Theory of Elastisity, Stability and Dynamics of Structures Common Problems

Trafford Publishing

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

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 are linear strains in the coordinate axes, respectively x, y z, and γ_xy, γ_yz, γ_zx are shear strains. They can be calculated by the relations:

ε_x = [partial derivative]u/[partial derivative]x, ε_y = [partial derivative]v/[partial derivative]y, ε_z = [partial derivative]w/[partial derivative]z

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.3)

The deformations at a point are often shown in tensor form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.4)

This tensor, and the stress tensor, which we are going to show below, both are symmetrical in regards to the leading diagonal. This means that for T_ε we have γ_yx = γ_xy, γ_zx = γ_xz γ_zy = γ_yz.

If we know the strains in three perpendicular directions, we are able to calculate the strain in arbitrary direction.

The strains in a body, or at a particular point of it, satisfy certain conditions, called equations of continuity of the strains or compatibility conditions. Thus, the equations of continuity of the strains appear as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.5)

We are going to show a proof of the first condition. Let us differentiate the forth equation in (1.3), i.e.:

γ_xy = [partial derivative]u/[partial derivative]y + [partial derivative]v/[partial derivative]x (1.6)

once in x and once in y. The result is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.7)

Taking the consideration that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.8)

we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.9)

1.2.3 Stress

A certain field of strains corresponds to a certain field of stresses. At point A(x, y, z) it has six components. Recorded as a vector, it appears as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.10)

where σ_x, σ_y, σ_z are normal stresses, with directions in coordinate axes, respectively x, y and z, τ_xy, τ_yz, τ_zx are shear stresses. Under the notation τ_xy we are going to consider the shear stresses in a plane with a normal, parallel to axis x, acting in direction of axis y. The stresses are shown in Figure 1.3.

The stresses at a point can be given in a tensor form as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.11)

Here τ_yx = τ_xy, τ_zx = τ_xz and τ_zy = τ_yz. Each column of the tensor (1.11) corresponds to a plane with a normal, parallel to one of the coordinate axes x, y or z, i.e. gives the stresses, acting upon it. Each row, itself, shows the stresses in one of the directions.

1.2.4 Stress at a point in arbitrary (random) direction.

If the stresses in three perpendicular planes at point A(x, y, z) are known, we can calculate the stress in a plane with arbitrary direction. Let us consider the tetrahedron, Figure 1.4. Its apex coincides with point A(x, y, z) and has altitude h. The normal to the base is parallel to the direction of which we are interested the stress. Let us express it with n, and the stress with [bar.p]ⁿ. The base area is denoted as F, and the directional cosines respectively as cos(n, x), cos(n, y) and cos(n, z).

The equation of equilibrium in x direction is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.15)

where P_x are volume forces in x. If h -> 0, the forth term vanishes. Then, after dividing by F, we obtain:

pⁿ_x = σ_x cos(n,x)+τ_yx cos(n,y)+τ_zxcos(n,z). (1.16)

If h -> 0, stresses towards the seat A(x, y, z), i.e. [bar.σ]_x -> σ_x, [bar.τ]_yx -> τ_yx and [bar.τ]_zx -> τ_zx.

Using the equations for equilibrium in y and z axes we obtain:

pⁿ_x = τ_xy cos(n,x)+σ_y cos(n,y)+τ_zycos(n,z). (1.17)

and

pⁿ_z = τ_xz cos(n,x)+τ_yz cos(n,y)+σ_zcos(n,z).. (1.18)

The stress pⁿ, acting at point A(x, y, z), we calculate from:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.19)

1.2.5 Equations of equilibrium

Let us consider the elementary parallelepiped with sides dx, dy and dz, Figure 1.5. All of the stresses and volume forces, acting upon it are shown. In the figure, the stresses with differential increase are written below the wavy line, for instance [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let us take only stresses, acting in coordinate direction x. Constituting the equation of equilibrium, we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.20)

Relation (1.20) we divide by dxdydz. The result is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.21)

By analogy, the equilibrium in other two coordinate directions leads to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.22)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.23)

We will give a proof of the reciprocity of the shear stresses, for instance τ_xz and τ_zx, assuming that they refer to the centre of differential volume.

If we sum now the moments, related to axis y, we will get that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.24)

or finally: τ_zx xzdxdydz = τdxdydz, i.e. τ_zx = τ_xz. Summation of moments towards axes x and z leads to τ_yz = τ_zy and τ_xy = τ_yx, respectively.

1.2.6 Principal stresses

Through point A(x, y, z) it is possible to draw three perpendicular planes, the stresses upon which correspond to the condition:

σ³ - I₁σ² - I₂σ - I₃ = 0, (1.25)

where:

I₁ = σ_x + σ_y + σ_z, (1.26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.28)

These planes are called principal planes, and the stresses principal stresses. We are going to denote these stresses as σ₁, σ₂ and σ₃. Typical for these planes is that, shear stresses are missing (zeros) upon them. and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.30)

where E is a modulus of elasticity, and ν - Poisson's ratio.

It is completely obvious, from expressions (1.29) and (1.30) that, the relation between strains and stresses is linear, and depends on material constants. We could show it in a matrix form as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.31)

1.3 Basic assumptions in Theory of elasticity

The basic assumptions, used in Theory of elasticity are:

• Assumption for continuity of structures. We have already mentioned that, this precondition allows the use of functions, defined as continuous in the field (the domain) of the researched body. These functions describe the displacements, strains and stresses.

• The bodies are perfectly elastic. It means that, after removal of forces, they completely restore their dimensions and initial shape, i.e. the shape before the stress effect. This property is connected with proportionality of strains and stresses, i.e. the relation between these quantities is linear. The modulus of elasticity E is a constant, and in this case we are talking about material linearity.

• The effects upon bodies cause small displacements. It means that, the strains and the stresses can be described in the initial, non-deformed configuration of the bodies. When this assumption is not valid the mathematical model gets quite complicated. Then we are talking about large displacements, or about geometrical non-linearity.

• Saint-Venant's principle is valid.

We are going to use the following additional assumptions:

• The bodies are isotropic and homogeneous. This can be expressed as: E = E_x = E_y = E_z.

• The initial strains or stresses are zeros, i.e. ε₀ = 0 and σ₀ = 0.

Chapter Two

2.1.1 Equilibrium of plane rectangular differential element

The differential equilibrium equations, describing the plane problems in Theory of elasticity are derived from a plane rectangular differential element with sides dx and dy, Figure 2.1. This element is cut from a plane disk 1 m in thickness. The components of volume forces and the components of the stresses act along each side. For convenience the sides are assumed to be parallel to the coordinate axes. Obviously, the generalized forces act also in directions parallel to the coordinate axes.

If we consecutively constitute the following three equilibrium conditions ΣX = 0, ΣY = 0 and ΣM_P = 0, where P coincides with the intersection point of the diagonals of the rectangular, we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.3)

We would like to remind that in the Mechanics, the above mentioned conditions are called First group conditions of plane body equilibrium.

(Continues...)

Excerpted from Theory of Elastisity, Stability and Dynamics of Structures Common Problemsby Konstantin Kazakov Copyright © 2012 by Konstantin Kazakov. Excerpted by permission of Trafford Publishing. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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