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Precast concrete load bearing wall panels Chandwani, Ramesh Hassanand 1970

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PRECAST CONCRETE LOAD BEARING WALL PANELS by RAMESH H. CHANDWANI B, Tech (Hons.) Indian I n s t i t u t e of Technology Bombay, I n d i a , June 1967 A THESIS SUBMITTED- IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the department of CIVIL ENGINEERING We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA December 1970 In presenting t h i s thesis i n p a r t i a l f u l f i l l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C i v i l Engineering The University of B r i t i s h Columbia, Vancouver 8, Canada Date December 22, 1970 ABSTRACT The object of t h i s thesis i s twinfold. F i r s t l y , to study and check the e f f e c t i v e width requirements recommended by d i f f e r e n t committees f o r the design of ribbed precast concrete load bearing walls. Secondly, to r a t i o n a l i z e the practice of the designing of the precast prefabricated components of any general polygonal shape. The recommendations regarding the minimum thickness of a thin wall, the e f f e c t i v e width, etc., have been s p e c i f i e d f o r some cases 2 in code books and other tentative s p e c i f i c a t i o n books , i n the form of r u l e s of thumb. In these r u l e s of thumb, several parameters which may be of s i g n i f i c a n c e , such as dimensions of the r i b i t s e l f , are not taken into account. A f i n i t e element approach has been adopted to investigate various combinations of these parameters, as well as the e f f e c t s of d i f f e r e n t boundary conditions. Similar problems arise also i n the cases of T-beams, L-beams and design of a i r c r a f t structures, i n which a stressed skin i s mounted on r i b s , which are assembled i n the form of a space frame. So f a r i n t e r a c t i o n curves have been made available i n some design books only for the prestressed concrete member having rectangular cross-sections and f o r any other shape, approximations are made, such as making a rectangular section having an equivalent area or having the same moment of i n e r t i a or section moduli!, etc. But t h i s practice seems very i r r a t i o n a l e s p e c i a l l y i n the case of precast components which are always produced i n a factory on a mass s c a l e . A computer program has been written which can give the i n t e r a c t i o n curve f o r the member of any polygonal shape. DRAFT 3: Of PCI Committee, 'Recommendations for Prestressed Bearing Wall Design.' 'Symposium Dn Precast Concrete Wall Panels.' Publications ACI, SP-11, Second Printing 1966, pp. 39-*t*t. TABLE OF CONTENTS CHAPTER Page I . INTRODUCTION 1 I I . EFFECTIVE DIMENSIONS: PROBLEMS TO BE ANALYZED 4 D e f i n i t i o n s 4 Clauses 5 I I I . ELASTIC ANALYSIS OF SECTIONS: THEORY AND PROGRAMMING 9 Plane S t r e s s Problem 9 P l a t e Bending Problem 9 S t a b i l i t y C o e f f i c i e n t s M a t r i x 12 Buckling Loads 17 IV. TESTING OF PROGRAMS 20 Plane S t r e s s Program 20 P l a t e Bending Program 21 V. INVESTIGATIONS AND RESULTS 28 (L/H) R a t i o 28 (Span/thickness) R a t i o 32 E f f e c t i v e Width of T-Wall 40 VI. DESIGN OF PRESTRESSED SECTIONS IN COMPRESSION 52 Assumptions Used i n the Program 5k S t r e s s / s t r a i n R e l a t i o n s h i p 54 C o m p a t i b i l i t y 56 E q u i l i b r i u m 57 B r i e f O u t l i n e of Program 57 Method of Computation 58 Whitney S t r e s s Block 59 Re s u l t s 60 V I I . CONCLUSION 71 D E S C R I P T I O N OF SYMBOLS L Total width of a th i n wall H Height of a wall panel t Thickness of a wall panel or an element P Concentrated load acting on the wall panel 2a 1 Length of a f i n i t e element 2b' Breadth of a f i n i t e element X, Y, Z Global coordinate system UJ Deflection i n Z-direction u, w, Rotations of element node about Y and X axes x y Ox, <jy Stresses i n X and Y directions ~Cxy Shear stress Q - J P ^ J J Stress at mid height of a thin wall Ccr C r i t i c a l stress Omi Maximum stress under a concentrated load i n wall with i n f i n i t e width Qmf Maximum stress under a concentrated load i n wall with f i n i t e width R Generalized forces vector r Generalized displacement vector )^ Eigen values k. Buckling constant D Flexure r i g i d i t y of a plate Poisson r a t i o f ^ Crushing strength of concrete f Yielding stress of s t e e l y Ultimate strength of s t e e l i i i ACKNOWLEDGEMENT I am indebted to my supervisor, Dr. N. D. Nathan, f o r his time, patience and invaluable assistance at every stage of t h i s work. F i n a n c i a l assistance i n the form of a research a s s i s t a n t -ship and free use of U.B.C.'s excellent computing f a c i l i t i e s i s g r a t e f u l l y acknowledged. Chapter I INTRODUCTION The use of precast concrete elements i n the building industry i s increasing d a i l y , f i r s t l y because of the better control i n factory conditions, which makes f o r e f f i c i e n t use of building materials, and, secondly, because of the economies associated with mass production. Precast concrete elements are proving very useful i n both i n d u s t r i a l and r e s i d e n t i a l buildings, wherever a l o t of elements are generally of the same size and shape; even greater e f f i c i e n c y can often be obtained by coupling precasting with prestressing. In the use of precast elements as load bearing wall panels, three questions are immediately confronted: a) What part of the section i s e f f e c t i v e i n carrying the loads? b) How can the i r r e g u l a r section which i s e f f e c t i v e be economically designed for a x i a l load and moment? c) What are the magnification e f f e c t s a r i s i n g from slenderness of the section, or what are the c r i t i c a l loads f o r s t a b i l i t y ? The object of t h i s thesis i s to investigate the f i r s t two questions. The t h i r d problem involving the c l a s s i f i c a t i o n of members into short ( f a i l u r e by crushing), intermediate ( f a i l u r e by p l a s t i c buckling), and long ( f a i l u r e by e l a s t i c buckling) classes, i s not considered as i t merits a complete study in i t s e l f . However, 2 i n the case of f l a t plates, used either as wall panels themselves or as parts of panels, the approach generally used i s simply to make them thick enough to ensure that they do not f a i l by buckling. This question has been studied. During the past few years, committees of the Grestressed Concrete I n s t i t u t e have been working on the formulation of s p e c i f i c a t i o n s f o r the design and construction of precast s t r u c t u r a l members. The present study of the e f f e c t i v e dimensions D'f sections consists of an inv e s t i g a t i o n of some of the recommendations made in reference 2. The number of parameters to be controlled by s p e c i f i c a t i o n s for walls (which are, of course, two-dimensional) i s large compared to that f o r unidimensional elements, such as beams and columns. This introduces some complexities into the problem. In s p i t e of i t s heterogeneous nature, concrete i s usually treated as an e l a s t i c material during analysis of a structure. Thus, in the past, d i f f i c u l t problems, such as f l a t - s l a b s or two-way slabs, have been designed by analogy with corresponding e l a s t i c structures. This approach i s even more appropriate in the case of prefabricated and prestressed structures, since prestressing makes concrete act more l i k e a homogeneous material. Thus, although the investigations described here r e f e r to a homogeneous i s o t r o p i c e l a s t i c material, they may provide a guide to the behaviour of corresponding concrete members. In f a c t , as w i l l be shown l a t e r , most of the r e s u l t s do confirm what has become standard practice i n the design of poured-in-place walls. Nevertheless, there i s no experimental evidence to support the r e s u l t s presented herein, and they should, at t h i s 3 stage, be considered merely as preliminary guidelines. In t h i s t h e s i s , most of the investigations are made using the f i n i t e element method''. The v e r s a t i l i t y of the method, with an equally v e r s a t i l e computer program, made i t possible to investigate the e f f e c t s of various parameters and t h e i r combinations on the behaviour of the structures. With the increasing popularity of precast prestressing methods, i t i s becoming more and more common to adopt i r r e g u l a r shaped concrete members. While a r c h i t e c t u r a l l y these shapes look very pleasing and r a t i o n a l , t h e i r s t r u c t u r a l design so f a r has not been done very r a t i o n a l l y . In standard design books, one would normally f i n d dimensionless i n t e r a c t i o n curves only f o r rectangular shapes, and so far a common practice has been to transform the given section into an 'equivalent rectangular section' of equal area s a t i s f y i n g some c r i t e r i o n , such as moment of i n e r t i a or section moduli! equal to those of the r e a l section. However, i t i s f e l t that since such precast components are always manufactured on a mass scale, i t i s quite reasonable, from an economic point of view, to do the analysis of the actual shape, and base the design on a more r a t i o n a l basis. With respect to the second of the three questions posed above, therefore, a computer program has been written to automatically f i n d the in t e r a c t i o n curve of a section of any shape. A comparison has also been made between the analysis done by using Whitney's Compression Block theory, and by using the actual s t r e s s / s t r a i n c h a r a c t e r i s t i c s of the concrete mix, to see whether preliminary designs could be made using that s i m p l i f i c a t i o n . Chapter II EFFECTIVE DIMENSIONS: PROBLEMS TO BE ANALYZED 2.1 Categories of U a l l Panels 2 In the draft s p e c i f i c a t i o n s , load bearing wall panels have been categorized i n the following three groups: 1. Walls designed as Columns. 2. Walls designed as Thin Walls. 3. Walls designed as Ribbed Walls. 2.2 D e f i n i t i o n s The following are the d e f i n i t i o n s of the three categories (Section 2.1). A. Walls designed as Columns A portion of prestressed concrete load bearing wall s h a l l be designed as a column i f i t has minimum dimensions of 6 i n . , i n each d i r e c t i o n , and i s reinforced with t i e s s a t i s f y i n g the requirements for columns. B. Walls designed as Thin Walls A wall i s termed a thin wall i f the d e t a i l i n g of the r e i n -forcement and the l i m i t a t i o n s of the sectional dimensions described i n A are not s a t i s f i e d , and the portion under consideration i s of uniform width. 5 C. Walls designed as Ribbed Walls A prestressed concrete load bearing wall i s c a l l e d a ribbed wall, i f the load to be c a r r i e d i s d i s t r i b u t e d between the whole or a part of the thin wall and the portion of the i n t e g r a l l y b u i l t - i n prestressed concrete section, which can be considered as a prestressed concrete column according to A. 2.3 Clauses from and Aspects of S p e c i f i c a t i o n s to be Investigated 2 The draft s p e c i f i c a t i o n s places certain apparently a r b i t r a r y l i m i t a t i o n s on dimensions and loads, which, i t i s the purpose of t h i s t h e s i s , to i n v e s t i g a t e . The following are some of the l i m i t a t i o n s : 2.3.1 I t i s recommended that f o r the walls designed as columns, the dimensions shown i n f i g . 2.1 should s a t i s f y the following conditions: The r a t i o s L/H 2/5 and t/L.,^275 (for concentrated load). P H P Fig. 2.1 6 Precast wall panels w i l l often be dimensioned for a r c h i t e c t -u r a l rather than s t r u c t u r a l reasons. The apparent intention of t h i s clause i s to ensure that a wall panel of unusual dimensions carrying a concentrated load can only be designed as a column i f that load i s , i n f a c t , d i s t r i b u t e d across the section to a s i g n i f i c a n t degree. The values quoted apparently represent an i n t u i t i v e a p p l i c a t i o n of the p r i n c i p l e of St. Venant. e f f e c t i v e length (in plan) should be the loaded length plus four times the thickness of the w a l l . This recommendation i s presumably based on the c r i t e r i o n of even d i s t r i b u t i o n of stresses within the s e c t i o n . 2.3.2 According to the code d r a f t , while designing a thin wall the load , 1 MM loaded length + kt F i g . 2.2 2.3.3 The draft recommends that the thickness of the load bearing thin walls should not be l e s s than 1/35 times the unsupported dimension of the w a l l . It i s assumed that t h i s i s based on the s t a b i l i t y c r i t e r i o n , and accordingly i t has been investigated. ( J 1 J 1 1 I 1 1 t 1 I I t I I 1 I 1 1 1 I ) r i b simply supported edge simply supported edge r i b . r : r / M I M f I f M t f t t I I M f F i g . 2.3 2.3.4 While designing load bearing ribbed walls, i t has been recommended that the e f f e c t i v e width to be taken into account be the width of the r i b plus s i x times the thickness of the wall on both sides of i t , i f i t i s a T-shaped wall, and on one side i f i t i s an L-shaped w a l l i P r i b T-shaped (a) L-shaped (b) F i g . 2.4 This has been thoroughly investigated for walls with d i f f e r e n t combinations of dimensions. 8 To study a l l these problems, they have been divided into two basic categories: 1. Plane Stress Problems. 2. Plate Bending Problems. Problems such as width to height r a t i o , or the e f f e c t i v e width of a ribbed wall, etc., f a l l into the f i r s t category, while those such as minimum thickness of a thin wall and others concerned with s t a b i l i t y c r i t e r i a are studied under the second category. 9 Chapter III ELASTIC ANALYSIS DF SECTIONS: THEORY AND PROGRAMMING As described e a r l i e r , From the analysis paint of view, we have two types of problems: 1. Plane Stress Problems. 2. Plate Bending Problems. In t h i s chapter, the assumptions made and the analysis pro-cedures adopted f o r each type of problems w i l l be discussed. 3.1 Plane Stress Problems Problems such as height to width r a t i o of thin walls or e f f e c t i v e width of the ribbed walls have been investigated by using a constant s t r a i n compatible trian g u l a r f i n i t e element as presented by Turner, Clough, Martin, Topp, et a l . A computer program f o r general triangular shaped elements was used. 3.2 Plate Bending Problems To investigate the r a t i o of free span to the thickness of the wall, a s t a b i l i t y analysis i s performed and theory of the rectangular 3 5 7 f i n i t e element i n bending ' ' i s applied. 5 A computer program using the s t i f f n e s s matrix of Melosh and 6 12 the s t a b i l i t y c o e f f i c i e n t matrices of Kapur ' has been written 10 to obtain the c r i t i c a l stresses f o r d i f f e r e n t combinations of the dimensions of walls. Some of the cases, however, have been taken B d i r e c t l y from the book by Timoshenko. F a m i l i a r i t y with the plane-strain f i n i t e element w i l l be assumed, but the theory of S t a b i l i t y C o e f f i c i e n t s matrix presented by Kapur, being l e s s f a m i l i a r , w i l l be reproduced here i n b r i e f . Ribbed members are modelled here simply by increasing the thickness of the f i n i t e elements within the r i b . Thus the analyses do not represent r i b s projecting on one side . It may be said that the r i b s are symmetrically placed with equal projections on either side of the plate section. Further than t h i s , however, the com-p a t i b i l i t y conditions implied at the edges of the r i b s are somewhat s p e c i a l . I t i s assumed that the ribbed section does not get distor t e d and continuity at the boundaries of the elements of d i f f e r e n t thicknesses i s maintained. P P r 71 r / ZL 7 7 A p| Structure analysed Y Actual ribbed wall Thick element i n r i b Neighbouring thin element i n wall F i g . 3.3 W U M i FRONT Y Y S IDE 2» Z Y 1 Y ri WW X' X PLAN F i g . 3.k 12 As i s known, i n plane s t r e s s f i n i t e element a n a l y s i s , every po i n t i n the s t r u c t u r e i s g iven only two t r a n s l a t o r y degrees of freedom. In f i g . 3.3, i t i s shown how po i n t s A and B always move t oge the r . The l i n e CBD remains und i s t o r t ed and normal to the plane so tha t po i n t s C and 0 always remain at the same l e v e l , and the same d i s t ance apa r t . As shown i n f i g . 3.4 ( three v iews ) , the s e c t i o n hi, X, Y, Z w i l l d i s p l a c e to p o s i t i o n W , X ' , Y ' , Z ' i n such a manner t ha t X ' Y 1 and W'Z* remain p a r a l l e l to XY and WZ and normal to the plane of the w a l l . 3.4 S t a b i l i t y C o e f f i c i e n t s Ma t r i x Cons ider a t y p i c a l element IM ( f i g . 3.5) w i th nodal p o i n t s i t J i k and 1. I t i s apparent tha t f o r t r an sve r se bending we w i l l have to cons ider three degrees of freedom (or three independent g ene r a l i z ed displacement components) at each node, a l a t e r a l d e f l e c t i o n W and two r o t a t i o n s about axes X and Y w i t h i n the plane of the p l a t e ( f i g . 3 .7) , hi, and hi, . Thus the gene ra l i z ed displacement vec to r at node 1 can be denoted by y Where Id , = Sw/<^x and hi, = Sus/dy (3.1) x v i , x r i " ' (3.2) 13 X P l a t e element F i g . 3.6 Degrees of freedom at each node F i g . 3.7 1*f and f o r the element (3.3) Then the displacement W(x,y) at any point i n the plate can be written as (3.4) Where A(x,y) i s a matrix giving the r e l a t i o n s h i p between the displacement W(x,y) and the generalized displacements r ^ . The p o t e n t i a l energy V a r i s i n g from stresses 0~k, 6y and Txy s due to forces applied i n the middle plane of the plate, which i s assumed inextensible, i s given by V/s a (t /2 ) JjfcxOu/dx)2 + tfyOui/^y)2 + 2rxyOw/ax)(2>w/ay)] dx dy (3.5) In matrix notation, using Cx txy 2xy Cfy (3.6) G r, N 15 (3.7) where (3.8) ue have U g = W)ffij? G T <r G r ^ dx dy (3.9) " (1/2) r N T K * N ; T N (3.10) where » t //G T cr G dx dy (3.11) i s the ' s t a b i l i t y c o e f f i c i e n t s matrix', for the plate element. S i m i l a r l y , we can derive the expression f o r p o t e n t i a l energy due to bending i t s e l f ( i n the absence of transverse loads, t h i s i s merely the s t r a i n energy of the plate) as "b = ( l / 2 ) V K N rN (3.12) where i s the s t i f f n e s s matrix of the element . 16 Therefore, the t o t a l p o t e n t i a l energy i s , V = (1/2) r , T KW r , + (1/2) r , T K'n r N - ( V 2 ) r N T KN rN ( 3' 1 3> Denoting the generalized force vector corresponding to the general-ized displacement vector r ^ by R^ j , the equilibrium equation fo r the element N, can be written as RN i KN + h N r N (3.14) With the help of a computer program, an equilibrium equation of the complete structure i s determined using these equations for each element. Thus, f o r the whole structure, the equation i s R = + K'] r (3.15) Where R and r are the generalized forces and displacements of i i the whole structure, and K, K are assembled from the , K ^ . Now, i f the s p a t i a l d i s t r i b u t i o n of a x i a l or in-plane loads i s f i x e d and the magnitude i s determined by an i n t e n s i t y f a c t o r F, then ii_ = F K1 (3.16) 17 where K„ i s a constant matrix. At c r i t i c a l loads the homogeneous equation (3.17) has a n a n - t r i v i a l s o l u t i o n . Equation (3.17) i s obtained from the c r i t e r i o n that at c r i t i c a l load the second v a r i a t i o n of po t e n t i a l energy changes from p o s i t i v e g d e f i n i t e to p o s i t i v e semidefinite . 3.5 Buckling Loads The c r i t i c a l load may be determined as the smallest root of the determinantal equation This determinant has the c h a r a c t e r i s t i c value parameter 'F' appearing i n elements o f f the main diagonal. This i s contrary to the most common formulations of the eigen value problems, such as the usual lumped mass vi b r a t i o n problems. I f desired, the s t a b i l i t y c o e f f i c i e n t s matrix presented here may be put i n diagonal form by -•I premultiplying by K when K + F K. I = 0 (3.18) -1 K + F I = 0 (3.19) 1 where I i s an i d e n t i t y matrix. 18 Since procedures for obtaining eigen-values obtain the highest values f i r s t , a more convenient form of t h i s equation i s H " 1 K 1 + (1/F) I = • (3.20) This r e s u l t s i n the f i r s t eigen-value being the highest value of 1/F or the lowest value of F, as i s required. A computer program i s available which gives the values of the required number of eigen-values. In the program, the problem of finding eigen-values was treated as follows: Writing (1/F) = .A = eigen-value, the equation can be put as K " 1 H 1 + /\ I = 0 (3.21) Let H = L L (3.22) Where ij_ i s lower trian g u l a r matrix and L_ i s upper tr i a n g u l a r matrix. Therefore K ~ 1 = L ~ 1 L ~ 1 Since K + F K, = 0 Therefore K r = - F r For a n o n - t r i v i a l solution I A l + L "1 K1 L ~1 = 0 (3.23) where i s a symmetric matrix. This f i r s t eigen value (^) gives the minimum value of the load at which the structure buckles. 20 Chapter IV TESTING OF PROGRAMS Before proceeding with the in v e s t i g a t i o n , two questions must be considered: i . Do the programs give adequate solutions f o r the problems of e l a s t i c i t y which they are designed to solve? i i . Do the solutions to the e l a s t i c i t y problems adequately represent the behaviour of reinforced or prestressed concrete wall panels? The f i r s t question can be answered by comparison of the solu-tions obtained by the programs with the known solutions f o r some problems which are solvable by t h e o r e t i c a l means. The second question can be studied by comparing the program solutions with r e s u l t s obtained experimentally by testing of laboratory models. Plane Stress Program The f i n i t e element program used f o r plane-stress problems w i l l . be referred to as RC 801. The s t i f f n e s s matrix for triangular elements used i n t h i s pro-it 5 gram has been proved s a t i s f a c t o r y ' , and i s used i n several commercial programs. I t i s known to give adequate r e s u l t s f o r the problems of e l a s t i c i t y . 21 An opportunity to t e s t the a p p l i c a b i l i t y of these r e s u l t s to concrete panels i s provided by the experimental i n v e s t i g a t i o n s 13 conducted by Smith . These t e s t s i n v o l v e d s t e e l frames i n f i l l e d u i t h concrete, as shown i n f i g . 4.1. The a n a l y t i c a l model used i s shown i n f i g . 4.2. The s t e e l frame was represented by t r i a n g u l a r elements with an approximately higher modulus of e l a s t i c i t y . The l o a d i n g and the boundary c o n d i t i o n s are e x a c t l y the same as used i n Smith's model. The p r i n c i p a l t e n sion at the c e n t r o i d of each element i s found and p l o t t e d ( f i g . 4.3). The crack p a t t e r n suggested by the l i n e s of p r i n c i p a l t e n sion of the a n a l y t i c a l s o l u t i o n matched the observed crack pattern of the t e s t specimen very w e l l . 4.3 P l a t e Bending Program: (For S t a b i l i t y A n a l y s i s ) Henceforth t h i s program w i l l be described as RC 901. RC 901 has been checked against the t h e o r e t i c a l b u c k l i n g 8 s t r e s s e s given by Timoshenko , and the r e s u l t s obtained by 6 12 Kapur ' . I t was found that as we increase the number of elements, the accuracy of the r e s u l t s i n c r e a s e s . The r e s u l t s obtained, using program RC 901, compare e x a c t l y w i t h Kapur's r e s u l t s as expected, since h i s s t a b i l i t y matrix has been used. In the f o l l o w i n g t a b l e s these r e s u l t s have been compared with the exact t h e o r e t i c a l values of the b u c k l i n g s t r e s s e s of p l a t e s w i t h d i f f e r e n t boundary c o n d i t i o n s as obtained by Timoshenko . The comparison i s made with reference to the value of 'k'. F j g , 4.1 This i s the model of i n - f i l l e d frame which was experimentally investigated by Smith (reference 13) and cracking pattern of the concrete found. Here the same frame i s analyzed by di v i d i n g the structure as shown i n F i g . ^ .2 and using Program RC 801. t i n u u n t u H I i i 11111 111 i i 11 n i i 11111 u i n i i t n u t u i i u i 4.2 Half of the actual structure shown in Fig. 4.1, showing the loading and the division into triangular elements. For Program RC 801, positive displacements are j — " and 1 = displacement possible 0 = displacement not possible 23 2.8H Zk F i g . P r i n c i p a l stresses and the p r i n c i p a l d i r e c t i o n s are shown at the centroid of each element as obtained from the Program RC 801. Tensile stresses are p o s i t i v e and cracks develop as the stresses exceed the permissible t e n s i l e strength of the concrete mix. This cracking pattern agrees with the pattern described by S m i t h 1 3 . 25 where and 2 k = C w 0 t = S t a b i l i t y Constant 2 ' 7TD 0 = E t 3 = F l e x u r a l R i g i d i t y 12 (1 -/«- 2) P l a t e under c r i t i c a l s t r e s s F i g . 4.4 cr Table U.1 Values of 'k' f o r a simply supported Bquare p l a t e under compression i n one d i r e c t i o n (./•= 0.25). Exact value of k = 4.0 GRID SIZE 3 by 3 6 by 6 10 by 10 12 by 12 k 3.645 3.885 3.965 3.978 er r o r i n percentage -8.88 -2.8 -1.0 - .58 26 Table U.2 Values of 'k1 f o r a square plate f i x e d at a l l sides under compression i n one d i r e c t i o n . Exact value: k = 10.07 GRID SIZE 4 by 4 6 by 6 8 by 8 10 by 10 k 9.285 9.615 9.780 9.888 error i n percentage -7.80 -4.56 -2.86 -1.79 Table 4.3 Values of ' k' for a rectangular plate (a/b = 0.6) f i x e d at Y = 0 and Y = b and simply supported at other sides. Exact value: k = 7.05 GRID SIZE 4 by 4 6 by 6 8 by 8 10 by 10 k 6.55 6.784 6.886 6.950 error i n percentage -7.24 -3.74 -2.34 -1.4 These r e s u l t s show that with s u f f i c i e n t l y f i n e subdivision, t h e o r e t i c a l e l a s t i c i t y problems can be solved with an acceptable accuracy. Unfortunately there do not appear to be any published r e s u l t s of experimental studies on s t a b i l i t y of wall panels, so 27 that no comparison has been made between the behaviour of precast concrete panels and e l a s t i c models with respect to s t a b i l i t y . 28 Chapter V INVESTIGATIONS AND RESULTS 2 5.1 Here, some of the main recommendations of the draft , as stated i n Chapter I I , w i l l be investigated by analyzing wall panels of d i f f e r e n t sizes with various boundary conditions. 5.2 To have a s t r u c t u r a l l y e f f i c i e n t component, i t i s necessary that stresses be evenly d i s t r i b u t e d within i t . It i s from t h i s point of view that the maximum r a t i o of length to height of a wall with concentrated load has been set at 0.4. To study t h i s , various wall panels of unit thickness and d i f f e r e n t length (L) to height (H) r a t i o s have been analyzed using the program RC 801. A concentrated load P i s applied as shown i n f i g . 5.1. Maximum stress at the mid height of each panel ((Tj^^) * s found and compared with the average stress (P/L), which should a c t u a l l y exist i f the stresses are evenly d i s t r i b u t e d at that section. The r a t i o O"^^/^/1-) ? 0 T each combination of parameters i s determined (Table 5.1). F i g . 5.2 shows the r e l a t i o n s h i p between the r a t i o s L/H and CK-*V( p/L) for d i f f e r e n t wall panels. It i s observed that f o r mid the r a t i o CT m i d /(P/L) = 1.0, the value of L/H » 0.4, as s p e c i f i e d i n the draft i s quite reasonable. 29 I-H Thickness t = Uni P F i g . 5 . 1 Typical thin wall panel Table 5.1 To determine the pe rm i s s i b l e r a t i o of 30 L/H f o r m a l l panels 1 # L f t . H f t . L/H CJmid k s i P k i p P/L k s i tfmid/(P/L) 1 4.0 7.0 0.57 0.02301 1.0 > 1.104 2 3 4.0 4.0 10.0 12.0 0.40 0.33 0.02085 0.02079 1.0 1.0 0.02083 1.00 0.998 4 4.0 14.0 0.286 0.02075 1.0 0.996 5 5.0 8.0 0.625 0.01921 1.0 1.153 6 5.0 10.0 0.5 0.0.1711 1.D 1.027 7 5.0 12.0 0.415 0.01667 1.0 0.01666 1.000 8 5.0 14.0 0.357 0.01651 1.0 0.9909 9 5.0 16.0 0.313 0.01632 1.0 0.9795 10 6.0 10.0 0.60 0.01571 1.0 r 1.132 11 12 6.0 6.0 13.0 15.0 0.46 0.40 0.01442 0.01401 1.0 1.0 0.01388 1.0389 1.0093 13 6.0 18.0 0.33 0.01385 1.0 0.9978 14 7.0 12.0 0.582 0.01334 1.0 1.1206 15 7.0 14.0 0.5 0.01262 1.0 1.060 16 7.0 18.0 0.39 0.01191 1.0 0.011904 1.0005 17 7.0 21.0 0.33 0.01190 1.0 0.9996 18 7.0 23.0 0.305 0.01185 1.0 0.9954 19 8.0 16.0 0.5 0.01103 1.0 1.058 20 8.0 19.33 0.42 0.01062 1.0 1.019 21 8.0 21.0 0.38 0.01042 1.0 0.010425 0.9995 22 8.0 23.0 0.35 0.01041 1.0 0.9986 23 8.0 25.0 0.32 0.01038 1.0 0.9956 24 9.0 20.0 0.45 0.00950 1.0 1.0260 25 26 9.0 9.0 23.0 25.0 0.39 0.36 0.00925 0.00920 1.0 1.0 0.009259 0.9990 0.9936 27 9.0 27.0 0.33 0.00915 1.0 0.9882 28 10.0 20.0 0.50 0.00888 1.0 1.066 29 10.0 24.0 0.416 0.00845 1.0 0.008333 1.014 30 10.0 27.0 0.37 0.00830 1.0 0.9960 31 I 10.0 30.0 0.33 0.00829 1.0 0.9948 31 - -. . . . . . - -- -- •--- - -- - - -- -- - -E -- — — - ! - — — - 4|L4-i~ - - - -- 1T/9 1 ll ... - -— - ----L 3 --— 'A i-... . . . 7 rr j .j uera _ - m: c -J- > -t Ii --> — I V - -t-El 3--"5 i < _ n lie. J • - i j — i i i i #• i d .50 C U C Li C U L 45 0 --_ -J _ /til - • --F i 1 - --t — IE I rs t r. 7" -t\ .scat the m n r a t l Or Of 1-1 T n e 30] inzs i s o •>.?_ i\~ re: ly: uner pnar z - ± -H-ir %B 9 h f i •ihi s [ "c ( It riLe r r i e n t s Id ne l i g h t nidrbv i - no re z TC 1.1 "j i .-r l e ss xr X I : nc?easi — I ~ i - i-1 -' n i im i bfat - - ----- -- - ri 3 e 3: 3( 3_i_re M\ su l,tsU are. -beJ - --- - . . . - ex i iCDUHO - - --- d 1 n m n h e -J . . . fho -1 m JI 1 1 U I I I ± V - • • . . . — - -— -... — _ . - ---. . . ----. . . -- - --- --£ -- - - -- -I -- _. _ : - -- - - - -- — — --u_. 32 5.3 As stated before, the recommendations specifying the minimum value of the thickness of a thin wall i s assumed to be based on a s t a b i l i t y c r i t e r i o n . I t states that the r a t i o of the minimum thickness to the distance between supporting or enclosing members should never be less than 1/35. 2 It should be noted that the draft does not mention the ef f e c t s of the boundary conditions or even the r a t i o of the two sides of the wall on t h i s s p e c i f i e d r a t i o . Here, using the theory of buckling of plates, plates with f i v e d i f f e r e n t kinds of boundary conditions have been studied and l a t e r these observations have been applied to the ribbed walls. The f i v e types of boundary conditions studied are: 1. Plates with sides x = • and x = a simply supported and sides y = 0 and y = b b u i l t - i n . 2. A l l sides simply supported. 3. Sides x = 0 and x = a simply supported, side y = 0 b u i l t - i n and side y = b f r e e . 4. Sides x = •, x = a and y = • simply supported and side y = b f r e e . 5. A l l sides b u i l t - i n . X b Icr-' Y F i g . 5.3 33 For d i f f e r e n t r a t i o s of a/b and the thickness of the plates, values of the s t a b i l i t y factor 'k' have been determined. Some of the values have been taken from reference 8, while others have been determined using program RC 901. Normally, load bearing concrete wall panels are made of high strength concrete. Here concrete with crushing strength of 5000 p s i i s used. In order to ensure that f a i l u r e does not occur by buckling, the dimensions of the plate should be such that the c r i t i c a l stress f o r buckling i s higher than 5000 p s i . Thus, using equation 5.1, and assuming a buckling stress of 7000 p s i , r a t i o s of b/t and a/t have been determined and tabulated. where < r c r = k 7 C E E o~cr 12 (1 -/«•*) (t/b)' Modulus of E l a s t i c i t y Poisson r a t i o 7000 p s i (5.1) 4.2 x 10 p s i 0.25 Therefore or and ( b / t r = 490.04 k ( b / t ) = 22.2 -fk (a/t) = (a/b) 22.2 f k The values of the r a t i o b/t have been plotted against the values of a/b i n f i g . 5.4. It can be observed i n t h i s figure that Table 5.2 Values of ' k ' and other parametr i c r a t i o s f o r a p l a t e of Type 1. a/b k b/t a/b k b/t 0.6 13.38 81.5 1.73 5.33 51.5 0.8 8.73 66.0 1.8 5.18 50.5 1.00 6.74 58.0 2.0 4.85 49.5 1.2 5.84 54.0 2.5 4.52 47.5 1.4 5.45 52.5 2.83 4.50 47.5 1.6 5.34 51.5 3.0 4.41 47.0 1.7 5.33 51.5 Table 5.3 Values of 1 k' and other parametr i c r a t i o s f o r a p l a t e of Type 2. a/b k b/t a/b k b/t 0.2 27.0 115 1.4 4.47 47.0 0.3 13.2 80.9 1.6 4.25 45.9 0.4 8.41 64.7 1.8 4.12 45.0 0.5 6.25 55.6 2.0 4.0 44.4 0.6 5.14 50.4 2.4 4.15 45.25 0.7 4.53 47.4 2.45 4.2 45.6 0.8 4.20 45.6 2.6 4.06 44.7 0.9 4.04 44.6 3.0 4.0 44.4 1.0 4.00 44.4 3.4 4.08 44.9 1.1 4.04 44.6 3.65 4.15 45.25 1.2 4.13 45.2 4.0 4.0 44.4 1.3 4.28 46.0 Table 5.4 Values of ' k ' and other parametr ic r a t i o s f a r a p l a t e of Type 3. a/b k b/t a/b k b/t 1.0 1.70 29.1 1.8 1.34 25.9 1.1 1.56 28.0 1.9 1.36 26.2 1.2 1.47 27.1 2.0 1.38 26.3 1.3 1.41 26.6 2.2 1.45 27.0 1.4 1.36 26.2 2.6 1.48 27.2 1.5 1.34 25.9 2.8 1.48 27.3 1.6 1.33 25.7 2.4 1.47 27.1 1.7 1.33 25.7 Table 5.5 Values of r a t i o s f o r 1 k' and other parametr i c a p l a t e of Type 4. a/b k b/t a/b k b/t 0.5 4.40 47.0 2.0 0.698 18.6 1.0 1.44 29.0 2.5 0.610 17.5 1.2 1.135 26.0 3.0 0.564 16.8 1.4 0.952 21.9 4.0 0.516 16.0 1.6 0.835 20.5 5.0 0.506 15.9 1.8 0.755 19.4 Table 5.6 Values of ' k ' and other parametr ic r a t i o s f o r a p l a t e of Type 5. a/b k b/t a/b k b/t 0.75 11.69 76.5 2.50 7.57 61.1 1.00 10.07 71.0 2.75 7.44 61.0 1.25 9.25 68.0 3.00 7.37 60.6 1.50 8.33 64.8 3.25 7.35 60.5 1.75 8.11 64.0 3.50 7.27 60.2 2.00 7.88 62.5 4.00 7.23 60.0 2.25 7.63 61.9 4.50 7.21 60.0 36 37 the recommended r a t i o i s not independent of e i t h e r the boundary c o n d i t i o n s or the r a t i o of the si d e s of the p l a t e . However, f o r l a r g e values of a/b, the values of b/t do not vary a p p r e c i a b l y . The minimum values of the r a t i o 'b/t', obtained from f i g . 5.4, are a c t u a l l y the maximum values that we should allow f o r 'b/t' i n order that the p l a t e may not reach the c r i t i c a l s t r e s s e s and buck l e . The r a t i o f o r d i f f e r e n t boundary c o n d i t i o n s i s : 48 f o r the type 1, 44 f o r the type 2, 25 f o r the type 3, 15 f o r the type 4 and 60 f o r the type 5. While d i s c u s s i n g the b u c k l i n g of a ribbed w a l l here, we a c t u a l l y r e f e r to the b u c k l i n g of the t h i n p o r t i o n of the w a l l . F i g . 5.5 shows the va r i o u s boundary c o n d i t i o n s : F i g . 5.5(a) shows a t h i n w a l l , which i s simply supported on two h o r i z o n t a l edges, and may be e i t h e r i . Simply supported on both the v e r t i c a l edges, or i i . Simply supported on t h i r d edge, with the f o u r t h edge f r e e , or i i i . Free on the other two edges; depending on the d e t a i l s of the attachment to neighbouring members. For a w a l l simply supported on a l l four edges the c r i t i c a l r a t i o of •span/thickness 1 i s 44, i f one of the edges i s made f r e e , the r a t i o becomes nearly h a l f . In f i g . 5.5(b), the t h i n p o r t i o n of the w a l l i s simply supported on two h o r i z o n t a l edges and e l a s t i c a l l y b u i l t - i n , i n t o the r i b s , on both the v e r t i c a l edges. This type of w a l l conforms with the p l a t e of the type 1, f o r which the r a t i o i s 48. However, there i s one 38 i i i t i t i n h t t t t m t t t i t t i r r F i g . 5.5(a) l l J i H l l t l M U M l l l l TTTTT H I I t t I I l l I M t i l t I It F i g . 5.5(b) f I H I T y U l t M j _ l l l U I \\\\W\\\\\\\\\\\ty TTTTT I M l I I I I I t f I M U I I M M i l I I t l f i F i g . 5.5(c) 39 d i f f e r e n c e . The p l a t e i s b u i l t - i n r i g i d l y , while here the w a l l i s b u i l t - i n i n t o e l a s t i c r i b s , which are not p e r f e c t l y r i g i d and do undergo e l a s t i c deformations. This reduction i n the s t i f f n e s s of the edge supports w i l l decrease the r a t i o . The r e d u c t i o n i n r a t i o , however, w i l l depend.on the dimensions of the r i b . The t h i c k n e s s of the r i b which i s adopted from an a r c h i t e c t u r a l point of view, i s g e n e r a l l y not more than two to three times that of the t h i n p o r t i o n of the w a l l . Considering the e l a s t i c behaviour D f such r i b s , the r a t i o 'span/thickness' may be taken as low as 40. Sometimes i t may a l s o be necessary to construct a load bearing w a l l between two already e x i s t i n g columns, which w i l l act l i k e r i b s . In such cases the boundary c o n d i t i o n s of the w a l l w i l l be q u i t e d i f f e r e n t and w i l l n e c e s s i t a t e f u r t h e r r e d u c t i o n of the r a t i o . F i g . 5.5(c) shows the end p o r t i o n of a continuous r i b b e d w a l l . Here, a l s o , both the h o r i z o n t a l edges are simply supported, while the l e f t - h a n d edge i s b u i l t - i n e l a s t i c a l l y and the right-hand edge may be f r e e or simply supported. I f t h i s edge i s f r e e , the w a l l w i l l conform with the p l a t e of the type 3, f o r which the r a t i o i s 25, but i f the edge i s simply supported, then the r a t i o w i l l be over 30. Thus, we observe that f o r most of the commonly encountered boundary c o n d i t i o n s (discussed above), the r a t i o of 'span/thickness' f a l l s w i t h i n the range 30 - 45. 2 In the d r a f t , the recommendations s p e c i f y the r a t i o to be 35. I t i s f e l t , however, that t h i s number i s too conservative f o r some boundary c o n d i t i o n s . Nevertheless, i t may be h e l p f u l to use the value 35 as a guide while doing p r e l i m i n a r y designs. But during 40 the f i n a l design c a l c u l a t i o n s , the engineer must use his d i s c r e t i o n and adopt a more appropriate value i n conformity with his boundary conditions. 5.4 In t h i s section the problem of the e f f e c t i v e width of a load-bearing ribbed wall i s studied. This i s a shear-lag type of problem, and has been investigated with the help of program RC 801. 2 As stated e a r l i e r , the draft recommends that the e f f e c t i v e width be twelve times the thickness of the thin portion of a T-wall, but i t does not mention about the e f f e c t s of the thickness of the thin wall or of the area of the r i b , etc. Here the e f f e c t s of such parameters have been examined. To explain the mode of analysis, a t y p i c a l case i s described here: A wall 6 i n . (t) thick, with a r i b 9 i n . by 9 i n . (1.5t x 1.5t), o v e r a l l height of 12 f t . and having an i n f i n i t e width i s analyzed: under a concentrated load P, and maximum stress at the mid height of the wall (cr .) i s determined. Then walls with the same dimen-mi sions but with the widths varying from 2t to 16t are analyzed and for each one, the maximum stress at the mid height section ((Jn,f) i s determined. The f i r s t parameter investigated here i s the thickness of the thin wall portion. As observed i n Table 5.7, three values of 't* have been tested 2 keeping the area of the r i b equal to 2.25t . For d i f f e r e n t values of the assumed widths, value of the r a t i o 1 0~_,/rr-.' have been mr mi determined and tabulated. We notice that the stress r a t i o s do Table 5.7 Showing the e f f e c t D f thickness of the thin wall portion on the stress r a t i o <<^ f/<rn i>. Thickness of wall t = k" t = 5" t = 6" Area of r i b 1.5t x 1.5t 1.5t x 1.5t 1.5t x 1.5t Width of thin wall STRESS RATIO (o- m f 2t 3.41 3.47 3.42 4t 2.54 2.55 2.5 at 1.40 1.43 1.39 10t 1.11 1.11 1.1 12t 1.058 1.06 1.06 I4t 1.053 1.048 1.051 X*. Yfc T y p i c a l viW X-t x. Y t , 42 Table 5.8 Ratios of Stresses (cr e/rr .) for d i f f e r e n t mf umi values of the areas of the r i b and the widths of the thin wall portion, showing the ef f e c t s of interchanging the dimensions of the r i b on the stress r a t i o . PAIR I PAIR II PAIR III Area of 1.5t 2t x 2.5t x 1.5t x 2.5t 3t x r i b ^ x 2t 1.5t 1.5t 2.5t x 3t 2.5t Width of STRESS RATIO ier Jcc- .) t h i n wall -mr umi 2t 2.82 2.82 2.75 2.75 1.785 1.786 4t 1.97 1.97 1.91 1.912 1.38 1.385 Bt 1.25 1.275 1.19 1.2 1.088 1.088 10t 1.11 1.125 1.081 1.082 1.068 1.069 12t 1.05 1.06 1.055 1.056 1.045 1.045 I4t 1.046 1.05 1.045 1.048 1.005 1.005 Table 5.9 Ratios of (cjpf/tJ^-) f ° r various values of the area of the r i b and the width of the thin wall portion. 1 i / / / Area of r i b 1.5t x 1.5t 1.5t x 2t 1.5t x 2.5t 2.5t x 2.5t 2.5t x 3.0t 3t x 3t Width of thin wall STRESS RATIO 2t 3.42 2.82 2.76 1.958 1.785 1.71 4t 2.51 1.97 1.91 1.42 1.38 1.28 8t 1.39 1.25 1.19 1.09 1.088 1.085 1Dt 1.1 1.11 1.081 1.071 1.068 1.066 12t 1.06 1.05 1.055 1.05 1.045 1.032 14t 1.051 1.048 1.045 1.025 1.005 1.003 45 - - _. ... - — -- --— - ... - ----... ... — - - _ -— d . . . -— - -- -- -- — . . . . . ---_ •-— •- _ — - - - . . . . — •-- -— - - - - - — - -... -1 ,4 - — --• -- - -. -- 1 1 -- - Ii -... - -- - - - -- ... - - - - ! - f - r -... ... - .j ~,—| -- - i - -• >-V J . - 1 [ - - 4 u k i r i n -— _. - -i i -i - H -- - i ---- -- - ... _ . . . ... -- - -- - - r - • ... j i r • -- -- 4 ... -1 1 - 1 - -Ii — -- - ---k . •' LTY - — --• * ! i r~i CT _LJ ri\ i 1 --- - - - -- -- r 1 -- / to --— - J -n i - -- - - - *-P * — / ...... h " -- r >-- - ... --I - • i _ . - -- - ... - - -- --n i t i - -- - i n -- i — 1 | -. . J -I — ... | -- -— - J - -- - -i / j 1 1 •--1~ . . . . - --11 - -- - - - - -- - j - -... -- - - - - -- - 1, — -4 -- - ---- --- — ... / -— -- - — - - - - - - - - \: - - - --• - ... . .. - _ - — - 1 -- - - . . . - --• u . d i < 3 h _ i . — - — ... - i - - -• d i ± i — i -1 3 *- 1 --— --... . . . — -- * -- — I: _ " - ... 1 - - -- -H - - -- i 4- _ _ --- _ . . - .i --- -i -L. i i xui U l . n . i . 1 .;..J !. - -.± ft -_ | --- — -- - - -... - t t f - - ... T j . . . --- --- ---1 1 I T T I T I kG not get affected by changing the thickness of the thin wall portion as long as the 'area of the r i b / t ' r a t i o ( = 2.25 here) remains the same. Another parameter investigated here i s the orientation of the dimensions of the cross-section of the r i b , that i s , whether the ' e f f e c t i v e width 1 depends on the breadth and width of the r i b or whether i t depends on the area of the r i b as a whole only. To check t h i s , three pairs of walls have been tested, and the r e s u l t s tabulated in Table 5.8. The curves for a t y p i c a l pair have been plotted i n the F i g . 5.6. From t h i s graph and from the table i t can be concluded that the interchanging of the width and the breadth dimensions of the r i b does not a f f e c t the stress r a t i o s , and only the area of the r i b as a whole a f f e c t them. Having concluded that the stress r a t i o s do not depend on the thickness of the thin wall portion or on the i n d i v i d u a l dimensions 2 of the r i b , but they depend on the r a t i o 'area of the r i b / t • only, an B f f o r t has been made tb investigate how the r a t i o 'area of the 2 r i b / t ' a f f e c t s the ' e f f e c t i v e width'. For various values of the area of the r i b and the width of the thin wall portion, values of the stress r a t i o (a~-/(r-.) have been determined and tabulated i n mf mi the Table 5.9. A t y p i c a l curve has been plotted in F i g . 5.7, and the "effective width 1 determined corresponding to the point at which the curve approaches the l i n e n~ e/tr~ • = 1»0, within 7.5% of i t s u mf mi asymptote. S i m i l a r l y , the ' e f f e c t i v e width' values f o r other sections have been determined corresponding to the r a t i o Omf/cTmi " 1 ' 0 7 5 -The so determined values of the r a t i o ' e f f e c t i v e width/t' and 47 Table 5.10 Variation of r a t i o s e f f e c t i v e uidth/t and e f f e c t i v e design uidth/t, with r i b a r e a / t 2 . Rib Area/t 2 E f f e c t i v e Uidth/t E f f e c t i v e Design Uidth/t 2.25 11.35 10.1 3.0 11.0 9.9 3.75 10.65 9.6 6.25 9.6 8.2 7-5 9.3 8.3 9.0 9.1 • 8.4 48 •—> Q I id J — 1 ( Ir u » T 1 H •r- 1 J -1 —^  J — i U r 1 _ rr 1 • H n 1 3 1 , D a I i / • ^  1 / - +-_r_ i / •j u n i 1 n i f ri +. i r U- •s I I ii 1 / n 1 / Q 1 I • » ti 1 ' 1 J / —i Jt h-1 / n-I •>. 1 / c 1 / 1 + i u LE I • • —t — a ( n T L _ / I * f\ i + J / 1 | / • f y / / - | t _ - r • r i r M * < c -> r i c I r r j r" c j i a i r / 1 \ V. UI J- L B 5 -* £ 3 31 • * T / 3 - d. XL 1 / L P T h w 4 ,4 .i )-r 49 2 the corresponding values of the 'area of the r i b / t • have been tabulated i n Table 5.10, and plotted i n F i g . 5.8. Clearly the material l y i n g outside the ' e f f e c t i v e width' so determined makes n e g l i g i b l e contribution to reducing the stress i n the stem. Therefore, any design method which rests upon e f f e c t i v e dimensions greater than these cannot r e f l e c t the truth, and thus a l i m i t of the e f f e c t i v e width has been defined. Obviously, however, the stress i s at a peak i n the stem and decreases sharply towards the outer l i m i t s of the section defined above. If the maximum stress i s l i m i t e d to the allowable value, much of the section w i l l be understressed ( F i g . 5.9). In view of the p l a s t i c r e d i s t r i b u t i o n which w i l l surely take place before f a i l u r e , i t seems reasonable to allow peak stresses higher than the allowable. Furthermore, p r a c t i c a l design procedures assume a uniform stress i n the section. Thus we calculate the load by assuming a uniform stress equal to the allowable value acting on some ' e f f e c t i v e ' section. The width of t h i s ' e f f e c t i v e ' section w i l l be termed the ' e f f e c t i v e design width'. In fact t h i s i s the parameter an engineer w i l l need while designing a load bearing ribbed w a l l . LJe seek, therefore, an e f f e c t i v e design width such that t h i s procedure y i e l d s a load which w i l l produce a peak stress somewhat higher than the allowable one i n the r e a l section. In order to calculate the ' e f f e c t i v e design width' of a section, the load that causes a 10% overstress, based on the calculated peak stress f o r a p a r t i c u l a r e f f e c t i v e width, i s assumed to give a uniform stress equal to the allowable strength of the material over an e f f e c t i v e design area, which in turn gives the ' e f f e c t i v e 50 Assumed stress d i s t r i b u t i o n f o r design purposes Actual stress d i s t r i b u t i o n f o r wall width = e f f e c t i v e width ~1iffeiptive des| width ign f e c t i v e Ljidth &h = Allowable strength of the material F i g . 5.9 51 design width'. In other words, by l e t t i n g the allowable stress uniformly d i s t r i b u t e d over the 'ef f e c t i v e design width', we allow the peak stress to exceed the allowable value by 10% ( F i g . 5.9). The values of the 'ef f e c t i v e design width 1 so found have been tabulated i n Table 5.10, and plotted in F i g . 5.8. It i s very important to c l e a r l y understand the difference between an ' e f f e c t i v e width' and an 'ef f e c t i v e design width'. An ' e f f e c t i v e design width' i s the one that a designer w i l l be interested i n f o r the determination of the load carrying capacity of a wall section, whereas an 'e f f e c t i v e width 1 i s that width of the section beyond which the contribution of the section i n a f f e c t i n g the stresses i n the r i b becomes n e g l i g i b l e . In order to use the graph in F i g . 5.7, f o r determining the value of the 'e f f e c t i v e design width' for a p a r t i c u l a r value of the 'area of 2 the Rib/t ' r a t i o , we should make B u r e that the actual width of the section i s not l e s s than the 'effective width' obtained from the same graph. Once the 'e f f e c t i v e design width' i s determined, the load carrying capacity of the section i s found by multiplying t h i s ' e f f e c t i v e design section' with the permissible strength of the material. I f the value of load i s less than the actual load acting on the section, then the section should be revised. If the actual width of the section i s l e s s than the 'e f f e c t i v e width' as obtained f o r that p a r t i c u l a r r a t i o of the 'area of the 2 Rib/t ', the section should simply be designed as a column using the code s p e c i f i c a t i o n f o r column design. 52 Chapter VI DESIGN OF PRECAST PRESTRESSED SECTIONS IN COMPRESSION After the shape of the ribbed section has been determined, (which may be T, L or any other polygonal shape), i t seems i r o n i c a l i f the section i s designed by making too many assumptions about the shape. The i n t e r a c t i o n curve for reinforced concrete columns, which has been extended to the cases of prestressed columns by Zia and 14 Guillermo , can be made a dimensionless curve only f o r simple sections, whose si z e and shape can be expressed by one or two parameters. Because of the i r r e g u l a r shapes of t y p i c a l precast prestressed wall sections, t h e i r i n t e r a c t i o n curves do not lend themselves to dimensionless p l o t t i n g . Nevertheless, various devices have been used tD achieve the s i m p l i c i t y of the former problem - for example, su b s t i t u t i o n i n the ca l c u l a t i o n s of the 'equivalent 1 rectangular section, which has a radius of gyration equal to that of the r e a l s e c t i o n . It seems, however, that such devices are misleading and unnecessary, and that, in f a c t , the number of assumptions i n the design procedure could and should be reduced rather than increased i n connection with these members. This contention i s based upon the following points: i . The in t e r a c t i o n curves of members such as prestressed T-sections are unusual in shape and are not well 53 represented by d r a s t i c s i m p l i f i c a t i o n s . In f a c t , the radius of gyration of the cross-section has no s i g n i f i c a n c e at a l l with respect to the i n t e r a c t i o n curve at ultimate strength. i i . The j u s t i f i c a t i o n f o r the use of such members as load bearing wall panels i s that they are mass produced in large numbers in precast plants. Therefore, the economic pressure f o r highly s i m p l i f i e d design procedures i s reduced. i i i . A consequence of point ( i i ) i s that large numbers of cases are l i k e l y to be designed i n advance as part of the promotional p o l i c y , and the use of computers i s strongly indicated. The need for s i m p l i f i c a t i o n i s immediately removed. i v . The labour involved i n c a l c u l a t i n g i n t e r a c t i o n curves even by hand i s not, i n f a c t , p r o h i b i t i v e i f large numbers of members are to be manufactured. v. Since there i s no established t r a d i t i o n f o r the design of prestressed concrete columns of unusual shape, an opportunity i s presented to introduce the most r a t i o n a l possible methods at the outset. In t h i s t h e s i s , a simple computer program which can compute the i n t e r a c t i o n curve f o r prestressed concrete sections of any shape with any arrangement of prestressing s t e e l and mild s t e e l , using the minimum number of assumptions, i s presented. The i n t e r a c t i o n curves obtained by t h i s program for some examples of highly asymetric members: single and double tees ( F i g . 6.1) are shown in Figs. 6.3 and 6.4. These i n t e r a c t i o n curves are obtained by using both the Whitney Stress Block theory and a more precise theory which uses 5k the actual s t r e s s / s t r a i n c h a r a c t e r i s t i c s of the concrete mix. Thus, a p p l i c a b i l i t y of the Whitney Stress Block theory i s invest-igated, since i t allows an immense s i m p l i f i c a t i o n i f hand ca l c u l a t i o n s have to be used. 6.2 Assumptions used i n the Program 6.2.1 S t r e s s / S t r a i n Relations Concrete: i t i s assumed that the actual s t r e s s / s t r a i n curve of the concrete i s known. This information i s fed into the com-puter by reading i n up to ten points on the curve (pairs of stress and s t r a i n values) including the l a s t one: the s t r a i n at which f a i l u r e i s deemed to have occurred. A subroutine then f i t s a polynomial to these points by a l e a s t squares method; a t h i r d , fourth or f i f t h order polynomial i s selected, according to which gives the l e a s t deviation from the given points, and subsequent c a l c u l a t i o n s are based upon t h i s polynomial s t r e s s / s t r a i n curve. If only the cylinder strength i s supplied, provision i s made for an appropriate previously-stored polynomial to be selected. The concrete i s assumed to carry no stress i n tension. This assumption could be modified i f required, but i t i s reasonable and usual i n connection with ultimate load ana l y s i s . The compression force i n the concrete section i s a c t u a l l y computed by numerical i n t e g r a t i o n : the area above the neutral axis i s divided into twenty areas of equal depth, and the average stress f o r that depth i s assumed to act over each area. In e f f e c t , then, the stepped s t r e s s / s t r a i n curve of F i g . 6.2 i s used. In 55 t h i s way, s i m p l i c i t y i s retained while allowing considerable f r e e -dom i n the shape of section to be handled. The A.C.I. Code assumes that f a i l u r e of concrete members occurs when the extreme f i b e r s t r a i n reaches .003. In the case . of rectangular beams or beam-columns, the load generally increases with s t r a i n u n t i l t h i s value. In certain circumstances, however, the load may reach a peak at a lower s t r a i n and begin to decline before t h i s value i s reached; i n the problem under discussion several factors are present which tend to cause t h i s r e s u l t : i . We are, here, generally concerned with high-strength concretes which usually have a more pronounced descending branch i n the . / i . 15,16 s t r e s s / s t r a i n curve ' . i i . We are concerned with wide, very shallow flanges that dominate the compression area of the member. When the neutral axis i s low i n the stem, the flange i s almost uniformly strained to the extreme f i b e r value; thus i f thBre i s a pronounced descending branch i n the s t r e s s / s t r a i n curve, the compression force w i l l begin to decline e s s e n t i a l l y with the extreme f i b e r s t r e s s . i i i . We do, indeed, have the neutral axis low i n the stem when the e c c e n t r i c i t y i s small i n a beam column member. In view of these f a c t s , i n t e r a c t i o n curves should be computed for a few values of s t r a i n between that corresponding to the peak concrete stress and that defined as ' f a i l u r e ' . The outermost curve may be accepted as defining the highest combination of loads that the member can sustain. 56 P r e s t r e s s i n g S t e e l : again, i t i s assumed that the a c t u a l s t r e s s / s t r a i n curve of the s t e e l can be s u p p l i e d by reading i n up to ten p o i n t s o f f the l a b o r a t o r y curve. ( I f only the u l t i m a t e s t r e n g t h i s read i n , the computer i s d i r e c t e d to use a curve s t o r e d f o r that v a l u e ) . The subroutine i s again used to f i t a polynomial to these p o i n t s , and the polynomial i s used without f u r t h e r m o d i f i c a t i o n s i n subsequent c a l c u l a t i o n s . Unprestressed reinforcement: the usual e l a s t i c p e r f e c t l y p l a s t i c curve i s assumed, with a modulus of the e l a s t i c i t y equal to 29,000, KSI and a y i e l d point as s u p p l i e d i n the input data. 6.2.2 C o m p a t i b i l i t y S e c t i o n s o r i g i n a l l y plane and normal to the n e u t r a l Burface are assumed to remain so; the i m p l i e d l i n e a r v a r i a t i o n of s t r a i n 15 i s borne out by experimental observation P e r f e c t bond i s assumed between concrete and reinforcement, both p r e s t r e s s e d and unprestressed, but p r o v i s i o n must be made f o r crBep and shrinkage of the concrete. In clause 1902 of the A.C.I. Code i t i s assumed that the immediate s t r a i n plus creep of the concrete i s .003 on the compression face - no mention i s made, i n the l a s t sentence of clause 1902(a), of any f u r t h e r creep allowance. The reason f o r t h i s i s presumably that i t i s u s u a l l y conservative to assume short-term l o a d i n g ; and at the extremes of pure a x i a l l o a d , or pure bending wi t h no com-pre s s i o n s t e e l , i t makes no d i f f e r e n c e to u l t i m a t e strength c a l c u l a t i o n s whether creep i s i n c l u d e d or not. Between these cases, the i n t e r a c t i o n curve f o r u l t i m a t e strength i s l i t t l e a f f e c t e d by 57 the rate of loading. In the case of prestressed concrete columns, i t i s again conservative to use a lower bound on stress losses. With t h i s i n mind, a 'conservative' - i . e . low-estimate of losses from a l l causes other than e l a s t i c shortening i s made. This leads to the force that exists i n the prestressing s t e e l corresponding to zero stress i n the concrete at the same location; t h i s value i s supplied to the computer with the input-data. Additional stresses are then computed on the basis of perfect-bond, plane sections remaining plane, and short-term moduli. (The case of unbonded tendons i s not considered). The foregoing procedure accounts for shrinkage and for creep under prestress. I f further creep under the action of long-term applied loads i s to be considered, i t can be allowed f o r by modifying the s t r e s s / s t r a i n curve f o r the concrete input as described above. Both the e f f e c t s of loss of prestress and of creep under applied load were investigated. 6.2.3 Equilibrium The forces i n the various materials are computed on the basis given above, and the equations of equilibrium are solved without further assumptions. 6.3 Br i e f Outline of Program A. Input: Points on the s t r e s s / s t r a i n curve, or the cylinder strength of the concrete. 58 P o i n t s on the p r e s t r e s s i n g s t e e l s t r e s s / s t r a i n curve, or the u l t i m a t e v a l u e . Y i e l d s t r e s s of the non-pres t res sed re in fo rcement . Co -o rd inates of corner po i n t s on c r o s s - s e c t i o n . C r o s s - s e c t i on may have any shape but corner p o i n t s are assumed to be j o i n e d by s t r a i g h t l i n e s . Co -o rd ina te s and areas of non-pres t res sed re i n fo r cement . Co -o rd ina te s of p r e s t r e s s i n g tendons. Forces or s t r a i n s i n p r e s t r e s s i n g tendons a f t e r a l l l o s s e s except e l a s t i c s ho r t en i n g . B. Method of Computation: The depth of the member i s d i v i d e d i n t o s i x or more equal p o i n t s , and a po in t on the i n t e r a c t i o n curve i s obta ined f o r the n e u t r a l a x i s p o s i t i o n e d at each of these depths i n t u r n , as f o l l o w s : The area above the n e u t r a l a x i s i s d i v i d e d i n t o 20 areas of equal depth. Assuming the s t r a i n at the top f i b e r s i s the va lue i n d i c a t e d at ' f a i l u r e ' by the concrete s t r e s s / s t r a i n curve, the s t r a i n i s computed at the mid depth of each of these 20 areas as w e l l as at the p r e s t r e s s i n g tendons and non-pres t res sed r e i n f o r c e -ments. In the case of the p r e s t r e s s i n g tendons, the s t r a i n i s computed to i n c l ude tha t which e x i s t e d when the surrounding con-c r e t e was u n s t r a i n e d . The f o r ce i n each of these elements i s then computed by re fe rence to the s t r e s s / s t r a i n curve of the r e l e van t m a t e r i a l , as w e l l as the moment of t ha t f o r c e about the x - a x i s of the co - o rd i na te system. In the case of the concrete segments, the p r o p e r t i e s of a segment are g iven by: 59 Area = (1/2) £ ( X j, + 1 - x ^ ) F i r s t moment of area about x-axis = (1/6) 2 ( x . + 1 - x l ) ( y 2 + 1 + y i + i y i - y*> where the sum extends, i n each case, over the number of corners of the segment. The case of uniform s t r a i n corresponding to the peak of the concrete s t r e s s / s t r a i n curve i s also considered, and a l l moments are then transferred to the p l a s t i c centroid of the section defined by that case. The process i s repeated for bending i n the opposite d i r e c t i o n . G. Output: the output consists of points on the i n t e r a c t i o n curve, and the depth to the p l a s t i c centroid. A p l o t t i n g routine can be used to draw the curve. The program was run on an IBM 360-67. It took 29 seconds to compile the program and 9 seconds of C.P.U. time to obtain the i n t e r a c t i o n curves f o r 17 double-tee sections. 6.1* Whitney Stress Block: The foregoing procedure can c l e a r l y be s i m p l i f i e d considerably i f the twenty segments of concrete area can be replaced by one through use of the Whitney stress block; the integration of the concrete force and i t s moment i s s i m p l i f i e d by t h i s means. If t h i s modification i s considered acceptable, the i n t e r a c t i o n curve can even be obtained by hand c a l c u l a t i o n within an acceptable time 60 i f a large number of s i m i l a r members i s under consideration. However, the p r i n c i p l e of the Whitney method i s that the i n t e g r a l s of area and moment of area under the r e a l concrete s t r e s s / s t r a i n curve are equal to those f o r the substituted rectangular area; but when i r r e g u l a r - and e s p e c i a l l y wide - flange-sections are used, the i n t e g r a l s are weighted by the l o c a l width of the section, and there i s a danger that t h i s fundamental hypothesis of the Whitney theory w i l l be upset. Therefore, a modified program was written to solve the problem using Whitney theory i n order to examine i t s a p p l i c a b i l i t y to wide-flange sections such as double and s i n g l e tees. 15 It has been found that the Whitney, method does give r e s u l t s that compare well with laboratory tests on rectangular sections. In order to l i m i t the present i n v e s t i g a t i o n to the e f f e c t of the shape of cross-section only, a concrete curve was chosen which gave close agreement with Whitney's theory f o r a rectangular cross-16 s e c t i o n . This curve, obtained from the 5000 p s i case i n reference but adjusted to give a peak stress of exactly 50.00 p s i , and cut o f f at a s t r a i n of .0032, was used to obtain a l l the reported r e s u l t s . 6.5 Results F i g . 6.3 shows the i n t e r a c t i o n curves f o r a family of s i n g l e -tee units representative of those which might be used as wall panels. A l l the curves r e f l e c t a minimum e c c e n t r i c i t y of 0.1 times the t o t a l depth of the section, i n accord with the A.C.I, recommendation for columns. It i s seen that the loads that can 61 be c a r r i e d at the minimum e c c e n t r i c i t y d i f f e r greatly depending upon whether the corresponding bending moment i s of such a character as to cause tension or compression in the flange of a T-section. In many cases the d e t a i l i n g may make i t clear that the bending moment can be i n one d i r e c t i o n only, but, i f the load i s t h e o r e t i c a l l y exactly at the p l a s t i c centroid, the lower value of load would have to be used. Various values of extreme f i b e r s t r a i n (.0023, .0025, .0032) fo r concrete were used. They gave r e s u l t s i n f a i r l y close agree-ment at large e c c e n t r i c i t i e s , but the curve f o r ultimate s t r a i n of .0032 dipped below the others at small e c c e n t r i c i t i e s , when the neutral axis was low i n the stem of the T-sections. The curves shown are f o r extreme f i b e r s t r a i n s of .0025, The asymmetry of the i n t e r a c t i o n curves f o r the single tees, the e f f e c t s of added strands, and of added depth in the section, are a l l evident i n F i g . 6.3. F i g . 6.4 shows a s i m i l a r set of curves f o r a family of double-tee sections. F i g . 6.5 shows the r e s u l t s obtained by use of Whitney's theory compared with those from the more precise method. The former i s conservative, considerably so at smaller e c c e n t r i c i t i e s . The e f f e c t i s more marked with respect tD the curves f o r lower s t r a i n s ; thus better agreement might be expected when concrete with a less steep descending branch in the s t r e s s / s t r a i n curve i s used. F i g . 6.6 shows the not unexpected r e s u l t that 'equivalent' rectangular sections are not successful i n representing the i n t e r -action curves for tee-sections. 62 F i g . 6.7 indicates the s l i g h t l y unconservative r e s u l t obtained when higher losses are assumed i n the prestressing tendons. In F i g . 6.8 the e f f e c t of very slow applied loading i s i l l u s -t r a t e d . In the long-term loading case, the usual s t r e s s / s t r a i n , curve i s used f o r the concrete, but the s t r a i n s are doubled to allow f o r creep. The r e s u l t i s to change the s t r a i n s i n the tendons; therefore, the ef f e c t on the i n t e r a c t i o n curve i s greater when the tendons are placed further from the p l a s t i c centroid. The case shown i n F i g . 6.8 i s thus the member of the previously considered family of sections i n which t h i s e f f e c t i s most pro-nounced. It i s seen that the assumption of short-term loading i s again generally conservative. 63 8'—D " • A " T i " to th e c e n t r o i d of 8 #k b a r s ' 3-1-^13 a p a i r of p r e s t r e s s i n g s t rands 2" c/c A t y p i c a l s i n g l e - T s e c t i o n Z"i_ H- S'-O"1 12 a p a i r of p r e s t r e s s i n t s t rands 2" c/c J " f 1" to the c e n t r o i d of 8 #U bars A t y p i c a l double-T s e c t i o n F i g . 6.1 - Str a i n -F i g . 6.2 Concrete s t r e s s / s t r a i n curve — S' - O i l — o - l 4 '•'"tr'i'b.'i TENS'I!DN'"IN: FLANGE 2DO • 4oo • j ;-j - • eoo C O M P R E S S I O N I N - F L A N G E ' ., 14...!. . i..H±[:.;::T- ^  . i U.:.L !4_I-,-L.,.J4. . ; .XJ.J : L.:. .4 '_ r i : . m f i r ! : 1 r 1 , 1 > s 1 s t« ! • • 1 _[* ! ' _LI . . C D . _ U i -M O M E N T . K O T ; 4-. * B O O K ' Fig 1.: G.3M In teract ion curves- f o r ^ t yp i c a l ; s i n q l e - t e e s i H + -• 4i.T_LJ.".)_ - H H - f T i f-M-L.; 1-:-. 1 -J..,.! J L. i .. i -Li 1. i . i 11 • ' T t t -;• - j - i • 4 . U J - 4 - U - M - 1 1 •| 4-• 4 . 'XL VT Irli^liN-frl^rF -i _! .4-1 -4 4 4 . . ] " . j _ i : i • 1 • i 1 ' * 1 '41 " • " T rr;"1 i:.i.i.'::4:a::i -f~-l 4 X L i J. _ L 4-Case Depth d Strands and depths from top sur face P l a s t i c c e n t r o i d depth from top su r face 1 12" 2 @ 4", 2 @ 6", 2 @ 8" 2,56" 2 18" 2 @ 4", 2 @ 6", 2 @ 8" 3.97" 3 18" 2 @ 10", 2 i 12", 2 @ 14" 3.61" 4 24" 2 @ 4, 2 @ 6 " , 2 @ 8" 5.70" 5 24" 2 § 4", 2 every 2" to 20" 4061" l - r 1 , )... :H:l ! I Vr _rr+T"r-.--r • - r - - r r - ' ' — * — t — i " l- -.u. 44- I 1 • ' It!- 11 • 1 i i i -Hi-TP 4:':.. ' - f h i 1 - ' - f - r l - f - . xlr L_ i_ . 1 > ' 1 -4 t-.f :J.'L"L|_ J . _ ...a . i T i j l - r I -.1 67 69 ,4 i n n i e S s a be !1 E i 1 L 3 B! SE <C t g 1 a g ; • LC q f- • J t 1 Ll 1 ' •h h r |[ IC E t r a T - i s H ; j E g g a 11 » t L 1 3 7 £V L. £ S e s <l !f it • , j u 1 £ S A 3 aj , ^  .1 C s n • r bi 3r IC 13 n g • \ / ? 5 H > s t i< a n j \ ,> 4 1 1 i[ [ // 1 -j 1 L, t c u • 1 c ;c 'C t i • n A \ f r c • 3 3d 1 F i c • . 4 f/ A // If IC )[ K 1 -p-iC c J ( 5 •4 : fc ll II II II Y. II 1 i i 1 1 k 1 | f 1 1 •f E I\I s I b [p i F L A l\l 3 E C M P R Bl 3 . [[ 111 1 1 l\l F VC ;E n N r j r r 7 + p Pi € 3 3 li i . L a L • / 1 E c • T J. 3! H •• 70 )l it •e r m _ i re c o * r -t t r C _-i g >; c rj • 1 \x If >r 1) r >r r n _c >i V t 'I L U n • L r ' k / / \, / *-r f n / A . )L U n n • 1 /j j J J , t I / 1 r •f in i \\ / T F 1 3 6. L 1 r 1 yf t 1 | 1J I r r n / (. JL u r / / / r i \ \ L v r r n 1 f ll LJ n A -.r n * > u j \ J ~y / • > •f I*' v, / fi r n k 11 11 V II Ll l I ) U J 0 J t J •r ^ ^ r T n f- -Ij \ w G IF] R E & ri W L IV i . u Ll - i IC M E N r 1 L. /- r - | i J L • b • 3 1 .1 f •b L D P Li JI 1L L r it r1 n i] .1 fc d 1 i\ 71 Chapter VII CONCLUSION 2 The Code d r a f t makes l i t t l e or no re fe rence to boundary c o n d i t i o n s , but i n the va r ious graphs presented here, one observes the e f f e c t s of boundaries and a l so the f a c t t ha t f o r genera l purposes i t i s not r e a l i s t i c to present a s i n g l e number f o r any of the parameters under i n v e s t i g a t i o n . However, as n o t i c e d i n Chapter V, some of the recommended r a t i o s may be used as ' a gu ide ' dur ing p r e l i m i n a r y c a l c u l a t i o n s . A lthough sub jec t to the usua l i n a c c u r a c i e s of numer ica l methods, the f i n i t e element methods used he re i n are f e l t to have prov ided s u f f i c i e n t l y accurate s o l u t i o n s to the va r i ou s e l a s t i c i t y problems, s i nce s u i t a b l y f i n e mesh s i z e s have been used. The w a l l panels are gene ra l l y p recas t i n a f a c t o r y r a t he r than cas t i n - s i t u . The problems of t r a n s p o r t a t i o n and handl ing l i m i t t h e i r s i z e s to some ex ten t . In most cases, the common type of boundary c o n d i t i o n s encountered a re : s imply supported on two h o r i z o n t a l edges and e i t h e r s t r u c t u r a l l y b u i l t - i n , or s imply supported on the other two v e r t i c a l edges. For such r e gu l a r types of l oad bear ing w a l l s , i t i s concluded t h a t : i . ' T o t a l U id th/He i gh t ' r a t i o should be 0.4 to a l l ow even d i s -t r i b u t i o n of s t r e s s e s i n the w a l l . i i . The r a t i o of ' Span/Thickness ' may be taken as 35 f o r p r e -l i m i n a r y c a l c u l a t i o n s , but c o r r ec ted to more app rop r i a te 72 value i n the range 30 - 40, depending on the boundary conditions. However, i f a/b<1.0, we should not use the values obtained from F i g . 5.4. Rather we should treat the wall as an i n f i n i t e l y wide one, and design i t by taking a unit s t r i p , otherwise the value of the thickness w i l l be very conservative, i i i . The e f f e c t i v e width of the wing of a ribbed wall does vary with the thickness of the r i b , and f a l l s between 9t - 12t f o r the most cases. I f the actual width of the wall i s l e s s than the e f f e c t i v e width, the wall section should be designed as a column. In Chapter VI, the assumptions made i n determining the i n t e r -action curve of a precast prestressed member of any general shape, are l e s s r e s t r i c t i v e than the assumptions generally employed i n design, where the s t r e s s / s t r a i n curves of the materials are i d e a l i z e d to a considerable extent. Therefore, there i s no reason to believe that these r e s u l t s are any less r e l i a b l e than any other computed predictions of behaviour i n prestressed concrete members. The speed and ease with which the r e s u l t s were obtained support the contention that no s i m p l i f i c a t i o n s are necessary when i n t e r -action curves are desired f o r mass produced elements of standard shapes. Furthermore, the sub s t i t u t i o n of an 'equivalent' section i s not j u s t i f i e d . It i s noteworthy that short-term loads and underestimates of prestress losses are conservative f o r these members under a x i a l loads as well as bending. BIBLIOGRAPHY 1. "Tentative Recommendations for the Design of Prestressed Concrete Columns". Reprinted from copyrighted Journal of the Prestressed Concrete I n s t i t u t e , Vol. 13, No. 5, October 1968. 2. Draft 3 of PCI Committee, "Recommendations f o r Prestressed Bearing Wall Design". 3. Zienkieuicz, O.C. and Cheung, K.K. "The F i n i t e Element Method i n Structural and Continuum Mechanics", McGrau H i l l Publishing Co. Limited, Reprinted 1970. 4. Clough, R.W. " F i n i t e Element i n Plane Stress Analysis". Proc. 2nd A.S.C.E. Conference on E l e c t r o n i c Computation,.Pittsburg, Pa., September 1960. 5. Melosh, R.J. "Basis for Derivation of Matrices for the Direct S t i f f n e s s Method". Journal of American  I n s t i t u t e of Aero. Astro., Vol. 1, July 1963. 6. Kapur, K.K. and Hartz, B.J. " S t a b i l i t y of Plates using the F i n i t e Element Method". Journal of Engineering  Mechanics, Div. Proc. A.S.C.E., A p r i l 1966. 7. Melosh, R.J. "A S t i f f n e s s Matrix for the Analysis of Thin Plates i n Bending". Journal of Aero. Science, Vol. 28, 1961. 8. Timoshenko, J . "Theory of E l a s t i c S t a b i l i t y " . McGrau H i l l , N.Y. 9. Nathan, N.D. Class notes of a course on Energy Theorem, given at U.B.C, Vancouver, Winter Term 1968. 10. Tocher, J.L. and Kapur, K.K. Comments on "Basis f o r Derivation of Matrices f o r Direct S t i f f n e s s Method". A.I.A.A. Pu b l i c a t i o n . 11. Zienkieuicz, O.C. and Cheung, K.K. "The F i n i t e Element Method f o r Analysis of E l a s t i c Isotropic and Orthotropic Slabs". Proceedings Inst, of C.E., Vol. 28, August 196L. 12. Kapur, K.K. "Buckling of Thin Plates using Matrix S t i f f n e s s Method". Ph.D. Thesis, University of Washington, 1965. 13. Smith, B.S. "Model Test Results of V e r t i c a l and Horizontal Loading of I n f i l l e d Frames". Journal of American  Cone. I n s t i t u t e , No. B, August 1968. 14. Z i a , P. and Guillermo, E.C. "Combined Bending and A x i a l Load i n Prestressed Concrete Columns". P.C.I. Journal, June 1967, pp. 52-58. 15. Hongnestad, E. f Hanson, IM.U. and McHenry, D. "Concrete Strength D i s t r i b u t i o n i n Ultimate Strength Design". December 1955, pp. 455-479. 16. Winter, G., Urquhart, L . C , O'Rourke, C.E., IMilson, A.H. "Design of Concrete Structures". Seventh E d i t i o n , McGrau H i l l , Neu York, 1964, p. 14. 

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