Elastic, Plastic and Yield Design of Reinforced Structures

By Patrick De Buhan, Jérémy Bleyer and Ghazi Hassen

Elastic, Plastic and Yield Design of Reinforced Structures presents a whole set of new results which have been published by the authors over the last 30 years in the field of continuum solid mechanics applied to the analysis and design of reinforced civil engineering structures.

The focus is on the development and application of up-scaling/homogenization methods in the design of such composite structures, with a special emphasis on the plastic behavior and ultimate strength of materials.

The specificity of the book is highlighted by at least two completely innovative concepts which lie at the very heart of the book’s originality: the elaboration of a fully comprehensive homogenization-based method for the design of reinforced structures (and not only materials), through the study of macroscopic behavior, and the development of a multiphase model for materials reinforced by linear inclusions, which considerably extends the range of applicability of the classical homogenization procedure.

• Sums up almost thirty years of original research in the field of mechanics applied to the analysis and design of reinforced civil engineering structures
• Focuses on the application of upscaling/homogenization methods to the design of civil engineering structures
• Highlights the elaboration of a fully comprehensive homogenization-based method for the design of reinforced structures (and not only materials), through the concept of macroscopic behavior
• Features development of a multiphase model for materials reinforced by linear inclusions, which considerably extends the range of applicability of the classical homogenization procedure.

• Language: English
• Publisher: Elsevier Science
• Release date: Jul 21, 2017
• ISBN: 9780081021132

Patrick De Buhan

Patrick de Buhan is Professor in Structural Mechanics (plasticity and yield design) at Ecole des Ponts ParisTech in France.

Book preview

Elastic, Plastic and Yield Design of Reinforced Structures

Patrick de Buhan

Jérémy Bleyer

Ghazi Hassen

Series Editor

Noël Challamel

Table of Contents

Cover

Title page

Copyright

Introduction

1: The Mixed Modeling Approach to Reinforced Structures

Abstract

1.1 Mechanics of reinforced systems in the mixed modeling framework

1.2 Application to the stability analysis of reinforced soil structures

1.3 Extensions and limitations of the mixed modeling approach

2: The Homogenization Approach: Reinforced Materials as Macroscopically Homogeneous Anisotropic Media

Abstract

2.1 Fundamentals of the periodic homogenization method

2.2 Macroscopic longitudinal elastic shear stiffness of a column-reinforced material

2.3 Elastoplastic analysis of column-reinforced soil foundations

2.4 Yield design homogenization applied to reinforced soil structures

3: Macroscopic Behavior of Materials Reinforced by Thin Highly Stiff/Resistant Linear Inclusions

Abstract

3.1 Macroscopic linear elastic behavior

3.2 Macroscopic strength properties

3.3 Macroscopic elastoplastic behavior

3.4 Concluding remarks: toward a multiphase approach

4: Mechanical Modeling of Reinforced Materials as Multiphase Systems

Abstract

4.1 Construction of the multiphase model by the virtual work method

4.2 Linear elastic behavior

4.3 Elastoplastic behavior of reinforced structures described as two-phase systems

4.4 Yield strength of reinforced materials as multiphase media

5: Applications of the Multiphase Approach Part 1: Static and Dynamic Stiffness of Piled Raft Foundations

Abstract

5.1 Evaluation of the static stiffness of piled raft foundations

5.2 Evaluating the dynamic horizontal and rocking impedance of pile groups

5.3 Vertical dynamic stiffness of square piled raft foundations

6: Applications of the Multiphase Approach Part 2: Load-Bearing Capacity and Stability Analysis of Reinforced Structures

Abstract

6.1 Multiphase model accommodating for elastoplastic interaction behavior

6.2 Stability analysis of reinforced structures using the multiphase yield design approach

7: Yield Design of Reinforced Concrete Beams, Plates and Shells

Abstract

7.1 Composite beams in membrane-bending interaction

7.2 Reinforced plates in bending

7.3 A proposed simplified extension to the yield design of shells

Bibliography

Index

Copyright

First published 2017 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

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© ISTE Press Ltd 2017

The rights of Patrick de Buhan, Jérémy Bleyer and Ghazi Hassen to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

Library of Congress Cataloging in Publication Data

A catalog record for this book is available from the Library of Congress

ISBN 978-1-78548-205-2

Printed and bound in the UK and US

Introduction

The process of reinforcing materials and structures in order to improve their performances is being widely used today in various branches of industry and civil engineering. Thus, owing to their excellent mechanical properties, and notably to their high stiffness-to-weight ratio, fiber-reinforced composite materials are becoming increasingly indispensable in aeronautical engineering. Likewise, their wear resistance along with their very high ultimate strength make them extremely attractive for mechanical engineering purposes. These materials are generally made up of fibers of various kinds (steel, glass, graphite, boron, etc.) embedded in homogeneous matrix material (usually an epoxy resin), which ensures their binding (Figure I.1).

Figure I.1 Fiber-reinforced composite material at different scales

In the field of civil engineering, reinforced concrete represents a well-known example of composite material in which the reinforcing inclusions (generally steel rebars) are compensating the lack of tensile resistance of plain concrete (Figure I.2). Finally, a large variety of reinforcement techniques have been developed since the middle of last century in the field of geotechnical construction from reinforced earth retaining walls or soil nailed excavations (Figure I.3(a)) to rock bolted tunnels and piled raft foundations (Figure I.3(b)).

Figure I.2 Reinforced concrete as a composite material

Figure I.3 a) Soil-nailed excavation and b) piled raft foundation of bridge pier

The ability to predict the essential features governing the behavior of such inclusion-reinforced materials, from the properties of their individual components, remains a major concern for engineers involved in the design of structures made up of such reinforced materials. A considerable amount of theoretical as well as experimental work has been devoted to this subject in recent decades, in connection with the ever increasing development of fiber-reinforced composites in various industrial applications and reinforcement techniques for concrete and geotechnical structures.

This book proposes a detailed and thorough review of more than 30 years of research dedicated to the modeling of reinforced materials in the continuum mechanics framework. It is more specifically focused on materials reinforced by:

– long linear inclusions placed in one (unidirectional composites: Figure I.4(a)) or several (multidirectional composites: Figure I.4(b)) directions;

Figure I.4 a) Unidirectionally and b) bidirectionally reinforced materials; c) reinforcement by two-dimensional inclusions

– two-dimensional (2D) inclusions (Figure I.4(c)), such as soils reinforced by geomembranes or rubber bearing pads strengthened by metal sheets or plates.

The strong heterogeneity of such composite reinforced materials, combined with the relatively high number of inclusions involved in the reinforcing process (up to several hundreds in some cases: see Figures I.2 and I.3, not to mention the situation of industrial fiber-reinforced composites: Figure I.1), makes it very difficult, if not impossible, to devise appropriate design-oriented calculations methods, in which the inclusions would be treated as individual elements embedded in the matrix material (epoxy resin, concrete, soil or rock).

Indeed, referring, for instance, to a finite element simulation of this kind of reinforced structures, where the cylindrical shape of the linear inclusions has to be taken into account, a fully three-dimensional (3D) analysis is required, with a locally refined mesh discretization in order to capture the rather complex interactions prevailing between the inclusions and the surrounding matrix, with sufficient accuracy. This would inevitably lead to oversized numerical problems, or at least to the elaboration of complex and sophisticated numerical tools, the use of which would hardly be compatible with a quick engineering design approach, notably for 3D configurations.

The objective of this book is to propose and develop alternative computational methods based on appropriate mechanical models of inclusion-reinforced materials, namely the "mixed modeling" approach, the homogenization method and the most recently developed and fully original multiphase model. Each of these different but somewhat complementary approaches are first presented in the context of a linear elastic behavior for the different constituent materials, then extended to the case of elastoplasticity, and finally implemented in the framework of the limit analysis/yield design theory. It is worth emphasizing that any other kind of material constitutive behavior, such as creep or damage, can be accommodated within such models, while their extension to large strain analysis could also be envisaged.

The book is thus organized as follows:

– Chapter 1 opens with the presentation of the so-called "mixed modeling" approach, which is none other than a formalization of the simple intuition, according to which the inclusions may be treated as one-dimensional (1D) (beams) or 2D (membranes/plates) structural elements plunged into a 3D matrix material (soil, rock, concrete). This approach, which is implicitly advocated by many classical engineering design methods, is particularly suitable for the simulation and design of reinforced structures involving a small number of thin highly stiff/resistant inclusions. Special attention is paid here to its implementation in the context of the kinematic approach of yield design, where it provides the theoretical background for efficient computer software. However, a serious limitation to the model is underlined in the situation of linear inclusions embedded in a 3D continuous matrix.

– The limitations of the above mixed modeling approach may be overcome by resorting to the homogenization concept presented in Chapter 2. Indeed, the homogenization method and related procedure developed in this chapter provide an efficient alternative to the mixed modeling approach in the situation of densely reinforced structures, and more specifically in the context of regularly distributed inclusions, where the reinforced material may be treated as a periodic heterogeneous material, and regarded as an equivalent homogeneous medium at the (macroscopic) scale of the whole structure. The periodic homogenization method is then developed in the context of elastic, then plastic and finally yield strength ultimate behavior, through the notions of macroscopic stiffness tensor and strength criterion. Several recent applications to the case of soils, and more specifically foundation soils, reinforced by columnar inclusions, where the mixed modeling approach is no more relevant, will be presented in details, leading to new significant results from an engineering standpoint.

– Chapter 3 is devoted to the specific, but frequently encountered situation of a matrix material periodically reinforced by a sufficiently dense array of thin linear inclusions made up of a material exhibiting much higher stiffness or strength characteristics than the matrix. The application of the previously presented periodic homogenization procedure leads to remarkably simple analytical formulations for both the elastic and yield strength macroscopic behavior of such reinforced materials perceived as homogenized anisotropic media. The obtained formulations may then be clearly connected to the mixed modeling approach mentioned in Chapter 1, in which the inclusions are regarded as 1D or 2D flexible structural elements that are undergoing axial forces only. It is shown how such closed-form expressions can then easily incorporated into computational design procedures relative to the elastic, plastic and ultimate failure analysis of reinforced structures.

– With a view to remedying some fundamental shortcomings of the homogenization method observed in Chapter 3 on a few illustrative examples, the concept of multiphase model applied to reinforced materials is then introduced and developed in Chapter 4. This model is based on the intuitive idea that the reinforced material can be appropriately described as the superposition of two continua, called phases, relating to the matrix material and the group of parallel reinforcing inclusions, respectively. It may be regarded as an extension of the homogenization method, in that it takes matrix–inclusion interactions into account in an explicit way, while allowing to incorporate shear and bending effects in the reinforcements. Boundary value problems involving such a multiphase description are then formulated in the context of elastoplasticity and yield design and it is shown in the already mentioned examples that boundary as well as scale effects may be accurately accounted for from employing this multiphase model.

– Chapters 5 and 6 are devoted to presenting some of the latest applications of the multiphase approach to the solution of geotechnical problems involving reinforcement systems:

– Chapter 5 is concerned with the evaluation of pile-reinforced foundations in terms of static as well as dynamic stiffness by means of a specifically devised finite element linear elastic computer code. It is thus clearly verified that under both static and dynamic loadings, the shear and flexural characteristics of the reinforcements play an important role in the case of lateral loading, whereas it remains negligible under vertical loading. Conversely, the interactions between the soil and the reinforcing inclusions may strongly influence the vertical impedance of the foundation, while they do not significantly affect its horizontal impedance.

– On the other hand, Chapter 6 presents some recent findings relative to typical problems in the field of geotechnical engineering: the evaluation of the load-bearing capacity of vertically loaded piled raft foundations in terms of elastoplastic settlements and the stability analysis of reinforced earth retaining walls and seismically loaded piled embankments. The detailed analysis of these case studies underline the excellent performance of the multiphase-based simulation tools regarding the production of reliable predictions and, on the one hand, the computational efficiency, on the other.

– Finally, Chapter 7 addresses some key issues concerning the limit analysis/yield design of composite structural members with a special emphasis on reinforced concrete beams, plates and shells. Referring to a homogenization procedure analogous to that implemented for 3D periodic heterogeneous media in Chapters 2 and 3, an axial-bending interaction strength condition is first derived for reinforced concrete beams, leading to a closed-form analytical expression in the case of thin reinforcing metal rebars. The method is generalized to reinforced plates where a macroscopic strength condition expressed in terms of bending moment tensor may be determined and incorporated into yield design calculations at the structure level. A simplified extension to homogeneous, multilayer and inclusion-reinforced shells is then proposed, where the resulting strength condition is expressed as a function of both membrane forces and bending moments, and the implementation of such a strength condition in the yield design kinematic approach making use of generalized yield lines is briefly outlined.

1

The Mixed Modeling Approach to Reinforced Structures

Abstract

Most civil engineering structures involving inclusion-reinforced materials, such as reinforced concrete beams and slabs or reinforced geotechnical structures, are commonly designed by resorting to an implicit mechanical model in which the material to be reinforced (concrete, soil, rock, etc.) is regarded as a three-dimensional (3D) continuous medium, whereas the reinforcing inclusions (steel bars, piles or geomembranes) are treated as one-dimensional (1D) (beams) or two-dimensional (2D) (plates) structural elements. Hence, the denomination of "mixed modeling" to which this chapter is devoted.

Keywords

1D–3D mixed modeling approach; Deformation patterns; Material elastic behavior; Matrix/reinforcement interface behavior; Mixed Modeling; Reinforced structures; Reinforced vertical excavation; Stability analysis; Yield design

Most civil engineering structures involving inclusion-reinforced materials, such as reinforced concrete beams and slabs or reinforced geotechnical structures, are commonly designed by resorting to an implicit mechanical model in which the material to be reinforced (concrete, soil, rock, etc.) is regarded as a three-dimensional (3D) continuous medium, whereas the reinforcing inclusions (steel bars, piles or geomembranes) are treated as one-dimensional (1D) (beams) or two-dimensional (2D) (plates) structural elements. Hence, the denomination of "mixed modeling" to which this chapter is devoted.

The general equations governing the mechanics of materials reinforced by 1D or 2D inclusions are demonstrated in section 1.1, where they are first implemented in the context of linear elasticity, then extended to the elastoplastic behavior and yield design (limit analysis) of such reinforced systems. Section 1.2 focuses on the stability analysis of reinforced soil structures based on the mixed modeling approach, notably developed in the context of the upper-bound kinematic method of yield design. Finally, section 1.3 presents some possible extensions of the mixed modeling approach to structures made up of reinforced materials, while some limitations of the model are pointed out.

1.1 Mechanics of reinforced systems in the mixed modeling framework

1.1.1 Outline of the mixed modeling approach

This approach is simply derived from the civil engineer’s basic intuition facing, for instance, the design of geotechnical earthworks or concrete structures reinforced by continuous linear inclusions. Two major characteristics are exploited to this end:

– the volume fraction of the reinforcing material is small (metal, for instance), if not very small, rarely exceeding a low percentage (< 10%);

– as a counterpart, the mechanical characteristics of the reinforcement, such as its stiffness and/or strength characteristics, are much higher than those of the surrounding matrix material (soil, rock or concrete) in which they are embedded by a factor that could be equal to several thousands (steel inclusions vs. soft soils).

As a result, owing to their strongly heterogeneous nature, such composite reinforced structures give rise to numerous difficulties, notably from a computational point of view, when trying, for instance, to set up reliable and efficient design methods for this particular kind of structures. The mixed modeling approach is specifically intended to circumvent such difficulties, by considering that the inclusions can be treated as 1D (or 2D in the case of reinforcing sheets or plates) structural elements, embedded in the matrix material described as a 3D continuous medium.

It appears that this kind of approach, which will now be developed in the context of a combination between structural and continuum mechanics, is underlying many civil engineering design methods. The detailed analysis of a simple illustrative example will help clarify the approach.

1.1.2 An illustrative example

The example under consideration is that of rectangular block of height 2H, width 2B and infinite length in the transverse direction, lying on a rigid substratum, while its top surface is subject to a compressive downwards vertical force Q (per unit transverse length) exerted by means of a rigid downwards moving punch, as shown in Figure 1.1. The block has been reinforced by a plate located at mid height of the block, along the Ox-axis, and perfectly adherent to the block matrix material. Gravity loads are neglected, lateral surfaces are stress free and the contact between the punch/substratum and the block is perfectly smooth.

Figure 1.1 Plane strain compression of a reinforced block

The above problem can therefore be treated as a plane strain problem in the Oxy plane, so that the system of internal efforts relative to the mixed modeling approach of the reinforced block is made up of:

– a stress field , defined at any point (x, y) of the rectangular domain Ω occupied by the block, located outside the reinforcement;

– a distribution of generalized stresses N (axial force), V (shear force) and M (bending moment) defined at any point of abscissa x along the 1D beam Δ representing the reinforcing plate. It is to be emphasized that, on account of the plane strain 2D framework adopted here, those generalized stresses are calculated per unit transverse length of the reinforcement in the out-of-plane direction Oz.

Consequently, any field of internal efforts of the composite structure denoted by:

   [1.1]

is statically admissible (SA) for the problem under consideration if it satisfies the set of following equations (see notations of Figure 1.1):

   [1.2]

in the matrix material, where equations [1.2(a)] and [1.2(b)–(c)] are the equilibrium equation and stress boundary conditions, respectively. As concerns the reinforcement, modeled as a straight beam, such equations become:

   [1.3]

where [*](x) denotes the discontinuity of a quantity (*) (here the stress tensor) in the matrix when crossing the reinforcement segment Δ at point x . Equation [1.3(c)] expresses the fact that both ends of the reinforcement are stress free.

It is to be observed that plays the role of an external load density for the reinforcement, while by way of reciprocity, the presence of the reinforcement induces a jump of the stress vector in the matrix. They reflect, from the equilibrium point of view, the mutual interaction prevailing between the reinforcement and the surrounding matrix.

These