deflections using energy methods conservation of energy: 9.1 work

Deflections using Energy Methods

Conservation of energy:

9.1 Work and Energy

Work done by external forces on a material point or a structure is converted to internal work and internal stored energy.

Example: (Spring)

(Ref Chapter 9)

Example: (Trusses)

(for conservative systems)

(for linear spring)

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.

Note:

(for linear material behavior of truss elements)

WorkEnergyMethods

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

Internal strain-energy stored

Axial (for truss members):•

Bending•

For General 3D problems:


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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 force/moment.Note: This principle provides 1 scalar equation for the whole structure

Examples:

Example: A= 0.1 m2 ; E = 210 Gpa ; P = 1KN

A B

C

3 m 4 m

5 m

WorkEnergyMethods

Example

Find displacements at A and D.

WorkEnergyMethods

A B

C

3 m

4 m

5 mA B

C

3 m4 m

5 m

Thought experiment: (Principle of Virtual Work using Virtual forces)

External Work

Internal Energy

External (virtual) Work done Total Internal Energy stored (due to virtual stresses)

A B

C

3 m4 m

5 m

This can also be used for horizontal displacement:

WorkEnergyMethods

Principle of virtual work (using virtual forces)

External virtual work done by

an assumed external virtual force

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.•

Note:

Proof:

Virtual Forces and StressesReal displacements and strains

From the principle of (real) work and (real) energy

linear / non-linear structures and elastic / inelastic structures.Note: Principle of Virtual Work is valid for

SUM of Internal virtual work done by

corresponding internal virtual stresses(while undergoing real displacements)

WorkEnergyMethods

For general 3D problems:

Consider any structure under loads:

Virtual Stress

Real Strain

Virtual Force

Real Displacement

Aside: Principle of Virtual Work (using virtual displacements):

To find an unknown force / reaction for equilibrium.

Virtual Displacement

Real Force

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.

Note:

WorkEnergyMethods

Method of Virtual Work for Trusses

Applications: (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 more elements:•

To find a specific displacement, apply a virtual force P' at that point.1.

Find virtual internal forces N '2.

Find real internal forces Na.Find real displacements "d" of individual members.b.

For real displacements "d":3.

Apply Principle of virtual work to calculate ∆:4.

WE' = UI'

Assumed Virtual External Force (P') x

UNKNOWN Real External Displacement (∆)

Virtual Internal forces (N')x

Known internal displacements (d)

d = α ∆T L

d = ∆l

In all these cases, the internal displacements d must be very smallcompared to the dimensions of the members.

Note:

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Example:

Real

Virtual

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Example:

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Method of Virtual Work for Beams

To find a deflection apply virtual force P' at that point and in that direction.

To find a slope/rotation apply virtual moment M' at that point and in that direction.

Hibbeler

Example:

WorkEnergyMethods

Example

WorkEnergyMethods

Axial•

Shear•

(Virtual Internal Work due to real Axial, Shear, Torsion, Temperature loads)Applications:

PureShear

Example : Consider shear in the previous example

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Example

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Temperature change across depth of the Beam•

Applications (Continued)

Torsion•

Example: Center deflection:

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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.

Stationary (extremum) Potential EnergyEquilibriumExample

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

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) or to calculate deflections (2nd theorem).

Note: This theorem can also be derived from the Principle of Stationary (extremum) Potential Energy.

Example

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Proof

Theorem 2:

For an linear-elastic structure with constant temperature and rigid supports:

This theorem can also be derived from the Principle of Stationary (extremum) Complementary Energy.

Note:

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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:

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Castigliano's (2nd) Theorem for Trusses

Solution Steps: To calculate ∆i

WorkEnergyMethods

Example

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Castigliano's Theorem for Beams & Frames


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Contributions of shear, axial, torsion and temperature can also be accounted for similarly.

Example:

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WorkEnergyMethods

deflections using energy methods conservation of energy: 9.1 work

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