Table Of ContentElastic 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
