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This book attempts to acquaint engineers who have mastered the essentials of structural mechanics with the mathematical foundation of their science, of structural mechanics of continua. The prerequisites are modest. A good working knowledge of calculus is sufficient. The intent is to develop a consistent and logical framework of theory which will provide a general understanding of how mathematics forms the basis of structural mechanics. Emphasis is placed on a systematic, unifying and rigorous treatment. Acknowledgements The author feels indebted to the engineers Prof. D. Gross, Prof. G. Mehlhorn and Prof. H. G. Schafer (TH Darmstadt) whose financial support allowed him to follow his inclinations and to study mathematics, to Prof. E. Klingbeil and Prof. W. Wendland (TH Darmstadt) for their unceasing effort to achieve the impossible, to teach an engineer mathematics, to the staff of the Department of Civil Engineering at the University of California, Irvine, for their generous hospitality in the academic year 1980-1981, to Prof. R. Szilard (Univ. of Dortmund) for the liberty he granted the author in his daily chores, to Mrs. Thompson (Univ. of Dortmund) and Prof. L. Kollar (Budapest/Univ. of Dortmund) for their help in the preparation of the final draft, to my young colleagues, Dipl.-Ing. S. Pickhardt, Dipl.-Ing. D. Ziesing and Dipl.-Ing. R. Zotemantel for many fruitful discussions, and to cando ing. P. Schopp and Frau Middeldorf for their help in the production of the manuscript. Dortmund, January 1985 Friedel Hartmann Contents Notations ........................................................... XII Introduction ........................................................ .

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
Unlike mathematicians who live happily in the realm of their intellectual creations and must never bring their symbols in contact with the rough outside world, the engineer identifies mathematical symbols with physical objects.
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1. Fundamentals

Abstract
We introduce in this chapter our notations and the principal equations of linear, first-order, structural mechanics.
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2. Work and Energy

Abstract
A spring is a very simple elastic element and, therefore, quite appropriate to acquaint us with the principles of structural mechanics.
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3. Continuous Beams, Trusses and Frames

Abstract
In the previous chapter we formulated the principles of virtual work and Betti’s principle for rather simple structures.
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4. Energy Principles

Abstract
We formulate in this chapter the energy principles of structural mechanics.
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5. Concentrated Forces

Abstract
In this chapter we shall formulate
  • the principle of virtual displacements
  • the principle of virtual forces
  • Bett’s Theorem
  • the principleeigenwork = int. energy
when the structural elements (bars, beams, Kirchhoff plates and elastic plates or bodies) are loaded with concentrated forces.
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6. Influence Functions

Abstract
The equations formulated in chapter 5 find many applications. Not because concentrated loads occur so often, they are rather fictitious, abstract quantities, but because concentrated loads are useful in the calculation of single displacements. This method is known as “the dummy-unit-load method”.
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7. The Operators A

Abstract
Up to now we were concerned with the differential equations which govern the displacement of the structural elements, as e.g. with the equation − Lu = p which governs the displacement of an elastic body. Now we focus on the systems of three equations which, originally, preceded the displacement equations. In the case of an elastic body this was the system
$$\matrix{\hfill {E(u) - E = 0} \cr \hfill {C[E] - S = 0} \cr \hfill { - div\,S = p} \cr}$$
(7.1)
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8. Shells

Abstract
In this chapter we will extend our approach to shells. As there are many, many different formulations for shells we had to decide for one particular model. We opted for Koiter’s model because the mathematical properties of this model are fully worked out. But as anyone who is familiar with shells will recognize all what is said in the following applies to different models (nearly) as well. The reader will, certainly, also realize that the mathematics of shells closely fits into the general picture.
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9. Second-Order Analysis

Abstract
If the equations of equilibrium are established using the geometry of the displaced structure then we speak of second-order analysis.
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10. Nonlinear Theory of Elasticity

Abstract
We extend in this chapter our formulations to the nonlinear theory of elasticity (geometric and physical nonlinearities) and the large displacement analysis of beams and plates (geometric nonlinearities)
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11. Finite Elements

Abstract
To model the behaviour of a structure the finite element method replaces the structure by a patch of finite elements which, compared with the real structure, can undergo only a limited number of states or modes, namely all those modes whose state variables are piecewise polynomials of maximum degree, say, k.
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Backmatter

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