimate moment and o ultimate curvature used i n the model w i l l not agree with those obtained using an exact method, such as Nathan's program. This follows.because hand c a l c u l a t i o n s would use the s i m p l i f i e d Whitney stress block assumptions for the concrete stress s t r a i n r e l a t i o n at ultimate. This i s p a r t i c u l a r l y true for tee sections where the error r e s u l t i n g from the stress block assumptions can become s i g n i f i c a n t (see Ref. 10). Values of Myb and <|>yb which are of s u f f i c i e n t accuracy may be obtainable by taking a f r a c t i o n of the ultimate moment and curvature, depending upon P / P . This o i s l e f t to further study. 46. 6. OTHER METHODS 6.1 Introduction The solutions obtained using the energy method and the two parabola moment-curvature model were lengthy and awkward to use as a design procedure. Moreover, i t was shown i n Chapter 5 that i n order to get a reasonably i n s e n s i t i v e model of the moment-curvature curve i t would be necessary to use a model c o n s i s t i n g of one s t r a i g h t l i n e and two parabolas. This would r e s u l t i n an even more complicated expression f o r the s o l u t i o n equation. It was, therefore, recognized that perhaps another method would lead to a simpler solution equation. Other possible methods include the Galerkin method, the l e a s t squares method, c o l l o c a t i o n and f i n i t e d i f f e r e n c e s . The Galerkin method can be reduced to the same expression obtained using the energy method and i s therefore not persued here. The method of c o l l o c a t i o n and the f i n i t e difference method are fundamentally d i f f e r e n t than the energy method i n that they are applied to the d i f f e r e n t i a l equations rather than to the f u n c t i o n a l , and attempts were made to apply these methods to the problem. 6.2 The Method of Collocation The mathematical technique known as the method of c o l l o c a t i o n i s described i n many calculus texts, for instance Boyce and DiPrima (15). It ' s a p p l i c a t i o n to e l a s t i c i t y problems i s b r i e f l y discussed by Sokolnikoff (16). For the slender column problem under consideration, the procedure i s as follows: 1. Assume a deflected shape for the column i n terms of one or more parameters which r e f l e c t the desired behaviour. 2. E s t a b l i s h a moment-curvature r e l a t i o n s h i p for the column. 47. 3. S u b s t i t u t e t h e above terms i n t o t h e g o v e r n i n g d i f f e r e n t i a l e q u a t i o n f o r t h e column a t s p e c i f i c p o i n t s a l o n g t h e column l e n g t h , one p o i n t f o r each p a r a m e t e r o f t h e d e f l e c t e d shape i n s t e p one. 4. Set t h e r e s i d u a l from t h e d i f f e r e n t i a l e q u a t i o n e q u a l t o z e r o a t each p o i n t . A s e t o f s i m u l t a n e o u s e q u a t i o n s f o l l o w s w h i c h can be s o l v e d f o r t h e unknown p a r a m e t e r s . The l o a d w h i c h c a u s e s i n s t a b i l i t y can be d e t e r m i n e d by s e t t i n g t h e d e t e r m i n a n t o f t h e c o e f f i c i e n t s from t h e s e t o f e q u a t i o n s e q u a l t o z e r o o r by r e c o g n i z i n g t h e c r i t e r i o n w h i c h causes t h e e q u i l i b r i u m s o l u t i o n t o "blow up". The g o v e r n i n g d i f f e r e n t i a l e q u a t i o n f o r a column u n d e r g o i n g s m a l l d e f l e c t i o n s i s : d 2 d 2 y d dy — ( E I — * ) + — ( P r * ) = 0 6.1) 2 * 2 dx dx dx^ dx F o r a c o n s t a n t a x i a l l o a d t h i s r e d u c e s t o d 2 f d 2 y d 2 y — ( E I — * ) + P — - = 0 (6.2) d x 2 d x 2 d x 2 But d 2 y / d x 2 i s c u r v a t u r e and E I d 2 y / d x 2 i s t h e moment, t h e r e f o r e t h e d i f f e r e n t i a l e q u a t i o n can be e x p r e s s e d a s : + P<j> = 0 (6.3) d 2M d x 2 The a c c u r a c y o f t h e method i s g e n e r a l l y dependent upon t h e number o f p a r a m e t e r s i n t h e assumed d e f l e c t e d shape which i s e q u a l t o t h e number o f p o i n t s a t w h i c h t h e c o l l o c a t i o n i s p e r f o r m e d . More p a r a m e t e r s r e s u l t s i n a more c o m p l i c a t e d e x p r e s s i o n , t h e r e f o r e some a c c u r a c y w i l l have t o be s a c r i f i c e d f o r t h e sake o f s i m p l i c i t y . F o r t h i s r e a s o n a two parameter f u n c t i o n was s e l e c t e d . I n o r d e r t o m a i n t a i n symmetry, t h e f u n c t i o n chosen must be even and must be a complete p o l y n o m i a l . The p r o c e d u r e p r e v i o u s l y a d opted, o f assuming an e x p r e s s i o n f o r t h e b e n d i n g moment and d e r i v i n g 48. there from a deflected shape, was used again. The bending moment was assumed to have the form: M = Ax"* + Bx3 + Cx 2 + Dx + E (6.4) which upon substitution of the boundary conditions yields: M = 16 (Mm-Me) [ © 2 - 2 © M ^ ] + DL [ £ ) - 5 ( ^ ) 2 + 8 (^) 3-4 ] + Me (6.5) The second derivative of the moment with respect to x becomes: d£M m 16 (Mm-Me) [ 2 . 1 2 ( X ) + 1 2 ^ ) 2 J + £ [ - 1 0 + 4 8 £)-48 ( J ) 2 ] (6.6) dx 2 L 2 L L L L L The two unknown parameters are then Mm and D. The collocation must be performed at two points in order to obtain two simultaneous equations in the two unknowns, and the logical points to choose would be the end point x = 0 and the mid-height, x = L/2. Evaluating equation 6.6 at these two points results in: 32(Mm-Me) 10D d2M and dx 2 d2M dx 2 (6.7) x=0 L 2 -16(Mm-Me) 2D L R 0 , = + — (6.8) L 2 L x=- L z It i s then necessary to evaluate the curvature <j> at the two points in order that equation 6.3 may be satisfied at the two points. In Chapter 5 i t was shown that the straight line and two parabola moment-curvature relation of Fig. 27 could adequately model real moment-curvature curves. The collocation method was therefore applied with this assumed model for the moment-curvature. The straight line portion of the model, when Mm < M v A ' c a n represent a linear elastic problem. In this case equation 6.3 becomes: 32(Mm-Me) 10D PMe n ,„ n x at x = 0, 1 — + — — = 0 (6.9) T 2 L A 49. L -16(Mm-Me) 2D PMm and at x = —, + ~— + —-— = 0 2 T2 L A Solving these two equations simultaneously r e s u l t s i n the following expression for the maximum mid-height moment: -48 . P> ' L 2 (6.10) .  A Mm (•48 _ 5P^ \ 2 A > Me (6.11) The c r i t i c a l load can be recognized by s e t t i n g the denominator equal to zero, which r e s u l t s i n : Per =9.6 — T.2 (6.12) The accuracy of t h i s equation can be recognized by comparing 9.6 to IT*. Substituting equation 6.12 into 6.11 r e s u l t s i n the following equation: rl+0-2 P/Pcr>| Mm = I—•, „ J Me (6.13) v 1-P/Pcr J A comparison of the moment magnification factor thus determined by equation 6.13 with the exact secant formula i s given i n Table 6.1. Table 6.1 Comparison of Moment Magnification Equation Obtained Using C o l l o r a t i c with Exact Secant Formula P/Pcr Equation 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.10 (•1+0 .2P/Pcr -j 1-P/Pcr 1.133 1.300 1.514 1.800 2.200 2.800 5.800 00 ~ ( * f e ) 1.137 1.310 1.533 1.832 2.252 2.884 6.058 CO Comparing these r e s u l t s with the r e s u l t s f o r the energy method and the ACI formula presented i n Table 3.1 indicates that the c o l l o c a t i o n equation i s not as accurate as the equation obtained from the energy method, but i s 50. more accurate than the ACI equation. Equation 6.12 then provides an upper l i m i t for the load on a column modelled using the str a i g h t l i n e and two parabola moment-curvature curve. At loads higher than t h i s , the column f a i l s due to i n s t a b i l i t y no matter how small the e c c e n t r i c i t y of load a p p l i c a t i o n i s . For loads below t h i s l e v e l , the i n s t a b i l i t y load w i l l depend on the e c c e n t r i c i t y of the loading. For the single s t r a i g h t l i n e and two parabola moment curvature model curve, there are f i v e d i f f e r e n t cases to consider when the load i s less than the c r i t i c a l load: Case 1 : Me < Mya, Mya < Mm < Myb Case 2 : Me < Mya, Myb < Mm Case 3 : Mya < Me < Myb, Mya < Mm < Myb Case 4 : Mya < Me < Myb, Myb < Mm Case 5 : Myb < Me, Myb < Mm The derivation for the f i r s t case where the end moment l i e s on the st r a i g h t l i n e and the maximum mid-height moment l i e s on the f i r s t parabola of the moment-curvature model curve i s presented as an example. For t h i s case; Me at x = 0, <|> = — (6.14) where A = Mya/ttya, and L , (Mm-Mya)2 , . r ,_. at x = -, <j> = - + <}>ya (6.15) 2 B where B = (Myb-Mya) 2/(<f>yb-<|>ya) . The two simultaneous equations obtained when the re s i d u a l from the governing d i f f e r e n t i a l equation (Eqn. 6.3) i s set equal to zero are; . t i - 0 , 3 2 ( M m - M 6 ) - + P ^ = 0 (6.16) L 2 L A L -16(Mm-Me) 2D P(Mm-Mya)2 PMya and at x = -, + — + — ~ + = 0 (6.17) 2 o L B A L 51. Solving equations 6.16 and 6.17 for the maximum mid-height moment r e s u l t s i n the following equation for Mm: Mm = + Mya - /[ +MyaJz - ~ Mya z - ~(Mya+—J (6.18) 5PL 2 5PL 2 PL 2 Substituting for <|>ya, and s e t t i n g Mb = 24B/5PL 2 r e s u l t s i n the following s i m p l i f i e d form of equation 6.18: / 9 BMe Case 1, Mm = Mya + Mb-/Mbz + 2Mb(Mya-Me) - Bc|)ya - — — (6.19) DA Solutions f or the other four cases were found i n a s i m i l a r manner, r e s u l t i n g i n the following equations; Case 2; , o CMe Mm = Mya + Mc - /Mcz + 2Mc(Myb-Me)-C<)>yb - 7^- (6.20) Case 3; . I 6B<bya (Mya-Me)z Mm = Mya + Mb - /Mbz + 2Mb(Mya-Me) - — g — - ~ (6.21) Case 4; o •tya C 9 Mm = Myb + Mc -/ Mc z + 2Mc(Myb-Me)-C(<|>yb+-Lg— )-~(Mya-Me) ^  (6.22) Case 5; . o 6Cd)yb (Myb-Me)^ Mm = Myb + Mc - /Mcz + 2Mc(Myb-Me) - — f — - = (6.23) where Mc = 24C/5PL 2. These equations are equilibrium equations and i f the load and e c c e n t r i c i t y are such as to cause the column to become unstable, then the terms within the square root become negative which implies that the maximum moment i s represented by a complex number. This then i s the "blowing up" c r i t e r i o n f o r these formulas. The procedure used to v e r i f y the accuracy of these formulas was as follows: 52. Solving equations 6.16-and 6.17 f o r the maximum mid-height moment r e s u l t s i n the following equation for Mm: 24B ,r2AB ->, 48 BMe 9 B, Me >, Mm = + Mya - /( +Mya J z - Mya z - — (Mya+— J (6.18) 5PL 2 5PL 2 PL 2 Substituting for cbya, and s e t t i n g Mb = 24B/5PL 2 r e s u l t s i n the following s i m p l i f i e d form of equation 6.18: . , BMe Case 1, Mm = Mya+Mb-/Mbz + 2Mb(Mya-Me) - Bttya - "g^ - (6.19) Solutions for the other four cases were found i n a s i m i l a r manner, r e s u l t i n g i n the following equations; . 0 CMe Case 2, Mm = Mya+Mc-/Mcz + 2Mc(Myb-Me)-C<|>yb - "g^ - (6.20) . , 6B<bya (Mya-Me)z Case 3, Mm = Mya+Mb-/Mbz + 2Mb(Mya-Me) - — g — - g (6.21) Case 4, Mm = Myb+Mc-/Mcz + 2Mc(Myb-Me)-C( <|>yb+—r~)-—(Mya-Me) z (6.22) . I 6Cd>yb (Myb-Me)z Case 5, Mm = Myb+Mc-/Mcz + 2Mc(Myb-Me) - f - " (6.23) where Mc = 24C/5PL 2. These equations are equilibrium equations and i f the load and e c c e n t r i c i t y are such as to cause the column to become unstable, then the terms within the square root become negative which implies that the maximum moment i s represented by a complex number. This then i s the "blowing up" c r i t e r i o n for these formulas. The procedure used to v e r i f y the accuracy of these formulas was as follows: 54. Table 6.2 Comparison of Solutions Obtained Using C o l l o c a t i o n with Solutions Obtained Using Nathan's Program for Both the Modelled and Real Moment-Curvature Curves - Tee Section P/Po =0.1 Solution Method Slenderness Ratio F a i l u r e Load (kips) Maximum End Moment (kip-feet) Maximum Mid-height Moment (kip-feet) C o l l o c a t i o n 158.7 163.8 179.1 Exact (CDC) Modelled 25 163.0 168.2 182. Exact (CDC) Real 163.0 170.3 204. C o l l o c a t i o n 158.3 140.7 172.3 Exact (CDC) Modelled 50 163.0 144.9 174. Exact (CDC) Real 163.0 141.3 158. C o l l o c a t i o n 161.4 122.1 144.7 Exact (CDC) Modelled 75 163.0 123.3 149. Exact (CDC) Real 163.0 128.2 149. C o l l o c a t i o n 161.0 110.7 130.5 Exact (CDC) Modelled 100 163.0 112.1 134. Exact (CDC) Real 163.0 115.8 145. Co l l o c a t i o n 160.6 101.7 123.9 Exact (CDC) Modelled 125 163.0 103.3 127 Exact (CDC) Real 163.0 103.4 141. Co l l o c a t i o n 160.5 93.1 120.2 Exact (CDC) Modelled 150 163.0 94.6 123. Exact (CDC) Real 163.0 90.9 136. 55. Table 6.3 Comparison of Solutions Obtained Using Co l l o c a t i o n With Solutions Obtained Using Nathan's Program f o r Both the Modelled and Real Moment-Curvature Curves - Tee Section P/PO = 0.5 Solution Method Slenderness Ratio F a i l u r e Load (kips) Maximum End Moment (kip-feet) Maximum Mid-height Moment (kip-feet) C o l l o c a t i o n 790.3 410.0 470.5 Exact (CDC) Modelled 25 814.9 422.8 475. Exact (CDC) Real 814.9 421.3 480. C o l l o c a t i o n 799.2 303.9 407. Exact (CDC) Modelled 50 814.9 309.9 425. Exact (CDC) Real 814.9 321.8 435. C o l l o c a t i o n 796.3 217.8 347.7 Exact (CDC) Modelled 75 814.9 222.9 365. Exact (CDC) Real 814.9 224.9 390. C o l l o c a t i o n 803.3 132.6 326.6 Exact (CDC) Modelled 100 814.9 134.5 345. Exact (CDC) Real 814.9 130.2 345. C o l l o c a t i o n 807.7 40.5 316.9 Exact (CDC) Modelled 125 814.9 40.9 330. Exact (CDC) Real 814.9 39.3 305. A l l Methods 150 Load exceeded c r i t i c a l load 56. Table 6.4 Comparison of Solutions Obtained Using Co l l o c a t i o n with Solutions Obtained Using Nathan's Program for Both the Modelled and Real Moment-Curvature Curves - Square Section P/Po = 0.5 Solution Method Slenderness Ratio F a i l u r e Load (kips) Maximum End Moment (kip-feet) Maximum Mid-height Moment (kip-feet) C o l l o c a t i o n Exact (CDC) Modelled 25 339.7 348.0 92.5 94.8 104.1 (Material f a i l u r e governs) 104.1 Exact (CDC) Real 348.0 93.8 104.1 Co l l o c a t i o n 342.9 73.4 98.9 Exact (CDC) Modelled 50 348.0 74.5 100. Exact (CDC) Real 348.0 71.0 93. C o l l o c a t i o n 339.5 50.7 83.5 Exact (CDC) Modelled 75 348.0 52.0 89. Exact (CDC) Real 348.0 51.5 87. C o l l o c a t i o n 342.9 29.7 77.5 Exact (CDC) Modelled 100 348.0 30.1 82. Exact (CDC) Real 348.0 28.7 82. C o l l o c a t i o n 345.4 7.0 74.8 Exact (CDC) Modelled 125 348.0 7.1 78. Exact (CDC) Real 348.0 7.1 68. A l l Methods 150 Load exceeds c r i t i c a l load 1. A single s t r a i g h t l i n e and two parabola moment-curvature model which resulted i n the best solutions when compared to the r e a l column r e s u l t s was obtained i n the manner described i n Section 5.3. 2. The e c c e n t r i c i t i e s corresponding to the maximum end moments determined by Nathan's program were calculated for various slenderness r a t i o s . 3. These e c c e n t r i c i t i e s and the parameters from the modelled moment-curvature r e l a t i o n were used i n the appropriate equation 6.19-6.23 with an a x i a l load that was increased u n t i l the load l e v e l was such that i n s t a b i l i t y f a i l u r e was predicted as previously described. 4. The f a i l u r e load thus obtained was compared to the s p e c i f i c load l e v e l which was used to obtain the moment-curvature model. If close agreement i s achieved then the parameters used to describe the model are j u s t i f i e d . The r e s u l t s of applying t h i s procedure to the column sections of F i g . 16 and F i g . 22 are shown i n Tables 6.2, 6.3 and 6.4 Two d i f f e r e n t load l e v e l s ; P/Po = 0.1 and P/Po = 0.5, are presented for the tee section ( F i g . 16) and one load l e v e l ; P/Po = 0.5 i s presented for the square section ( F i g . 22). The slenderness r a t i o s were varied from 25 to 150 i n steps of 25. The maximum mid-height moments thus determined were within approxi-mately 12% of the maximum mid-height moments for the r e a l columns. Therefore, i n order to maintain an o v e r a l l accuracy of 20%, which would be a considerable improvement over the ACI method, the moment-curvature model must be determined with s u f f i c i e n t accuracy that the o v e r a l l error does not increase by more than about 8%. The f a i l u r e loads predicted by the c o l l o c a t i o n method were within approximately 6% of the f a i l u r e load l e v e l 57. s p e c i f i e d for the r e s u l t s obtained using Nathan's program and the modelled and r e a l moment-curvature curves. Thus the formulas obtained by the c o l l o c a t i o n method (equation 6.12 and 6.19-23) can adequately p r e d i c t r e a l column behaviour provided a reasonably accurate model can be obtained f o r the r e a l column moment-curvature r e l a t i o n s h i p . 6.3 The F i n i t e Difference Method The f i n i t e difference method i s another mathematical technique which can be applied to solve p a r t i a l d i f f e r e n t i a l equations. For d e t a i l s of the method, the reader i s re f e r r e d to publications on applied mathematics such as Cairns (17) and Buckingham (18). The accuracy of the method increases with the number of points at which the difference equations are written. For the slender column problem under consideration, the f i n i t e d i f f e r e n c e method can be applied to the d i f f e r e n t i a l equation 6.3 and the n o n - l i n e a r i t i e s can once again be taken into account by taking a single s t r a i g h t l i n e and two parabola moment-curvature curve as an approximation to the r e a l curve. For the stra i g h t l i n e portion, where the solution represents the li n e a r e l a s t i c case, writing c e n t r a l difference equations at the column end and at the mid-height r e s u l t s i n the following two equations •Mm-2Me (L/ 2 ) 2 a t x - 0 , ( - M m - 2 M e + M m ) + ^  = 0 (6.24) and at x = L/2, + ™ S = 0 ( 6 . 2 5 ) (1/2)2 A It so happens that equation 6.24 i s independent of the maximum mid-height moment Mm. Solving equation 6.25 for Mm r e s u l t s i n ; 8Me/L2 Mm = x (6.26) 8/L 2 - P/A from which the c r i t i c a l load becomes; Per = — (6.27) L 2 58. Substituting equation 6.27 into equation 6.26 res u l t s i n the following equation: Mm = 1-P/Pcr M e ( 6 , 2 8 ) The moment magnification factor thus determined i s i d e n t i c a l to the ACI formula. Comparing ir^ to 8 i n equation 6.27 indicates that the accuracy of the c r i t i c a l load thus determined would not be very good. However, increasing the number of points at which the difference equations are to be written r e s u l t s i n some problems. I f an addit i o n a l point or points are taken such that the points are equally spaced along the member, then i t i s not known on which portion of the moment-curvature model curve such points would l i e . An a l t e r n a t i v e would be to write the equations at the y i e l d points Mya, <|>ya and Myb, <(>yb where applicable, as well as at the end and mid-height. The unknown i n the equation then becomes the distance at which the y i e l d moment occurs. However, looking at Case 4 from the previous section as an example, where Mya < Me < Myb, and Myb < Mm, i f the loca t i o n of the y i e l d point, Myb, on the column was very close to either the end moment or the maximum mid-height moment, the i n c l u s i o n of the extra difference equation would have l i t t l e e f f e c t on the accuracy. An attempt was made to solve t h i s problem by adding addi t i o n a l points halfway between the end moment and the y i e l d moment and halfway between the y i e l d moment and the maximum mid-height moment. The r e s u l t i n g equations became very complex and would therefore r e s u l t i n a formula which would be impractical for design purposes. As a r e s u l t a p r a c t i c a l solution using f i n i t e differences was unobtainable. 59. 7. CONCLUSIONS 7.1 Summary The purpose of t h i s thesis was to attempt to obtain a workable design formula which would solve the slender prestressed column problem where f a i l u r e was i n i t i a t e d by i n s t a b i l i t y rather than by material f a i l u r e . Three d i f f e r e n t approximate methods were applied to the problem; the energy method, the method of c o l l o c a t i o n and the f i n i t e difference method. Solutions were obtained for the energy method assuming a parabolic (sing l e parameter) shape f o r the moment diagram. The n o n - l i n e a r i t i e s of the problem were introduced by the assumption of a moment-curvature r e l a t i o n s h i p c o n s i s t i n g of two parabolas. This model for the moment-curvature curve could not adequately represent a l l column load and slenderness r a t i o combinations, but would be suitable i n a considerable number of cases. The r e s u l t i n g equations predicted the maximum mid-height moment and the c r i t i c a l load very accurately when solutions were compared to those obtained using an 'exact' computer method with the same moment-curvature assumptions. The equations obtained, however, were very lengthy and were thus unsuitable for a design formula. Attempts to simplify the equations were only p a r t i a l l y successful and the equations thus obtained were only marginally easier to use. The s e n s i t i v i t y of the solutions to changes i n the modelled moment-curvature shape was investigated, and i t was found that the s e n s i t i v i t y increased with load and with slenderness, thus a heavily loaded slender column was very se n s i t i v e to va r i a t i o n s i n the model. A model moment-curvature r e l a t i o n s h i p c o n s i s t i n g of one s t r a i g h t l i n e followed by two parabolas was proposed and i t was shown that t h i s model could adequately represent a l l load and slenderness r a t i o combinations. Moreover, i t was 60. shown that i f the slope of the straight l i n e portion of the model accur-ately matched the slope of the i n i t i a l portion of the r e a l moment-curva-ture r e l a t i o n s h i p , the s e n s i t i v i t y of the l o c a t i o n of the y i e l d moment terminating the s t r a i g h t l i n e portion was considerably reduced. A two parameter assumed moment diagram was used to obtain a solution using the c o l l o c a t i o n method. The single s t r a i g h t l i n e and two parabola moment-curvature curve was used to introduce the appropriate n o n - l i n e a r i -t i e s into the problem. The r e s u l t i n g equations were not as accurate as the energy method, however, they were s u f f i c i e n t l y accurate for a design formula. The equations obtained were much shorter and easier to apply than those obtained using the energy method and i t would appear f e a s i b l e to use these equations i n a workable design method. An attempt was also made to apply a f i n i t e difference s o l u t i o n to the problem, but, i t was shown that for a mesh of two, the solutions were lacking i n accuracy. For a denser mesh, i t was determined that the solu-t i o n would be very complex and thus unsuitable for a working design formula. 7.2 Recommendations The c o l l o c a t i o n method resulted i n the most suitable equations for use as design formulas, but, the accuracy of the method depends on a close agremeent between the single straight l i n e and two parbola moment-curva-ture model and the r e a l moment-curvature curve. Additional research i s required to determine the appropriate method(s) by which an acurate moment-curvature model can be obtained from the r e a l column properties. F i n a l l y i t should be mentioned that the complete development of such a method may be a rather academic exercise since computers are becoming more and more av a i l a b l e i n the design o f f i c e and exact computer solutions can e a s i l y be obtained at l i t t l e cost. 61. BIBLIOGRAPHY 1. MacGregor, J.G., " S t a b i l i t y of Reinforced Concrete Building Frames," State of Art Paper No. 1, Technical Committee 23, Proceedings of  the Internaional Conference on Planning and Design of T a l l  B u ildings, V o l . 3, American Society of C i v i l Engineers, New York, 1973, pp. 517-536. 2. Chandwani, R., and Nathan, N.D., "Precast Prestressed Sections Under A x i a l Load and Bending," PCI Journal, V o l . 16, No. 3, May-June 1971, pp. 10-19. 3. Nathan, N.D., "Slenderness of Prestressed Concrete Beam-Columns," PCI Journal, V ol. 17, No. 6, Nov.-Dec, 1972, pp. 45-47. 4. MacGregor, J.G., Breen, J.E., and Pfrang, E.O., "Design of Slender Concrete Columns," Journal of the American Concrete I n s t i t u t e , Vol. 67, January 1970, pp. 6-28. 5. Johnston, B.G.(Ed.), Guide to S t a b i l i t y Design C r i t e r i a f o r Metal Structures, 3rd Ed., John Wiley & Sons, New York, 1976. 6. Nathan, N.D., " A p p l i c a b i l i t y of ACI Slenderness Computations to Prestressed Concrete Sections," PCI Journal, V ol. 20, No. 3, May-June 1975, pp. 68-85. 7. MacGregor, J.G. and Hage, S.E., " S t a b i l i t y Analysis and Design of Concrete Frames," Proceedings, ASCE. V o l . 103, No. ST10, Oct. 1977, pp. 517-554. 8. Timoshenko, S.P. B u l l . Polytech. Inst., Kiev, 1909. 9. Timoshenko, S.P. and Gere, J.M., Theory of E l a s t i c S t a b i l i t y , McGraw-H i l l , New York, 1961. 10. Park, R., and Paulay, T., Reinforced Concrete Structures, John Wiley, New York, 1975. 11. MacGregory, J.G., Oelhafen, V.H., and Hage, S.E., "A Re-examination of the EI Value for Slender Columns," Reinforced Concrete  Columns, (SP.50), American Concrete I n s t i t u t e , Detroit 1975, pp. 1-40. 12. Nathan, N.D., Unpublished Report to the Column Committee of the Precast Concrete I n s t i t u t e , 1981. 13. PCI Design Handbook - Precast Prestressed Concrete, 2nd Ed., Prestressed Concrete I n s t i t u t e , Chicago, 1978, pp. 14. CEB/FIP Manual of Buckling and I n s t a b i l i t y , Construction Pres., New York, 1978. 62. 15. Boyce, W.E., and DiPrima, Elementary D i f f e r e n t i a l Equations and Boundary Value Problems, 3rd Ed., John Wiley & Sons, New York, 1977. 16. Sokolnikoff, I.S., Mathematical Theory of E l a s t i c i t y , 2nd Ed., McGraw-Hill, New York, 1956. 17. Cairns, E.J., Mathematics for Applied Engineering, Prentice H a l l , New Jersey, 1967. 18. Buckingham, R.A., Numerical Methods, Pitman and Sons; London, 1957. F i g . 2 Load Moment Interaction Diagram Indicating Material F a i l u r e and S t a b i l i t y F a i l u r e -p 01 I ft • H 4J C QJ e o s 90 i 80 -I 70 60 50 40 A 30 A 20 10 Range of moment/curvature f or slenderness r a t i o = 25 Range of moment/curvature f o r slenderness r a t i o = 25 Curvature ( i n . - 1 x 10~ 4) F i g . 3 Typical Moment Curvature Relationships 2700 0 200 400 600 800 1000 End Moment (kip - feet) F i g . 4 Slender Column Interaction Curves for Square Prestressed Concrete Column (Ref. End Moment (kip - feet) CTi F i g . 5 Slender Column Interaction Curves for Prestressed Double Tee Section (Ref. 12) 67. F i g . 6 Ty p i c a l Moment Diagram f o r a Pin-ended Column with Equal End Moments 50 H w Moment (kip - feet) F i g . 7 Real Moment Diagram for a Square Prestressed Concrete Column with P/PQ= 0.05 and a Slenderness Ratio of 25 Compared to a Parabolic Moment Diagram. 17 Me 18 19 20 21 22 23 24 25 26 Mm 27 Moment (kip - feet) F i g . 8 Real Moment Diagram for a Square Prestressed Concrete Column with P/P0-- 0.05 and a Slenderne Ratio of 150 Compared to a Parabolic Moment Diagram. Moment (kip - feet) ,^ o F i g . 9 Real Moment Diagram for a Square Prestressed Concrete Column with P/P 0 = 0.7'. and a Slenderness Ratio of 75 Compared to a Parabolic Moment Diagram. 110 1 Curvature ( i n ~ l x 10 - 4) F i g . 10 T y p i c a l Moment-curvature Relationships f o r a Prestressed Concrete Column Subjected to D i f f e r e n t A x i a l Loads. Curvature F i g . 11a Straight Line Moment-curvature Relationship 73. F i g . 12a B i l i n e a r Moment-curvature Relationship F i g . 12b Two Parabola Moment-curvature Relationship 74. dx: dy< x=L dv_ dx x=0-I— F l e x u r a l shortening L Note: dy dx w 5 = -dy x=0 dx x=L -2Me dy_ dx x=0 (with small angle assumptions) F i g . 13 Work Due to External Load on an E c c e n t r i c a l l y Loaded Pin-ended Column. 60 80 100 120 140 160 180 200 210 220 230 240 260 Maximum Mid-height Moment (kip - feet) F i g . 14 Typ i c a l Energy Solution f o r the Maximum Mid-height Moment as Determined for a Parabolic Moment-curvature Relationship for a Column Loaded at Constant E c c e n t r i c i t y . r-80 located on p l a s t i c centroid F i g . 16 Prestressed Concrete Tee Section; 400 PSI Prestress 250 H i , i 1 1 1 , 1 1 0 cj)y 5 10 15 20 25 30 Curvature ( i n - 1 x 10 - 4) F i g . 17a Real Moment-curvature Curve and Two Parabola Moment-curvature Model for Prestressed Tee Section (Fig. 16); P / P q = 0.1. 600 Real moment-curvature r e l a t i o n F i g . 17b Real Moment-curvature Curve and Two Parabola Moment-curvature Model f o r Prestressed Tee Section (Fig. 16); P/P = 0.5. 10 80. Fig.."18 Parabolic Moment Diagram With Y i e l d Point 0 0.1 0.2 0.3 0.4 0.5 Beta 0,.'2 -, 0.16 -J 0 0.1 0.2 0.3 0.4 0.5 'Beta 19a and 19b Beta Functions f o r Two Parabola Energy Solution 82. 0.5 - i 0 0.1 0.2 0.3 0.4 0.5 Beta 0.1 - l ' 0.08-1 0.06 H Beta F i g . 19c and 19d Beta Functions for Two Parabola Energy Solution 83. 84. 1.0 -, 0 0.1 0.2 0.3 0.4 0.5 Beta 0.5 -, 0.4 H 0 0.1 0.2 0.3 0.4 0.5 Beta F i g 19g and 19h Beta Functions f o r Two Parabola Energy Solution 85. 0.2 -, 0.16H F i g . 19i and 19j 86. 0.2 -. 0.16 H 0 0.1 0.2 0.3 0.4 0.5 Beta F i g . 19k Beta functions for Two Parabola Energy Solution 87. 0.1 +j 0.075 H XI 4-1 0.05 H 0.025 -I T T T 0 0.1 0.2 0.3 0.4 0.5 Beta F i g . 20a and 20b Beta Functions f o r Non-Dimensionalized Two Parabola Energy Solution. 88. X i ro o:o8i n 0.065 _ J 0.048 J 0.032 -J 0.016 -4 0.5 4-> Q) XI 0.279 0.224 0.168 0.112 -I 0.056 H 0.2 0.3 Beta F i g . 20c and 20d Beta Functions f o r Non-Dimensionalized Two Parabola Energy Solution. 89. F i g . 20e and 20f Beta Functions f o r Non-Dimensionalized Two Parabola Energy Solution. 90. 0.197' —i 0.156 0.118 J 0.079 J 0.039 J 1 1 1 0 0.1 0.2 0.3 0.4 0.5 Beta 0.33' 0.264 J 0.198 J 0.132 0.066 0.0 Fi g . 20g and 20h Beta Functions f o r Non-Dimensionalized Two Parabola Energy Solution. 91 . 0.13 _ 0.104 -H I O H is ~ 0.078 -0.052 -J 0.026 -\ 0 0 0.1 0.2 0.3 0.4 0.5 Beta Beta F i g . 20i and 20j Beta Functions f o r Non-Dimensionalized Two Parabola Energy Solution. F i g . 20k Beta Function f o r Non-Dimensionalized Two Parabola Energy Solution. 93. F i g . 21a and 21b Lower Order Polynomials F i t t e d to Beta Function Curves 1) and 2 ) . 94. F i g . 21c and 21d Lower Order Polynomials F i t t e d to Beta Function Curves 3) and 4). 95. •Prestressing Strand, As = 0.197 i n co 12" F i g . 22 Square Prestressed Section; 400 PSI Prestress. 28 Curvature (in x 10" ) F i g . 23a S e n s i t i v i t y - Two Parabola Model f o r Tee Section (Fig. 16), P/P = 0.1, L/r = 25. F i g . 23b S e n s i t i v i t y - Two Parabola Model f o r Tee Section (Fig. 16), P/P = 0.1, L/r = 75. F i g . 23c S e n s i t i v i t y - Two Parabola Model for Tee Section (Fig. 16), P/P = 0.1, L/r = 150. 600-, •Limits for 10% v a r i a t i o n i n maximum end moment when the y i e l d curvature i s varied at constant y i e l d moment = 360 k i p - f e e t . Curvature (in 1 x 10 4) F i g . 24a S e n s i t i v i t y - Two Parabola Model for Tee Section (Fig. 16), P/P Q = 0.5, L/r = 25. 600 -I Curvature (in : x 10 ) F i g . 24b S e n s i t i v i t y - Two Parabola Model f o r Tee Section (Fig. 16), P/P = 0.5, L/r = 75. 600 0 1 2 3 4 6 Curvature (in 1 x 10 ) F i g . 24c S e n s i t i v i t y - Two Parabola Model f o r Tee Section (Fig. 16), P/P Q = 0.5, L/r = 150 .'(Model shown does not adequately represent r e a l moment-curvature for t h i s load and slenderness r a t i o ) . 70 -. 60 /Limit f o r le s s than 10% v a r i a t i o n i n maximum end moment •;when the y i e l d moment i s varied at constant y i e l d curva-t u r e = 0.000145 50 40 30 20 10 -1 -4 Curvature (in x 10 ) o F i g . 25a S e n s i t i v i t y - Two Parabola Model f o r Square Section (Fig. 22), P/P 0 = 0.1, L/r (Material f a i l u r e ) . • y " ; . . ' : " 7 '.: = 25 70 -, Curvature (in x 10 4) F i g . 25b S e n s i t i v i t y - Two Parabola Model f o r Square Section [ (Fig. 22), P/P = 0.1, L/r = 75. 70 "I F i g . 25c. S e n s i t i v i t y - Two Parabola Model for Square Section (Fig'. 22), P/P = 0.1, L/r = 150 4-> V ft •H X 4-> a a) e o 2 3 Curvature ( i n - 1 x 10-4) F i g . 26a S e n s i t i v i t y - Two Parabola Model f o r Square Section (Fig. 22), P/P = 0.5, L/r = 25 (Material f a i l u r e ) . o 110 H o f i 1 1 1 1 1 1 r 0 1 2 . 3 4 Curvature (in x 10 ) F i g . 26b S e n s i t i v i t y - Two Parabola Model for Square Section (Fig. 22), P/P = 0.5, L/r = 75. o CTi 110 F i g . 26c S e n s i t i v i t y - Two Parabola Model for Square Section (Fig. 22), P/P Q = 0.5, L/r = 150. (Model shown does not adequately represent r e a l moment-curvature r e l a t i o n for t h i s load and slenderness r a t i o . ) 4 Limit f o r less than 10% v a r i a t i o n i n maximum end moment when eit h e r Mya or Myb i s varied at constant <f>ya = 0.000114 i n - 1 and <|>yb = 0.000250 X: i n " 1 . Curvature (in x 10 28a S e n s i t i v i t y - One Straight Line and Two Parabola Model f o r Square Section (Fig. 22), P/P = 0.5, L/r = 25 (Material "failure). •Limits f o r 10% v a r i a t i o n i n maximum end moment when <f>ya i s varied at constant Myb = 97 k i p - f e e t . Curvature (in x 10 ) F i g . 28b S e n s i t i v i t y - One Straight Line and Two Parabola Model f o r Square Section (Fig. 22), P/P =0.5, L/r = 75. f—1 M O 110 1 0 1 2 3 4 -1 -4 Curvature (in x 10 ) F i g . 28c S e n s i t i v i t y - One Straight Line and Two Parabola Model f o r Square Section (Fig. 22), P/P Q = 0.5, L/r = 125. 4-1 Q) m ft •rH M 4-1 C CL) £ O .Limits for 10% v a r i a t i o n i n maximum end moment when <))yb i s varied at constant Myb = 170 k i p - f e e t . "l 1 1 Curvature (in--'- x 10 - 4) One Straight Line and Two Parabola Model f o r Tee Section (Fig. 16), P/P = 0.5, F i g . 29 S e n s i t i v i t y L/r = 150. n o i 0 1 2 3 4 Curvature (in x 10 ) F i g . 30a S e n s i t i v i t y - One Straight Line and Two Parabola Model f o r Square Section (Fig. 22) , P / P q = 0.5, L/r = 75. F i g . 30b S e n s i t i v i t y - One Straight Line and Two Parabola Model f o r Square Section (Fig. 22) , P / P q =0.5, L/r = 125. Curvature (in 1 x 10-4) F i g . 30c S e n s i t i v i t y - One Straight Line and Two Parabola Model f o r Tee Section (Fig. 16), P/P = 0.1, L/r =150. ° 

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