VIRTUAL WORK PRINCIPLES
Prepared by: Dr. Oscar Victor M. Antonio, Jr.
Institute of Civil Engineering
University of the Philippines
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After this lesson, the students should be able to:
Understand and define the Principles of Virtual Work;
Formulate the virtual work equations (internal and external) using
Virtual Displacement and Virtual Force Principles.
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Outline
Definition of Virtual Work
Principle of Virtual Displacements
Principle of Virtual Forces
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Recall: Definition of Work
The work, dW, of a force F acting through a change in displacement
dD in the direction of that force is the product FdD.
Over a total displacement D1 the total work W is
ò
D1
0
FdD
Note that the force applied is a static (gradual) load.
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Definition of Virtual Work
A fundamental theory in the mechanics of bodies which allows us to
solve determinate and indeterminate structures and to calculate
their deflections.
It states that “For a body subjected to a system of forces in
equilibrium, the work due to a virtual displacement is zero.”
dW = 0
Conversely, it can be stated as “A body is in equilibrium under the
action of a system of forces if the virtual work is zero for every
independent virtual displacement. “
The word ‘virtual’ means ‘having the effect of, but not the actual
form of, what is specified.
Different ways can be used to impose virtual work, without worrying
about how it might be achieved in the physical world.
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Definition of Virtual Work
Strain Energy and Work Done
Strain energy is the amount of energy stored in a structural
member due to deformation caused by an external load
For example, consider a simple spring:
F
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Definition of Virtual Work
Strain Energy and Work Done
Strain energy is the amount of energy stored in a structural
member due to deformation caused by an external load
For example, consider a simple spring:
F
y
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Definition of Virtual Work
Strain Energy and Work Done
We can see that as it is loaded by a gradually increasing force, F, it
elongates. We can graph this as:
Load
F+dF
F
dW
W
y
y +dy
Displacement
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Definition of Virtual Work
Strain Energy and Work Done
An increase in the force of a small amount, dF results in a
small increase in deflection, dy
The work done during this movement is the average force
during the course of the movement, times the displacement
undergone
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Definition of Virtual Work
Strain Energy and Work Done
Thus, the increase in work associated with this movement is:
dW
As δ y → 0 , we get:
dW → F ⋅ dy
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Definition of Virtual Work
Strain Energy and Work Done
The total work done when a load is gradually applied from 0
up to a force F is the summation of all such small increases in
work, i.e.:
W
This represents the work done during the elongation of the spring
This work is stored in the spring and is called strain energy which is
denoted U
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Definition of Virtual Work
Strain Energy and Work Done
As we know that the work done is the area under this graph,
then the work done by the load in moving through the
displacement
– the External Work Done, We - is given by:
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Definition of Virtual Work
Strain Energy and Work Done
We can also calculate the strain energy, or Internal Work
Done, WI , by:
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Definition of Virtual Work
Strain Energy and Work Done
Also, since F = ky , we then have:
But this is the external work done, We . Hence we have:
The external work done by external forces moving through
external displacements is equal to the strain energy stored in
the material
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Definition of Virtual Work
Includes two distinct principles:
1. Principle of Virtual Displacements:
Virtual work is the work done by the actual forces acting on
the body moving through a virtual displacement.
This means we solve an equilibrium problem through
geometry
→ Important in the formulation of stiffness equations, which
imply deformable behavior
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Definition of Virtual Work
Includes two distinct principles:
2. Principle of Virtual Forces:
Virtual work is the work done by a virtual force acting on the
body moving through the actual displacements.
This means we solve a geometry problem through equilibrium
→ Basis of the direct formulation of element flexibility
relationships
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Principle Virtual Displacements
Work done
dF
change in work done
(McGuire et al., 2000)
since the second term is very small
compared to the first term, the change in
work can be approximated as
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Principle Virtual Displacements
dF
(McGuire et al., 2000)
The same form of relationship is obtained if it is imagined that the displacement
increment occurs without the action of a corresponding force increment
were
dD
→ virtual displacement
dW0 = F0 (dD) → virtual work
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Principle Virtual Displacements
A virtual displacement is a displacement that is only imagined to
occur.
Requirements:
• virtual displacements must be small enough such that the
force directions are maintained (unchanged)
• virtual displacements within a body must be geometrically
compatible with the original structure.
- geometrical constraints (i.e. supports) and member continuity
must be maintained
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Principle Virtual Displacements
Application of the principle to deformable bodies requires that
the total virtual work of a system be clearly defined. →
distinction between external and internal work
Example
FBD of joints and elements
(McGuire et al., 2000)
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Principle Virtual Displacements
The forces on a joint and the counterpart member force must
be equal and opposite in direction.
Equilibrium condition of element 12
or
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Principle Virtual Displacements
Equilibrium conditions of joints 1 and 2
Consider the effect of virtual displacements du1 and du2 of
joints 1 and 2. The total virtual work is
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Principle Virtual Displacements
The first term represents the virtual work of the applied loads
The second term represents the virtual work of the internal
forces acting on the joints
Thus
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Principle Virtual Displacements
From the equilibrium conditions of joints 1 and 2
therefore
δW = δWext - δWint = 0
For a deformable structure in equilibrium under the action of a
system of applied forces, the external work due to an admissible
virtual displaced state is equal to the internal work due to the
same virtual displacements.
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Principle Virtual Displacements
Axial behavior
(McGuire et al., 2000)
=
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Principle Virtual Displacements
Axial behavior
since δWext = δWint
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Principle Virtual Displacements
Torsional behavior
torsional strain
(McGuire et al., 2000)
real
virtual
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Principle Virtual Displacements
Torsional behavior
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Principle Virtual Displacements
Flexural behavior
flexural strain
real
(McGuire et al., 2000)
virtual
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Principle Virtual Displacements
Flexural behavior
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Principle Virtual Displacements
General formula for Internal Work
General formula for External Work
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Principle Virtual Forces
(McGuire et al., 2000)
The virtual force system produces external complementary virtual
work, dW*ext, and internal complementary virtual work, dW*int.
were
dF
→ virtual force
dW*ext = dFD0 → virtual work
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Principle Virtual Forces
A virtual force is a force imagined to be applied and is then
moved through the actual deformations of the body, thus causing
virtual work.
Requirement:
virtual forces must form an equilibrium set of their own
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Principle Virtual Forces
Equations of Equilibrium
Axial
(McGuire et al., 2000)
→
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Principle Virtual Forces
Equations of Equilibrium
Torsion
(McGuire et al., 2000)
→
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Principle Virtual Forces
Equations of Equilibrium
Flexure
(McGuire et al., 2000)
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Principle Virtual Forces
Equations of Equilibrium
Flexure
(McGuire et al., 2000)
→
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Principle Virtual Forces
The general expression for the internal complementary virtual
work dW*int can be established by referring to the stress strain
diagram
The internal complementary virtual work in a complete body
of volume (vol) is
→
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Principle Virtual Forces
The strains and displacements in a deformable system are compatible
and consistent with the constraints if and only if the external
complementary virtual work is equal to the internal complementary
virtual work for every system of virtual forces and stresses that satisfies
the conditions of equilibrium.
Axial member
Torsional member
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Principle Virtual Forces
Flexural member
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Reference
McGuire, W. , Gallagher, R.H., and Ziemian, R.D. (2000), Matrix Structural Analysis Second Edition.
John Wiley & Sons, Inc., New York, USA.
https://www.wallpapertip.com/s/university-of-the-philippines/, UP Oblation Wallpaper
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