1、 Obb2949 054395b 92T W The Design of Two-way Slabs Editor T. C. Schaeffer international SP-183 0bb2949 0543957 bb W DISCUSSION of individual papers in this symposium may be submitted in accordance with general requirements of the AC1 Publication Policy to AC1 headquarters at the address given below.
2、 Closing date for submission of discussion is November 1, 1999. Ail discussion approved by the Technical Activities Committee along with closing remarks by the authors will be published in the MarcWApril 2000 issue of either AC1 Structu ral Journal or AC1 IWgxids Journal depending on the subject emp
3、hasis of the individual paper. The Institute is not responsible for the statements or opinions expressed in its publications. Institute publications are not able to, nor intended to, supplant indi- vidual training, responsibility, or judgment of the user, or the supplier, of the information presente
4、d. The papers in this volume have been reviewed under Institute publication proce- dures by individuals expert in the subject areas of the papers. Copyright O 1999 AMERICAN CONCRETE INSTITUTE P.O. Box 9094 Farmington Hills, Michigan 48333-9094 All rights reserved including rights of reproduction and
5、 use in any form or by any means, including the making of copies by any photo process, or by any electronic or mechanical device, printed or written or oral, or recording for sound or visual reproduction or for use in any knowledge or retrieval system or device, unless permission in writing is obtai
6、ned from the copyright proprietors. Printed in the United States of America Editorial production: Jane D. Carroll Library of Congress catalog card number: 99-61474 m O662949 0543958 7T2 m PREFACE At ACIs 1996 fall convention in New Orleans, La., ACI-ASCE Joint Committee 421, Design of Reinforced Sla
7、bs, sponsored two technical sessions. The focus of the morning session was “Design of Two-way Slabs using Elastic Frame Analogies,” and the afternoon session concentrated on “Design of Two- Way Slabs using Theorems of Plasticity.” The sessions were moderated by Hershel1 Gill and Thomas C. Schaeffer.
8、 This AC1 Special Publication consists of 10 papers, all of which were presented at the sessions in New Orleans. The current AC1 3 18 Building Code specifically addresses two methods for the design of two-way slabs. These methods are the Equivalent Frame Method, and the Direct Design Method. However
9、, the Building Code also “.permits a designer to base a design directly on fundamental principles of structural mechanics provided it can be demonstrated explicitly that all safety and serviceability criteria are satisfied.” The papers contained in this volume should give the designer an overview of
10、 some of the different analysis and design techniques that are currently being used. Committee 421 would like to thank all of the authors and presenters for their contributions to the two technical sessions and to this volume. We would also like to thank the reviewers of the original manuscripts, as
11、 well as AC1 staff for their assistance. T. C. Schaeffer Editor Obb2949 0543959 639 CONTENTS CONCEPT AND BACKGROUND OF ELASTIC FRAME ANALOGIES FOR by S. Simmonds . 1 TWO-WAY SLAB SYSTEMS DESIGN FOR SEVERE DYNAMIC LOADS by S. Woodson and T. Krauthammer . 17 DESIGN FOR PUNCHING SHEAR IN CONCRETE by S.
12、 Megaily and A. Ghali . 37 DEVELOPMENT IN YIELD LINE THEORY FOR SLABS by W. Gamble 67 USING THEORUMS OF PLASTICITY HISTORY AND CONCEPT by S. Simmonds . 77 STRIP METHOD FOR FLEXURAL DESIGN OF TWO-WAY SLABS by S. Alexander 93 PLANE-FRAME ANALYSIS APPLIED TO SLABS by W. Gamble . 119 DETAILING FOR SERVI
13、CEABILITY by D. Rogowsky 13 1 DESIGN AND CONSTRUCTION OF TWO-WAY SLABS FOR DEFLECTION CONTROL by A. Scanlon . 145 STRIP DESIGN FOR PUNCHING SHEAR by S. Alexander 161 V 0bb2949 05439b0 350 SP 183-1 Concept and Background of Elastic Frame Analogies for Two-way Slab Systems by S. Simmonds Synopsis: The
14、 justification for using elastic frame analogies to determine design moments in two-way slab systems is discussed. A brief history of two-way reinforced concrete slab design leading to the current code procedures is presented. This history includes a description of the various elastic frame analogie
15、s that have existed in past codes, the reasons for changes and the research leading to improved frame analogies. This is followed by a critical review of the Equivalent Frame Method in the current code with suggestions for improving and simplifying provisions for elastic frame analogies in future co
16、des. Keywords: analysis; design; elastic frames; history; reinforced concrete slabs 1 D Obb2947 05437b3 297 2 Simmonds Sidney H. Simmonds, currently Professor Emeritus, University of Alberta was for many years Secretary of the Canadian Concrete Code Committee A23.3. He also served on AC1 Committees;
17、 118 - Computers (Chairman 1979-83), 120 - History (Chairman 1991-95), 318F - dc on Slabs, 334 - Shells, 340 - Handbook, 421 - Slabs, and was a founding member and first President of the Alberta AC1 Chapter. WHY AN ELASTIC FRAME ANALOGY? Two-way slab systems are a common structural component in rein
18、forced concrete construction. If asked how they design these slabs, many designers in North America would answer I use a computer program. If pressed as to the methodology incorporated in the program they would likely respond elastic frame analogy. Why an elastic frame analogy? Traditionally, in rei
19、nforced concrete design, one uses a linear elastic theory to determine design parameters and then proportions members using an ultimate strength procedure. The justification for this apparent anomaly is that by designing for moments determined from elastic theory the amount of moment redistribution
20、at service load conditions will be minimized thereby ensuring that serviceability requirements will generally be satisfied. Except for special cases such as deep beams or sudden changes in cross section where elastic theory is not applicable, this technique has served us well. To apply a similar pro
21、cedure to the design of two-way slab systems it is necessary to have a means of obtaining an elastic analysis. As early as 18 11, Lagrange proposed an elastic theory for thin slabs which requires determining a fiinction that will satis both a fourth-order differential equation and the boundary condi
22、tions. Solutions using this approach have been successful only for slabs with the simplest idealized boundary conditions, generally panels with non-deflecting boundaries. This method has been used to develop design procedures for slabs with beams between all supports. It was the need to provide a si
23、mple elastic analysis for column supported two-way slab systems that led to the concept of an elastic frame analogy. Even today, although a number of ingenious techniques to obtain solutions for two-way systems have been proposed, for example Ang (1) and, more recently, numerical solutions based on
24、finite element or finite difference techniques, none have proved practical for routine ofice use. Hence the continuing interest in elastic frame analogies. WHAT IS AN ELASTIC FRAME ANALOGY? The concept behind the use of elastic frame analogies is that satisfactory values for the design moments and s
25、hears in two-way slab systems can be obtained by considering a portion of the slab-column structure to form a design frame that can be analyzed as a plane frame. a) define the analogous plane frame including assigning member stiffness The process consists of three parts: = 0543qb2 Design of Two-way
26、Slabs 3 b) analyze frame with appropriate loading to obtain maximum frame moments, and c) distribute frame moments laterally across the corresponding critical sections of the slab. Frame analogies can be used for both gravity and lateral loading on slab- column structures. The basic approach for def
27、ining the geometry of the analogous elastic plane frames has remained essentially unchanged through various codes. The structure is considered to be made up of analogous or equivalent frames centered on the column lines taken longitudinally and transversely through the building, see Fig. 1. Each fra
28、me consists of a row of columns or supports and slab-beam strips bounded laterally by the centerline of the panel on each side of the centerline of the columns or supports. Frames adjacent and parallel to an edge are bounded by that edge and the centerline of the adjacent panel. Each frame may be an
29、alyzed in its entirety, or for vertical loading each floor or roof with attached columns may be analyzed separately. Success in applying this analogy depends on the appropriate apportioning of stiffness to the members of the frame so that the elastic analysis of the two- dimensional frame will appro
30、ximate that of the non-linear three-dimensional slab- beam-column system. This problem is made more complex by a fundamental assumption in the analysis of plane frames that does not apply to slab-column systems. In a typical plane frame analysis it is assumed that at a beam-column connection all mem
31、bers framing into that joint undergo the same rotation as shown in Fig. 2(a). For slabs supported by columns this assumption is valid only locally at the column. Portions of the slab laterally removed from the column will rotate lesser or greater amounts depending on the geometry and loading pattern
32、s as shown in Fig. 2(b). Furthermore, actual slab systems crack even under service loading, especially near the face of the column resulting in locally reduced stiffness. To account for the differences in behavior of the actual slab-column system and the idealized plane frame, it is necessary to mod
33、i the stiffness of the frame elements. Unfortunately, the modifications required to the member stiffness for lateral loading differ fiom those for gravity loading. The definition of the analogous frame, the apportioning of stiffness and the rules for the lateral distribution of design moments across
34、 the slab have evolved through successive codes. To follow this evolution, it is helpful to review the history of the development of two-way slab construction. TWO-WAY SLABS AND FLAT SLABS Since the 1971 AC1 Code, the term two-way slab refers to all slab systems reinforced for flexure in more than o
35、ne direction with or without beams between supports. The term flat slab is not used. Prior to 1971, the term two-way slab referred only to those slabs with beams between supports along all sides of each panel and the term flat slab referred to slabs without beams between supports but could have colu
36、mn capitals andlor drop panels. The need for the distinction in earlier codes was because of the different genesis of the two slab types and the resulting differences in design rules. The elastic frame method was initially 9 Obb2949 05439b3 ObT W 4 Sirnrnonds developed for two-way slabs without beam
37、s (flat slabs). In the remainder of this paper the term flat slab is used as defined above when discussing design rules prior to 1971. EARLY HISTORY OF SLAB DESIGN Reinforced concrete flat slabs were invented in the sense that they were not a logical extension of elastic theory or construction pract
38、ice. Credit for this invention is generally given to C. A. P. Turner who constructed his first mushroom slab (a reinforced concrete slab supported on columns with flared column capitals) for the five-story C. A. Bovey Building in Minneapolis in 1906. Lacking a rational analysis, the validity of his
39、design was verified with a load test. So successful was this slab that almost immediately competitors were constructing slabs using various proprietary methods. Since there was no generally accepted procedure for analyzing such slabs, it is not surprising that the amount of reinforcement required va
40、ried considerably fiom design to design. A comparison of the amounts of reinforcement required in an interior panel by six different design procedures made by McMillan (2) in 1910 showed that some designs required four times as quch steel as others. In an attempt to reconcile these difference in des
41、igns, many of the slabs that were load tested had measurements of the strains in the reinforcement. Moments in the slab were computed from these steel strains using a straight line expression. These tests did not resolve the differences in design procedures. In 1914, Nichols (3) examined the statics
42、 of a uniformly loaded interior panel of a slab without beams with square panels extending infinitely in both directions. In his original paper, he considered only a quarter of the panel but in the closure to his paper he considered as a free body the half panel designated as A, B, C, and D in Fig.
43、3. From symmetry no shears or twisting moments exist on faces B, C, and D but bending moments exist on all faces. He assumed that the shear forces on face A are uniformly distributed. Denoting the sum of the moments of all vertical forces about x-x as M, yields the simplified expression (1) WL 2c 8
44、3L Mo = -(1-)2 where W is the total load on the panel. The difference between this expression and the exact expression is less than 1% for values of CL smaller than 0.3. Nichols concluded that for equilibrium this must be the sum of the bending moments on faces C and B plus the components about x-x
45、on face A. While this analysis does not give the actual moment at any point or even across any section, it does provide a criterion against which proposed design moments could be evaluated. Since many of the designs that successfully passed load tests used moments that were significantly lower than
46、this sum, his paper evoked a spirited discussion that was over five times the length of the original paper. While some of the discussions applauded his analysis others, including Turner, questioned even the validity of applying statics to two-way slabs. O662949 05439h4 TTb Design of Two-way Slabs 5
47、Those who were opposed to Nichols analysis referred to the results of slab tests. Values of the total moment obtained from steel strain readings for six slabs representative of the many slab tests reported in terms of the total panel static moment, Mo, were Purdue test slab J 0.59 Mo Purdue test sla
48、b S 0.74 Mo Bell Street Warehouse 0.40 M, Western Newspaper Union 0.72 Mo Sanitary Can Building 0.30 Mo Shonk Building 0.38 M, This apparent disagreement between the requirements of equilibrium and the results of tests was a dilemma that was perplexing to engineers and code writers. In 1917, the Joi
49、nt Committee on Concrete and Reinforced Concrete included principles of design for flat slabs in their Final Report (5). Influenced by Nichols logic but unable to ignore the results of the load tests, they compromised by adopting the form of Nichols expression but arbitrarily reduced the coefficient and hence the magnitude of the total panel moment by recommending the expression 2c 3L Mo = 0.107WL (1 - -)2 However, the approved 1920 AC1 Building Code (6) defied statics even more by firther reducing the coefficient to yi
