Deflections using Energy Methods - Purdue University

1 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.

2 (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 ; P = 1 KNABC3 m4 m5 m WorkEnergyMethods Page 4 ExampleFind displacements at A and D. WorkEnergyMethods Page 5 ABC3 m4 m5 mABC3 m4 m5 mThought experiment: (Principle of Virtual Work using Virtual forces)External WorkInternal EnergyExternal (virtual) Work doneTotal Internal Energy stored (due to virtual stresses)ABC3 m4 m5 mThis can also be used for horizontal displacement: WorkEnergyMethods Page 6 Principle of virtual work ( using virtual forces)External virtual work done by an assumed external virtual forceONE scalar equation => Gives displacement / rotation at ONE point.

3 Need to repeat analysis for different points. Can be used to calculate displacements at some key points. Note: Proof:Virtual Forces and StressesReal displacements and strainsFrom the principle of (real) work and (real) energylinear / non0linear structures and elastic / inelastic : Principle of Virtual Work is valid for SUM of Internal virtual work done by corresponding internal virtual stresses(while undergoing real displacements) WorkEnergyMethods Page 7 For general 3D problems:Consider any structure under loads:Virtual StressReal StrainVirtual ForceReal DisplacementAside: Principle of Virtual Work ( using virtual displacements):To find an unknown force / reaction for DisplacementReal ForceThe principle of virtual work using virtual displacements is good for finding forces in equilibrium.

4 It does not give displacements directly. If we can express unknown displacements in terms of corresponding unknown forces, then this method can be used to find those with be done later in the stiffness method and it also forms the basis of the Finite element : WorkEnergyMethods Page 8 Method of Virtual Work for TrussesApplications: (Types of internal real displacements of individual members)External Loading: Temperature changes in 1 or more elements: Fabrication errors / changes in length of 1 or moreelements: To find a specific displacement, apply a virtual force P' at that virtual internal forces N ' real internal forces real displacements "d" of individual real displacements "d" Principle of virtual work to calculate '= UI'Assumed Virtual External Force (P') x UNKNOWN Real External Displacement (E)Virtual Internal forces (N')xKnown internal displacements (d)d= ETLd= ElIn all these cases, the internal displacements dmust be very smallcompared to the dimensions of the : WorkEnergyMethods Page 9 Example:RealVirtual WorkEnergyMethods Page 10 Example.

5 WorkEnergyMethods Page 11 Method of Virtual Work for BeamsTo find a deflection apply virtual force P' at that point and in that find a slope/rotation apply virtual moment M' at that point and in that : WorkEnergyMethods Page 12 Example WorkEnergyMethods Page 13 Axial Shear (Virtual Internal Work due to real Axial, Shear, Torsion, Temperatureloads)Applications: PureShearExample : Consider shear in the previous example WorkEnergyMethods Page 14 Example WorkEnergyMethods Page 15 WorkEnergyMethods Page 16 Temperaturechange across depth of the Beam Applications(Continued)Torsion Example: Center deflection : WorkEnergyMethods Page 17 Principle of Stationary (Extremum) Potential EnergyGiven a structure with loads and boundary conditions, and a set of possibly infinitely many (kinematically admissible) deformed configurations, then the actual deformed (equilibrium) configuration is one that extremizes (minimizes/maximizes) total potential Energy of the (extremum) Potential EnergyEquilibriumExampleNote: In order to use this principle to calculate Deflections for beams, we need to be able to express the total potential Energy of the system in terms of displacement functions y(x) and then minimize it with respect to y(x).

6 There are Methods called Variational Methodsthat can do that. WorkEnergyMethods Page 18 Castigliano's TheoremsTheorem 1: For an elastic structure (linear / non0linear) 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) or to calculate Deflections (2nd theorem).Note: This theorem can also be derived from the Principle of Stationary (extremum) Potential WorkEnergyMethods Page 19 ProofTheorem 2: For an linear0elastic structure with constant temperature and rigid supports:This theorem can also be derived from the Principle of Stationary (extremum) : WorkEnergyMethods Page 20 Corollary: (theorem of least work)The redundant reaction components of a statically indeterminate structure are such that they make the internal work (strain Energy ) a can also be used to calculate redundant reactions in indeterminate : WorkEnergyMethods Page 21 Castigliano's (2nd) Theorem for TrussesSolution Steps.

7 To calculate i WorkEnergyMethods Page 22 Example WorkEnergyMethods Page 23 Castigliano's Theorem for Beams & Frames(for linear moment0curvature relationship) WorkEnergyMethods Page 24 Contributions of shear, axial, torsion and temperature can also be accounted for : WorkEnergyMethods Page 25 WorkEnergyMethods Page 26