Structural Analysis with Finite Elements

The finite element method has become an indispensible tool in structural analysis, and tells an unparalleled success story. With success, however, came criticism, because it was noticeable that knowledge of the method among practitioners did not keep up with success. Reviewing engineers complain that the method is increasingly applied without an understanding of structural behavior.

Often a critical evaluation of computed results is missing, and frequently a basic understanding of the limitations and possibilities of the method are nonexistent. But a working knowledge of the fundamentals of the finite element method and classical structural mechanics is a prerequisite for any sound finite element analysis. Only a well trained engineer will have the skills to critically examine the computed results.

Finite element modeling is more than preparing a mesh connecting the elements at the nodes and replacing the load by nodal forces. This is a popular model but this model downgrades the complex structural reality in such a way that—instead of being helpful—it misleads an engineer who is not well acquainted with finite element techniques.

The object of this book is therefore to provide a foundation for the finite element method from the standpoint of structural analysis, and to discuss questions that arise in modeling structures with finite elements. What encouraged us in writing this book was that—thanks to the intensive
research that is still going on in the finite element community—we can explain the principles of finite element methods in a new way and from a new perspective by making ample use of influence functions. This approach should appeal in particular to structural engineers, because influence functions are a genuine engineering concept and are thus deeply rooted in classical structural mechanics, so that the structural engineer can use his engineering knowledge and insight to assess the accuracy of finite element results or to discuss the modeling of structures with finite elements.

Just as a change in the elastic properties of a structure changes the Green’s functions or influence functions of the structure so a finite element mesh effects a shift of the Green’s functions. We have tried to concentrate on ideas, because we considered these and not necessarily the technical details to be important. The emphasis should be on structural mechanics and not on programming the finite elements, and therefore we have also provided many illustrative examples.

Finite element technology was not developed by mathematicians, but by engineers (Argyris, Clough, Zienkiewicz). They relied on heuristics, their intuition and their engineering expertise, when in the tradition of medieval craftsmen they designed and tested elements without fully understanding the exact background. The results were empirically useful and engineers were grateful because they could suddenly tackle questions which were previously unanswerable. After these early achievements self-confidence grew, and a second epoch followed that could be called baroque: the elements became more and more complex (some finite element programs offered 50 or more elements) and enthusiasm prevailed. In the third phase, the epoch of “enlightment” mathematicians became interested in the method and tried to analyze the method with mathematical rigor. To some extent their efforts were futile or extremely difficult, because engineers employed “techniques” (reduced integration, nonconforming elements, discrete Kirchhoff elements) which had no analogy in the calculus of variations. But little by little knowledge increased, the gap closed, and mathematicians felt secure enough with the method that they could provide reliable estimates about the behavior of some elements.

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REINFORCED CONCRETE DEEP BEAMS

This book is designed as an international reference work on the behaviour, design and analysis of reinforced concrete deep beams. It is intended to meet the needs of practising civil and structural engineers, consulting engineering and contracting firms, research institutes, universities and colleges.

Reinforced concrete deep beams have many useful applications, particularly in tall buildings, foundations and offshore structures. However, their design is not covered adequately by national codes of practice: for example the current British Code BS 8110, explicitly states that ‘for design of deep beams, reference should be made to specialist literature’. The major codes and manuals that contain some discussion of deep beams include the American ACI Building Code, the draft Eurocode EC/2, the Canadian Code, the CIRIA Guide No. 2, and Reynolds and Steedman’s Reinforced Concrete Designer’s Handbook. Of these, the CIRIA Guide No. 2: Design of Deep Beams in Reinforced Concrete, published by the Construction Industry Research and Information Association in London, gives the most comprehensive recommendations.

The contents of the book have been chosen with the following main aims: (i) to review the coverage of the main design codes and the CIRIA Guide, and to explain the fundamental behaviour of deep beams; (ii) to provide information on design topics which are inadequately covered by the current codes and design manuals: deep beams with web openings, continuous deep beams, flanged deep beams, deep beams under top and bottom loadings and buckling and stability of slender deep beams; (iii) to give authoritative reviews of some powerful concepts and techniques for the design and analysis of deep beams such as the softened-truss model, the plastic method and the finite element method.

The contributing authors of this book are so eminent in the field of structural concrete that they stand on their own reputation and I feel privileged to have had the opportunity to work with them. I only wish to thank them for their high quality contributions and for the thoroughness with which their chapters were prepared.

I wish to thank Mr A.Stevens, Mr J.Blanchard and Mr E.Booth of Ove Arup and Partners for valuable discussions, and to thank Emeritus Professor R.H.Evans, C.B.E., of the University of Leeds for his guidance over the years. Finally, I wish to thank Mrs Diane Baty for the much valued secretarial support throughout the preparation of this volume.

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Discussion of “Unloading and Reloading Stress-Strain Model for Confined Concrete” by Junichi Sakai and Kazuhiko Kawashima

by Asad Esmaeily, A.M.ASCE; Steven D. Hart; and`Brandy Gaitan

In this paper, the authors, Sakai and Kawashima, propose a “comprehensive . . . model . . . that takes into account the effect of repeated unloading and reloading and partial loading.” This model was evaluated by conducting several tests on concrete cylinders confined by carbon-fiber-reinforced polymer
CFRP material.

Three standard 150 mm
300 mm
6 in.
12 in. concrete cylinders were cast and cured for 28 days in a moist curing room. Two strain gauges with a length of 50 mm
2 in. were placed
longitudinally on the central part of the specimens on opposite sides. The unconfined compressive strength of the specimens was 40.7 MPa
5.9 ksi at the time of testing. Confinement was provided by two layers of CFRP attached by an epoxy adhesive, which is different from conventional confinement by steel as used in the authors’ original research. Testing was conducted by using a closed loop servocontrolled material testing system with a maximum capacity of 667 kN
150 kips. Since the envelope for CFRP confined concrete does not have the descending branch after a peak point as observed for conventionally reinforced cases, all specimens were initially loaded to 614 kN
138 kips at a rate of 62 kN
14 kips per minute to achieve sufficient plastic strain for a reasonable evaluation of the unloading/reloading paths.

There was a creep-hold of 108 min at 133 kN
30 kip load level for one of the specimens. At the load level of 614 kN
138 k, the specimen with a creep-hold was subjected to three complete unloading and reloading cycles at a rate of 124 kN
28 kips per minute, the next specimen had similar cycles but at a rate of 186 kN
42 kips per minute, and the last one was loaded monotonically to 614 kN
150 kips to establish the envelope curve.

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Analytical Sensitivity of Plastic Rotations in Beam-Column Elements

by Michael H. Scott

Abstract: Analytical sensitivity equations for the plastic rotation of beam-column finite elements are derived for reliability and optimization algorithms in structural engineering and for the assessment of plastic rotation sensitivity to uncertain design parameters and modeling assumptions. The plastic rotation is defined by elastic unloading of element forces in a basic system, which makes the corresponding sensitivity computations applicable to most material nonlinear beam-column formulations available in the literature. The analytical response sensitivity is verified by finite differences then applied to a first-order reliability analysis of a steel subassemblage where the performance function places a limit on plastic rotation.

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