Virtual Work & Energy Methods. External Energy-Work Transformation

Transcription

1 External Energy-Work Transformation

2 Virtual Work Many structural problems are statically determinate (support reactions & internal forces can be found by simple statics) Other methods are required when the problem is statically indeterminate Two alternative solution schemes often used: Virtual work (able to deal with conditions that are not in the elastic range) Energy method (can provide solution of complex problems)

3 Virtual work by a rigid body Total work External work (4.6) Internal work All particles in body move with same displacement Work done on A1 is opposite in sign to work done on A2 Virtual work done on A1 and A2 is zero Similarly total virtual work of entire body is zero Hence, for a rigid body (4.7) All work comes from external work applied on body

4 Virtual work by a deformable body Total work External work (4.6) Internal work Points inside body do NOT move equal distance. If body is in equilibrium, action of external force on every particle is in equilibrium. Virtual work done by forces on particle is zero. Hence, for a deformable body in equilibrium (4.8) This is applicable even if the body is not linearly elastic

5 Virtual work by internal force system (1) Load system is a combination of axial force, shear force, bending moment and torsion. Axial force (4.12) Shear force (4.17) Bending moment (4.21) Torsion (4.22)

6 Virtual work by internal force system (2) The internal virtual work in response to external load distributed over body is (4.25) Virtual work done by internal moment of a hinge (in some cases it may be convenient to impose a virtual rotation where a moment acts)

7 Virtual work by external force system The virtual work done by complete external force system acting at a load position is concentrated force bending moment torque distributed force displacement rotation twist angle For a structure comprising a number load positions (4.24) (4.17)

8 Sign of virtual work External tensile load P applied on body AB in (a) produces internal force N Suppose virtual displacement is made by moving B to B Virtual work done by applied load P is positive as displacement is in the same direction to line of action Virtual work done by internal force N is negative as displacement is in opposite direction to line of action (4.23) ElasticMachine1.wmv ElasticMachine4.wmv

9 Virtual force systems Virtual work can be found by actual forces moving through imposed virtual displacements Principle applies for any set of forces in equilibrium and any set of displacements It is also possible to specify the forces as virtual and the displacements as actual Actual external and internal displacements can be related through virtual forces

10 Energy Methods Elasticity method involves employing (i) equations of equilibrium, and (ii) compatibility in stress/strain relationships Energy methods are useful for Rapid approximation where exact solutions do not exist Statically indeterminate problems Two major energy methods Total complimentary energy Total potential energy

11 Elastic Energy - Archery When an archer pulls the bow, the deformed structure stores elastic energy. The stored energy is released through the arrow which travels at a high speed towards its target.

12 Elastic Energy Parallel Bars The parallel bars is an Olympic gymnastic event. The ability of the gymnast to propel to heights is dependent on how well he can harness the stored elastic energy of the deformed bar.

13 Strain & Complimentary Energy (1) For member subjected to increasing load P, work is stored as strain energy (4.1) Complimentary energy has no physical meaning; a convenient mathematical quantity (4.2) Complimentary energy obeys law of conservation of energy

14 Strain & Complimentary Energy (2) Differentiating Eq. (4.1) and (4.2) wrt y and P respectively If function is represented by When n = 1 (4.5) Strain & complimentary energies are interchangeable in linearly elastic member

15 Prosthetic Foot The first versions of prosthetic feet for amputees resulted in unusual gaits. It was discovered that they were not energy storing. Current prosthetic feet are made from carbon fiber composites and are able to provide high amounts of elastic energy storage.

16 Strain Energy of Loaded Members Straight bar under axial load P U = strain energy, L = length of bar, A = cross sectional area, E = modulus of elasticity. U = 2 PL 2 AE Member under shear stress τ G = shear modulus of elasticity. U = τ2 2G Circular bar under torsion T G = shear modulus of elasticity, L = length of bar, J = polar moment of area U = TL 2 2GJ Beam under bending moment M E = modulus of elasticity, L = length of beam, I = moment of area U = 2 M L 2EI LiquidMetalElasticity.wmv

17 Stationary Value of Total Complementary Energy For elastic system equilibrium supporting forces and real displacements, the principle of virtual work gives (5.6) This represents increment in complementary energy; the first term for internal forces, the second term for external loads. Thus (5.7) (5.8) The total complementary energy has a stationary value if the elastic body is in equilibrium under action of applied forces. This principle is useful for (i) deflection, (ii) solution of statically indeterminate structures

18 Application to deflection problems Suppose the objective is to find deflection Δ2 of load P2 of k members supporting loads P1, P2 Pn. The total complimentary energy of framework is (5.9) extension force From the principle of stationary value of total complementary energy (5.10) Hence the deflection can be found using (5.11)

19 Application to statically indeterminate problems In statically determinate structures, internal forces are found uniquely by simple equilibrium equations In statically indeterminate problems, an infinite number of internal force or stress distributions may satisfy the conditions of equilibrium The true force system must satisfy the condition of either Compatibility of displacement Total complimentary energy having a stationary value

20 Shape Memory Alloy Shape memory alloys (SMAs) are metals that "remember" their original shapes. SMAs are useful for such things as actuators which are materials that "change shape, stiffness, position, natural frequency, and other mechanical characteristics in response to temperature or electromagnetic fields" Shape memory metals was used on the Sojouner rover that landed in Mars. Electrical heating of a NiTi wire was used to remove dust from the solar panels which affected power efficiency.

21 Aero-elasticity Aeroelasticity studies the interactions between the inertial, elastic, and aerodynamic forces that occur when an elastic body is exposed to a fluid flow. Static aero-elasticity is responsible for Divergence when the elastic twist of the wing suddenly becomes theoretically infinite, typically causing the wing to fail Control reversal control surfaces reverse their usual functionality (e.g. the rolling direction with a given aileron movement is reversed) Dynamic aero-elasticity is responsible for Flutter is the harmonic motion caused by positive feedback between the body's deflection and forcing exerted by fluid flow Due to the need to handle indeterminate conditions, energy methods are often used in calculations.

22 Aero-elastic Wing Testing Fighter aircraft that have aero-elastic wings have the ability to change shape in flight. This create the aerodynamics needed to create sudden turns. The ability of the wing to withstand the stresses are important. In testing a system of stands, jacks, and instrumentation will be used to determine the strain and deflection under various loading conditions.