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)
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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
3m
A
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4m
5m
B
Example
Find displacements at A and D.
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Thought experiment: (Principle of Virtual Work using Virtual forces)
C
C
3m
4m
3m
A
5m
4m
B
A
5m
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
3m
4m
A
5m
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B
B
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)
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
Real Displacement
Virtual Stress
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.
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Method of Virtual Work for Trusses
WE' = UI'
Assumed Virtual External Force (P')
x
UNKNOWN Real External Displacement (∆)
Virtual Internal forces (N')
x
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.
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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:
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Example
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Applications:
(Virtual Internal Work due to real Axial, Shear, Torsion, Temperature loads)
• Axial
• Shear
Pure
Shear
Example : Consider shear in the previous example
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Example
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Applications (Continued)
• Torsion
• Temperature change across depth of the Beam
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.
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.
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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
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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.
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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
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Example
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Castigliano's Theorem for Beams & Frames
(for linear moment-curvature relationship)
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Contributions of shear, axial, torsion and temperature can also be accounted for similarly.
Example:
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