Elastic analysis of slab structures To the memory of my father Prof. Nicolae Neg,rutiu MECHANICS OF SURFACE STRUCTURES Editors: W. A. Nash ani G. Ai, Oravas P. Seide, Small elastic deformations of thin sh ells. 1975. ISBN 90-286-0064-7 V. Pane, Theories of elastic plates. 1975. ISBN 90-286-0104-x T. L. Nowinski, Theory of thermoelasticity with applications. 1978. ISBN 90-286-0457-x S. Lukasiewics, Local loads in plates and shells. 1979. ISBN 90-286-0047-7 V. Fih, Statics, formfinding and dynamics of air-supported membrane structures. 1983. ISBN 90-247-2672-7 Yeh Kai-yuan, Progress in applied mechanics. ] 986. ISBN 90-247-3249-2 R. Negrutiu, Elastic analysis of slab structures. 1986 IS B N 90-247-3367-7 Elastic analysis of slab structures by Radu Negrutiu 1987 MARTINUS NIJHOFF PUBLISHERS a member of the KLUWER ACADEMIC PUB LISHERS GROUP DORDRECHT / BOSTON / LANCASTER Distributors for the United States and Canada: Kluwer Academic Publishers, 190 Old Derby Street, Hingham, MA 02043, USA for the UK and Ireland: Kluwer Academic Publishers, MTP Press Limited, Falcon House, Queen Square, Lancaster LA 1 1 RN, UK for all other countries: Kluwer Academic Publishers Group, Distribution Center, P.O. Box 322, 3300 AH Dordrecht, The Netherlands for socialist countries: Editura Academiei, Calea Victoriei 125, R-79717, Bucharest, Romania Library of Congress Cataloging in Publication Data Negrutiu, Radu. Elastic analysis of slab structures. (Mechanics of surface structures; 7) Rev. translation of: Analiza elastica a structurilor cu dale. Bibliography: p. Includes index. 1. Slabs. 1. Title. H. Series: Mechanics of surface structures; v. 7. TA660.S6N4313 1987 624.1 772 86-12538 ISBN-13:978-94-01O-8064-4 e-ISBN-13:978-94-009-3501-3 DOl: 10.1007/978-94-009-3501-3 Book information This book is the revised and updated English version of the Romanian book Analiza elastica a structurilor cu dale, published in 1976 by Editura Academiei, Calea Victoriei 125, Bucure§ti 79717. Copyright © 1987 by Martinus Nijhoff Publishers, Dordrecht and Editura Academiei, Bucharest. Softcover reprint of the hardcover 1st edition 1987 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publishers, Martinus Nijhoff Publishers, P.O. Box 163, 3300 AD Dordrecht, The Netherlands and Editura Academiei, Calea Victoriei 125, R-79717, Bucharest, Romania. Contents Preface ix 1. Introduction 1 1.1 Aim of the book 1 ].2 State of the art. Main re8earch trend8 6 1.2.1 Theoretical approaches of plates 6 1.2.2 Classical approaches to flat slabs with capitals 12 1.2 ..3 Modern approaches to flat RlabR without capitals 30 1.2.4 Early studieR on surface couples. The effective plate width . . . . . . '. . . . ., 54 1.2.5 Early approaches to the elastic analysis of slab stluctures 70 1.3 Scope of the book . . . 84 2. New mathematical models lor the deflected middle surface of plates .... . . . . . 89 2.1 Prelt:minr.1'i(s . . . . . . . . . . . . . . . .. 89 2.1.1 Baiiic atltlumptions . . . . . . . . . . . . . 89 2.1.2 General computation formulae . . . . . . .. 91 2.1.3 Characteristics of the categories of plates inves- tigated . . . . . . . . . . . . . . . . . .. 92 2.1.4 Characteristics of the types of design loads .. 94 :2.2 Equation8 of the middle 8u1:fare of plate8 deflected by load8 of type P, M and q . . . . . . . . . . 95 2.2.1 The infinitely long plate . . . . . . .. 95 2.2.1.1 The locally distributed load P . .. 95 2.2.1.2 The surface couple of vector ill Oy 97 2.2.1.3 The surface couple of vector .l~l 1! O.v . 98 2.2.1.4 The unifOlmly distributed load "q .. 100 2.2.2 The rectangular plate with two parallel free edges . . . . . . . . . . . . . . . 101 2.2.2.1 The locally distributed load P . . . . 101 v Contents Ii 2.2.2.2 The surface couple of yector _1l! 0.'1. 104 Ii 2.2.2.3 The surface couple of vector .ill Ox. 105 2.2.2.4 The uniformly diKtributed load q . . 108 2.2.3 The reetangular plate simply supported along the entire boundar v ... . . . . . . . . . 110 2.2.3.1 The locally' distributed load P . . . . 110 i: 2.2.3.2 The surface couple of vector M ON. 111 2.2.3.3 The surface couple of vector _ZII ii 0:1'. 113 2.2.3.4 The uniformly distributed load q . 11:") 2.3 Algorithms for compnting elastic displacements, sec tional stresses and boundarN reactions . . . . . . 115 2.3.1 Systematization of numerical computations . IV, 2.3.2 Expressions for dimensionless coefficients . .. 118 2.3.2.1 The infinitely long plate . . . . . . 119 2.3.2.2 The rectangular plate "'ith two parallel free edgeK . . . . . . . . . . . . . 124 2.3.2.3 The rectangular plate simpl~' supported along the entire boundary . . 132 3. Behaviour of plates subjected to a surface couI_le . . . . 137 3.1 Aim of th(} numerical studN. Computation parameters and grid . . .. . . . . . . . . . . . . . . . . 137 3.2 Factors hifl~lencing the magnit~lde and distribution of elastic displacements of plates snbjected to a surface couple . . . . . . . . . . . . . . . . . . . . . 141 3.2.1 Magnitude and distribution of elastic displace- ments along the plate -width . . . . . . . . 142 3.2.2 Magnitude and distribution of elaKtic displace ments along the plate length . . . . . . . . 152 3.3 Factors infl~lencing the magnitude and d1'stribution of stresses in plates snbjected to a snrface couple . . . 157 3.3.1 Magnitude and distribution of bending moments along the plate width . . . . . . . . . . . 158 3.3.2 Magnitude and distribution of bending mo- men ts along the plate length ..... 174 3.3.3 Magnitude and distribution of ton,ional mo- ments . . . . . . . . . . . . . . . . 180 3.3.4 Influence of Poisson's ratio on the values of bending and torsional moments .. . . . . . 185 3.4 Accuracy of nttmer1:cal computations . . . . . . . . 190 3.4.1 Influence of the number of summed terms in the series on the accuracy of numerical computa- tions ..... . . . . . . . . . . . . . . 190 vi Oontents :3.4.2 Checking the accuracy of compu ta tions by use of the reciprocity theorem of unit displacements in the case of plates . . . . . . . . . . . . 196 3.4.3 Checking the accuracy of computations by reference to the results obtained by other au- thors . . . . . . . . . . . . . . . . . . . . 200 4. Elastic plates as structural members . . . . . . . . . 203 4.1 Slab structures stlbjected to tmnsverse loads. Effect of M- and P-type loads . . . . . . . . . . .' . . . . 203 4.1.1 Svmmetrical and an tisvmmetrical sets of surface couples M . . . . .. . . . . . . . 205 4.1.1.1 The infinitely long plate . . . . . . . 206 4.1.1.2 The square plate .. . . . . . . . . 219 4.1 2 Symmetrical and antisymmetrical sets of locall;v distributed forces P . . . . . . 224 4.1.2.1 The infinitely long plate . . . . . . 225 4.1.2.2 The square plate . . . . . . . . . . 233 4.1.3 Associated loads of type 21{ and P applied on partiall~T symmetrical or unsymmetrical slab structures . . . . . . . . . . . . . . . . . . 236 4.1.3.1 Unsymmetrical sets of couples M and of locallv distributed forces P . . . . 237 4.1.3.2 Sets of ·couples vM and forces vP of different value and sign . . . . . . . 244 4.2 Slab structures su.bjected to lateml loads. Evaluation of the effective slab width . . . . . . . . . . . . . 248 4.2.1 The effective width of the plate subjected to only one surface couple ~~t . . . . . . . . . 249 4.2.1.1 Definition. Calculation procedure . .. 249 4.2.1.2 Comparative "\'alues of the effective plate width ............... 252 4.2.2 The effective width of the plate subjected to sets of surface couples lit associated in a simple row ..... . . . . . . . . . . . . . . 256 5. A new method lor the elastic analysis 01 slab structures 261 5 .1 Use of the general force method. Transverse and in-plane loads . . . . . . . . . . . . . . . . . . . . . 264 5.1.1 General scheme . . . . . . . . . . . . . . 264 5.1.1.1 Basic systems and condition equations 264 5.1.1.2 Calculation of the elements included in the matrices d and D in the general case 268 5.1.1.3 Systematization recommended in the case of associated unknowns .... 270 vii Contents .-; .1.1.4 Expressions for the elastic displacements and stresses in the initial statically in determinate slab structure ...'.. 272' ;1.1.2 Influence of the relative slab-column stiffnesR on the value of displacements and Rtres:-;eR in slabfl and columnfi 274 5.1.2 .1 TranflVer:-;e loaufl :.; 77 fi.1.2.'3 In-plane loadfi 298 ;}.1.:3 Effect of failure of :-;upportfi on ;slab structure~ 302 ;}.2 Use of the general displacement method. Tn-plane loads :312 5.2.1 Ba~ic flcheme .............. ~112 5.2.2 Ul'le of the design procedure in the analYl'lil'l of symmetrical multistorieu Rlab Rtructurel'l . . . :314 6. Introduction to the dynamic analysis of slah structures 319 6.1 Calculation elements for the dynamic analysis (d slab structures . . . . . . . . . . . . . . . . . . :31 9 6.1.1 Elementfl for the stuUY of free vibration:-; . ::120 6.1.2 ElementR for the stud}T of forced vibration:') :327 6.2 Behav/:01lr of slab strllct~lres subjected to dynamic ac- tion. Comparative numerical studies . . . . . . 83:) 6.2.1 Determination of natural circular frequencic;s :33;}, 6.2.2 Dynamic uefleeted shape and streSRel'l induced by disturbing foree:-; . . . . . . . . . . . . 841 6.2.:3 Comparative dynamic analYfii:-; of a lllulti"toried Rlab fitlucture 846 Conclusions :3;-;4 Appendix . :357 Relel'ence;; . 408 4n Author index 417 Suhje~~t inl~x viii Preface Any practitioner who takes his profession in earnest, such that daily work is not a heavy duty but part of their life, will recognize in this book the rigorousness of the analysis and the comprehensive presentation of the problems. This professional attitude is solely able to make the research and design engineer deal with strength structures and their behaviour. Indeed, the computational means that are nowadays available permit the numerical computation of whatever problem; the pro gram libraries are extremely rich and programs themselves have developed intensively. Howeyer, though computers are available at any moment without restrictions on the frequency with which they are employed, they finally impoYerish the creative compe tency of the civil engineer. Thus, he will calculate increasingly more while devising increasingly less. He will draw less and less on the experience gained in devising and implementing bearing structures because the computational process can be repeated as often as desired over a minimum time-period by means of the available programs. \Ve note that nowadays structures are no longer investigated or economically designed to comply with the requirements of the topic of interest. :Much to the contrary, the solutions are chosen so as to comply with the capabilities of the programs. A bearing structure lives as is prescribed by its initial con structive data. However, the numerical computations that hold pre-eminence today, are only a later confirmation by static analysis means of the fact that the Hizes, stiffness and other items have been well chosen. A correctly built structure can be obtained only if one acquires full knowledge of the distribution of stresses in the over-all structureH and in each of the strength members. Of course, it is essential to have a clear idea of the structural behaviour under stress and strain and of the structural response to loads. ix Preface It is in this sense that, Dr. N egrutiu's work should be under stood. So far, this is the most comprehensive research of the behaviour of plates subjected to locally distributed couples, defined as surface couples. Let us note the preliminary theoretical approach which gives ;,;olutions to the fundamental equation of plates under thiH new type of load. Owing to the rigid column-slab connection, the study of surface coupleH is of outstanding importance whenever Hlab structureH having an arbitrary location of columns are subjected to uniformly or locally distributed loads or to horizontal in-plane forces. An independent chapter revolves about the dynamic beha viour of slab structures, i.e. the response of such structures to dynamic loads as may be induced, for instance, in Heismic zones. To the familiarized reader, the pertinent diagrams for the variation of the static quantities and the numerical data tabu lated for the sectional stresses and elastic dhiplacemenb, furnish a clear image of the behaviour of slab structures under load. It is my hope that this book will find wide audience among engineers, particularly among the younget't. Ilarlsruhe, ~1I1arch 1985 KLAUS STIGLAT x