Indian Politics
Module 1
Energy Methods in Structural Analysis
Version 2 CE IIT, Kharagpur
Lesson 2
Principle of Superposition, Strain Energy
Version 2 CE IIT, Kharagpur
Instructional Objectives
After reading this lesson, the student will be able to 1. State and use principle of superposition. 2. Explain strain energy concept. 3. Differentiate between elastic and inelastic strain energy and state units of strain energy. 4. Derive an expression for strain energy stored in one-dimensional structure under axial load. 5. Derive an expression for elastic strain energy stored in a beam in bending. 6. Derive an expression for elastic strain energy stored in a beam in shear. 7. Derive an expression for elastic strain energy stored in a circular shaft under torsion.
2.1 Introduction
In the analysis of statically indeterminate structures, the knowledge of the displacements of a structure is necessary. Knowledge of displacements is also required in the design of members. Several methods are available for the calculation of displacements of structures. However, if displacements at only a few locations in structures are required then energy based methods are most suitable. If displacements are required to solve statically indeterminate structures, then only the relative values of EA, EI and GJ are required. If actual value of displacement is required as in the case of settlement of supports and temperature stress calculations, then it is necessary to know actual values of E and G . In general deflections are small compared with the dimensions of structure but for clarity the displacements are drawn to a much larger scale than the structure itself. Since, displacements are small, it is assumed not to cause gross displacements of the geometry of the structure so that equilibrium equation can be based on the original configuration of the structure. When non-linear behaviour of the structure is considered then such an assumption is not valid as the structure is appreciably
Energy Methods in Structural Analysis
Version 2 CE IIT, Kharagpur
Lesson 2
Principle of Superposition, Strain Energy
Version 2 CE IIT, Kharagpur
Instructional Objectives
After reading this lesson, the student will be able to 1. State and use principle of superposition. 2. Explain strain energy concept. 3. Differentiate between elastic and inelastic strain energy and state units of strain energy. 4. Derive an expression for strain energy stored in one-dimensional structure under axial load. 5. Derive an expression for elastic strain energy stored in a beam in bending. 6. Derive an expression for elastic strain energy stored in a beam in shear. 7. Derive an expression for elastic strain energy stored in a circular shaft under torsion.
2.1 Introduction
In the analysis of statically indeterminate structures, the knowledge of the displacements of a structure is necessary. Knowledge of displacements is also required in the design of members. Several methods are available for the calculation of displacements of structures. However, if displacements at only a few locations in structures are required then energy based methods are most suitable. If displacements are required to solve statically indeterminate structures, then only the relative values of EA, EI and GJ are required. If actual value of displacement is required as in the case of settlement of supports and temperature stress calculations, then it is necessary to know actual values of E and G . In general deflections are small compared with the dimensions of structure but for clarity the displacements are drawn to a much larger scale than the structure itself. Since, displacements are small, it is assumed not to cause gross displacements of the geometry of the structure so that equilibrium equation can be based on the original configuration of the structure. When non-linear behaviour of the structure is considered then such an assumption is not valid as the structure is appreciably