Deflections using Energy Methods

[Pages:26]Deflections using Energy Methods

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)

(Ref Chapter 9)

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)

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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:

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Internal strain-energy stored

? Axial (for truss members):

For General 3D problems: ? Bending

(for linear moment-curvature relationship)

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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:

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

C

3 m

4 m

A

5 m

B

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Example Find displacements at A and D.

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

C

C

4 m 3 m

3 m 4 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

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Principle of virtual work (using virtual forces)

External virtual work done by

an assumed external virtual force

SUM of Internal virtual work done by

corresponding internal virtual stresses

(while undergoing real displacements)

Virtual Forces and Stresses

Real displacements and strains

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.

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For general 3D problems: Consider any structure under loads:

Virtual Force

Virtual Stress

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

To find an unknown force / reaction for equilibrium.

Real Strain

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.

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