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Deflections Using Energy Methods (Ref Chapter 9) 9.1 Work and Energy

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Deflections Using Energy Methods (Ref Chapter 9) 9.1 Work and Energy Deflections using Energy Methods (Ref Chapter 9) 9.1 Work and Energy Conservation of energy: (for conservative systems) Work done by external forces on a material point or a structure is converted to internal work and internal stored energy. Example : (Spring) (for linear spring) Example : (Trusses) Note : • There are multiple (infinitely many) "paths" that the external load can be applied along i.e. 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 i.e. depends only on the initial and final states. (for linear material behavior of truss elements) WorkEnergyMethods Page 1 Example : (Beams) (for linear moment -curvature relationship) Note : External work done by a single force or a single moment on a linear system: Note : External Work done by a force or moment in the presence of another existing force or moment: WorkEnergyMethods Page 2 Internal strain -energy stored • Axial (for truss members): For General 3D problems: • Bending (for linear moment -curvature relationship) WorkEnergyMethods Page 3 Principle of (Real) work and (Real) energy Real external work done = Real internal energy stored (for conservative systems) Note : This principle provides 1 scalar equation for the whole structure => only 1 unknown displacement/rotation can be solved for 1 applied force/moment. Examples : 2 Example : A= 0.1 m ; E = 210 Gpa ; P = 1KN C 3 m 4 m A 5 m B WorkEnergyMethods Page 4 Example Find displacements at A and D. WorkEnergyMethods Page 5 Thought experiment : (Principle of Virtual Work using Virtual forces) C C 3 m 4 m 4 m 3 m A 5 m B A 5 m B External Work Internal Energy External (virtual) Work done Total Internal Energy stored (due to virtual stresses) This can also be used for horizontal displacement: C 3 m 4 m A 5 m B WorkEnergyMethods Page 6 Principle of virtual work (using virtual forces) External virtual work done SUM of Internal virtual work done by by an assumed external virtual force corresponding internal virtual stresses (while undergoing real displacements) Real displacements and strains Virtual Forces and Stresses Note : • ONE scalar equation => Gives displacement / rotation at ONE point. • Need to repeat analysis for different points. • Can be used to calculate displacements at some key points. Proof : From the principle of (real) work and (real) energy Note : Principle of Virtual Work is valid for linear / non -linear structures and elastic / inelastic structures. WorkEnergyMethods Page 7 For general 3D problems : Consider any structure under loads: Virtual Force Virtual Stress Real Displacement Real Strain Aside : Principle of Virtual Work (using virtual displacements) : To find an unknown force / reaction for equilibrium. Virtual Displacement Real Force Note : The principle of virtual work using virtual displacements is good for finding forces in equilibrium. 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 displacements. This with be done later in the stiffness method and it also forms the basis of the Finite element method. WorkEnergyMethods Page 8 Method of Virtual Work for Trusses WE' = UI' Assumed Virtual External Force ( P') Virtual Internal forces ( N') x x UNKNOWN Real External Displacement ( ∆) Known internal displacements ( d) 1. To find a specific displacement, apply a virtual force P' at that point. 2. Find virtual internal forces N ' 3. For real displacements " d": a. Find real internal forces N b. Find real displacements " d" of individual members. 4. Apply Principle of virtual work to calculate ∆: Applications : (Types of internal real displacements of individual members) • External Loading: • Temperature changes in 1 or more elements: d = α ∆T L • Fabrication errors / changes in length of 1 or more elements: d = ∆l Note : In all these cases, the internal displacements d must be very small compared to the dimensions of the members. WorkEnergyMethods Page 9 Example : Real Virtual WorkEnergyMethods Page 10 Example : WorkEnergyMethods Page 11 Method of Virtual Work for Beams To find a deflection apply virtual force P' To find a slope/rotation apply virtual moment M' at that point and in that direction. at that point and in that direction. Hibbeler Example : WorkEnergyMethods Page 12 Example WorkEnergyMethods Page 13 Applications : (Virtual Internal Work due to real Axial , Shear , Torsion , Temperature loads) • Axial • Shear Pure Shear Example : Consider shear in the previous example WorkEnergyMethods Page 14 Example WorkEnergyMethods Page 15 WorkEnergyMethods Page 16 Applications (Continued) • Torsion • Temperature change across depth of the Beam Example : Center deflection: WorkEnergyMethods Page 17 Principle of Stationary (Extremum) Potential Energy Given 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 system. Example Equilibrium Stationary (extremum) Potential Energy Note : 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 Π in terms of displacement functions y(x) and then minimize it with respect to y(x). There are methods called Variational Methods that can do that. WorkEnergyMethods Page 18 Castigliano's Theorems 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) . Theorem 1 : For an elastic structure (linear / non -linear) with constant temperature and rigid supports: Note : This theorem can also be derived from the Principle of Stationary (extremum) Potential Energy. Example WorkEnergyMethods Page 19 Theorem 2 : For an linear -elastic structure with constant temperature and rigid supports: Proof Note: This theorem can also be derived from the Principle of Stationary (extremum) Complementary Energy. 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 minimum. This can also be used to calculate redundant reactions in indeterminate structures. Example : WorkEnergyMethods Page 21 Castigliano's (2nd) Theorem for Trusses Solution Steps : To calculate ∆i WorkEnergyMethods Page 22 Example WorkEnergyMethods Page 23 Castigliano's Theorem for Beams & Frames (for linear moment -curvature relationship) WorkEnergyMethods Page 24 Contributions of shear, axial, torsion and temperature can also be accounted for similarly. Example: WorkEnergyMethods Page 25 WorkEnergyMethods Page 26 .
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