Virtual Work & Energy Methods External Energy-Work ... · PDF fileVirtual Work & Energy...

Virtual Work & Energy Methods

External Energy-Work Transformation


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)


Virtual work by a rigid body

(4.6)

Hence, for a rigid body

All particles in body move with same displacementWork done on A1 is opposite in sign to work done on A2Virtual work done on A1 and A2 is zeroSimilarly total virtual work of entire body is zero

All work comes from external work applied on body

(4.7)

Total work

External work Internal work


Virtual work by a deformable body

(4.6)

Hence, for a deformable body in equilibrium

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.

This is applicable even if the body is not linearly elastic

(4.8)

Total work

External work Internal work


Virtual work by internal force system (1)

(4.12)

Torsion

Load system is a combination of axial force, shear force, bending moment and torsion.

(4.22)

(4.17)

Axial force

Shear force

Bending moment (4.21)


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)


Virtual work by external force system

The virtual work done by complete external force system acting at a load position is

(4.24)

(4.17)

For a structure comprising a number load positions

displacement rotation twist angle

concentrated force bending moment torque distributed force


Sign of virtual work

(4.23)

External tensile load P applied on body AB in (a) produces internal force NSuppose 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 actionVirtual work done by internal force N is negative as displacement is in opposite direction to line of action

08-01-ElasticMachine1.wmv08-02-ElasticMachine4.wmv


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


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


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.


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.


Strain & Complimentary Energy (1)

(4.1)

Complimentary energy has no physical meaning; a convenient mathematical quantity

For member subjected to increasing load P, work is stored as strain energy

Complimentary energy obeys law of conservation of energy

(4.2)


Strain & Complimentary Energy (2)

(4.5)If function is represented by

Differentiating Eq. (4.1) and (4.2) wrt y and P respectively

Strain & complimentary energies are interchangeable in linearly elastic member

When n = 1


Prosthetic Foot

Current prosthetic feet are made from carbon fiber composites and are able to provide high amounts of elastic energy storage.

The first versions of prosthetic feet for amputees resulted in unusual gaits. It was discovered that they were not energy storing.


Strain Energy of Loaded Members

U P LAE

=2

2

UG

=τ2

2

U T LGJ

=2

2

U M LEI

=2

2

Straight bar under axial load PU = strain energy, L = length of bar, A = cross sectional area, E = modulus of elasticity.

Member under shear stress τG = shear modulus of elasticity.

Circular bar under torsion TG = shear modulus of elasticity, L = length of bar, J = polar moment of area

Beam under bending moment ME = modulus of elasticity, L = length of beam, I= moment of area

08-03-LiquidMetalElasticity.wmv


Stationary Value of Total Complementary Energy

The total complementary energy has a stationary value if the elastic body is in equilibrium under action of applied forces.

This represents increment in complementary energy; the first term for internal forces, the second term for external loads. Thus

For elastic system equilibrium supporting forces and real displacements, the principle of virtual work gives

(5.6)

(5.7)

(5.8)

This principle is useful for (i) deflection, (ii) solution of statically indeterminate structures


Application to deflection problems

From the principle of stationary value of total complementary energy

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)

(5.10)

(5.11)

Hence the deflection can be found using

extension force


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


Shape Memory AlloyShape 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.


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.


Aero-elastic Wing TestingFighter 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.

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