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Slenderness effects in prestressed concrete columns Alcock, William John 1976

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SLENDERNESS EFFECTS IN PRESTRESSED CONCRETE COLUMNS WILLIAM JOHN ALCOCK B.A.Sc. (1972) The University of Toronto A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of '• CIVIL ENGINEERING . . Y/e accept this thesis as conforming to the required standard The University of British Columbia October, I976 © William John Alcock, 1976 In presenting the thesis in partial fulfilment of the require-ments for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I. further agree that; permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. W.J. Alcock Department of C i v i l Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 October, 1976 i ABSTRACT. The purpose of t h i s i n v e s t i g a t i o n was to compare the behaviour of r e a l prestressed concrete columns to the predictions of a mathematical model. A previously developed computer programme, based on the mathematical model, had suggested several problems which needed examination. The programme indicated that, i n some circumstances, an unstable equilibrium configuration could occur. The existence of t h i s unstable loading path meant that a snap-through type of buckling was a p o s s i b i l i t y . To check these hypotheses, six T-shaped prestressed concrete columns were constructed and tested at the University of B r i t i s h Columbia. In most instances, experimental observations c l o s e l y matched the predictions of the mathematical model. The computed and observed peak loads compared well and the presence of an un-stable equilibrium path was confirmed. Unfortunately, attempts to measure curvatures and to compare them with mathematically obtained values were unsuccessful. No s a t i s f a c t o r y explanation f o r t h i s problem was found. Having established the v a l i d i t y of the mathematical model through the experimental programme, an examination of snap- — through buckling was made. I t was concluded that prestressed concrete columns are not prone to snap-through buckling, although s u f f i c i e n t a d d i t i o n a l energy applied to a column might r e s u l t i n a jump from a stable equilibrium configuration to an unstable one. i i Acknowledgements. The author would l i k e to express his appreciation to his advisor, Dr. N.D. Nathan f o r his encouragement and prompt assistance i n the preparation of t h i s t h e s i s . In addition, the author would l i k e to thank the s t a f f of the C i v i l Engineering S t r u c t u r a l Laboratory at the Un i v e r s i t y of B r i t i s h Columbia f o r t h e i r help i n the construction and t e s t i n g of the concrete columns used i n the experimental programme. Their s k i l l and encouragement was much appreciated. The author i s also g r a t e f u l to Con-Force Products Ltd., Richmond, B.C. and Joe Harrison, i n p a r t i c u l a r , f o r construction of the concrete laboratory t e s t columns. The co-operation of Wire Rope Industries Ltd., Burnaby, B.C. i s also acknowledged. Without f i n a n c i a l assistance from the National Research Council and Prestressed Concrete I n s t i t u t e , t h i s thesis would not have been possible. F i n a l l y , the author i s deeply indebted to Denize Robertson f o r typing and a s s i s t i n g i n preparing t h i s t h e s i s . i i i TABLE OF CONTENTS. Page Abstract i Acknowledgements i i L i s t of Figures v Notation v i i i Chapter It Introduction 1 Chapter 2: The Mathematical Model 11 i 2.1 Assumptions 11 2.2 Load-Moment Interaction Curve 11 2.3 Potential Energy of Load 12 2, h Load Moment Curvature Relationships, 13 Energy of Load Contours 2.5 Column Deflection Curves 2.6 End Moment vs Mid-Height , Moment Curves ' 2.7 Load-Mid-Height Moment Curves 17 Chapter 3: The Experimental Programme 19 3*1 Design Constraints 19 3.2 Testing Machine Calibration 19 3.3 Column Dimensions} Details of 20 End Bearings 3. J+ . Formwork, Casting and Material 21 Testing 3.5 Measurement of Prestress 22 3.6 Instrumentation 23 3.6.1 Curvature Measurement 23 3.6.2 Calibration of Linearly Varying 28 Displacement Transducers 3 .6 .3 Monitoring Stabilitys Load-Axial 28 Deflections Graphs i v Page 3.6.4 Data Recording System 29 ' 3.7 Conduct of the Testsi 30 3.7.1 Prior to Loading 30 3.7.2 Loading Procedures 31 Chapter 4: Results from the Testing Programme 32 4.1 Observations During Testing 32 4.2 Concrete Stress-Strain Curves 33 4.3 Load-Moment Interaction. Curves 34 4.4 Moment-Curvature Relationships 35 4.5 Case Study: Column 2: Factors Affecting 36 Moment-Curvature Relationships 4.5.1 Concrete Stress-Strain Curves 36 4.5.2 Prestressing Force 38 4.5.3 Effect of Built-in Deformation 39 4.6 Summary 41 Chapter 5*« Investigations into Snap-Through 43 Buckling 5.1 Potential Energy of the Column 44 5.2 End Moment vs..Potential Energy Curves 46 5.3 Load vs. Potential Energy Curves 47 5.4 Load vs. Energy Difference Curves 47 Conclusions 49 Figures 51-103 Bibliography 104 V LIST OF FIGURES. Figure Page la S t r a i n Diagram 51 lb Short Column Interaction Curve 51 2a S t r a i n Diagrams 52 2b Load-Moment-Curvature Relationships 52 2c Moment-Curvature Relationship f o r Load P o 53 2d u Load Energy Contours Due to Compressive Shortening of the P l a s t i c Centroid 5^ 3a Column D e f l e c t i o n Curve 55 Column D e f l e c t i o n Curves f o r Various Mid-height Moments 55 k End Moment vs. Mid-height Moment Curves 56 5 Load vs. Mid-height Moment Curves 57 6 Load-Deflection Curves 58 7a Column Dimensions 59 7b End Plates and End Reinforcing 60 7c End Bearings 61 8 S t r e s s - S t r a i n Curve f o r 3/8 Inch Diam. 7-Wire Pre s t r e s s i n g Strand 62 9 Load-Mid-height Moment Interaction Curves 63 10a Curvature Measurement using C a n t i l e v e r System 64 10b Arrangement of Cantilevers in: s t r a i g h t Line along Web of Column 65 10c Approximation to Determine Curvature over Length of T y p i c a l C a n t i l e v e r 65 11a Device f o r Obtaining Column D e f l e c t i o n Curves 66 l i b Carriage Device at Bottom End of S t e e l Tube R a i l 66 v i Figure Page 11c S t e e l Tube R a i l and P u l l e y System 67 l i d X-Y P l o t t e r Used to Obtain Column 67 D e f l e c t i o n Curve 12a-e Measured Column D e f l e c t i o n Curves 68-72 I3a-c Crack Patterns i n Concrete Columns 73 13d V i s i b l e Curvature i n Column P r i o r 7^ To F a i l u r e 14a-b Compression F a i l u r e of the Flange 75 15 Load-Axial Shortening Curve f o r 76 Column 4 16 Concrete Stress S t r a i n Curves 7 7 - 79 17 End vs. Mid-height Moment Curves 80-82 18 Load-Mid-height Moment Curves 83-84 19 Observed and Predicted Curvatures 8 5 - 89 20 Moment-Curvature Relationships f o r 90 Column 2 21 Load-Mid-height Moment Curves f o r 91 Column 2 £ = 0.0020 and £ = 0.0031 . 0 o 22 Load-Moment Interaction Curves f o r 92 Various Concrete F a i l u r e S t r a i n s 23a S t r a i n and Stress D i s t r i b u t i o n f o r 93 Various Shapes of Cross-Section 23b Ultimate S t r a i n as a Function of Cross-section and P o s i t i o n of Neutral Axis 24a S t r e s s - S t r a i n Curves f o r Various S t r a i n $k Rates of Concentric Loading 24b S t r e s s - S t r a i n Relationship i n Flexure . 9 ^ k 25 Load-Mid-height Moment Curves f o r 8 and 95 16k Prestress Force: Column 2 k 26 Moment Curvature Relationships f o r 8 and 95 16k Prestress Force: Column 2 V l l Figure Page 27 Load Mid-height Moment Curves for 96 +0.1 and - 0 . 2 Inch Maximum Built-i n Deformations: Column 2 28 Moment Curvature Relationships for 97 +0.1 and - 0 . 2 Inch Maximum Built-i n Deformations: Column 2 29 Shortening Along the Line of Loading 98 of the Column Due to Bending 30 Potential Energy of Equilibrium 99 Configurations 31 End Moment vs. Potential Energy Curves 1°° for 6n = 0 . 0 0 2 8 and S „ 1 + = 0 . 0 0 3 5 : 0 Column 4 u l t 32 .. End Moment vs. Potential.Energy Curves 1°* for £ Q = 0 . 0 0 2 0 and £ult ~ ° » 0 0 2 5 33 Load-Potential Energy Curves 102 34 . Load-Energy Difference Curves 1°3 t • • • V l l l LIST OF SYMBOLS. Definition Cross-sectional area of prestressed column. Column deflection curve (deflections measured from plastic centroid to line of axial thrust). Depth of cross section. F i r s t derivative of y with respect to x where x is measured along thrust line and y i s distance from thrust line of the CDC. Second derivative of y with respect to x. Eccentricity of plastic centroid of a cross section from line of axial thrust. Young's Modulus for concrete. 28 day compressive strength of a standard 6 inch diameter, 12 inch long concrete cylinder (ksi). Peak compressive strength on concrete stress-strain curve (ksi). Length of half wave of cosine curve. Length of column. Elastic axial shortening of column under compressive axial load. Moment at any point along the length of a column. Axial load on column. Euler buckling load. Short distance along thrust line of a column. distance of CDC from thrust line at a distance x from mid-height of column. Same as dy/dx. ix Definition Same as d y/dx . Maxium lateral deflection of column. Slope ( = dy/dx = y*). Strain. Strain at extreme fibre of column section in compression. Strain at extreme fibre of column section in tension or extreme fibre with least compressive stress. Compressive strain of concrete at material failure. Compressive strain at peak stress on concrete stress-strain curve. difference in extreme fibre strains Curvature ( = = y" for circular dx curvature). Moment magnification factor used in American Concrete Institute Building Code (318-71) Formulas 10.7 and 10.8 for slenderness effects in reinforced concrete columns. Radius of curvature ( = ^ ). 1 1, Introduction In recent years, the lack of a r a t i o n a l method f o r designing f o r slenderness e f f e c t s i n prestressed concrete columns and load bearing wall panels has caused some d i f f i c u l t y to engineers. In general, i t has probably been the p r a c t i c e to apply the A.C.I, procedure, although that method was derived f o r reinforced concrete members and i s not n e c e s s a r i l y applicable to prestressed elements of the types commonly i n use. With the advent of e f f i c i e n t computers, engineers no longer need to resort to design procedures based on doubtful analogies. Since Euler's time, considerable e f f o r t has been devoted to column analysis and design. One of the most noteworthy of papers on column buckling was presented by Von Karman (1) i n 1910. Von Karman's theory was based on the actual s t r e s s -s t r a i n r e l a t i o n of the column material and used numerical integration of angular rotations along the column length, as well as moment-curvature r e l a t i o n s h i p s , to obtain a column d e f l e c t i o n curve (CDC). Being both general and non-linear, i t was, u n t i l recently, a laborious method of obtaining CDC's, fo r that reason, attempts to use simple deflected shapes to approximate the r e a l CDC have prevailed f o r some time. Many of the d i f f i c u l t i e s encountered i n determining the buckling load of concrete columns arose from the highly non-linear concrete s t r e s s - s t r a i n curve and the cracking of concrete. For s t e e l , the problem was somewhat simpler. For example, Ketter.Kaminsky and Beedle (2) obtained moment-curvature relationships f o r s t e e l wide-flange columns by assuming a b i - l i n e a r s t e e l s t r e s s - s t r a i n curve. S e l e c t i n g 2 various stress d i s t r i b u t i o n s f o r str a i n s beyond the e l a s t i c l i m i t , combinations of load, moment and curvature were derived. From these combinations, graphs of moment-curvature and load-curvature were plotted f o r cases when the s t e e l had: i ) reached the e l a s t i c l i m i t i i ) yielded through the flange i i i ) yielded to i / 4 depth iv) yielded to 1/2 depth v) yielded to 3/4 depth For a selected load, one could then p l o t the moment-curvature r e l a t i o n s h i p . A t r i a l - a n d - e r r c r numerical approach was employed to determine column d e f l e c t i o n curves f o r a pin-ended column: a deflected shape was assumed and moments computed at points along the column. From the moment-curvature curves, curvature values were interpolated and these integrated to get a new deflected shape. I f the assumed curve was correct, the de-f l e c t i o n values would be i d e n t i c a l to the assumed values. .Comparison of the r e s u l t s was made with a sine curve assumption and a p a r t i a l cosine curve suggested by Osgood and Westergaard (3) f o r s t e e l columns. In 1958 Broms and Veest (4) used the assumption that the de f l e c t i o n of reinforced concrete columns could be approximated by part of a cosine curve of wavelength 2L, such that the de-f l e c t i o n shape was y=y m cos 1 ~ . For a column of length 1=1^+12 the end e c c e n t r i c i t i e s would be e,=ym cos ^ 1 and e 0 = y ~ cos r^2. 1 Jm L 2 •'m L Then the curvature would be (A) which at maximum moment yields* 3 £ - y « < E > 2 - € * - * i m d Equilibrium equations which solve for P and M in terms of ^4 " £1 w e r e established. P and £^ - £^ values were selected varying) and the moment at mid-span M determined. Then y m = w a s found and substituted in (B) to solve for y. m IT ii Finally, the terms for y, e^ and e 2 were arranged to obtain: y = T ( a r c cos -=- + arc cos — ) (C) L L ym ym which could be differentiated to get: ym d(|) = c o t ^ i l + cotZh ( D ) TT dy ^ = ^ Y Jm T T I L L Solution of the equation (D) yielded the c r i t i c a l slenderness length 1 and corresponding values of 1^  and 1 2. Broms and Veest used a graphical solution to solve the equation. With the advancement of computers, attempts to use. approximating curves based on sine or cosine functions have been succeeded by numerical methods. Chang and Ferguson (5) for instance, began with Hognestad's (6) stress-strain relation for concrete in flexure and an idealized bi-linear stress-strain curve for steel to obtain column deflection curves for reinforced concrete columns. They derived equilibrium equations for load and moment in terms of edge strains for rectangular columns and developed moment-curvature relations for various loads. For convenience in computer programming, the non-linearity of the moment-curvature curves was overcome by dividing the curves into small segments to which f i r s t and second order polynomials were fi t t e d . Knowing the curvature of a column section, i t was possible to integrate, f i r s t to obtain the slope, then the de f l e c t i o n of the column. Integration was performed using Simpson's rule and the trapezoidal r u l e . For large scale a p p l i c a t i o n or f o r non-rectangular sections, Chang and Ferguson's technique might prove impractical. Nevertheless, they did observe reasonable comparison between t h e o r e t i c a l and tested column d e f l e c t i o n curves. Since 1965» numerous methods have been suggested f o r determining the strength of prestressed concrete columns. R.Itaya (7) reported a simple approach based on s t a t i c e q u i l i b -rium and geometrical compatability and discussed modifications to the I963 A.C.I, code based on the code provisions f o r reinforced concrete. S t a b i l i t y e f f e c t s were not discussed. K.J.Brown (8) used equilibrium equations and a parabolic concrete s t r e s s - s t r a i n r e l a t i o n f o r f " c = 0.85f' c and £ ult = 180xl0~5 to obtain the ultimate strength load-moment i n t e r a c t i o n curve f o r prestressed concrete columns. He tested 73 rectangular prestressed sections himself, and used r e s u l t s of 3 prestressed and 162 regular reinforced concrete columns tested by others to compare with his t h e o r e t i c a l predictions. Since most of the columns f a i l e d above or just below the balanced point on the ultimate strength i n t e r a c t i o n curve, moment magnification was discussed but i n s t a b i l i t y e f f e c t s were not encountered. This w i l l be enlarged upon below. Kabaila and H a l l (9) investigated buckling of prestressed concrete columns using a mathematical model applicable to rectangular sections only. Using a concrete s t r e s s - s t r a i n r e l a t i o n reported by Rusch (10), an equation f o r the prestressing s t e e l s t r e s s - s t r a i n curve, and a l i n e a r s t r a i n d i s t r i b u t i o n , equations f o r load and moment (PtM) at any section were found. The column was divided i n t o segments of f i n i t e length and a central difference formula used to obtain the column d e f l e c t i o n curve. Results from t h i s mathematical model were compared with tests on 30 prestressed columns tested by H.R.Brown and A.S.Hall (11) and with two cosine assumptions f o r the deflected shape of the column. Good comparisons were found with experimental r e s u l t s f o r prestress up to O.30 f * c . I t was also found that the cosine assumption (y = y mcos ^- ) always underestimated the buckling load while y = e + ( y m - e ) c o s ^ overestimated the buckling load. As an i n t e r e s t i n g side e f f e c t , i t was also discovered that the i n i t i a l tangent modulus of e l a s t i c i t y of concrete was dependent on the prestressing force sustained by the concrete, probably due to creep e f f e c t s . This phenomenon was investigated by B r e t t l e (12). T.Y.Lin and F.R.Lakhwara (13) modelled slenderness e f f e c t s i n p a r t i a l l y prestressed concrete columns assuming both b i - l i n e a r concrete and b i - l i n e a r prestressing s t e e l s t r e s s -s t r a i n curves. By f i x i n g the ultimate concrete s t r a i n at 0.003 and varying the depth to the neutral axis, a closed form, so l u t i o n was obtained f o r the load-moment combinations to cause f a i l u r e . Knowing the s t r a i n d i s t r i b u t i o n at f a i l u r e , a mid-he ighti curvature was obtained. This mid-height curvature was then used to f i t a cosine wave curve as the deflected shape of the column. A technique s i m i l a r to the mathematical model used i n t h i s thesis was proposed by Z i a and Moreadith (14). Assuming a l i n e a r s t r a i n d i s t r i b u t i o n and using equilibrium equations f o r forces and moments on that section, moment curvature curves were derived by varying the values of the concrete edge s t r a i n s (and the load). A f t e r construction of these moment curvature curves, numerical integration was c a r r i e d out with two known boundary conditions: i ) the slope must vanish at mid-height i i ) the d e f l e c t i o n at the ends must vanish Beginning with an assumed d e f l e c t i o n and known slope (^y/dx=0) at midheight, the slope and d e f l e c t i o n at various points along the 1 column were calculated successively. Calculations continued u n t i l the d e f l e c t i o n was equal to zero. This was done f o r various assumed mid-span d e f l e c t i o n s , and the longest column length obtained was selected as the c r i t i c a l column length. An attempt to determine the ultimate strength i n t e r a c t i o n curve using the equivalent rectangular stress block permitted by the A.C.I. Building Code (318 - 63) and using e u l 1 - = 0.003 was made by Z i a and Guillermo (15). However, Nathan, (16) pointed out that t h i s method can lead to anomolies when the area of concrete on the compression side of the neutral axis i s of i r r e g u l a r shape. For t h i s reason, as Nathan suggested, i t would be a mistake to introduce an i r r a t i o n a l approach to the design of prestressed concrete columns when the opportunity of using a r a t i o n a l method existed at the outset. In I968, S.Aroni (17) reported t e s t s on 36 prestressed pin-ended rectangular columns i n which e f f e c t s of varying the e c c e n t r i c i t y of load, i n i t i a l prestress.and slenderness r a t i o 7 were examined. Aroni's method of analysis was b a s i c a l l y s i m i l a r to previously reported analyses. A column was f i r s t divided into " f i n i t e elements" of chosen segment length. Two equations of equilibrium were written to handle each of seven combinations of concrete s t r a i n d i s t r i b u t i o n with four unknowns ( P t ^ . ^ i y ) • An i n i t i a l deflected shape of the column was assumed from which A£ at mid-height and hence curvature could obtained. Concrete stresses were found from the s t r a i n s using Hognestad's stress-, s t r a i n curve. Knowing the curvature at mid-span, a recurrence formula ( a c t u a l l y a Taylor ser i e s expansion) was then used f o r numerical integration along the length of the column to get the column d e f l e c t i o n curve. F i n a l l y the required boundary condition f o r the l a s t segment was checked, that i s , y m = e, the end e c c e n t r i c i t y . I f t h i s was not s a t i s f i e d the. assumed mid-height A 6 was changed and the whole procedure repeated. In many ways Aroni's i t e r a t i v e technique was s i m i l a r to that suggested by Ketter, Kaminsky, and Beedle ( 2 ) . Aroni's method also shows some likeness to the mathematical model employed i n t h i s t h e s i s . However, instead of determining moment-curvature relationships f o r various load values before using the recurrence formula, Aroni calculated them at each stage i n the numerical integration. I f h i s i n i t i a l assumption f o r the deflected shape of the column was wrong, a l l the other curvature values along the column would also be i n c o r r e c t , leading to mis-matched end e c c e n t r i c i t i e s (between assumed and calculated CDC's). Hence he was forced to re-evaluate the deflected shape by t r i a l - a n d - e r r o r . In t h i s t h e s i s , the curvature at mid-span f o r a known d e f l e c t i o n i s correct from the 8 outset; thereafter a l l other combinations of moment and curvature calculated along the column are also correct within the accuracy of the i n i t i a l assumptions. Therefore, no t r i a l -and-error or i t e r a t i o n procedures are necessary. Development of the computer model used i n t h i s thesis originated with Chandwani and Nathan (18). A programme was i n i t i a l l y devised to compute the i n t e r a c t i o n curve f o r pre-stressed concrete sections of any shape with any arrangement of prestressing and mild s t e e l . Results from the computer model were compared with Whitney's theory (19) and i t was found that the "equivalent" rectangular sections were not successful i n representing i n t e r a c t i o n curves f o r T-sections. Later, the computer programme was extended by Nathan (20) to calculate load-moment-curvature rel a t i o n s h i p s and column d e f l e c t i o n curves. Some l i m i t e d experimental v e r i f i c a t i o n with tests on rectangular reinforced concrete sections by Chang and Ferguson (5) was- observed. The mathematical model was also used f o r comparison with the current A.C.I. Building Code (318-71) s p e c i f i c a t i o n s f o r prestressed concrete columns by Nathan (21). The A.C.I, design procedure uses moment magnifiers to handle slenderness e f f e c t s . The magnification fa c t o r i s : g - 1 where The question immediately a r i s e s : "What does one use f o r EI?" Formulas 10.7 or 10.8 i n the A.C.I, code imply a l i n e a r moment-curvature r e l a t i o n s h i p , (as, indeed, does the basic form of the 9 magnification factor) which i s quite reasonable f o r loads above the "balanced value", when f a i l u r e i s governed by compression. This i s generally the case f o r concrete sections with conventional r e i n f o r c i n g where the "balanced load" i s at a r e l a t i v e l y low percentage of the maximum pure a x i a l load. However, f o r prestressed concrete sections with a wide compression flange, the "balanced load" i s a much higher percentage of the maximum load, and the range of t e n s i l e f a i l u r e , and, therefore, d u c t i l e behavior, i s greatly increased. Using the a n a l y t i c a l model, i t has been found that f o r slender prestressed concrete sections with wide compression flanges, the d u c t i l i t y can lead to an i n s t a b i l i t y type of f a i l u r e before material f a i l u r e occurs i f the end moments are s u f f i c i e n t l y large. In c e r t a i n circumstances t h i s i n s t a b i l i t y f a i l u r e might occur under load and end moment conditions which should have been safe according to the A.C.I, formulae. Nathan (21) also pointed out that under these conditions, there exists a d e f i n i t e p o s s i b i l i t y that snap-through type buckling can occur even before the predicted point of i n s t a b i l i t y , so that the A.C.I, formulae may be doubly unconservative i n t h i s region. I t i s d i f f i c u l t to predict the point at which snap-through may occur since i t depends upon accidental disturbances and imperfections, or small unanticipated l a t e r a l loads; nevertheless, some i n d i c a t i o n of the p o t e n t i a l danger of snap-through may be obtained by studying the differences i n the po t e n t i a l energies associated with the stable and unstable loading paths. Since a l l the previous conjectures r e l y on a mathematical 1 0 model, esta b l i s h i n g the v a l i d i t y of the model i s e s s e n t i a l . With some minor modifications, and the addition of subroutines to calculate the p o t e n t i a l energy of the column, the computer programme used i n t h i s thesis i s the same as that developed by Nathan and Chandwani. The purpose of t h i s thesis i s , therefore, to choose a case which severely stretches the assumptions of the mathematical model and to compare the a n a l y t i c a l s o l u t i o n with tests on r e a l columns. Having determined the r e l i a b i l i t y of the mathr ematical model, comments can then be made on implications of the l a t t e r f o r the s t a b i l i t y of prestressed concrete columns. In p a r t i c u l a r , the question of snap-through buckling can be examined with somewhat more confidence. 11 2. The Mathematical Model. 2.1 Assumptions: i ) Bernoulli's assumption: plane sections remain plane and normal to the neutral surface, i i ) a b i - l i n e a r mild s t e e l s t r e s s - s t r a i n curve, i i i ) a well defined concrete s t r e s s - s t r a i n curve applicable to both a x i a l load and f l e x u r a l behavior. iv) a known s t r e s s - s t r a i n curve f o r the prestressing s t e e l . v) perfect bond between mild or prestressing s t e e l and concrete. 2.2 Load-Moment Interaction Curve: Short Column. The computation begins with determination of the material f a i l u r e l i m i t s f o r a short column (that i s , with no slenderness e f f e c t s ) . The extreme f i b r e i n compression i s given the f a i l u r e value of concrete s t r a i n (£ ui^)« A neutral axis depth i s selected and the l i n e a r s t r a i n diagram i s then completely defined.(Fig la) The concrete s t r e s s - s t r a i n curve i s constructed by f i t t i n g a smooth curve to a c o l l e c t i o n of experimentally obtained points, or, a l t e r n a t i v e l y , the programme contains a default polynomial which may be adjusted i n terms of chosen values of c y l i n d e r strength ( f ' c ) i peak s t r a i n ( £ Q ) t and ultimate s t r a i n (£UT_-t)« The concrete stress d i s t r i b u t i o n implied by the previously described s t r a i n diagram i s then known; the resultant force and moment about the p l a s t i c centroid are numerically integrated by subdividing the compression area into narrow s t r i p s p a r a l l e l to 12 the neutral axis, to each of which i s a t t r i b u t e d a constant stress. Likewise the s t r a i n s i n the prestressed and/or mild s t e e l reinforcement can be obtained from the s t r a i n diagram. The prestressing s t e e l s t r e s s - s t r a i n curve can also be fed i n using point values, or by using a default polynomial included i n the programme. Summing a l l the loads and moments across the section gives the f a i l u r e conditions f o r that section. For example, point 1 on the i n t e r a c t i o n curve of F i g . l b might be determined by the s t r a i n configuration 0-1 of F i g . l a with r o t a t i o n (j^, or point 5 of F i g . l b by 0-5 of F i g . l a with r o t a t i o n (f)^, Discrete l i m i t s f o r load, moment and r o t a t i o n are thus acquired and a smooth curve can be f i t t e d to them. The curve so generated i s referred to as the "Short Column Interaction Curve". 2.3 P o t e n t i a l Energy of Load. For each s t r a i n d i s t r i b u t i o n associated with the load-moment i n t e r a c t i o n curve, the s t r a i n at the p l a s t i c centroid can be r e a d i l y found. When the centroidal s t r a i n i s m u l t i p l i e d by the t o t a l load on the section, the loss of p o t e n t i a l energy per unit length due to compressive shortening of the column i s known. Contributions to p o t e n t i a l energy due to curvature of the axis, r o t a t i o n of the end moments, and s t r a i n energy are obtained separately f o r a member of given length, once the deflections, slopes and curvatures are completely defined f o r a given column d e f l e c t i o n curve. 13 2.4 Load Moment Curvature Relationships, Energy of Load Contours. The moment curvature relationships can "be determined by a similar procedure. It w i l l be realized that the curvature i s given by the extreme fibre strain divided by the neutral axis depth, ie. i t is the angle of the strain diagram as shown in Fig. 2a. Thus for a given value of curvature, each value of neutral axis depth defines a possible strain diagram. When the stresses are computed and integrated as in section 2.2, each neutral axis depth yields a pair of corresponding values of load and moment for the given curvature. When these values are plotted on the load moment diagram, a smooth curve can be drawn through them to give a curvature contour as on Fig. 2b. Only the points f a l l i n g within the short column interaction curve are valid. Pfrang, Siess, and Sozen (22) obtained contours of curvature in a similar manner for reinforced concrete columns. From these contours, the moment-curvature relation for any value of the load can be extracted as shown in Fig. 2c. As described in section 2.3t i t i s possible to calculate the loss of potential energy per unit length caused by compress-ive shortening of the column. Thus,: for a chosen curvature, each value of neutral axis depth defines a strain distribution from which the strain at the plastic centroid can be determined. Each neutral axis depth, therefore, yields a pair of corresponding values of load and load energy per unit length for the given curvature. Again, when these values are plotted on the load-load energy diagram, smooth contours are obtained (see Fig. 2d). 14 2.5 Column De f l e c t i o n Curves. A n i n f i n i t e l y long column under a given a x i a l load may occupy an i n f i n i t e number of equilibrium configurations. A l i n e a r s o l u t i o n y i e l d s waves whose amplitude i s a r b i t r a r y but whose length depends on the load. When material n o n - l i n e a r i t i e s are included, both the wave length and amplitude are a r b i t r a r y but related. The wave forms are symmetrical about the point of maximum d e f l e c t i o n from the thrust l i n e so that the shape of a quarter wave-length i s representative of the entire configuration. This configuration i s c a l l e d a "Column Deflection Curve" (CDC). Galombos (23) suggested the following numerical method f o r the determination of C D C S J Let the distance along the thrust l i n e be x. Let the displacement of the CDC from the thrust l i n e be y. Expanding y(x) i n a Taylor series about a point xQt y ( x Q + ax) = y ( x Q ) + y ' ( x 0 b x + §y"(x 0)Ax 2 + S i m i l a r l y , expanding y*(x) i n a Taylor series about x. y ' ( x 0 + AX) = y*(x 0) + y"(x Q)Ax + |y"(x Q)Ax 2 + ...... Reca l l i n g that y* i s the slope (©0 of the CDC and y" i s a good approximation to the curvature (0), one obtains, by • truncating the series a f t e r the second deri v a t i v e of y: y ( x Q + AX) = y(x Q) + C X ( X Q ) A X + -|-<})(X0)AX2 ( A ) ( x Q +AX) = c x(x 0) + ( J ) ( X 0 ) A X (B) Truncation of the approximation at the second d e r i v a t i v e i s equivalent to assuming constant ( c i r c u l a r ) curvature over AX. With relationships ( A ) and ( B ) , one can proceed to f i n d the CDC f o r a pin-ended column bent i n single curvature as followst 15 i ) pick a value of the a x i a l load f o r which the CDC i s sought. i i ) enter the graph of the short column i n t e r a c t i o n curve and obtain the maximum moment (which w i l l occur at mid-length) and the corresponding r o t a t i o n , and energy of loading (per unit length). Material f a i l u r e occurs at t h i s combination of load and moment, i i i ) the mid-length displacement (y) of the column i s just the mid-length moment divided by the a x i a l load} the slope (©0 at mid-length i s zero by symmetry of the deflected shape. iv) now calculate y ( x Q + £>x) and °<(x0 + AX) f o r a new point a distance AX along the thrust l i n e from the mid-length using the Taylor series expansion, v.) knowing the d e f l e c t i o n at t h i s new point, c a l c u l a t e the moment, enter the contours oft a) curvature to obtain a new value of § . b) load energy to obtain the load energy per unit length at that l o c a t i o n . v i ) calculate a new y and cx further along the thrust l i n e . Continue i n t h i s way along the thrust l i n e u n t i l the half-length of the column i s exceeded or the CDC crosses over the thrust l i n e , v i i ) e s t a b l i s h a family of CDC's by decreasing the mid-height moment incrementally from the maximum value and determining a new CDC f o r each increment. Note that f o r each segment along the CDC's, values of decreased p o t e n t i a l energy of load per unit length 1 6 due to a x i a l shortening, are now a v a i l a b l e . In r e a l i t y , the deflected shape of a column increases with load i n a process reversed from that described above. For example, r e f e r r i n g to F i g . 3b f o r a column of length L with load P at an end e c c e n t r i c i t y e^, the mid-length moment would be P y ^ For a s l i g h t l y higher end moment, Pe 2,the mid-length moment would be Py 2» A - t e-j» maximum end e c c e n t r i c i t y possible has been reached, although the mid-length moment Py^ i s less than the moment Py^ which would r e s u l t i n material f a i l u r e . In f a c t , i f the end moment could be backed down to Pe^, the mid-length moment would s t i l l increase to Py^. Material f a i l u r e would occur when the mid-height moment reached Py^, even though the end moment would be only Pe^. From the preceding discussion on CDC's, i t can be seen that f o r a given load, and end moment, there may exis t two values of the mid-length moment f o r which the column i s i n equilibrium. For example, end moments Pe 2 and Pe^ (having the same numerical value) lead to two d i f f e r e n t mid-length moments Py 2 and P y / +. The energy of load per unit length due to a x i a l compression i s now availa b l e at each segment of the CDC. When t h i s i s integrated numerically along the length of the column, the t o t a l energy of loading at the p l a s t i c centroid i s obtained. F i n a l l y i t i s necessary to determine the. s t r a i n energy i n the column, loss of energy due to the applied end moment r o t a t i n g through the end slope, and energy losses from the shortening of the column due to bending. These l a s t three terms necessary to determine the p o t e n t i a l energy of the column w i l l be discussed 1 7 i n section 5 . 1 . 2 . 6 End Moment vs. Mid-height Moment Curves. By p l o t t i n g the end moment versus mid-length moment f o r a selected a x i a l load (say of section 2 . 5 ) » curves such as those of F i g . 4 are obtained f o r a column of p a r t i c u l a r length. Other a x i a l loads y i e l d s i m i l a r curves. Generally, the higher the a x i a l load, the higher i s the peak end moment on each curve. A column whose e c c e n t r i c i t y of loading i s known can now be examined. Let that e c c e n t r i c i t y be e. Fcran a x i a l load P^, the end moment would be P^e. Entering P^e as the ordinate on F i g . 4 one finds only one corresponding value of mid-length moment from the curve f o r P^. Selecting a s l i g h t l y higher load say P 2, the end moment would be P 2 e ' ^ o r w h i ° h " f c w o corresponding mid-height moments would be obtained. By increasing the a x i a l load, one w i l l eventually f i n d a load f o r which only one mid-length moment i s possible. Increasing the a x i a l load also increases the slenderness e f f e c t so that the mid-length moment grows at a f a s t e r rate than the end moment. For loads above the peak value no mid-height moments can be found. Hence, the maximum load and end moment that t h i s p a r t i c u l a r column could sustain would be P^ and P^e respectively, 2 .7 Load-Mid-height Moment Curves. Returning to the load-moment diagram on which the short column i n t e r a c t i o n curve i s shown, one may now use the data from section 2 . 6 to p l o t a graph of load vs. mid-length moment, on the same axes. In the previous discussion, i t was found that only one value of mid-height moment occurred f o r load P^, two for loads P 2 and P^, and one f o r P^. I f these and values f o r other loads up to P^ are plotted on t h i s diagram, a curve such as OB of F i g . 5 i s obtained. The"line of no slenderness e f f e c t " simply gives the end moment conditions f o r corresponding a x i a l loads. Arrows on the curve follow the d i r e c t i o n of the curve as the column i s compressed. The same column with a much smaller end e c c e n t r i c i t y might produce the curve OA with no associated downward path. Slenderness e f f e c t s r e s u l t i n g from the smaller end e c c e n t r i c i t y are much reduced; f o r t h i s case, f a i l u r e occurs above the "balanced point" by compression. Note that f o r the column whose cross-section, length and ec c e n t r i c i t y of loading produce the curve OB, there i s a rather long descending branch of the load-moment curve beyond the peak load, i n d i c a t i n g an i n s t a b i l i t y type of f a i l u r e . Material f a i l u r e of the section would not occur u n t i l the load-mid-height moment curve intersected the short column i n t e r a c t i o n curve at B. Since the i n s t a b i l i t y case i s one which severely stretches t h e . c a p a b i l i t i e s of the mathematical model, the experimental portion of t h i s thesis was aimed at d u p l i c a t i ng t h i s type of f a i l u r e with tests on r e a l columns. 19 3. The Experimental Programme. 3.1 Design Constraints. To obtain a prestressed concrete column suitable f o r t e s t i n g within the constraints of the s t r u c t u r a l laboratory at the University of B r i t i s h Columbia, several factors were considered: i ) the column required a large compression flange i n order to produce a r e l a t i v e l y high balanced point on the short column i n t e r a c t i o n curve, so that the i n s t a b i l i t y f a i l u r e mode could be ensured, i i ) the column had to be s u f f i c i e n t l y long to develop the prestressing force of a 3/8 inch diameter 7 wire strand which was the smallest strand a v a i l a b l e at the Con-Force Products Ltd. precast concrete plant where the columns would be cast. Since the computer model does not handle loss of prestress due to t r a n s f e r at ends of the column,these e f f e c t s had to be minimized, i i i ) the web had to be s u f f i c i e n t l y t h i c k to prevent s p a l l i n g of the concrete due to t r a n s f e r of the pre-s t r e s s i n g force, iv) the t a l l e s t mechanical screw type t e s t i n g machine i n the s t r u c t u r a l laboratory could accomodate specimens not exceeding 12 feet i n length. 3.2 Testing Machine C a l i b r a t i o n . For columns subject to an i n s t a b i l i t y type of f a i l u r e , the mathematical model predicts that the column w i l l continue to d e f l e c t l a t e r a l l y , once the peak load has been reached, even 20 Further, i f the slope of the descending branch of the load-axial d e f l e c t i o n curve f o r the column became equal to the t e s t i n g machine load-deflection curve, premature f a i l u r e of the column might occur. To ensure that t h i s would not happen, a load d e f l e c t i o n curve f o r the 200,000 l b . Tinius Olsen t e s t i n g machine was obtained as follows. The t e s t i n g machine loading head was set 10 feet above the weighing table (10 feet being the anticipated length of column to be tested). A hydraulic jack was positioned on the weighing table so that load could be applied to the loading head of the t e s t i n g machine by the jack. As the load was applied, the d e f l e c t i o n between the weighing table and the head was measured. The load-deflection curve o b t a i n e d i s s h o w n i n F i g . 6 along w i t h a l o a d - a x i a l d e f l e c t i o n - c u r v e c a l c u l a t e d f o r t h e columns prior, Xo R e s t i n g . 3.3 Column Dimensions; D e t a i l s of End Bearings. Dimensions and d e t a i l s of the 12 inch wide, 10 foot long T-shaped pin-ended column selected f o r t e s t i n g are i l l u s t r a t e d i n F i gs. 7a to 7c. The single prestressing strand was positioned 5 inches, from the bottom edge of the web, approximately 0.125 inches above the calculated p o s i t i o n of the p l a s t i c centroid. The axis of loading was through the flange, I.75 :inches -above the p l a s t i c centroid. Each column was cast with §• inch thick machined s t e e l plates b u i l t i n at the ends. Three number 3 r e i n f o r c i n g bars were welded to the inside of the plate and extended 6 inches l o n g i t u d i n a l l y into the flange of the column; one number 4 also extended 15 inches into the web to prevent l o n g i t u d i n a l cracking 21 caused by stress t r a n s f e r from the ends of the prestressing strand. Closed s t i r r u p s of number 7 r e i n f o r c i n g wire, hooked around the l o n g i t u d i n a l r e i n f o r c i n g within the f i r s t 6 inches of the column ends, were also used to prevent s p a l l i n g due to the stress t r a n s f e r . The b u i l t i n s t e e l end plates were used as mounting blocks f o r a d d i t i o n a l J inch thick machined s t e e l bearing p l a t e s . A \ inch deep groove was machined into t h i s a d d i t i o n a l plate* to accomodate a 12 inch long, 1 inch diameter r o l l e r bearing. The second plate was then bolted to the f i r s t and c a r e f u l l y adjusted i n p o s i t i o n so that the axis of the r o l l e r would be at the desired 1.75 inch e c c e n t r i c i t y from the calculated p l a s t i c scentroid of the section. Bearing seats on both the weighing table and loading head consisted of 4 inch wide, 3 inch deep, 13 inch long machined st e e l blocks. A \ inch deep groove was machined l o n g i t u d i n a l l y on each block to c l o s e l y f i t the one inch diameter r o l l e r . The lower bearing seat was bolted to a j/b inch t h i c k plate which could be adjusted i n p o s i t i o n to seat the column correctly by adjusting the lower bearing seat on the weighing table. 3.4 Formwork, Casting and Material Testing. The s i x T-shaped columns used i n the t e s t i n g programme were a l l made at the ConForce Products Ltd. precasting plant i n Richmond B.C. Formwork f o r the column was constructed of j/k inch thick plywood with a dimensional tolerance of - 1/16 inch. Dimensional accuracy of the forms was checked p r i o r to casting. The columns were cast two at a time, l a i d end-to-end i n a 28 22 foot 8| inch s t r e s s i n g bed. A f t e r the concrete had been placed, each column was cured with heat f o r approximately 15 hours before the forms were removed. Standard 6 inch diameter, 12 inch long concrete cylinders were made at the same time as the concrete was placed; cylinders were kept with the columns and cured at the same time. A f t e r the i n i t i a l curing, the forms were stripped and the prestressing strand released from the bulkheads. Cylinders were tested to ensure that the concrete had s u f f i c i e n t strength to sustain the prestress. Both the columns and the cylinders were stored under cover but open to the a i r f o r the duration of the period before t e s t i n g . Compressive strength tests were c a r r i e d out on the cy l i n d e r s at as close a time as possible to the date of the column t e s t , usually within a day. The f i r s t column was tested 57 days a f t e r casting and the l a s t a f t e r 123 days. Cylinders could not be tested on the same day as the columns as the same t e s t i n g machine was required f o r both t e s t s . A mechanical screw-type t e s t i n g machine was necessary i n order to obtain the downward slope of the concrete s t r e s s - s t r a i n curve. Measured c y l i n d e r strengths were used i n the computer programme to simulate the behaviour of the column as c o l s e l y as possibl e . S t r e s s - s t r a i n curves f o r the prestressing s t e e l were determined i n the t e s t i n g laboratories of Wire Rope Industries Ltd., 3185 Grandview Hwy., Burnaby, B.C. (see F i g . 8). 3.5 Measurement of Prestress. Computed load-mid-length moment curves f o r the column are shown i n F i g . 9 f o r a 1.75 inch e c c e n t r i c i t y and expected f i n a l 23 prestress forces of 11, 12 and 13 kips assuming f* = 5 k s i . , £Q - 0.002 and € u l i - - 0.003. At the time of casting, the pre-stress force was determined from measured elongations. J^ fo instrumentation was subsequently used to measure loss of pre-stress i n the strand p r i o r to t e s t i n g , but estimates of losses were made using methods suggested by Libby (24). A preliminary estimate showed the f i n a l prestressing force i n the columns would be about 12 kips. Referring to F i g . 9» i t can be seen that an error of - 1 kip from the estimated 12 kips prestress force r e s u l t s i n an error of only - 2.8$ i n the p r e d i c t i o n of the peak load. Since the primary purpose of t e s t i n g the columns was to e s t a b l i s h the existence of the unstable downward path of the load-mid-length moment curve, such an error was considered of minor importance. 3.6 Instrumentation. 3.6.1 Curvature Measurement. I t w i l l be r e c a l l e d that the a n a l y t i c a l procedure begins with the determination of the moment-curvature r e l a t i o n s h i p f o r various load l e v e l s and then goes on to p r e d i c t the column behaviour. Thus, i t was desired to check the computed moment-curvature v a r i a t i o n and then to compare the o v e r a l l behaviour of the column with the predicted performance. Two independent methods of curvature measurement were employed on the t e s t columns. The f i r s t system measured curvature assuming c i r c u l a r curvature over the same segment length as was used i n the Taylor series expansion of the mathematical model. This consisted of 2k a series of aluminium cant i l e v e r s arranged i n a s t r a i g h t l i n e down one side of the web, p a r a l l e l to the column axis. The cantilevers were fastened to aluminium brackets, which, i n turn, were bolted to short aluminium bars epoxied to the concrete surface at 7h inch spacing, t h i s being the segment length used i n the computer programme. The t i p d e f l e c t i o n f o r each cantilever, was measured by a l i n e a r l y varying displacement transducer mounted on the aluminium bracket adjacent to i t , F i g s . 10a.and 10b. i l l u s t r a t e t h i s system. Since the magnitude of the rotations were extremely small, differences i n d e f l e c t i o n measurement due to r o t a t i o n of the transducer holder were neglected. A c a l i b r a t i o n model was constructed to simulate rotations on the column'and i t was found that small changes i n the angle at which the transducer core entered the transducer had l i t t l e e f f e c t . Loose f i t t i n g cores were used to prevent binding of the core i n the transducer. Referring to F i g . 10c, calcuation of curvature using these devices proceeds as follows: i ) the curved segment of the column i s represented by the l i n e OA1, assuming c i r c u l a r curvature, i i ) the d e f l e c t i o n measured by the l i n e a r varying d i s -placement transducer i n the adjacent bracket i s the distance A'B, where A'B i s normal to the curve at A'. Because the r o t a t i o n between adjacent segments on the curve i s always small, one may argue that A'B i s very c l o s e l y approximated by AB. iv) using a Taylor series expansion truncated a f t e r the t h i r d term, one obtains: 25 y(x + x) = y Q + AX + | — i f AX but: and J hence: and since ^^ -4 i s good approximation to the curvature d x .. ^ _ 2AB then : (J) = —-^ x Thus the curvature over AX i s twice the c a n t i l e v e r t i p d e f l e c t i o n divided "by the square of the segment length. The second system was intended to measure the deflected shape of the column. Attempts to measure deflections were made on the f i r s t column tested using d i a l gauges equally spaced 7| inches apart down the length of the column. To avoid observing each gauge at every load stage, photographs were taken of the gauges. This proved unsatisfactory f o r several reasons: i ) i t was d i f f i c u l t to read the gauges from the photos despite use of accurate equipment, i i ) i t was time consuming to in t e r p r e t the readings f o r each gauge p o s i t i o n on the column, i i i ) the range of the gauges was only 1 inch, but the columns deflected as much as 2 inches at midspan. Therefore, the gauges had to be repositioned h a l f way through the tes t at a c r i t i c a l stage when the column was becoming unstable, iv) the r e s u l t s of the gauge readings were not immediately a v a i l a b l e during the t e s t i n g . CD = y Q 2 26 For the second and subsequent column t e s t s , another method of d e f l e c t i o n measurement was employed. The upper end of a 2-inch square rectangular s t e e l tube was fi x e d to the loading head of the t e s t i n g machine; the lower end was connected to the weighing table with Schneeberger bearings which permitted only v e r t i c a l motion of the s t e e l tube ( F i g . 11a). An aluminium carriage with r o l l e r bearings could be propelled up and down the s t e e l column with a motorized pulley system. A l i n e a r varying displacement transducer with a - 1 inch range was attached to the carriage and the core of the transducer spring loaded so that i t was bearing upon and normal to the centre l i n e of concrete column flange. The t i p of the transducer core was equipped with a t e f l o n runner. To reduce the roughness of the concrete surface, a masking tape track ran along the centre l i n e of the flange ( F i g . l i b ) . When the carriage was run up and down the column, the displacement transducer produced a continuous voltage output which could be ca l i b r a t e d to p l o t the d e f l e c t i o n of the column as the abscissa on an x-y p l o t t e r , with the po s i t i o n of the carriage along the column being used as the ordinate. To determine the p o s i t i o n of the carriage, the top pulley was equipped with a threaded axle extension. As the pul l e y rotated, a threaded socket was driven i n , or out, by the turning axle. The displacement of the socket could then be measured with a l i n e a r varying displacement transducer fi x e d to the s t e e l column. Output voltage from the transducer was c a l i b r a t e d as the ordinate on the x-y p l o t . F i g . 11c i l l u s t r a t e s the pulley transducer mounted at the top of the s t e e l tube. 27 P r i o r to t e s t i n g , short wires were attached at known locations transverse to the path of the carriage on the flange of the concrete column, to produce c a l i b r a t i n g marks on the x-y p l o t . These wires were removed before the undeflected shape of the column was obtained. T y p i c a l l y , i t took about 3 seconds to run the carriage up and down the column to produce an x-y p l o t . F i g . l i d i s a photograph of the x-y p l o t t e r and t y p i c a l curves measured by the device. The transducer measuring the column deflections was accurate to within - 0.005 inches. Using the experimentally obtained d e f l e c t i o n graphs, curvatures were calculated as follows. The ordinate of the graph was divided into the same segment lengths as were used i n the mathematical model; deflections were then obtained at discrete points along the column f o r each load stage. I t was found that numerical d i f f e r e n t i a t i o n of de f l e c t i o n s at d i s c r e t e points would require measurement accuracy of - 0.00001 inches to obtain accurate curvatures. Therefore, instead of numerical d i f f e r e n t i a t i o n , a computer l i b r a r y subroutine was used to f i t a curve to the d e f l e c t i o n measurements. Estimates of the accuracy associated with each measurement could be included i n the programme and the curve selected which best f i t t e d the experimental values within the s p e c i f i e d tolerances. The computer programme also provided f i r s t and second derivatives f o r the curve so f i t t e d ; the second derivative was used as a good approximation to the curvature of the column. Curves f i t t e d to experimental values are i l l u s t r a t e d i n F i g s . 12a to 12e f o r loads ranging from.5 to 35 kips, on the f i v e column tests reported. 28 3.6.2 C a l i b r a t i o n of Line a r l y Varying Displacement Transducers. Linear varying displacement transducers with a - 0.1 inch d e f l e c t i o n range were used to measure the c a n t i l e v e r t i p de f l e c t i o n s . For c a l i b r a t i o n , each transducer, complete with i t s connecting wires, was mounted i n a brass c a l i b r a t i o n stand. The transducer core was fastened to a threaded crankshaft on the c a l i b r a t i o n stand. One complete turn of the crank d i s -placed the core 0.025 inches. A Hewlett Packard power supply with d i g i t a l voltage controls capable of producing s i x v o l t s continuous DC power accurate to within - 0.0001 v o l t s supplied the input voltage to the transducers. A d i g i t a l voltmeter with four s i g n i f i c a n t d i g i t s was used to monitor both the input and output voltages. To c a l i b r a t e the transducer, the core was moved to positions i ) + 0.025 i i ) + 0.050 i i i ) + 0.075 iv) - 0.25 v) - 0.050 v i ) - 0.075 inches from the p o s i t i o n of approximately neutral output voltage. The output voltages were recorded at each p o s i t i o n of the core. This procedure was repeated three times f o r each transducer and the average output voltage determined f o r each p o s i t i o n . Later, a computer programme was used to f i t a smooth curve through the c a l i b r a t i o n values. 3.6.3 Monitoring S t a b i l i t y ; Load-Axial D e f l e c t i o n Graphs. The s t a b i l i t y condition of the column could be observed on 29 an a x i a l load vs. a x i a l d e f l e c t i o n graph plo t t e d on a second x-y recorder. A l i n e a r l y varying displacement transducer measured the a x i a l shortening by recording the v e r t i c a l motion of the s t e e l tube (on which carriage travelled) r e l a t i v e to the weighing table. A second transducer mounted inside the T i n i u s Olsen t e s t i n g machine was c a l i b r a t e d to read the applied load. Output from both these transducers produced l o a d - d e f l e c t i o n plots such as that of F i g . 13. Neutral equilibrium of the columns was reached when the slope of the curve became zero; unstable equilibrium was observed when the slope became negative. 3*6,k Data Recording System. At each load stage, output voltages from the displacement transducers were scanned by a Vidar d i g i t a l voltmeter, scanner and recording system. Data stored by the system included; i ) clock time (day, hour, minutes, seconds), i i ) output voltage from the load measuring transducer, i i i ) seven output voltages from c a n t i l e v e r t i p d e f l e c t i o n measurement. iv) output voltage from the a x i a l d e f l e c t i o n measuring transducer, v) the input voltage to each transducer. The accuracy of the Vidar d i g i t a l voltmeter was checked by comparison with an equally accurate voltmeter. No error was observed. A computer programme was developed to convert the voltages recorded on magnetic tape to the desired information using the c a l i b r a t i o n data f o r each transducer. 30 For several t e s t s , a D i g i t a l PDP11 mini-computer was connected to the Vidar recording system and used to convert the transducer voltages to curvatures and de f l e c t i o n s , while t e s t i n g was i n progress. The mini-computer tended to slow down the rate of t e s t i n g since i t s p r i n t e r was not s u f f i c i e n t l y f a s t . L i b r a r y programmes on the univ e r s i t y ' s main IBM 370 computer were also more accurate than those used on the mini-computer. 3.7. Conduct of the Tests. 3.7.1 P r i o r to Loading. The procedure followed f o r each column i s described belows i ) a l l e l e c t r i c a l equipment was switched on 4 hours i n advance of t e s t i n g to ensure that the l i n e a r varying displacement transducers had reached a stationary voltage output while the core remained i n a fix e d p o s i t i o n . i i ) the transducers were positioned i n t h e i r holders to make maximium use of t h e i r range of d e f l e c t i o n measurement. i i i ) the a x i a l load and a x i a l d e f l e c t i o n transducers were ca l i b r a t e d . iv) the transducers mounted on the carriage and pulley . system were ca l i b r a t e d and the i n i t i a l undeflected shape of the column was determined using the x-y p l o t t e r . v) a small load ( t y p i c a l l y 5 kips) was applied to the column then removed, to ensure correct seating of the end bearings. 31 3 * 7 , 2 . Loading Procedures: i ) a slow rate of d e f l e c t i o n , i n the range of 0 ,0025 to 0.005 i n . per min., of the loading head was selected. Load stages f o r instrument readings were chosen at every 5 kips from 0 kips to 35 kips; readings were taken without i n t e r r u p t i n g the loading rate, i i ) at loads beyond 35 kips, readings were taken at every additonal 2 kips or whenever p r a c t i c a l , u n t i l the peak load had been reached. Thereafter, as many readings as possible were made as the load descended and the defle c t i o n s increased, . i i i ) once the l a t e r a l d e f l e c t i o n of the columns had increased to the point that s u f f i c i e n t data had been acquired, the t e s t was stopped and the loading head of the t e s t i n g machine raised u n t i l there was no load on the column. Several columns were tested to material f a i l u r e , A t y p i c a l column t e s t took about one hour to complete. 32 4. Results from the Testing Program. 4.1. Observations During Testing. With the exception of the f i r s t column t e s t , the behavior of the columns showed remarkable s i m i l a r i t y . In t e s t i n g the f i r s t Column, cracking originated on the web about three feet from the top of the column. Although other cracks appeared throughout the length of the specimen, the f i r s t crack grew most rapidly; thereafter, a l l the curvature tended to be concentrated at that section. I t would appear that t h i s was an unusually weak section on that p a r t i c u l a r column so that i t reached a peak load of 35 kips while the remaining columns peaked at loads between 40 and 45 kips. For t h i s reason, and because of v d i f f i c u l t i e s i n i n t e r p r e t a t i o n of the d e f l e c t i o n guages from photographs, r e s u l t s from the f i r s t t e s t have not been included i n t h i s t h e s i s . The remaining f i v e columns a l l developed reasonably symmetrical crack d i s t r i b u t i o n s throughout t h e i r c e n t r a l regions. Cracks d i d not begin to appear u n t i l the peak load was approached within one kip. These began as f i n e h a i r - l i n e cracks approx-imately equally spaced eight to twelve inches apart. As the l a t e r a l d e f l e c t i o n s increased, these cracks grew quickly while further cracking was i n i t i a t e d toward the ends of the column. A l l the cracks began at the outer f i b r e of the web and t r a v e l l e d inward normal to the axis of the column. About three inches from the outer f i b r e of the web, the cracks branched diagonally and developed a Y shape. This pattern can be seen i n F i g . 13a. F i g . 13b and 13c i l l u s t r a t e the symmetrical crack d i s t r i b u t i o n . F i g . 13d gives some i n d i c a t i o n of the amount of curvature 33 sustained by the columns before material f a i l u r e occurred. Material f a i l u r e was usually preceeded by one of the cracks i n the central regions of the column opening wide. F a i l u r e of the concrete by compression i n the flange adjacent to the wide crack would soon follow. F i g . 14a. and 14b. i l l u s t r a t e f a i l u r e of the concrete i n the flange. During the course of the experimental programme, i t was fortunate that only one crack passed beneath the epoxied trans-ducer holders; that occurred on the second column beneath a transducer located 22^ inches below mid-height. A t y p i c a l load-axial d e f l e c t i o n curve recorded f o r the columns i s shown i n F i g . 15. From t h i s graph i t i s evident that the shape of the unstable load d e f l e c t i o n curve f o r the column did indeed approach that of the t e s t i n g machine curve. F i g . 6 obtained during the preparatory stages was incorrect as i t measured only the centroidal shoretening and neglected r o t a t -ional d e f l e c t i o n due to the e c c e n t r i c i t y of the load. Further-more, F i g . 6 was calculated using the approximate r e l a t i o n PL A^ = A g . This was subsequently refined by using the s t r a i n s at the p l a s t i c centroid of each column segment i n the computer programme. This r e s u l t i s also shown on F i g . 15. 4.2. Concrete Stress-Strain Curves. The concrete s t r e s s - s t r a i n curves obtained i n the laboratory on standard 6 inch diameter, 12 inch long c y l i n d e r s , and subsequently used i n the computer model are shown i n F i g s . 16a to I6e. The peak load ranged from approximately 5.04 to 5.50 kips, with £ n values of 0.0028 to 0.0031 and f a i l u r e s t r a i n s fcmaxt_from 0.0035 to 0.0040 f o r compression tests of f i f t e e n to t h i r t y minute duration. F a i l u r e of the cylinders occurred when the slope of the concrete load d e f l e c t i o n curve became equal to the slope of the t e s t i n g machine unloading curve. F i g . 16b shows a modified concrete s t r e s s - s t r a i n curve the use of which w i l l be discussed i n section 4.5. 4 .3. Load-Moment Interaction Curve. Using the method discussed i n Section 2 . 6 , end moment vs. mid-length moment curves were plotted f o r columns 2,3,4,5 and 6 using the experimentally obtained concrete and s t e e l s t r e s s -s t r a i n curves. These graphs (Figs, 17a to I7e) were subsequently used to produce graphs of load vs. mid-length moment f o r the mathematical model f o r a 1 .75 inch end e c c e n t r i c i t y . The load-mid-length moment curves are shown i n Fi g s . 18a to 18d f o r columns 3 .^»5t6; the curves f o r column 2 are discussed i n section 4.5. The corresponding experimentally acquired load-mid-length moment curves are also plotted on Figs. 18a to 18d, f o r comparison with the computer models. The experimental curves were deter-mined by measuring the t o t a l midspan e c c e n t r i c i t y (which included the 1.75 inch end e c c e n t r i c i t y plus the measured mid-span deflection) f o r various loads, using the experimental, column d e f l e c t i o n curves (Figs. 12b to 12e). The mid-length moment was then simply the t o t a l midspan e c c e n t r i c i t y multiplied by the load f o r a p a r t i c u l a r stage of the t e s t . The experimental and mathematically obtained load-mid-height moment curves are c l o s e l y correlated, although the 35 mathematical curves underestimate the peak load and curvature s l i g h t l y . More importantly, the experimental curves exhibit the same downward path a f t e r the peak load has been reached, confirming the existence of the unstable condition predicted by the computer model. 4.4 Moment-Curvature Relationships. In the mathematical model, combinations of moment and curvature f o r a chosen load can be found by i n s e r t i n g the load as the ordinate on the load-moment-curvature graph ( F i g . 2b) to produce a curve such as F i g . 2c by f i n d i n g i n t e r s e c t i o n points on the contours of curvature. The computer model can output these values f o r selected loads. Thus, knowing the load, and entering the midheight moment as the ordinate on a curve such as F i g . 2c, the corresponding curvature i s obtained. Experimentally, curvatures were observed from the seven c a n t i l e v e r devices spanning the c e n t r a l portion of the column from 22| inches above to 30 inches below mid-height. In addition, experimental values of curvature were found by d i f f e r e n t i a t i n g the experimental column d e f l e c t i o n curves at the same seven corresponding positions along the column by the procedure described i n section 3.6.1, These values are plotted f o r each column i n Figs. 19a to I 9 e , showing the range of experimental values and t h e i r average, as well as the curvatures predicted f o r the same moments by the computer model. The experimental attempts to measure curvature do not compare well with predicted values, despite good agreement i n the o v e r a l l behaviour of the columns. In view of the poor 36 agreement, a study was made of column 2 to determine how factors such as the shape of the concrete stress-strain curve, the value of the prestressing force, and b u i l t - i n deformities might affect the moment-curvature relationships. These effects are discussed in the next section. 4.5 Case Study; Column 2;Factors Affecting Moment-Curvature Relationships. 4.5.1 Concrete Stress Strain Curves. Examination of Figs. 19a to 19e suggests that the mathematically obtained moment-curvature relationships lead to larger curvatures than those measured experimentally. This would imply that the mathematical model was not as s t i f f as the real one. Since the only factors involved in determining the stiffness of the column were the steel and concrete stress-strain curves, i t was suspected that the strains obtained i n the concrete cylinder tests were excessively high, especially since the commonly quoted strain at peak stress i s £Q = 0.0020. To study the effect of reducing £ Q to 0.0020, the strains of the original concrete stress-strain curve were reduced proportionally so that the peak strain became 0.0020 and £ u l t became 0.0025. When this was done, the agreement between experimental and computed values was much improved (Fig. 20a) compared to the original values (Fig. 20b). Unfortunately, changing the concrete stress-strain curve resulted in a computed load-mid-moment curve with a much higher peak load (Fig. 21a). The original load mid-moment curve is illustrated in Fig, 21b for comparison. 37 At f i r s t i t was thought that snap through buckling might account for the lower peak load obtained for the experimental model as opposed to the computer model with the modified concrete curve. Then agreement between computed and predicted moment-curvature relationships, could be obtained, and at the same time the discrepancy between curves could be accounted for. However, for reasons to be discussed in section 5*4, this proved unlikely to be the case, so snap-through buckling was discarded as a possibility and the modified stress-strain curve assumption was abandoned. To study the effect of varying the concrete failure strains, the short column interaction curves were re-computed based on the modified concrete stress-strain curve having £ Q = 0.0020 but using values of from 0.22$ to 0,3k%, The results were similar regardless of whether the original or modified curves were used. The curves obtained are shown in Fig. 22. The experimental loads at material failure for the six test columns were found to range from 15 to 22 kips. For that load range, i t is evident that the failure strain of the concrete would have l i t t l e significance since a l l the short column interaction curves are nearly co-incident. It is interesting that for T sections, the failure strain ^ u l t which produces the largest envelope of possible load-moment combinations, (which would presumably be reached by a real short column) is not much greater the concrete strain £ at peak stress. This fact was also observed by H. Rusch (25), who pointed out that the failure strain of concrete in flexure is highly dependent on the shape of the section. Rusch's observations are illustrated in Fig 23. 3 8 Another factor which might affect the concrete stress-strain distribution i s related to the fact that stress-strain relations in axial compression and flexure are not the same, partly due to the influence of the stress gradient in flexure, and even more due to differences in the time rate of application of strain. When the time rate of application of strain i s considered, the effect i s to increase the strain at peak stress, while peak stress is reduced slightly, relative to the short duration cylinder test. Fig. 24 shows schematically how this might happen according to Rusch. Rusch also found that stress gradients effects were negligible. The above discussion suggests that the concrete stress-strain curves should be modified by increasing SQ, however, this would serve to make the computer model less s t i f f than before and the correlation of moment and curvature would become worse. 4.5.2 Prestressing Force. Fig. 2 5 shows the computed load-mid-height moment curve for column 2 using both an 8 kip and a 16 kip f i n a l prestressing force. The correlation between these curves and the experimentally obtained curve is not as good as that in which the prestress force was calculated to be 1 3 . 3 5 kips, based on the original estimate of the prestress losses. In addition, there is poor agreement between experimental and calculated curvatures (Fig. 26) especially i f an 8 kip prestress force is used. It is unlikely, therefore, that a gross error in estimating the f i n a l prestress force would account for the unsatisfactory agreement between observed and predicted curvatures. 39 4.5.3 Effect of Built in Deformation. No column can be constructed without b u i l t - i n imperfections or non-homogeneity. Despite attempts to manufacture the columns used in the test programme as accurately as possible, i t i s not unlikely that lateral deformations in the plane of the web existed prior to testing. Such deformations could be caused by inaccuraries in the formwork, and in placing the concrete in the forms, slight misplacement of the prestress, or perhaps by additional curvatures resulting from creep effects due to the slight eccentricity on the prestress. Any known eccentricities would be taken into account in the programme. To study the effect of b u i l t - i n deformations, the computer model was modified as follows: i) The moment-curvature relationships were calculated as before, assuming no change in the material and sectional properties of the column, i i ) An estimate", was made of b u i l t - i n deflections along the column. i i i ) For a particular load, the mid-height moment was due to the sum of the b u i l t - i n deformation and l a t e r a l deflection caused by bending of the column. The curvatures in the next segment, a distance AX along the column,, were calculated using the Taylor series expansion as described in section 2.5 with the total moment at mid-height, iv) The deflection of a point a distance AX along the column from mid-height was calculated using the previously computed curvature. This was the deflection 4 0 at the point due to bending of the column alone; to this was added the b u i l t - i n deformation at the point. The total deflection could then be used to determine the moment sustained by the column at that cross-section. v) The curvature in the next segment was based on\ the total moment obtained at the previous point on the column. vi) The calculation was continued in this way along the the length of the column u n t i l the thrust line was crossed or the column length exceeded. The modified computer programme could now produce end moment vs. mid-length moment curves which included i n i t i a l deformations. From these, new load-mid-height moment curves could also be obtained. By examining the column deflection curves, slopes and curvatures, i t was found that the curvatures in the -midspanV;;-:' regions. of ..the column were only slightly affected by small b u i l t -in deformations, provided the load was stable, and less than about 9 0 $ of the peak load. This can be understood by studying the moment curvature curves such as Fig, 2 c for various loads. Under stable load conditions i t i s found that one i s always dealing with the steepest portion of the moment curvature curve. Thus, a small additional moment due to an imperfection in the column straightness does not significantly change the curvature obtained. The curvatures only begin to increase rapidly with small changes in the moment when one i s approaching the peak load. It i s in this range 41 that built in deformations have the greatest effect. A b u i l t - i n deformation, in the plane of bending w i l l result i n a reduced peak load i f i t i s in the same direction as the bending. However, in the lower range of stable loads, i t will: not affect the curvatures to any great extent., On the above basis, i t i s , therefore, believed that b u i l t -in deformations cannot account for the discrepancy between observed and predicted curvatures. For further confirmation, a study of column 2 was made using a 16 kip prestressing force and varying the b u i l t - i n deflection. For a b u i l t - i n midspan deflection of 0.1 inches in the same direction as bending occurred, there was excellent agreement between observed and calculated load-mid-height curves (Fig. 27a); the same correlation was not found with moment curvature relationships (Fig. 28a). A good correlation with moments and curvatures could be found assuming 0.2 inches maximum built in deflection which opposed the direction of bending, (Fig. 28b), but this led to poor agreement with load-mid-length moment curves (Fig. 27b). 4,6 Summary. The over-all behavior of the concrete columns closely matched the predicted behavior as witnessed by the load mid-height moment curves. Attempts to compare the observed curvatures with calculated values proved unsuccessful despite examination of various factors which might have affected the moment-curvature relationships. It can only be concluded that either: i) The attempts to measure curvatures in the laboratory were unsuccessful. 42 i i ) The o v e r - a l l b e h a v i o r of the columns was not s i g n -i f i c a n t l y a f f e c t e d by wide v a r i a t i o n s i n moment c u r v a t u r e r e l a t i o n s h i p s w h i l e the columns remained s t a b l e . 43 5« Investigations into Snap-Through Buckling. 5.1 Potential Energy of the Column. The computer programme used to obtain column deflection curves was easily adapted to determine the potential energy of the column. It w i l l be recalled that the method of determining the column deflection curves involved a Taylor series expansion in which the slopes and curvatures were calculated for equally spaced segments along the length of a column. For each column length studied, the end moment and slope were also determined. For uni-axial bending, there are four terms necessary (neglecting ; higher order effects) in the potential energy expression for a column. They are: i) Loss of energy of the load caused by shortening of the column due to compressive strain along the plastic centroid. i i ) Loss of energy of the load resulting from the column ends moving closer together as the column bends lat e r a l l y . i i i ) Loss of energy of the end moments (about the plastic centroid) rotating through the end slopes, iv) Strain energy in the column (energy gain). In the computer, model, each of the above terms i s resolved as follows; i) The energy loss per unit length due to compressive shortening of the column i s determined by the method described in sections 2.3 and 2.4 from the strain distribution of each segment along the column length. 44 By integrating the energy loss per unit length for each segment along the length of the column, the total energy loss due to compressive shortening i s obtained. i i ) Referring to Fig. 29, energy loss of the load due to bending of the column i s found: d l = )j dx 2 + dy 2 (from Binomial Expansion.) approximating yields: , 2 d l - dx ~ £• (jg) dx from which the total shortening along the column length would be: A l -The loss of potential energy of the load would, there-fore, be: P.E. - P o / * ( S ) 2 *< Since the slopes( j^) are known for each segment along the column length, the above integral can be easily solved numerically, i i i ) The energy loss due to the end moments rotating through the end slopes is a simple multiplication performed after each column deflection curve i s obtained. iv) Strain energy of the column: 45 At any cross-section along the length of the column,the strain distribution determines the curvature at that location : as well as the resisting forces and moments on the section. The strain energy per unit length at each section i s , therefore, the area under the moment-curvatute curve with the curvature ranging from zero to the value determined by the strain d i s t r i b -ution. Expressing the above in integral form, the strain energy for a segment of length 'dx' i s obtained: (where 0 Q i s the total curvature at the section under consider-ation) . The above expression for dU can then be integrated along the length of th<=; column to obtain the total strain energy: Addition of the above four terms yields the total potential energy of the column for a given load and column deflection curve. When several column deflection curves are studied for a column of given length and applied load, a graph of the potential energy of the column can be made. From Fig. 3 0 i t can be seen that only one column deflection curve leads to a relative minimum of potential energy corresponding to stable equilibrium. Likewise, the column deflection curve which pro-duces an unstable configuration corresponds to a relative maximum on the potential energy diagram. U = M($) d($) dx 46 5.2 End Moment vs. Potential Energy Curves, In the computer model, the f i r s t column deflection curve is found for a selected load using the failure moment at mid-height. Thereafter, the mid-height moment is decreased increm-entally, resulting in a new column deflection curve for each increment. Each column deflection curve produces a unique combination of end and mid-height moments which can be plotted to obtain a smooth curve as described in section 2.6. Similarly, each column deflection curve produces a single combination of end moment and potential energy for a given column length. A plot of these values results in curves such as those of Fig. 31. As in section 2.6, a column whose eccentricity of loading is known may now be studied. The end eccentricity and load determines the end moment to be used as the ordinate on Fig. 31. The curves in Fig. 31 were derived for column 4. As an example, for a 1.75 inch end eccentricity of a 25 kip load, the end moment would be 43.75 inch-kips or 3.65 foot-kips. A horizontal line through 3.65 ft-kips on Fig. 31 intersects the potential energy curve for the 25 kip load at two locations (- 1.25 inch-kips and +2.25 inch-kips). Following the above procedure the potential energy of the column under various loading conditions can be found. Fig. 31 was based on a concrete stress-strain curve having £ = 0.0028 and £ ... = 0.0035. Similar curves were derived for 0 ult column 4 having concrete with £Q = 0.0020 and ^ u l^. = 0.0025 (see Fig. 32). 47 5.3 Load vs. Potential Energy Curves. The combinations of load and potential energy obtained in section 5«2 may now be plotted in the same way as load-mid-height moment curves were found (section 2.7). Fig. 33a illustrates the load-mid-height moment curves for column 4 for concretes having £ = 0.0028 and £ = 0.0020 as well o o as the experimental results. Fig. 33° shows the corresponding computed load-potential energy curves. Arrows indicate the direction of the potential energy curve as the column i s loaded, reaches neutral equilibrium, and i s then unloaded down the un-stable equilibrium path. In the stable range, the potential . energy decreases with increasing load. Once the peak load has been reached, the load decreases while the potential energy increases rapidly along the unstable path. 5.4 Load vs. Energy Difference Curves. If one considers the difference in potential energy between the stable and unstable equilibrium conditions under a given load, some indication as to the amount of energy required to cause snap-through, buckling may be found. These energy difference curves are plotted in Fig. 34b for comparison with load-mid-height moment curves for column 4 (Fig. 34a). At the outset of the experimental programme, i t was believed that such curves would slope downward, convex when viewed from the origin. This would mean the energy difference curve would approach the zero energy position asymptotically, when nearing the peak load. If this were the case, a very small amount of externally applied energy might cause a snap-through 48 to unstable equilibrium at a somewhat lower load than the theoretical peak. Fig. 3^b, however, shows that the potential energy difference curves do not approach zero energy difference asymp-to t i c a l l y as surmised. In fact, for loads just below the peak load, there is a substantial difference in the potential energy between the stable and unstable equilibrium paths. The shape of the load-energy difference curves would, therefore, suggest that the columns are not prone to snap-through buckling. Of course, i f sufficient additional energy was provided to jump from the stable to the unstable path, a snap-through might s t i l l occur. Finally, the energy difference curves computed for column 4 for concrete with £ Q = 0.0020 indicate that considerable energy would be necessary to cause snap-through buckling at a load equal to the experimentally observed value. For this reason the hypothesis that the concrete stress-strain curves were in error, that the observed moment-curvature relationships were correct, and that the observed load-moment relationship reflected snap-through buckling, was abandoned. CONCLUSIONS. The objective of this thesis v/as to test the validi t y of a mathematical model which attempted to predict the s t a b i l i t y of prestressed concrete columns. Six real prestressed concrete columns were tested in the structural laboratories at the University of Bri t i s h Columnbia and their behaviour was compared to the predictions of a computer programme based on the math-ematical model. The computed behaviour was closely matched by the real columns. In particular, load versus mid-height moment curves determined by computer approximated the same curves obtained on the test columns favourably. The existence of a downward sloping unstable loading path on the load-mid-height moment curve was found in both the mathematical model and on the test columns. Having confirmed that the mathematical model did indeed approximate real behaviour satisfactorily, the po s s i b i l i t y of a snap-through type of buckling was examined. Differences in the potential energies associated with the stable and unstable loading paths for the column were determined, and a graph of the energy difference as a function of load was prepared. I n i t i a l l y i t was thought that the energy difference between the two equil-ibrium paths would approach zero asymptotically as the c r i t i c a l load was approached. However, results from the computer programme indicate that the columns were not prone to snap-through buckling. It should be noted, though, that sufficient additional energy applied to the column near the c r i t i c a l load might s t i l l cause a snap-through. In general, the instrumentation used to monitor the conditions of the test columns "behaved well. Attempts to measure the curvatures of the columns during the testing, and then to compare the curvatures to computer-predicted values were unfort-unately, unsuccessful. No satisfactory explanation for the poor comparison could be found, although the following two reasons were suggested* i) the deflection measurements necessary to obtain the curvatures were too small to measure accurately despite use of linearly varying displacement trans-ducers and precise monitoring equipment, i i ) the curvatures measured were perhaps correct but the overall behaviour of the column was not significantly affected by wide variations in moment-curvature relationships while the column remained stable. Further research in this f i e l d might be directed toward adapting the mathematical model to handle various conditions of end f i x i t y in prestressed columns. Development of an accurate method of measuring curvature on real concrete members would also be useful in both reinforced and prestressed concrete _ , .• research. ,51 .Tension S i d e STRAIN DIAGRAM F i g . l a . SHORT COLUMN INTERACTION CURVE F i g . l b . STRAIN DIAGRAMS F i g . 2 a . Moment LOAD-MOMENT.-CURVATURE RELATIONSHIPS F i g . 2 b . 54 o cvj-r Fig.2d. LOAD ENERGY CONTOURS DUE TO COMPRESSIVE SHORTENING OF THE PLASTIC CENTROID COLUMN DEFLECTION CURVE F i g . 3 a . - — y 3 ^ - -) ^ - — y 5 J COLUMN DEFLECTION CURVES FOR VARIOUS MID-HEIGHT DEFLECTIONS F i g . 3 b . 56 Note: P 1<P 2<p,<P i Mid-height Moment END MOMENT vs. MIDHEIGHT MOMENT CURVES Fig.4. LOAD vs. MID-HEIGHT MOMENT CURVES F i g . 5 . Axial Deflection (inches) LOAD-DEFLECTION CURVES Fig.6. Note : a l l dimensions in inches. | inch thick steel end block cast into column. 1 inch diameter r o l l e r bearing , .\ inch V thick bearing plate 1 2 i!7 Axis of Loading = 6 1.75 Plastic Centroid \ 4.875 3/8 inch diam. 7-wire strand ? (A = 0.08 in. ) Fig.7a. COLUMN DIMENSIONS #4 r e i n f o r c i n g bar ( l e n g t h =15 i n c h e s ) F i g . 7 c END BEARINGS ( Test date t Feb. 10, 1976. Cross-sectional area : 0,080 in.' Failure stress : 265 k.s.i. Failure strain t 6.0 % 0.2 0.4 0.6 0.8 1.0 1.2 Percent Strain (in./in.) Fig.8. STRESS-STRAIN CURVE for 3/8 INCH DIAM. 7-WIRE PRESTRESSING STRAND 1.4 ON ro Moment ( k i p s - f t . ) LOAD-MID-HEIGHT MOMENT INTERACTION CURVES F i g . 9 . 64 F i g . l O a . : C u r v a t u r e m e a s u r e m e n t u s i n g c a n t -i l e v e r s y s t e m . E a c h t r a n s d u c e r m o u n t e d i n a l u m i n u m h o l d e r m e a s u r e s t i p d e f l e c t i o n o f a d j a c e n t c a n t i l e v e r . Fig.10b. : A r r a n g e m e n t o f c a n t i l e v e r s i n s t r a i g h t l i n e a l o n g w e b o f c o l u m n . C a n t i l e v e r s m e a s u r e d c u r v -a t u r e i n c o l u m n f r o m 22g i n c h e s a b o v e t o 30 i n c h e s b e l o w m i d h e i g h t . ( T h e f i f t h t r a n s d u c e r f r o m t h e b o t t o m i s a t m i d h e i g h t . ) 65 SCHEMATIC DIAGRAM : CANTILEVER CURVATURE MEASURING DEVICES (not to scale) Y I — X x + x o APPROXIMATION USED TO DETERMINE CURVATURE OF TYPICAL SEGMENT O-A Fig.10c. 66 F i g . i l a . Illustrates device for obtaining col-umn deflection curves. Lower end of steel tube is connected to concrete column with Schneeberger bearings. Transducer core i s e l a s t i c a l l y load-ed to bear upon masking tape track. Pulley sys-tem i s powered by elect-r i c d r i l l . Second trans-ducer measures motion of cantilever connected to steel tube, thereby det-ermining axial deflection of the concrete column. F i g . l i b . This photo-graph shows the carriage at the bottom end of the steel tube r a i l . The masking tape track along the centre line of the flange of the concrete column i s clearly v i s i b l e . 67 F i g . l i e . The 2-inch square steel tube r a i l and pulley system can be seen just to the right of the concrete column. The circular object at the top i s a mounting bracket for the transducer monitoring the rotation of the top pulley. F i g . l i d . Illustrates use of x-y plotter to obtain column deflection curves. Curves are bunched togeth-er under stable load cond-itions but spread apart as late r a l deflections increase rapidly under unstable load conditions. F i g . 1 2 a . in COLUMN DEFLECTION CURVES COLUMN 3 LOAD ( k i p s ) 0.0 20.0 40.0 60.0 80.0 100.0 120.0 THRUST LINE (INCHES) F ig .12b. 20.0 COLUMN DEFLECTION CURVES COLUMN 4 LOAD ( k i p s ) 43.0 40.0 THRUST 60.0 80.0 LINE (INCHES) Fig.12c. LOO.O 120.0 -N3 O COLUMN DEFLECTION CURVES COLUMN 5 LOAD ( k i p s ) 0.0 20.0 40.0 60.0 80.0 100.0 120.0 THRUST LINE (INCHES) F i g . l 2 d . COLUMN DEFLECTION CURVES COLUMN 6 LOAD (kips) ,0.0 20.0 40-0 60.0 80.0 100.0 120.0 THRUST LINE (INCHES) Fig,12e. 7 3 F i g . 1 3 a . Illustrates crack pattern developed by column after peak load has been reached. Notice branch of ' Y' -shape beginning about 3 inches from edge of web. F i g . 1 3 b . Illustrates the symmetrical crack d i s t r i b -ution in central portions of the column. Crack spacing ranges from 8 to 1 2 inches. Transducers are spaced 7k inches apart. 74 F i g . 1 3 c This photo-graph also shows the equally spaced crack pattern in central regions of the column. Fig.13d. The view from the bottom of the column shows the amount of curvature the column was capable of sus-taining before failure of the concrete occurred in compression. Fig.14a. Figs.l4a. and 14b. show compression failure of the concrete flange at the ultimate lateral deflection. These are photographs of the second column tested. Fig.14b. LOAD-AXIAL SHORTENING CURVE FOR COLUMN 4 ' I 1 1 1 : 1 0.025 . 0 . 0 5 0 0.075 0.100 0.125 0.150 0.175 A x i a l S h o r t e n i n g (Inches) F i g . 15 F i g . 1 6 a . F i g s . 1 6 a . a n d 1 6 b . F i g . 1 6 b . CONCRETE STRESS STRAIN CURVES ^ Fig.16c. Fig.l6d. Figs. 16c. and I6d. CONCRETE STRESS STRAIN CURVES co Fig.l6e. F i g . l 6 f . F i g s . I6e. and I6f. CONCRETE STRESS STRAIN CURVES Fig.17a. Fig.17b. F i g s . 17a. and 17b. END v s . MID-HEIGHT MOMENT CURVES 81 2 3 k 5 6 7 8 9 10 Mid-height Moment ( k i p s - f t . ) Fig.17c. - A 1 . 2 3 4 5 6 7 8 9 10 Mid-height Moment ( k i p s - f t . ) F i g . l 7 d . F i g s . 17c. and -17d. END v s . MID-HEIGHT MOMENT CURVES 82 Mid-height Moment ( k i p s - f t . ) F i g . l ? e . END v s . MID-HEIGHT MOMENT CURVE 83 o ve-to ft • H a o o -3-COLUMN 3 • experimental computed 0 e Mid-height Moment (kips-ft.) Fig.18a. Mid-height Moment (kips-ft.) Fig.18b. Figs. 18a. and 18b. LOAD-MID-HEIGHT MOMENT CURVES 84 voT 0 2 4 6 8 10 Mid-height Moment (kips-ft.) -Fig.18c. 2 4' 6 8' Mid-height Moment (kips-ft.) Fig.l8d. Figs. 18c. and 18d. LOAD-MID-HEIGHT MOMENT CURVES o 00' • H w ft • H . « O + > e o o 0 A Average (Cantilever Devices) O Average (Observed Column Deflection Method) Range of Observed Values for Each System • Computer Predicted Curvatures £ 20' i—A - l l -O J z _ I - £ H - C M • • 151 £ - l o k A o-i k ^ 3 0 . -25r k D ^ 35* * COLUMN 2 Curvature (in.~* x 10""-5) 8 10 12 14 Fig.19a. OBSERVED and PREDICTED CURVATURES CO A Average (Cantilever Devices) O Average (Observed Column Deflection Method) HRange of Observed Values for Each Syst.em^ ^_ •' Computer Predicted Value 1 1 <-* ^~ 35' -AH—O-z. -A-4=e—i 20 k 151 25' 30 Q COLUMN 3 1 J 1 1 4 6 8 10 Curvature (in.~* x 10~-*) Fig.19b. OBSERVED and PREDICTED CURVATURES o 00 A Average (Cantilever Devices) O Average (Observed Deflection Method) I 1 Range of Observed Values for Each System D Computer Predicted Curvatures o \o I—N=£j—h I 1 / •301 A. 35' O—B-• t to ft • H -P c o a o o o C M ± 1 l-A-1 ^ 10 15' k 20' 25' O • 5k DO V • COLUMN 4 H 14 4 6 «8 . 10 Curvature (in.~ x 10"°) Fig.19c. OBSERVED and PREDICTED CURVATURES 12 co o 00 A Average (Cantilver Devices) O Average (Observed Column Deflection Method) 1 Range of Observed Values for Each System • Computer Predicted Curvature o c • H I W P< • H O 6 o S o C M I — A H I—AH A i 15-t-Al 10 I — 0 - 1 • • 20 k • I A H I A -H V L I—OH k. 25 30' mi • COLUMN 5 - O — 35 ' • 6 8 Curvature (in." 1 , x 10~^) 10 Fig.i9<i. OBSERVED and PREDICTED CURVATURES 12 14 CO oo o CO A Average (Cantilever Devices) O Average (Observed Column Deflection Method) 1 Range of Observed Values for Each System D Computer Predicted Curvature o i CO • H O •P c CD S o o CM - A -+ = © — A = H Z H GA= Z _ 3 0 k H QA I 1 z . — 2 5 * _ -MS) I I ^ 20 k -O—I Z 1 5 k hAl K3 Ov ^ — 1 0 k - ^ • 35' H • COLUMN 6 mf 4 6 « 8 10 Curvature i n . " x lO"-*) Fig.l9e. OBSERVED and PREDICTED CURVATURES 12 CO VO » CQ ft •H 4 s C <D CM s o MOMENT-CURVATURE RELATIONSHIPS '• f o r C0LUMN~2 NO -T -prestress force = 13«35 max. b u i l t - i n a A d e f l ' n . = 0 i n . concrete* u0 0.0020 A3 O A O ADO \o' I CO ft A l O A O A Cantilever system o Measured CDC system e o Computer prediction §2™ o + -prestress force = 13.35 -max. b u i l t - i n d e f l ' n . = 0 i n . -concrete* £ Q = 0.0031 A O • A O > k 6 8. 10 -12 Curvature ( i n . ~ x 10"*->) Fig.20a. 14 A O • A O A O A O • A C a n t i l e v e r system G Measured CDC system • Computer p r e d i c t i o n _i8 12< Curvature ( i n x 10 J ) Fig.20b. 12 Ti O 91 o ve-to o • H Xi o ^ o CM experimental v -computed using: k y,«v.«V,-/ -prestress force = 13»35 / -max. bu i l t - i n / defl'n. = 0 in. -concrete : £ 0=0.002 2 4 6 8 Mid-height moment (kips-ft.) Fig.21a. o VO ^ O to^-• H o ^ o CM -experimental -computed using: ^ . ,V?X -prestress force = 13.35 e -max. b u i l t - i n defl'n = 0 in. -concrete ; 0 2 4 6 8 Mid-height Moment (kips-ft.) Fig.21b. Figs. 21a. and 21b. LOAD-MID-HEIGHT MOMENT CURVES 9 2 <5> Moment ( k i p s - f t . ) F i g . 2 2 . LOAD-MOMENT INTERACTION CURVES FOR VARIOUS CONCRETE FAILURE STRAINS 93 SHAPE OF CROSS-SECTION 0 0.5 1.0 0 0.5 1.0 0_ 05 _I.O i STRESS DISTRIBUTION STRAIN DISTRIBUTION F i g . 23a. Strain and stress distribution at ultimate strength after I hr, ic' — 3000 psi at 56 days Fig .2 3 . 1.0 :o.5 r r- 1—' f /s \. \\ // 1 \ \ \ / ! 0.005 per hr. 1 / y\ \ 1.0 r —,—,——, f 1 / | 0.003 per h r / | / ^ \ / j 0.001 per hr. j ' / 1 ' f 1 ' i i i > • 0.002 0.004 0.002 Concrete strain, 0.004 Concrete strain, E Fig. 24b, Determination of stress-strain relationship in flexure (schematic only) !ieft) Stress-strain curves for concentric compression and various strain rates right) Stress-strain relationship for eccentric compression after I hr of loading at constant strain rates Fig.24. 95 o ve-to o -cf rt o ^ o C M experimental computed using: 16 kip' prestress 8 kip prestress -max. b u i l t - i n d e f l ' n = ( T i n . •-concrete; = 0.0028 e / 2 4 6 8 Mid-height Moment ( k i p s - f t . ) Fig. 2 5 , LOAD-MID-HEIGHT MOMENT CURVES 10 VO -P to PH •H O C M 6 o s -prestress force: -max. b u i l t - i n d e f l ' n = 0 i n . 161 A O • -concrete : £ Q=0.©028 A O A G A O • X A O A O ) Q • X • A Cantilever system O Measured CDC system • Computer p r e d i c t i o n , prestress force =.16 X Computer prediction. prestress force = 8 Fig.26. t + 10 12 14 16 Curvature ( i n . x 10~->) MOMENT-CURVATURE RELATIONSHIPS 18 96 o VO experimental computed u s i n g : . <fo / - p r e s t r e s s f o r c e = 16 . 0 \ , y -max. b u i l t - i n * ^ o CO ft • H a o tl o CM d e f l ' n = +0.1 i n . " / •concrete : / 0 2 4 6 8 Mid-height Moment ( k i p s - f t . ) F i g . 2 7 a . 10 o vO CO o ft -=r • H M a o hi o CM experimental •computed u s i n g : . - p r e s t r e s s f o r c e = 1 6 . 0 -max. b u i l t - i n d e f l ' n = - 0 . 2 i n . -co n c r e t e : 0 0LUMN 2 2 4 6 8 Mid-height Moment ( k i p s - f t . ) 10 F i g . 2 7 b . F i g s . 2 7 a . and 2?b. LOAD-MID-HEIGHT MOMENT CURVES «H + -L I w ft • H C o MOMENT-CURVATURE RELATIONSHIPS fo r COLUMN 2 -prestress force = 16 -max. b u i l t - i n d e f l ' n = +0.1 i n . -concrete : 0=0.0031 AO a o A O • A O • A O A O A Cantilever system o Measured CDC system 2 • Computer p r e d i c t i o n vo -r w ft o 2 4 6 8 10 _ 12 Curvature (i n . " " 1 x l O - 5 ) Fig.28a, 14 -crestress force = 16 -max. b u i l t - i n d e f l ' n = -0.2 i n . concrete t Q=0.0031 A O O &0 • A 0 A o • A oD A Cantilever system o Measured CDC system • Computer p r e d i c t i o n + 4-8 2 4 6 Curvature ( i n . " 1 - x 10"^) Fig.28b. 10 12 -5> 14 VO 98 P X D e t a i l A Fig.29. SHORTENING ALONG THE LINE OF LOADING of the COLUMN DUE TO BENDING 99 • I. I I I I. I. I . . • • — I .1.1 ... • Moment F i g . 3 0 . - 2 - 1 0 1 P o t e n t i a l Energy Fig.31. r 2 3 ( k i p s - i n . ) 1 o o Fig.33a. LOAD-MID-HEIGHT MOMENT CURVES " Fig.33b. LOAD-POTENTIAL ENERGY CURVES o O ~r-ft M a o ^ o CM A J experimental curve B » computed curve f o r £ Q=0.0028 C » computed curve f o r e0=0.0020 B : computed curve f o r fQ=0.0028 omputed curve 0.0020 •\ K — I — : f 2 4 6 8 Mid-height Moment ( k i p s - f t . ) Fig.34a. LOAD-MID-HEIGHT MOMENT CURVES o 1 2 3 4 5 Energy Difference Between Stable & Unstable Equilibrium (kips-inches) Fig.34b. LOAD-ENERGY DIFFERENCE CURVES o 104 BIBLIOGRAPHY. 1) . Karman, T.V., "Untersuchungen uber Knickfestigkeit," Mitteilungen 'uber Forschungsarbeiten auf dem Gebiete des  Ingenieurwessens. No. 81, Berlin Germany, 1910. 2) Ketter R.L., Kaminsky, E.L., and Beedle, L.S., "Plastic Deformation of Wide Flange Beam Columns," Transactions, American Society of C i v i l Engineers. V. 1 2 0 , 1 9 5 5 f p.1028. 3) Westergaard, H.M., and Osgood, W.R., "Strength of Steel Columns," Transactions. American Society of Mechanical Engineers, Vol. 5 1 , 1928 ,pp. 6 5 - 8 0 . 4) Broms, B., and Viest, I.M., "Ultimate Strength Analysis of Long Hinged Reinforced Concrete Columns," Journal of the  Structural Division. American Society of C i v i l Engineers, Proceedings, Vol. 84, 1958, pp. I 5 I O-I - I 5 I O - 3 8 . 5) Chang, W.F., and Ferguson, P.M., "Long Hinged Reinforced Concrete Columns," Journal of the American Concrete Institute, Proceedings, V. 60, No. 1 , Jan I 9 6 3 , pp.1 - 2 6 . 6) Hognestad, E., "A Study of Combined Bending and Axial Load in Reinforced Concrete Members," Bulletin No, 3 9 9 , Engineering Experiment Station, University of I l l i n o i s , Urbana, 111., 1 9 5 1 . 7) Itaya, R., "Design and Uses of Prestressed Concrete Columns," Journal of the Prestressed Concrete Institute, June, I965, p. 69. 8) Brown, K.J., "The Ultimate Load Carrying Capacity of P.C. Columns under Direct and Eccentric Loading," C i v i l Engineering  and Public Works Review. A p r i l , May and June, I 9 6 5 . 9) Kabaila, A.P. and Hall, A.S., "The Analysis of Instability of Unrestrained Prestressed Concrete Columns with End . Eccentricities," Symposium on Reinforced Concrete Columns, Special Publication, American Concrete Institute, I 9 6 6 , pp. 157-178. 10) Rusch, H., "Researches Toward a General Flexure Theory for Structural Concrete," Journal of the American Concrete  Institute. Vol. 57, July i 9 6 0 , p. 1-28. 11) Brown, H.R., and Hall, A.S., "Tests on Slender Prestressed Concrete Columns," Symposium on Reinforced Concrete Columns, Special Publication American Concrete Institute, 1966,- pp. 1 7 9 - 1 9 2 . 12) Brettle, H.J., "Effect of Variable Concrete Modulus of Elas t i c i t y on the Deflection of Prestressed Beams," Australian  Journal of Applied Science (Melbourne), Vol, 1 1 , No. 3, March, I 9 6 0 , pp. 9 2 - 1 0 7 . 105 13) Lin, T.Y., and Lakhwara, T.R., "Ultimate Strength of Eccentrically Loaded Pa r t i a l l y Prestressed Columns," Journal  of the Prestressed Concrete Institute, Vol. 11, No, 3, June 1 9 6 6 , pp. 37-49. 1 4 ) : Zia,p., and Moreadith, F.L., "Ultimate Load Capacity of Pre-stressed Concrete Columns," Journal of the American Concrete  Institute. Vol. 63, July 1961T. 15) Zia, P., and Guillermo, E.C., "Combined Bending and Axial Load in Prestressed Concrete Columns," Journal of the Pre-stressed Concrete Institute, June 1967$ pp. 52-58. 1 6 ) Nathan, N.D., "Discussion of the paper by Paul Zia and E.C. Guillermo: Combined Bending and Axial Load in Prestressed Concrete Columns," Journal of the Prestressed Concrete  Institute. December 1967f pp. 8 4 - 8 7 . 17) Aroni, S., "Slender Prestressed Concrete Columns," Journal of the Structural Division, Proceedings of the American Society  of C i v i l Engineers. A p r i l 1 9 6 8 , pp. 875-903. 1 8 ) Chandwani, R., and Nathan, N.D., "Precast Prestressed Sections under Axial Load and Bending," Journal of the Prestressed  Concrete Institute. May-June 1971f pp. 1 0 - 1 9 . 1 9 ) Whitney, C.S., "Plastic Theory in Reinforced Concrete Design," Transactions of the American Society of C i v i l Engineers. Vol. 107, 1942, pp. 251-326. 2 0 ) Nathan, N.D., "Slenderness. of Prestressed Concrete Beam-Columns," Journal of the Prestressed Concrete Institute, Vol. 1 7 , No. 6, November-December 1972, pp. 45-57. 2 1 ) Nathan, N.D., "Applicability of ACI Slenderness Computations to Prestressed Sections," Journal of the Prestressed Concrete  Institute. May-June 1975. pp. 68-85. . 2 2 ) Pfrang, E.O., Siess, CP., Sozen, M.A.,-"Load-Moment-Curvature Characteristics of Reinforced Concrete Cross Sections," Journal American Concrete Institute, July, 1 9 6 4 , P P . 763-770. 23) Galombos, T.V., "Column Deflection Curves," Lecture No. 9. Plastic Design of Multi-Story Frames, Fritz Engineering Lab. Report No. 273-20, Lehigh University, I965. 2 4 ) Libby, J.R., Modern Prestressed Concrete Design, Principles and Construction Methods, Van Nostrand Reinhold Ltd., T o r o n t o , 1 9 7 1 t P P 4 1 - 5 9 . 2 4 ) Rusch, H.," Researches Toward a General Flexural Theory for Structural Concrete", Journal of the American Concrete  Institute, Proceedings, July i960, pp. 1 - 2 8 . 

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