Page 1
Module
1
Energy Methods in
Structural Analysis
Version 2 CE IIT, Kharagpur
Page 2
Module
1
Energy Methods in
Structural Analysis
Version 2 CE IIT, Kharagpur
Lesson
1
General Introduction
Version 2 CE IIT, Kharagpur
Page 3
Module
1
Energy Methods in
Structural Analysis
Version 2 CE IIT, Kharagpur
Lesson
1
General Introduction
Version 2 CE IIT, Kharagpur
Instructional Objectives
After reading this chapter the student will be able to
1. Differentiate between various structural forms such as beams, plane truss,
space truss, plane frame, space frame, arches, cables, plates and shells.
2. State and use conditions of static equilibrium.
3. Calculate the degree of static and kinematic indeterminacy of a given
structure such as beams, truss and frames.
4. Differentiate between stable and unstable structure.
5. Define flexibility and stiffness coefficients.
6. Write force-displacement relations for simple structure.
1.1 Introduction
Structural analysis and design is a very old art and is known to human beings
since early civilizations. The Pyramids constructed by Egyptians around 2000
B.C. stands today as the testimony to the skills of master builders of that
civilization. Many early civilizations produced great builders, skilled craftsmen
who constructed magnificent buildings such as the Parthenon at Athens (2500
years old), the great Stupa at Sanchi (2000 years old), Taj Mahal (350 years old),
Eiffel Tower (120 years old) and many more buildings around the world. These
monuments tell us about the great feats accomplished by these craftsmen in
analysis, design and construction of large structures. Today we see around us
countless houses, bridges, fly-overs, high-rise buildings and spacious shopping
malls. Planning, analysis and construction of these buildings is a science by
itself. The main purpose of any structure is to support the loads coming on it by
properly transferring them to the foundation. Even animals and trees could be
treated as structures. Indeed biomechanics is a branch of mechanics, which
concerns with the working of skeleton and muscular structures. In the early
periods houses were constructed along the riverbanks using the locally available
material. They were designed to withstand rain and moderate wind. Today
structures are designed to withstand earthquakes, tsunamis, cyclones and blast
loadings. Aircraft structures are designed for more complex aerodynamic
loadings. These have been made possible with the advances in structural
engineering and a revolution in electronic computation in the past 50 years. The
construction material industry has also undergone a revolution in the last four
decades resulting in new materials having more strength and stiffness than the
traditional construction material.
In this book we are mainly concerned with the analysis of framed structures
(beam, plane truss, space truss, plane frame, space frame and grid), arches,
cables and suspension bridges subjected to static loads only. The methods that
we would be presenting in this course for analysis of structure were developed
based on certain energy principles, which would be discussed in the first module.
Version 2 CE IIT, Kharagpur
Page 4
Module
1
Energy Methods in
Structural Analysis
Version 2 CE IIT, Kharagpur
Lesson
1
General Introduction
Version 2 CE IIT, Kharagpur
Instructional Objectives
After reading this chapter the student will be able to
1. Differentiate between various structural forms such as beams, plane truss,
space truss, plane frame, space frame, arches, cables, plates and shells.
2. State and use conditions of static equilibrium.
3. Calculate the degree of static and kinematic indeterminacy of a given
structure such as beams, truss and frames.
4. Differentiate between stable and unstable structure.
5. Define flexibility and stiffness coefficients.
6. Write force-displacement relations for simple structure.
1.1 Introduction
Structural analysis and design is a very old art and is known to human beings
since early civilizations. The Pyramids constructed by Egyptians around 2000
B.C. stands today as the testimony to the skills of master builders of that
civilization. Many early civilizations produced great builders, skilled craftsmen
who constructed magnificent buildings such as the Parthenon at Athens (2500
years old), the great Stupa at Sanchi (2000 years old), Taj Mahal (350 years old),
Eiffel Tower (120 years old) and many more buildings around the world. These
monuments tell us about the great feats accomplished by these craftsmen in
analysis, design and construction of large structures. Today we see around us
countless houses, bridges, fly-overs, high-rise buildings and spacious shopping
malls. Planning, analysis and construction of these buildings is a science by
itself. The main purpose of any structure is to support the loads coming on it by
properly transferring them to the foundation. Even animals and trees could be
treated as structures. Indeed biomechanics is a branch of mechanics, which
concerns with the working of skeleton and muscular structures. In the early
periods houses were constructed along the riverbanks using the locally available
material. They were designed to withstand rain and moderate wind. Today
structures are designed to withstand earthquakes, tsunamis, cyclones and blast
loadings. Aircraft structures are designed for more complex aerodynamic
loadings. These have been made possible with the advances in structural
engineering and a revolution in electronic computation in the past 50 years. The
construction material industry has also undergone a revolution in the last four
decades resulting in new materials having more strength and stiffness than the
traditional construction material.
In this book we are mainly concerned with the analysis of framed structures
(beam, plane truss, space truss, plane frame, space frame and grid), arches,
cables and suspension bridges subjected to static loads only. The methods that
we would be presenting in this course for analysis of structure were developed
based on certain energy principles, which would be discussed in the first module.
Version 2 CE IIT, Kharagpur
1.2 Classification of Structures
All structural forms used for load transfer from one point to another are 3-
dimensional in nature. In principle one could model them as 3-dimensional elastic
structure and obtain solutions (response of structures to loads) by solving the
associated partial differential equations. In due course of time, you will appreciate
the difficulty associated with the 3-dimensional analysis. Also, in many of the
structures, one or two dimensions are smaller than other dimensions. This
geometrical feature can be exploited from the analysis point of view. The
dimensional reduction will greatly reduce the complexity of associated governing
equations from 3 to 2 or even to one dimension. This is indeed at a cost. This
reduction is achieved by making certain assumptions (like Bernoulli-Euler’
kinematic assumption in the case of beam theory) based on its observed
behaviour under loads. Structures may be classified as 3-, 2- and 1-dimensional
(see Fig. 1.1(a) and (b)). This simplification will yield results of reasonable and
acceptable accuracy. Most commonly used structural forms for load transfer are:
beams, plane truss, space truss, plane frame, space frame, arches, cables,
plates and shells. Each one of these structural arrangement supports load in a
specific way.
Version 2 CE IIT, Kharagpur
Page 5
Module
1
Energy Methods in
Structural Analysis
Version 2 CE IIT, Kharagpur
Lesson
1
General Introduction
Version 2 CE IIT, Kharagpur
Instructional Objectives
After reading this chapter the student will be able to
1. Differentiate between various structural forms such as beams, plane truss,
space truss, plane frame, space frame, arches, cables, plates and shells.
2. State and use conditions of static equilibrium.
3. Calculate the degree of static and kinematic indeterminacy of a given
structure such as beams, truss and frames.
4. Differentiate between stable and unstable structure.
5. Define flexibility and stiffness coefficients.
6. Write force-displacement relations for simple structure.
1.1 Introduction
Structural analysis and design is a very old art and is known to human beings
since early civilizations. The Pyramids constructed by Egyptians around 2000
B.C. stands today as the testimony to the skills of master builders of that
civilization. Many early civilizations produced great builders, skilled craftsmen
who constructed magnificent buildings such as the Parthenon at Athens (2500
years old), the great Stupa at Sanchi (2000 years old), Taj Mahal (350 years old),
Eiffel Tower (120 years old) and many more buildings around the world. These
monuments tell us about the great feats accomplished by these craftsmen in
analysis, design and construction of large structures. Today we see around us
countless houses, bridges, fly-overs, high-rise buildings and spacious shopping
malls. Planning, analysis and construction of these buildings is a science by
itself. The main purpose of any structure is to support the loads coming on it by
properly transferring them to the foundation. Even animals and trees could be
treated as structures. Indeed biomechanics is a branch of mechanics, which
concerns with the working of skeleton and muscular structures. In the early
periods houses were constructed along the riverbanks using the locally available
material. They were designed to withstand rain and moderate wind. Today
structures are designed to withstand earthquakes, tsunamis, cyclones and blast
loadings. Aircraft structures are designed for more complex aerodynamic
loadings. These have been made possible with the advances in structural
engineering and a revolution in electronic computation in the past 50 years. The
construction material industry has also undergone a revolution in the last four
decades resulting in new materials having more strength and stiffness than the
traditional construction material.
In this book we are mainly concerned with the analysis of framed structures
(beam, plane truss, space truss, plane frame, space frame and grid), arches,
cables and suspension bridges subjected to static loads only. The methods that
we would be presenting in this course for analysis of structure were developed
based on certain energy principles, which would be discussed in the first module.
Version 2 CE IIT, Kharagpur
1.2 Classification of Structures
All structural forms used for load transfer from one point to another are 3-
dimensional in nature. In principle one could model them as 3-dimensional elastic
structure and obtain solutions (response of structures to loads) by solving the
associated partial differential equations. In due course of time, you will appreciate
the difficulty associated with the 3-dimensional analysis. Also, in many of the
structures, one or two dimensions are smaller than other dimensions. This
geometrical feature can be exploited from the analysis point of view. The
dimensional reduction will greatly reduce the complexity of associated governing
equations from 3 to 2 or even to one dimension. This is indeed at a cost. This
reduction is achieved by making certain assumptions (like Bernoulli-Euler’
kinematic assumption in the case of beam theory) based on its observed
behaviour under loads. Structures may be classified as 3-, 2- and 1-dimensional
(see Fig. 1.1(a) and (b)). This simplification will yield results of reasonable and
acceptable accuracy. Most commonly used structural forms for load transfer are:
beams, plane truss, space truss, plane frame, space frame, arches, cables,
plates and shells. Each one of these structural arrangement supports load in a
specific way.
Version 2 CE IIT, Kharagpur
Beams are the simplest structural elements that are used extensively to support
loads. They may be straight or curved ones. For example, the one shown in Fig.
1.2 (a) is hinged at the left support and is supported on roller at the right end.
Usually, the loads are assumed to act on the beam in a plane containing the axis
of symmetry of the cross section and the beam axis. The beams may be
supported on two or more supports as shown in Fig. 1.2(b). The beams may be
curved in plan as shown in Fig. 1.2(c). Beams carry loads by deflecting in the
Version 2 CE IIT, Kharagpur
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