Section 3.1
3.1 Plane Problems
What follows is to be applicable to any two dimensional problem, so it is taken that
yz = xz = 0 , which is true of both plane stress and plane strain.
3.1.1
Governing Equations for Plane Problems
To recall, the equations g
Section 3.2
3.2 The Stress Function Method
An effective way of dealing with many two dimensional problems is to introduce a new
unknown, the Airy stress function , an idea brought to us by George Airy in 1862.
The stresses are written in terms of this new
Section 4.1
4.1 Cylindrical and Polar Coordinates
4.1.1
Geometrical Axisymmetry
A large number of practical engineering problems involve geometrical features which
have a natural axis of symmetry, such as the solid cylinder, shown in Fig. 4.1.1. The
axis
Section 4.2
4.2 Differential Equations in Polar Coordinates
Here, the two-dimensional Cartesian relations of Chapter 1 are re-cast in polar
coordinates.
4.2.1
Equilibrium equations in Polar Coordinates
One way of expressing the equations of equilibrium in
Section 4.3
4.3 Plane Axisymmetric Problems
In this section are considered plane axisymmetric problems. These are problems in
which both the geometry and loading are axisymmetric.
4.3.1
Plane Axisymmetric Problems
Some three dimensional (not necessarily p
Section 4.4
4.4 Rotating Discs
4.4.1
The Rotating Disc
Consider a thin disc rotating with constant angular velocity , Fig. 4.4.1. Material
particles are subjected to a centripetal acceleration a r = r 2 . The subscript r indicates
an acceleration in the r
Section 6.1
6.1 Plate Theory
6.1.1
Plates
A plate is a flat structural element for which the thickness is small compared with the
surface dimensions. The thickness is usually constant but may be variable and is
measured normal to the middle surface of the
Section 6.2
6.2 The Moment-Curvature Equations
6.2.1 From Beam Theory to Plate Theory
In the beam theory, based on the assumptions of plane sections remaining plane and that
one can neglect the transverse strain, the strain varies linearly through the thi
Section 6.3
6.3 Plates subjected to Pure Bending and Twisting
6.3.1
Pure Bending of an Elastic Plate
Consider a plate subjected to bending moments M x = M 1 and M y = M 2 , with no other
loading, as shown in Fig. 6.3.1.
M1
y
M2
M2
M1
x
Figure 6.3.1: A pla
Section 6.4
6.4 Equilibrium and Lateral Loading
In this section, lateral loads are considered and these lead to shearing forces V x , V y , in the
plate.
6.4.1
The Governing Differential Equation for Lateral Loads
In general, a plate will at any location
Section 6.5
6.5 Plate Problems in Rectangular Coordinates
In this section, a number of important plate problems will be examined using Cartesian
coordinates.
6.5.1
Uniform Pressure producing Bending in One Direction
Consider first the case of a plate whic
Section 6.6
6.6 Plate Problems in Polar Coordinates
6.6.1
Plate Equations in Polar Coordinates
To examine directly plate problems in polar coordinates, one can first transform the
Cartesian plate equations considered in the previous sections into ones in
Section 6.7
6.7 In-Plane Forces and Plate Buckling
In the previous sections, only bending and twisting moments and out-of-plane shear
forces were considered. In this section, in-plane forces are considered also. The in-plane
forces will give rise to in-pl
Section 6.8
6.8 Plate Vibrations
In this section, the problem of a vibrating circular plate will be considered. Vibrating
plates will be re-examined again in the next section, using a strain energy formulation.
6.8.1
Vibrations of a Clamped Circular Plate
Section 6.9
6.9 Strain Energy in Plates
6.9.1
Strain Energy due to Plate Bending and Torsion
Here, the elastic strain energy due to plate bending and twisting is considered.
Consider a plate element bending in the x direction, Fig. 6.9.1. The radius of cu
Section 6.10
6.10 Limitations of Classical Plate Theory
The validity of the classical plate theory depends on a number of factors:
1. the curvatures are small
2. the in-plane plate dimensions are large compared to the thickness
3. membrane strains can be
Section 7.1
7.1 Vectors, Tensors and the Index Notation
The equations governing three dimensional mechanics problems can be quite lengthy.
For this reason, it is essential to use a short-hand notation called the index notation1.
Consider first the notatio
Section 7.2
7.2 Analysis of Three Dimensional Stress and Strain
The concept of traction and stress was introduced and discussed in Part I, 3.1-3.5. For
the most part, the discussion was confined to two-dimensional states of stress. Here, the
fully three d
Section 1.1
1.1 The Equations of Motion
In Part I, balance of forces and moments acting on any component was enforced in order
to ensure that the component was in equilibrium. Here, allowance is made for stresses
which vary continuously throughout a mater
Section 1.2
1.2 The Strain-Displacement Relations
The strain was introduced in Part I: 3.6. Expressions which relate the displacements of
material particles to the strains for a continuously varying strain field are derived in what
follows.
1.2.1
The Stra
Section 1.3
1.3 Compatibility of Strain
As seen in the previous section, the displacements can be determined from the strains
through integration, to within a rigid body motion. In the two-dimensional case, there are
three strain-displacement relations bu
Section 2.1
2.1 One-dimensional Elastostatics
Consider a bar or rod made of linearly elastic material subjected to some load. Static
problems will be considered here, by which is meant it is not necessary to know how the
load was applied, or how the mater
Section 2.2
2.2 One-dimensional Elastodynamics
In rigid body dynamics, it is assumed that when a force is applied to one point of an
object, every other point in the object is set in motion simultaneously. On the other hand,
in static elasticity, it is as