The content of the book is based on the lectures on the theory of elasticity, stability, and dynamics of structures. The importance of these disciplines in the preparation of young structural engineers for work in the practice cannot be overemphasized. The university training in such fundamental discipline must seek to build a strong foundation and to illustrate the application of the used methods to practical engineering problems. The solution of a structural engineering problem usually consists of three basic steps: the simplification to such a state of idealization that it can be expressed in allegorical or geometrical form, the solution of this mathematical form, and the interpretation of the results of the solution in terms of the engineering needs. By successive illustration of these three steps in the solution of each problem, the student must be led and encouraged to approach the solution of his own engineering problems in a similar way or in similar manner with a desired degree of accuracy in the final result.

Theory of Elastisity, Stability and Dynamics of Structures Common Problems

By Konstantin Kazakov

Trafford Publishing

Copyright © 2012 Konstantin Kazakov
All right reserved.
ISBN: 978-1-4669-6862-2

Contents

Preface.............................................................................................................................................................................................ixChapter 1. Basic relations in Theory of elasticity..................................................................................................................................................1Chapter 2. Plane problems in Theory of elasticity...................................................................................................................................................17Chapter 3. Method of trigonometric series...........................................................................................................................................................25Chapter 4. Theories for bending of thin and moderate thick elastic plates...........................................................................................................................37Chapter 5. The Method of double trigonometric series in bending of thin elastic plates (Navier method)..............................................................................................49Chapter 6. Introduction to the linear theory of thin elastic shells.................................................................................................................................59Chapter 7. Introduction to the Finite element method (FEM)..........................................................................................................................................71Chapter 8. Stability of elastic structures..........................................................................................................................................................85Chapter 9. Buckling of frame structures. Application of the force method and the displacement method in the buckling of frame structures............................................................93Chapter 10. Buckling of structures, modeled by one-dimensional finite elements. Geometrical stiffness matrix........................................................................................99Chapter 11. Dynamics of structures. Single-degree of freedom system.................................................................................................................................105Chapter 12. Multi-degree of freedom systems.........................................................................................................................................................121Chapter 13. Systems with large number of degrees of freedom.........................................................................................................................................135Chapter 14. Introduction to seismic mechanics. Spectrum Method. The Finite elements method in the seismic response of structures. Direct integration of the equations of motion.....................143References..........................................................................................................................................................................................157

Chapter One

Basic relations in Theory of elasticity

1.1 Theory of elasticity subject

The Theory of elasticity is engaged in research of the behavior of elastic bodies. A body is called elastic when it is capable of restoring its initial shape and dimensions, after the forces or reasons causing strains have been eliminated. A variety of materials could be assumed as elastic, up to a certain stress limits. That shows a presence of proportionality between strains and stresses. The relations between these quantities, i.e. strains and stresses, to the mentioned limits of stresses, are known as Hooke's law (Robert Hooke 1636-1703). In practice, the implemented proportionality is an idealization that leads to significant computational relieves.

1.2 Basic concepts and relations

In multitude of problems the atomic, discreet in nature, structure of the material can be ignored. The researched bodies are assumed to be continuous. In this way, the quantities are described by functions, defined as continuous in the body domain. This approach lies at the root of the Theory of elasticity.

Let us recall known from Strength of materials concepts, quantities, and their symbols (notations), in reference to one-dimensional problems.

Strains ε in one-dimensional are: the relation between the geometrical changes of dimension to the value of this dimension. They can be assumed as relative changes, and they are non-dimensional. They are related to stresses σ by material characteristics, called modulus of elasticity:

E = σ/ε. (i)

Another important material characteristic is the Poisson's ratio. It shows the relation between the strains, perpendicular to the direction of stresses and the parallel ones (parallel to the direction of stresses), i.e.:

ν = ε_y/ε_x = ε_z/ε_x (ii)

Besides the modulus of elasticity, we use modulus of shear strains:

G = τ/γ. (iii)

The relation between the two modulii is:

G = E/2(1 + ν). (iv)

Let us define three basic quantities, which are going to be used in Theory of elasticity, and to show the relations between them.

1.2.1 Displacement

When one body is subjected to certain effect, for instance, system of forces, then its points get displaced. Displacements we call: the changes of the position of body's points, due to some effect (force) applied upon the body. In the general case, the displacement can be expressed as a sum of two displacements—as an ideal rigid body displacement, and relative displacement of the points. Later on we are going to be interested only in relative displacements because they are the ones that caused strains and stresses.

Let us use one example, Figure 1.1. The body shown in the figure is supported in a way that displacements as an ideal rigid body are not possible, the body is restrained. Point A of the body Ω is shifted after loading, and its new position (location) is point [bar.A], Figure 1.1.

The displacement at point can be expressed through its components in three orthogonal axes, for instance the Cartesian coordinate system Oxyz. These components are the projections of the displacement to corresponding axes, Figure 1.2. Then the vector, containing these values, scalars, can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.1)

where u (x, y, z) is displacement in axis x, v(x, y, z) - displacement in axis y, and w (x, y, z) - displacement in axis z.

1.2.2 Strain

When the distance between two points of the body changes, due to the loading, we assume that the body gets deformed. Strains can be expressed if the displacements are known. Because of the body is treated as continuous, strains are continuous functions as well.

In the case of three-dimensional deformation, the vector of strains ε(x, y, z) at point A(x, y, z) of the body contains six components:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.2)

where ε_x, ε_y and ε_z...