Transcription of Deflections using Energy Methods - Purdue University
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Deflections using Energy Methods - Purdue University
Deflections using Energy MethodsConservation of Work and EnergyWork done by external forces on a material point or a structure is converted to internal work and internal stored : (Spring)(Ref Chapter 9)Example: (Trusses)(for conservative systems)(for linearspring)There are multiple (infinitely many) "paths" that the external load can be applied along built up from 0 to P1, P2, P3(as long as all paths satisfy equilibrium at all points on each path).For conservative system of forces, total work done is path independent depends only on the initial and final : (for linear material behavior of truss elements) WorkEnergyMethods Page 1 Example: (Beams)(for linear moment0curvature relationship)Note: External work done by a single forceor a single momenton a linearsystem:Note: External Work done by a force or moment in the presence of another existing force or moment: WorkEnergyMethods Page 2 Internal strain0energy storedAxial(for truss members): Bending For General 3D problems:(for linear moment0curvature relationship) WorkEnergyMethods Page 3 Principle of (Real) work and (Real) Energy (for conservative systems)Real external work done = Real internal Energy stored=> only 1 unknown displacement/rotation can be solved for 1 applied : This principle provides 1 scalar equation for the whole structure Examples:Example: A= m2; E = 210 Gpa.
Castigliano's Theorems Theorem 1 : For an elastic structure (linear / non -linear) with constant temperature and rigid supports: Castigliano (an Italian railroad engineer) published 2 theorems of Work and Energy that allow us to either calculate unknown forces / reactions in indeterminate structures (1st theorem)
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