Energy Methods in Structural Analysis
Version 2 CE IIT, Kharagpur
Principle of Superposition, Strain Energy
Version 2 CE IIT, Kharagpur
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.
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 distorted. In this lesson two of the very important concepts i.e., principle of superposition and strain energy method will be introduced.
2.2 Principle of Superposition
The principle of superposition is a central concept in the analysis of structures. This is applicable when there exists a linear relationship between external forces and corresponding structural displacements. The principle of superposition may be stated as the deflection at a given point in a structure produced by several loads acting simultaneously on the structure can be found by superposing deflections at the same point produced by loads acting individually. This is Version 2 CE IIT, Kharagpur
illustrated with the help of a simple beam problem. Now consider a cantilever beam of length L and having constant flexural rigidity EI subjected to two externally applied forces P1 and P2 as shown in Fig. 2.1. From moment-area theorem we can evaluate deflection below C , which states that the tangential deviation of point c from the tangent at point A is equal to the first moment of the M area of the diagram between A and C about C . Hence, the deflection u below EI C due to loads P1 and P2 acting simultaneously is (by moment-area theorem),
u = A1 x1 + A2 x 2 + A3 x3
where u is the tangential deviation of point C with respect to a tangent at A . Since, in this case the tangent at A is horizontal, the tangential deviation of point Version 2 CE IIT, Kharagpur
C is nothing but the vertical deflection at C . x1 , x2 and x3 are the distances from point C to the centroids of respective areas respectively.
⎛ L L⎞ x2 = ⎜ + ⎟ ⎝2 4⎠
2L L + 32 2
P2 L2 A1 = 8 EI Hence,
P2 L2 A2 = 4 EI
( P1 L + P2 L) L 8EI
P2 L2 2 L P2 L2 ⎡ L L ⎤ (P1 L + P2 L) L ⎡ 2 L L ⎤ + + + ⎢3 2 + 2 ⎥ 8EI 3 2 4 EI ⎢ 2 4 ⎥ 8 EI ⎣ ⎦ ⎣ ⎦
After simplification one can write,
P2 L3 5 P1 L3 + 3EI 48EI
Now consider the forces being applied...
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