Energy Methods in Structural Analysis: Introduction - Civil Engineering Notes - Civil Engineering (CE)

 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