Energy Methods in Structural Analysis
Version 2 CE IIT, Kharagpur
Version 2 CE IIT, Kharagpur
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 useconditions 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.
Structural analysis and design is a very old art and is known tohuman 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 yearsold), 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. Theywere 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 alsoundergone 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 bepresenting 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 3dimensional in nature. In principle one could model them as 3-dimensional elastic structure and obtainsolutions (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 greatlyreduce 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...
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