Note: Before elaborating about the scope and the main directions of the
course, several conventions used in the text are described. Throughout the
entire body of this textbook, the first instance usage of important terminology
necessary to establish a base technical vocabulary is shown in boldface. Additionally, a selection of
the most important definitions are enclosed in a box and presented using the italic font.
1.1 What is a
Deformable Body?
The entire domain of Mechanics of
Materials is concerned with developing the mathematical methodology necessary to
completely define the behavior of the deformable
solid body.
The mathematical description of
the physical changes that take place, the modification of volume and shape,
when the threedimensional body is subjected to exterior actions, requires that
two Cartesian orthogonal coordinate
systems be employed: (1) a global coordinate system OXYZ, being considered as fixed in threedimensional space, and (2)
a local coordinate system oxyz,
rigidly attached to the threedimensional body. Consequently, a material point
pertinent to the threedimensional body can then be defined by two position
vectors, each one relative the two coordinate systems. A schematic
representation of the threedimensional deformable body subjected to various
actions and the Cartesian orthogonal coordinate systems described above are
shown in Figure 1.1. The exterior actions are shown as concentrated vectors
acting in some particular points (points 1 and 2), but in general the exterior
actions have a more complex nature. The symbols attached to some material
points (points 4, 5 and 6) of the threedimensional body are the schematic
representation of the supports or,
more appropriately, constraints.
Figure1.1 Schematic
Illustration of a ThreeDimensional Solid Deformable Body
Changes in the volume or shape
are generically referred to as deformations.
Consequently, any point of the threedimensional body moves after the
application of the exterior action from its initial position to a new position
in space. For example the material point located in the position marked with
the numeral _{} deforms into the new
position_{}. The deformation of the solid body can be small or large,
but it plays a vital role in the analysis of the behavior the threedimensional
solid under the given exterior actions.
1.2 Geometrical
Classification of the Deformable Body
In terms of geometry, any solid
body located in threedimensional space is represented by its volume and
exterior surface. For this study the exterior surface is considered continuous
without any holes or interruptions. The volume is characterized by three
dimensions (length_{}width _{} and height _{}) or by one dimension and two ratios. For example, if the
length_{}, typically the dimension of greatest magnitude, is retained,
than the ratios width to length _{}and height to length _{} fully define the body.
The following categories of bodies can be defined as:
(a)
Member
or Beam is a threedimensional solid
body which has the length _{} much larger than the
other two dimensions. To be more précised the major dimension _{}can be described by a curve called the member longitudinal axis. By intersecting the
threedimensional volume with a normal plane to the longitudinal axis a surface
is obtained. This surface is the beam crosssection.
Usually the beam is defined:
_{} (1.1)
The
crosssection represented by a single surface is called compact, while the surface which has a central hole is called tubular. A special type of tubular
crosssection is the thinwalls
crosssection characterized by very thin wall thicknesses.
The analytical description of the
beam longitudinal axis as a curve in space or plane curve can further
differentiate this category into spatial and plane curved members. The plane
members are linear (beam) or curved (arch) members.
Due to the
simplicity of manufacturing, the linear member, known in the engineering
practice under the generic name of the beam, is the most commonly employed
structural element. The behavior of the linear beam is the main subject of this
course. Examples of members are shown in Figure 1.2.
Figure
1.2 Examples of Member Types
(a)
RectangularLinear, (b) CircularLinear and (c) CircularCurved
(b) Plate or Slab is a
threedimensional solid body which has two of its three dimensions much larger
than the third one. The smallest dimension, usually called the thickness_{}, is replacing the height in the original definition. The
geometry of the plate median plane
or neutral plane, the plane
separating the thickness of the plate into two equal parts, can be represented
as a plane or a curved surface. The plate having a curved median plane is
called shell. By definition a plate
(slab) has the following ratios:
_{} and _{} (1.2)
where the dimensions _{}and_{}are the length and width of the median plane, respectively.
If the thickness _{} is very small
comparison with the other two dimensions, the plate is called membrane.
The plate is also a widely used
structural component, but its behavior is the subject of another course.
Examples of plates are depicted in Figure 1.3
Figure
1.3 Plate Types
(a)
Slab (Plane), (b) Cylindrical Shell and (c) Parabolic Shell
(c) Block is a threedimensional solid body which has all of its three
dimensions comparable.
Figure 1.4 Isolated
Foundation
In structural engineering this
type of solids are encountered with predilection in all kind of buildings and
machine foundations. An isolated foundation supporting a structural column is
illustrated in Figure 1.4.
Note: It is important to
emphasize that the geometrical characterization of the solid deformable body is
also reflected in the mathematical model used in the description of each body
type.
1.3 Exterior
Action Classification
The exterior action contributing
to the deformation of the threedimensional solid can of mechanical or a thermal
nature. In the engineering practice, these actions are labeled under the
generic name of loads. The
mechanical load is the direct result of the interaction of the
threedimensional deformable solid under investigation with other solids,
liquids or gases. For example the wind action is mechanical load induced by the
hydrodynamic pressure of the air movement. Similarly, the liquid contained in
the reservoir impinges on the walls and a hydrostatic pressure results.
In general a mechanical load is
considered as continuous function of two variables_{}, where _{}the position vector of the material is point where the
function is described and _{}is the time.
For the case of linear beam type
solid, the case of interest for this course, the spatial variable is described
only by the onedimensional position vector and the function can be written as_{}. The mechanical load is a vectorial function, which is characterized by direction and intensity.
The classification of the
mechanical loads is conducted base on two criteria, both related with the intensity:
(a) the time dependency and (b) the spatial variation.
Definition 1.2
If the intensity of the applied load changes with time the load is
called a dynamically applied load
or a dynamic load. As a result
the inertial forces, conforming to Newton’s
law, are induced and must be considered in the equations of equilibrium.
If the intensity of the applied load does not change in time the
load is called a statically applied
loads or a static load. In
this case there are no inertial forces to be included in the equations of
equilibrium.


The classification of the
mechanical and thermal action as dynamic or static has only theoretical merit,
because, any action is to some extent of dynamic nature. In fact, in some new
textbooks the static loads are called quasidynamic
loads in order to incorporate them
in the same theory as the dynamic loads.
The spatial variability of the
function _{}can be theoretically defined by any continuous mathematical
function. In engineering practice, these functions are typically limited for
simplicity to the following: constant function, linearly varying function and
parabolically varying function. For the rare case of more complicated spatial
variability the concept of piecewise–constant
(stepped) or piecewiselinear
functions may be employed.
Examples of mechanical loads
commonly used in the analysis of the linear plane beams are shown in Figure
1.5.
Figure 1.5 Line
Mechanical Loads
(a) Concentrated, (b)
Uniformly Distributed, (c) Linearly Distributed, and (d) Parabolically
Distributed
In the reality all mechanical
loads are applied on the surface of the solid body, which can be small or
large. The application surface degenerates into a line segment in the theoretical
case of the linear solid (beam). The situation when the area of the load
application surface or the length of load application segment is small relative
to the overall area or length of the threedimensional body suggests the
theoretical definition of the concentrated
load.
Obviously the concentrated load
can be static or dynamic in nature. This engineering simplification can be
easily accommodated when dealing with the beam case, but creates some
theoretical difficulties when the other two categories of deformable bodies
(plates and blocks) are investigated.
A special category of loads are
the body forces.
A special type of the body force
frequently used in structural engineering is the selfweight of the structural element. Because the gravitational
acceleration is considered constant, this load looses its dynamic nature and
can be represented as a static mechanical load with constant magnitude.
Note: Theoretically, it is very important to differentiate between
the pure mechanical loads and body forces, even from a practical point of view,
some types of body forces are treated as mechanical loads. The selfweight is a
typical example.
Thermal loads can be categorized
in a manner identical to that of the mechanical loads, as time dependent or
time independent. In engineering practice, the thermal loads are defined as expansioncontraction thermal loads and
gradient thermal loads.
For the case of the linear member
(beam) the expansioncontraction thermal load manifests when the temperature in
the beam varies only along the length of the beam (mathematically, this means
that the intensity of the load depends only on the variable_{}). In contrast, the thermal gradient load manifests only when
the temperature varies only trough the thickness of the beam.
In general the thermal loading is
time dependent (transient), but is
reasonably represented as steadystate
where the intensity is presumed to be constant.
All of these categories of loads
will be used and studied indepth in the lectures concerned with the
development of the linearly plane beam theory.
The deformation of the solid body
subjected to various types of loads modifies the internal equilibrium of the
body by forcing the threedimensional solid to move from its position at rest
to a new equilibrium position. The analysis of the local effect of these
deformations conducts to the theoretical concepts of the stress and strain distribution
in the deformable body, the two essential concepts of the Mechanics of
Materials.
This textbook is primarily
orientated towards the study of stress and strain distributions proper for the
linearly plane beam type solids. These concepts will be introduced in the
following lectures together with the constitutive
law, the functional relation between stress and strain.
1.5 Engineering
Aspects: Analysis, Verification, Optimization and Design
The problems solved by the
theoretical approaches which will be developed during the instruction time span
of this course can be organized in three categories: (a) analysis, (b) design
and (c) optimization.
The engineering activities
conducted to determine the deformations, and the stress and strain distribution
of a deformable solid body, when the loads and the geometry of the body are
known, is generically called analysis.
This activity will be continuously emphasized throughout the entire length of
the course.
The analysis activities precede
the verification activities. The
object of the verification activity is to check the maximum stress distribution
and deformations, calculated during the analysis, against some established
limiting values called allowable stress
and deformation, respectively. This
aspect is also emphasized in the following chapters.
If the loads acting on the
deformable body and the allowable related to the stress distribution and
deformation have been established, various types of solids can be analyzed and
found proper to the allowable limits. Therefore the described problem is
indeterminate and additional mathematical conditions must be imposed for the
complete solution to be obtained. These additional criteria are called optimization criteria. The minimum
weight criterion is an example. The optimization is a difficult mathematical
problem which is beyond the scope of this text book. However, for a great many
engineering applications, optimization can be achieved without mathematical
complexity by a simple trialanderror analysisverification iterative process,
drawing on the judgment, experience and creativity of the structural engineer.
The three previously described
engineering activities encompass the complete engineering effort from initial
concept to final design. In engineering practice, the loads are established
according to the functionality of the structural element, while the geometrical
characteristics are established through an iterative scheme involving analysis
and verification activities. The allowable limits (criteria) are established
and standardized for the material used to manufacture the structural element. Design is the generic name of the
activity for which a structural engineer is trained and tested.
1.6 Applications
of the Mechanics of Materials
Applications of the theory of
deformable solid bodies, especially beams and plates, can be found in our daily
life. In the past, the more sophisticated structures were designed by a careful
assemblage of simple structural elements which could be analyzed and verified.
Since 1960, with the development and increased usage of the digital computer, the Mechanics of
Materials methods evolved and were transcribed employing scientific programming
languages (FORTRAN, etc.). This way the modern computer programs or computer
codes were born. In today engineering practice a number of commercial computer codes, such as NASTRAN, ANSYS, ABAQUS and GTSTRUDL, are extensively employed.
These computer codes, sometimes referred to as ComputerAidedEngineering (CAE) codes, have the capability of conducting
the analysis as well as the verification activities for the investigated
structure.
Creation of threedimensional
models and drawings of the complex structures are conducted today utilizing the
ComputerAidedDesign (CAD) codes.
Many of these codes have the capability to idealize the constructed model and
submit this model directly or with minimal analyst intervention for analysis
and verification.
The analytical methods used in
the development of the commercial computer codes are beyond the scope of this
introductory course in Mechanics of Materials, but will be an integral part of
the educational process of the following years.