Table Of ContentCISM COURSES AND LECTURES
Series Editors:
The Rectors of CISM
Sandor Kaliszky -Budapest
Horst Lippmann -Munich
Mahir Sayir - Zurich
The Secretary General of CISM
Giovanni Bianchi-Milan
Executive Editor
Carlo Tasso -Udine
The series presents lecture notes, monographs, edited works and proceedings in the field of
Mechanics, Engineering, Computer Science and Applied Mathematics.
Purpose of the series in to make known in the international scientific and technical
community results obtained in some of the activities organized by CISM, the International
Centre for Mechanical Sciences.
INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
COURSES AND LECfURES - No. 332
EVALUATION OF GLOBAL
BEARING CAPACITIES OF STRUCTURES
EDITED BY
G. SACCHI LANDRIANI
POLYTECHNIC OF MILAN, MILAN
and
J. SALEN<;ON
ECOLE POLY TECHNIQUE, PALAISEAU
SPRINGER-VERLAG WIEN GMBH
Le spese di stampa di questo volume sono in parte coperte da
contributi del Consiglio Nazionale delle Ricerche.
This volume contains 143 illustrations.
This work is subject to copyright.
Ali rights are reserved,
whether the whole or part of the material is concerned
specifically those of translation, reprinting, re-use of illustrations,
broadcasting, reproduction by photocopying machine
or similar means, and storage in data banks.
© 1993 by Springer-Verlag Wien
Originally published by Springer Verlag Wien-New York in 1993
In order to make this volume available as economically and as
rapidly as possible the authors' typescripts have been
reproduced in their original forms. This method unfortunately
has its typographical limitations but it is hoped that they in no
way distract the reader.
ISBN 978-3-211-82493-1 ISBN 978-3-7091-2752-0 (eBook)
DOI 10.1007/978-3-7091-2752-0
PREFACE
This book collects the lectures given at CISM during June 1992, devoted to the
evaluation of global bearing capacities of structures. The course aimed at presenting a
unified frame for yield design, limit analysis and optimal design theories.
After a brief introduction on the basic results and the general comments on the
validity of each theory from the practical point of view, typical and illustrative applications in
various domains are given, in order to point out their versatility.
Based upon simple arguments of convex analysis, the theory of yield design makes
possible to determine the global bearing capacities of a structure from the sole knowledge of
the local strength of the constitutive materials.
Soil mechanics problems are considered, such as stability analysis of reinforced
structures, where classical limit analysis cannot be considered as adequate. Efficient methods
have been derived from the theory of yield design, some of them making use also of the theory
of homogenization.
On a different scale, composite materials with long fibers as reinforcement are
considered with the purpose of determining their strength capacities. Interesting results have
been obtained through this approach, in good agreement with experiments.
In the second part of the book the "classical" limit analysis theory is presented and it
is underlined how its use makes it possible to determine the global bearing capacities of a
class of structure, on the basis of the local strength and the plastic associated flow rule of the
constituent material. Structural mechanics problems are considered, such as the limit state of
steel structures or reinforced concrete plates and shells, where classical limit analysis can be
fully applied. Efficient methods are derived from the combined use of limit analysis and
convex analysis theories.
In the third part a general formulation of optimal design theory is proposed.
Particular emphasis is given to this method as a tool making it possible to determine the best
distribution of the employed material in a structure, with a given layout, to achieve a
minimum total weight at prescribed limit plastic load or buckling load. In such a way, optimal
design problems can be conceived as inverse problems in comparison to the yield design and
limit analysis formulation.
Finally some optimal design problems for structures with given layout and
· prescribed elastic compliance are presented. The aim of these contributions is to point out
some analogies existing between some plastic and elastic problems. Also, opportunity is taken
for proposing some numerical applications.
The unifying frame in which yield design, limit analysis and optimal design theories
have been presented is intended to be a clarifying reasoning tool leading to various new
relevant developments, the necessity of experiments being of course the ultimate and
unavoidable validation test.
Giannantonio Sacchi Landriani
Jean Salen~on
CONTENTS
Page
Preface
Yield Design: A Survey of the Theory
by J. Salenf:on ...................................................................................................................... 1
Application of the Yield Design Theory to the Mechanics of Reinforced Soils
by P. de Buhan .................................................................................................................. 45
Application of Homogenization Theory and Limit Analysis to the Evaluation
of the Macroscopic Strength of Fiber Reinforced Composite Materials
by A. Taliercio .................................................................................................................... 91
Rigid-Plastic Analysis and Design
by M. Save ....................................................................................................................... 121
Limit Design: Formulations and Properties
by C. Cinquini ................................................................................................................... 159
Optimal Design of Fiber-Reinforced Orthotropic Bodies
by G. Sacchi Landriani and M. Rovati ........................................................................... .189
Optimization of Fiber Orientation and Concentration in Composites
by J. Thomsen and N. Olhoff ........................................................................................... 221
Topology Optimization of Structures Composed of One or Two Materials
by J. Thomsen .................................................................................................................. 237
On Cad-Integrated Structural Topology and Design Optimization
by N. 0/hoff, M.P. Bends(Je and J. Rasmussen ............................................................... 255
YIELD DESIGN: A SURVEY OF THE THEORY
J.
Salen~on
Ecole Polytechnique, Palaiseau, France
ABSTRACT
The theory of Yield Design is based upon the obvious necessary condition for the
stability of a structure that the equilibrium of that structure and the resistance of its
constituents should be compatible. The static approach of the yield design theory
proceeds directly from this condition, leading to lower estimates of the extreme loads.
The kinematic approach is derived by dualizing the static approach through the
principle virtual work, thus ensuring full mechanical consistency. The treatment of a
classical example illustrates these arguments. Present and possibly future domains of
practical applications of the theory are reviewed, including the full adequacy between
the Yield Design Theory and the Ultimate Limit State Design concept of safety.
2 J. Salen~on
1.-SOME HISTORICAL LANDMARKS
The concept of yield (or limit) design is most popular among civil engineers who
have made an extensive use of it for centuries when designing structures or
earthworks. As a matter of fact, one usually refers to Galilee's study [1] of the
cantilever beam as the first written explicitation of the concept : the maximum load the
beam may withstand is derived from the only knowledge of the strength of its
constituents, namely the fibers (Fig.1).
Figure 1 -Galileo's analysis of the cantilever beam.
Coulomb's celebrated memoir [2] appears then as the reference where such
problems as the compression of a column, the stability of a retaining wall or of a
masonry arch, etc., are considered from the same point of view combining the
equilibrium of the structure and the resistance of its constituents. Papers by Heyman
[3-7] and by Delbecq [8, 9] may help getting a comprehensive view of Coulomb's
analysis and of the subsequent works (e.g. Mery [1 0]. Durand-Ciaye [11, 12], ...)
concerning masonry works.
As regards soil mechanics, the original Coulomb analysis has been followed by
others in the same "spirit" : the general equilibrium, in terms of resulting moments
and resulting forces, of a bulk of soil, defined in the considered earthwork by one or a
few parameters, is checked under the condition that the soil cannot withstand stresses
outside its strength criterion usually defined as a Coulomb criterion. Among many, one
must quote the Culmann method [13]. and Fellenius' famous analysis for slope stability
where the bulks of soil whose equilibrium is checked are limited by circular lines.
When considering such circular lines in the case of a frictional soil, the reasoning can
Yield Design: A Survey of the Theory 3
only be carried out by introducing complementary assumptions as in the slices method
for instance [14-17). Rendulic [18] introduced the use of logspirals instead of
circular lines in that case, which removes the difficulty and preserves the full
significance of the analysis as it will be shown later on. Analogous analyses have been
performed in order to investigate the stability of surface foundations.
For what concerns the stability of surface foundations and the determination of
their bearing capacity, the limit equilibrium methods (as classified by Chen [19) for
instance) have also been used, especially in the case of plane problems : the equations
were first established by Massau [20) and KOtter [21, 22], and important
developments may be found in the famous books by Sokolovski [23-25] and Berezancew
[26). Since the limit equilibrium equations for plane problems are homologous to the
equations appearing in the study of plane strain problems for perfectly plastic
materials obeying Tresca's yield criterion with associated flow rule (27], they are
often associated with the concept of plasticity : in other words they are improperly said
to rely on the assumption of a rigid perfectly plastic soil obeying Coulomb's criterion
with associated flow rule. As a matter of fact, it should be clear to everybody that limit
equilibrium methods are but another technique to perform the reasoning of the yield
design theory, and that they do not require any more assumption about the constitutive
law of the considered soil than the data of its strength criterion.
Another classical domain of application for the yield design theory concerns the
analysis of metallic plates and reinforced concrete slabs, with the use of Johansen's
criterion and yield line theory [28, 29] and other methods, as explained for instance in
Save and Massonnet (30).
Recently the combination of the homogeneization theory with the yield design
theory has opened the access to new types of investigations and has already led to
interesting results such as the determination of the "macroscopic" homogenized
strength criterion of a composite based upon the data of the strength criteria of its
constituent materials, including the interfaces between them, which is to be introduced
in the homogenized yield design analysis of a structure made of that composite material.
From the practical point of view the domain of relevance ranges from mechanical
engineering with, for instance, the "classical" long fiber/epoxy matrix composites
[31, 32) to civil engineering where soils reinforced by inclusions such as nails,
