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CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
1
Principle of Virtual Work A. Introduction The principle of virtual work is a very powerful mechanism for deriving matrix
structural analysis techniques. As we will see, there are two approaches to
application of the principle of virtual work. The first involves virtual work
done by real forces acting through virtual displacements and the second
involves the work done by virtual forces acting through real displacements.
The purpose of this chapter in the notes is to introduce the fundamental
concepts related to the principle of virtual work and then implement the
principle of virtual displacements to solve some common structural analysis
problems.
B. Strain Energy Strain energy is a fundamental concept needed to formulate “energy methods”
of analysis. The energy methods are then a very powerful mechanism for
deriving matrix structural analysis procedures. Consider the prismatic bar of
length L shown below.
As the applied loading F is increased (gradually) from 0 to F, the bar will
stretch. The load deformation response of the rod is illustrated in the figure
below (McGuire, et al 2000).
L
F F
Δ
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
2
The work done by the force as it is gradually moves through the deformations
can be expressed as,
1
0
W F dΔ
= ⋅ Δ∫ (1)
The work is nothing more than the area under the load-deformation response
curve. The strain energy is defined as the energy absorbed by the bar
during the loading process. A restatement of equation (1) with this new
definition is,
1
0
U strain energy W FdΔ
≡ = = Δ∫ (2)
As one requires the bar to assume a deformed configuration, the bar will store
energy. If the bar is allowed to return to its original length, the energy may or
may not be released. If the material _____________, all the energy
_________________________________________. We should all recall the
classical paper clip bending example. As we yield the paperclip, some of the
energy that we impart from the bending is lost in the form of heat generation.
Permanent (or set) deformation in the bar depends upon whether or not the
material’s proportional limit has been exceeded. The figure below illustrates
the concept of recoverable and irrecoverable strain energy.
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
3
The irrecoverable strain energy is sometimes called inelastic strain energy.
Sometimes the inelastic energy is relied upon to dissipate input energy. An
example of this is the hysteretic energy dissipation counted on for structures
subjected to seismic loading.
If the axial loading on the rod is applied at a magnitude that imparts a state
of stress below the material’s proportional limit, the material’s response
remains elastic and all the energy stored is __________________________.
The load deformation response for this case is shown below,
The strain energy for this case is given by,
(3)
Irrecoverable strain energy (dissipated heat - inelastic) Recoverable strain energy (elastic)
F
F
F 1
2
F 1
F 1
Starting Position
1ΔsetΔ
setΔ
1Δ
Δ
Recoverable strain (elastic
F
F
F 1
2
F2 F 2
Starting 2Δ
2ΔΔ
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
4
This loading application results in the axial rod behaving as an _________
______________. The rod stores energy when stretched (or compressed) and
releases _______ the energy when the force is removed. This is our first
introduction to the _____________________________ of analysis. The
concept of stored energy and spring behavior will become very important in
deriving stiffness matrices for elements in this course.
Let’s now take a look at how the strain energy may be computed using
some of our knowledge from mechanics of materials. Using equation (3), the
assumption of linear elastic behavior, and our knowledge of the expression for
axial deformation of a rod, the strain energy of the axial deformation can be
written as,
21
2 2FL F LU W FAE AE
⎡ ⎤= = =⎢ ⎥⎣ ⎦ (4)
A little algebraic manipulation allows us to rewrite equation (4) as,
2 2 2
2 2 2 2F L AE AEU
L LA E
⎡ ⎤ Δ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦ (5)
If we continue our algebraic exercise, we can expose the nature of the axial rod
to behave as an elastic spring. Using equation (5) we have,
( )2
21 1 12 2 2 2
AE AEU k FL LΔ
= = ⋅ ⋅ Δ = Δ Δ = Δ (6)
C. Strain Energy Density If we would like to examine strain energy at a more fundamental level, we can
look at the strain energy that exists in a _________________________ of
material from which the axial rod is composed. We will define the strain
energy density as ________________________________________________.
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
5
The strain energy density allows us to focus our strain energy computations at
the material level rather than the element (e.g. rod, beam, etc…) level. Let’s
consider an axial rod composed of material with a stress-strain diagram below.
The strain energy density acts on a unit volume of material. Therefore, we
must sum up the strain energy density over the entire volume of the rod. Using
calculus, the summation is done via integration,
1 10
U udV d dVε
σ ε⎡ ⎤⎢ ⎥= =⎢ ⎥⎣ ⎦
∫ ∫ ∫ (7)
Assuming linear elastic material behavior during the response, the integration
inside the [ ] gives,
21 1 1 1
0 0
12
d E d Eε ε
σ ε ε ε ε= =∫ ∫ (8)
Plugging equation (8) into (7) gives,
212U E dVε⎡ ⎤= ⎣ ⎦∫ (9)
If we make the assumption that the strain energy density is constant throughout
the element volume (valid since loading results in constant strain and the rod is
prismatic), the volume integration can be replaced with integration over length
and area,
σ
F F
L
A,
1σ
σ
ε1ε ε
1dσ
1dε
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
6
2 212 2
L A
AEU E dA dL Lε ε⎡ ⎤= =⎣ ⎦∫ ∫
We can use the definition of strain from mechanics of materials giving,
2 2
212 2 2
AE AEU L kL LΔ Δ⎛ ⎞= = = Δ⎜ ⎟
⎝ ⎠
Therefore, we started with the material level and calculus and got the same
strain energy as that obtained using the entire rod from the beginning. Much of
our analysis will begin from the strain energy density level and will evolve to
the stiffness matrices for our structural analysis.
D. Complementary Energy A second concept that is very useful for our analysis purposes is the notion of
complementary energy. The complementary energy will help us to differentiate
between the principle of virtual forces and the principle of virtual displacements
that will define our analysis approaches. When the flexibility method is
employed, the complementary energy density is defined. The c.e.d. is shown on
the stress-strain diagram below.
We will assume that the axial rod is still our “target application”.
σ
1σ
ε1ε1dε
1dσ
ε
σ
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
7
As shown above, the complementary energy density is the area above the
stress-strain diagram. This energy (sometimes called the stress energy) is
obtained by summing up the complementary energy density over the volume of
the axial rod.
* *1 1
0
U u dV d dVσ
ε σ⎡ ⎤
= = ⎢ ⎥⎢ ⎥⎣ ⎦
∫ ∫ ∫ (10)
Assuming linear elastic material behavior and constant complementary energy
density over the axial rod volume, equation (10) becomes,
( ) ( )22 2
* 12 2 2
F F LU AL AL UE E A EA
σ⎡ ⎤ ⎡ ⎤= ⋅ = = =⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ (4)
Thus, if we have linear elastic material behavior, the strain energy is equal to
the complementary strain energy.
The complementary work can be defined in a similar manner as the work.
Let’s again consider our axial rod problem and load deformation response.
The complementary work can then be written as,
(11) Carrying out the integration results in,
Complementary Work(elastic)
F
F
2
F2 F 2
Starting Position
Δ2Δ
2Δ
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
8
(12) E. Principle of Virtual Work Attaining correct solutions through structural analysis requires that three basic
conditions be satisfied:
1. The structure and all its components must be in __________________.
2. There must be _____________________________________________
throughout the structure (called compatibility).
3. The stress strain relationships for the material must be satisfied (i.e. the
constitutive relationships for the materials must be satisfied).
In this course, we will develop solutions that implicitly contain the basic
conditions, but do no explicitly apply them. The energy method we will use to
develop these solutions is the Principle of Virtual Work.
The method of virtual work can take two distinct avenues that depend upon
the choice for the virtual quantities. These choices are described below.
Virtual Displacements:
Using virtual displacements leads to the very common matrix analysis
technique called the ______________________________________. The
formulation of this method of analysis is linked to work and strain energy.
Virtual Forces:
Application of virtual forces is the ____________ of application of
principle displacements (recall strain energy and complementary strain
energy). Choosing virtual forces in the formulation results in
_________________________________ that can be used to solve
structural analysis problems.
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
9
Therefore, when we talk applying the principle of virtual work, we will talk
about application of the principles of virtual forces and virtual displacements.
The principle of virtual displacements will lead us to the stiffness method of
analysis and the principle of virtual forces will lead us to the flexibility method.
F. Principle of Virtual Displacements As a starting point to our analysis, let’s begin by considering a general force, F,
that is gradually applied to an axial rod. This will remain consistent with all our
discussion prior to this point in the lecture notes. This applied force will cause
a deformation Δ in the direction of the force. Assuming linear elastic behavior,
the load deformation response can be illustrated as (McGuire, et al 2000),
We should all appreciate that the work done by the force up to the level
( )0 0,F Δ is the area beneath the load deformation diagram. This can be
expressed in integral form as it is above.
Let’s now consider a very small movement dΔ from the position 0Δ . It
should be noted that at 0Δ the structure is in equilibrium. This small
deformation will now be allowed to take place. The work done during this
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
10
small deformation can be written using the load deformation response and an
understanding that work is the area under the load deformation response.
Therefore, an incremental work quantity can be defined as,
(13) If we recognize that multiplying two small quantities together results in a
smaller quantity, equation (13) reduces to,
(14) We already have established that the structure was in equilibrium in the
deformed configuration 0Δ . Therefore, the small deformation will not be
considered as real and we will call this small deformation _________________.
Therefore, we can now say that the real force 0F will do __________________
through a small virtual displacement. Mathematically, this can be written as,
( )0W Fδ δ= Δ (15)
Equation (15) is the basis for the principle of virtual work formulated using
virtual displacements.
G. Virtual Forces Applying the principle of virtual work using virtual forces is a second approach
that can be used in deriving methods for matrix structural analysis. The virtual
force formulation is the dual of the virtual displacement formulation. Consider
a small virtual force as shown below (McGuire, et al 2000).
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
11
As indicated above, we have a fixed real displacement 0Δ that has a virtual
force Fδ applied. Taking advantage of the knowledge that the small triangular
area above the load deformation response is negligible, we can write the
_________________________________________ as,
( )*0W Fδ δ= Δ (16)
At this point, it is often constructive to compare the two approaches:
Virtual Displacements:
Virtual Forces:
Hopefully one can now appreciate the dual nature of the two approaches.
H. Virtual Work – Rigid Bodies and Particles As a quick introduction to application of the virtual work principle, let’s
consider a particle with forces applied as shown below.
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
12
The particle P is going to undergo a virtual horizontal displacement to the new
configuration P′ . If we define the direction cosines for the force i to be iλ ,
equilibrium of the particle in the X-direction requires that,
3 3
1 1
cos 0i iX i iXi i
F Fλ θ= =
= =∑ ∑ (17)
Recall that the direction cosine for X-direction filters out the X-component of a
vector.
To get back to our virtual work discussion, let’s now assume that the
particle undergoes a virtual displacement in the X-direction. The virtual work
done by the real forces as this virtual displacement occurs can be easily written
using our direction cosines. Therefore, careful accounting for all sources of
virtual work gives,
The above expression can be written in a condensed form as,
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
13
( )3
1i iX X
i
W Fδ λ δ=
⎡ ⎤= Δ⎢ ⎥
⎣ ⎦∑ (18)
We must realize that if we are to solve equilibrium problems using the
principle of virtual work, we have to satisfy equilibrium. Relating equation
(18) to (17) gives us the indication that for equilibrium to be satisfied, the terms
in the [ ] needs to be zero. The virtual displacements can’t be zero. Therefore,
if we have equilibrium the following must hold,
(19) Equation (19) is a statement of the ___________________________________.
The principle can be stated in words as follows:
For a particle in equilibrium subjected to a system of
forces, the work done by the forces acting through a
virtual displacement is ZERO.
It should be noted that we have only considered the X-direction in our
discussion to this point.
If we have a general problem, we should expect that equilibrium in ALL
pertinent directions must be adhered to. Therefore, we need to apply or write
multiple virtual work equations for more general problems. We will discuss
this throughout the course. This statement should give us some indication how
we might solve rigid body or particle problems using the principle of virtual
work. An outline of the procedure can be given.
1. Determine all virtual displacements possible in the system considered.
Pay particular attention to those that are independent and those that can
be written in terms of others.
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
14
2. Compute the virtual work done by all forces as each independent virtual
displacement takes place.
3. Apply the principle of virtual work and group terms to create the
necessary equations of equilibrium for the system.
Our quick statement of the principle of virtual work above can actually be
made stronger and therefore, greater applicability to structural analysis
problems can be attained. Therefore, the principle of virtual work can be
written as:
A particle (or rigid body) is in equilibrium under
the action of a system of forces if the virtual work
is ZERO for every independent virtual
displacement.
We will now apply the principle of virtual work to solve equilibrium problems.
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
15
Example 1 – Application of Principle of Virtual Work
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
16
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
17
Example 2 – Application of the P.V.W. (McGuire, et al 2000)
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
18
I. Virtual Work – Deformable Bodies While assuming a body is rigid is perfectly adequate for determining reactions,
it is not sufficient to compute reactions for statically indeterminate structures,
nor is it sufficient to compute the deformations along members or for structural
systems. Therefore, we must extend our discussion to deformable bodies, or
elements that bend, stretch, or twist. Consider the axial rod below with two
kinematic degrees of freedom at points 1 and 2.
A, E A, E x, u
P1P2
1 2 3
LL
Instead of requiring that the rod is rigid, let’s now assume that the rod can
stretch or compress.
We will now take apart the axial rod into several pieces. Two of the pieces
will be the nodes at 1 and 2 (where each node is assumed to have one kinematic
degree of freedom – horizontal displacement) and two will be the members 12
and 23. This disassembly is shown below.
1 2
31 2 2
The subscripts have meaning. The first number is the member number and the
second number is the number corresponding to the kinematic degree of freedom
assumed. The reaction 23F has been temporarily omitted. Let’s let each
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
19
kinematic degree of freedom experience a virtual displacement in what we
will define to be the positive direction (to the right). The virtual work done
during this deformation is,
[ ] [ ]1 11 1 2 12 22 2W P F u P F F uδ δ δ′ ′ ′= + + + + (20)
It should be noted that node 3 is restrained and therefore, can undergo no virtual
deformation. Furthermore, all work done is positive for the nodal virtual
displacement.
Recalling our previous use of the principle of virtual work, we will assume
that the nodes (particles for our present use) are in equilibrium. Therefore, 1uδ
and 2uδ in equation (20) can be anything admissible (i.e. they are arbitrary). If
we have equilibrium, then the virtual work done during these virtual
deformations must be zero. Therefore, equation (20) implies two nodal
equilibrium equations.
We can further expand upon the notion of virtual work by separating the
work done by the internal member forces acting through the virtual
displacement and the work done by the external forces applied to the structure.
The forces at the nodes are related to the forces applied at the ends of the
members through Newton’s 3rd Law. Therefore, we can write the following,
11 11 12 12 22 22F F F F F F′ ′ ′− = − = − = (21)
Substituting equations (21) into (20) gives,
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
20
Thus, equilibrium of the deformable body can be written as,
0ext intW W Wδ δ δ= − = (22)
Equation (22) allows us to state the converse of the principle of virtual work.
A linear elastic structure subjected to a system of forces is in
equilibrium if the virtual work done by the externally applied
forces is equal to the virtual work done by the internal
member forces for any compatible virtual displacement state.
The concept of compatible deformations is very important and we will discuss
this further.
General application of the principle of virtual work will require that we
deal with stress-strain diagrams and internal energy. Thus, we will look at a
slightly more general derivation of the principle in the following. Consider an
axial rod subjected to tension force, the associated free body diagrams, and a
horizontal kinematic degree of freedom at node 2.
1
2
2
2
1
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
21
Each component in the figure above is in equilibrium: (a) the entire structure;
(b) node 2; and (c) the member 12. This is our first assumption: we have a
linear elastic structure in equilibrium.
Node 2, which has the only kinematic degree of freedom now undergoes a
virtual displacement from the current equilibrium state. The virtual work done
during this deformation is,
Since we have equilibrium of the node (i.e. it is a particle),
From mechanics of materials, we know the following,
12 1F Aσ′ =
We can go on to rewrite the virtual work as,
2 2 1 2W P u A uδ δ σ δ= −
We should have some appreciation that the virtual deformation of node 2 will
cause virtual strain in the elastic rod. Therefore, we can write the following,
2
1
2 2 1 1
X
X X
uL
W P u AL
δδε
δ δ σ δε
=
= −
The volume of the axial rod is AL and therefore, we can rewrite the virtual
work as,
2 2 1 1X XW P u dVδ δ δε σ= − ∫ (23)
Equation (23) implies a slightly more general form of the virtual work.
However, it should be noted that it remains a summation of internal and
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
22
external virtual work such that equilibrium of the deformable body is defined
as,
2 2 1 1
0X X
ext int
W P u dV
W W
δ δ δε σ
δ δ
= −
= − =∫ (24)
Equations (23) and (24) include our old friend: the virtual strain energy density
1 1X Xδε σ .
At long last, we can formally state the Principle of Virtual Work as we
will use it in our course.
If a linear elastic structure is in equilibrium under a
system of forces, then for ANY COMPATIBLE virtual
displacement state away from this equilibrium state, the
external virtual work is equal to the virtual strain energy
(i.e. the internal virtual work).
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
23
Example 3 – Virtual Work for a Deformable Body
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
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CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
25
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
26
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
27
J. Commentary on Displacements The preceding discussion illustrated that the computation of the external virtual
work and internal virtual strain energy involved arbitrary virtual displacements
or strains. One should be a bit unsettled by the term: arbitrary. What exactly is
arbitrary? Are there any conditions that will cause problems? In general, we
need to be a little bit more specific than just saying arbitrary virtual
displacements.
Our previous discussion also made it apparent that equilibrium could be
satisfied for all virtual displacements (we can enforce the “ = 0” condition and
therefore make the virtual displacements “work”). As it turns out, we can be
fairly liberal in our choices for virtual and real displacements as long as we
satisfy some conditions with each choice. Virtual displacements give rise to
virtual strains (as we have seen),
uδ δε⇒
When we choose a form for the virtual displacement, it must be
_______________________ __________________________________. That
is to say, the choice must do the following.
• It must satisfy the displacement boundary conditions.
• There can not be any discontinuities (rips or tears) in the material or
member.
The exact solution gives rise to the actual displacements, while an approximate
solution gives rise to real displacements. As a result, the principle of virtual
work will give rise to very good (albeit approximate) solutions for our structural
analysis problems. The nice thing is, we can make displacement assumptions
that are “convenient”, and therefore, get good solutions with reasonable effort.
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
28
K. Internal Virtual Work Expressions Most “real” structures (no pun intended) are subjected to a variety of
deformations: axial (stretching or compressing); twisting; and bending (about
one or two axes). Thus, we must be prepared to handle all these deformation
types when analyzing structures. Furthermore, real structures are subjected to
different types of loads: concentrated loads; linearly varying loads; uniformly
distributed loads, etc…. Therefore, we must develop expressions that can be
used to compute the internal virtual work and external virtual work for the
variety of situations that we expect to encounter. This is the motivation for this
section.
K.1 Axial Deformations Let’s quick consider the deformations that occur in the axial rod. We will
assume that this deformation component is stretching and or compressing along
the length. We can derive internal virtual work expressions through
consideration of a differential segment.
The virtual work done by the internal forces is,
The virtual strain and real internal force in the axial rod are given by,
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
29
( )x x x
d uF A
dxδ
δε σ= =
Plugging these into the virtual work expression above gives,
int x xW Adxδ δε σ=
We must now sum up all the internal work done within the member. Therefore,
0 0
L L
int x x x xW Adx EA dxδ δε σ δε ε= =∫ ∫
Rewriting the expression above using the derivative form of the strains gives,
( )
0
L
intd u duW EA dx
dx dxδ
δ⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦∫ (25)
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
30
Example 4 – Choice of Displacement Form Using Principle of Virtual
Work
Consider the axial rod with loading shown below.
,y v
,x u
L
( )q x P L= P
1 2
Compute the actual or exact displacement at point 2 using the differential
equation of equilibrium. Then consider the following choices for real and
virtual displacements and solve the problem using the principle of virtual work:
Case A:
real displacement takes the same form as the actual displacement
virtual displacement takes the same form as the actual displacement
Case B:
real displacement takes the same form as the actual displacement
virtual displacement takes the following form;
2xu uL
δ δ⎛ ⎞= ⎜ ⎟⎝ ⎠
Case C:
real displacement takes the same form as the actual displacement
virtual displacement takes the following form;
2sin2
xu uL
πδ δ⎛ ⎞= ⎜ ⎟⎝ ⎠
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
31
Case D:
real and virtual displacements take the following form;
2sin2
xu uL
π⎛ ⎞= ⎜ ⎟⎝ ⎠
and 2sin2
xu uL
πδ δ⎛ ⎞= ⎜ ⎟⎝ ⎠
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
32
Solution
We’ll first compute the exact solution to the problem for later use. Start this
solution through consideration of a Free Body Diagram.
x
,y v
,x u PP L
L x−
xP
L
Horizontal equilibrium of the free body yields,
( )
( )
0
0
0
x
x
FPP L x PL
PP L x PL
=
− + − + =
= − + =
∑
Plugging into the integral derived previously in the Modeling Deformations
chapter of the lecture notes gives,
( ) 21 1
1 1 10 02
xx
x
P L L x P xu dx x xAE AE L
−⎡ ⎤ ⎛ ⎤= = − +⎜⎢ ⎥ ⎥
⎝ ⎦⎣ ⎦∫
2
22x
P xu xAE L
⎛ ⎞= −⎜ ⎟
⎝ ⎠
2
42x
P xu xAE L
⎛ ⎞= −⎜ ⎟
⎝ ⎠ (1e)
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
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Now, we can develop an expression that describes the displacement at any point
x along the axial rod in terms of the displacement at the end of the rod (call it
point 2).
2
234
2 2x L
P L PLu u LAE L AE=
⎛ ⎞= = − =⎜ ⎟
⎝ ⎠
223AEP uL
⎛ ⎞= ⎜ ⎟⎝ ⎠
(2e)
Plugging equation (2e) into (1e) gives,
2
21 43
x xu uL L
⎡ ⎤⎛ ⎞ ⎛ ⎞= − ⋅⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
The actual displacement along the length of the axial rod in the example
considered varies quadratically.
Case A:
Assume that the real and virtual displacements along the rod can be written as,
2
21 43
x xu uL L
⎡ ⎤⎛ ⎞ ⎛ ⎞= − ⋅⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
(REAL)
2
21 43
x xu uL L
δ δ⎡ ⎤⎛ ⎞ ⎛ ⎞= − ⋅⎢ ⎥⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ (VIRTUAL)
The work done by external loads is computed using,
20
L
EXTPW P u u dxL
δ δ δ⎛ ⎞= + ⋅ + ⋅⎜ ⎟⎝ ⎠∫
Plugging in the assumed form for the virtual displacement and integrating
gives,
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
34
2
2 20
1 43
L
EXTP x xW P u u dxL L L
δ δ δ⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞= + ⋅ + ⋅ − ⋅⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦
∫
2149EXTW P uδ δ= ⋅ ⋅ (3e)
The virtual work done by internal forces can be found using equation (25),
( )0
L
INTd u duW EA dx
dx dxδ
δ⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦∫ (25)
The necessary derivatives can be taken,
22
4 23 3
du x udx L L
⎡ ⎤= − ⋅⎢ ⎥⎣ ⎦ ( )
22
4 23 3
d u x udx L Lδ
δ⎡ ⎤= − ⋅⎢ ⎥⎣ ⎦
Plugging into equation (25) gives,
2 22 20
4 2 4 23 3 3 3
L
INTx xW u EA u dx
L L L Lδ δ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= − ⋅ − ⋅⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦∫
Carrying out the rather cumbersome integrations in MathCAD gives,
2 22827INT
AEW u uL
δ δ= ⋅ ⋅ ⋅ (4e)
The principle of virtual work demands the following be true if the axial rod is in
equilibrium,
EXT INTW Wδ δ=
Therefore, plugging equations (3e) and (4e) into the principle gives,
2 2 214 289 27
AEP u u uL
δ δ⋅ ⋅ = ⋅ ⋅ ⋅
The virtual displacements are arbitrary, but not zero, and therefore,
232
PLuAE
= ⋅
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
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35
The above equation is a restatement of the exact expression given in equation
(2e).
Case B:
The real and virtual displacements are now assumed to take the following form;
2
21 43
x xu uL L
⎡ ⎤⎛ ⎞ ⎛ ⎞= − ⋅⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
2xu uL
δ δ⎛ ⎞= ⋅⎜ ⎟⎝ ⎠
It should be noted that both of these are admissible in that they satisfy the
boundary conditions for the problem. However, the real displacements now
take a nonlinear form and the virtual displacements are assumed to be linear.
The virtual work done by real external forces now changes because the form of
the virtual displacement has changed. We can now write the virtual work done
by real external forces as the following;
2 20
L
EXTP xW P u u dxL L
δ δ δ⎡ ⎤⎛ ⎞⎛ ⎞= + ⋅ + ⋅ ⋅⎢ ⎜ ⎟ ⎥⎜ ⎟⎝ ⎠⎝ ⎠⎣ ⎦
∫
Carrying out the integration yields,
232EXTW P uδ δ⎛ ⎞= ⋅⎜ ⎟
⎝ ⎠ (5e)
The virtual work done by real internal forces can be computed using equation
(25) after taking the necessary derivatives;
22
4 23 3
du x udx L L
⎡ ⎤= − ⋅⎢ ⎥⎣ ⎦ ( )
21d u
udx Lδ
δ⎛ ⎞= ⋅⎜ ⎟⎝ ⎠
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
36
2 220
1 4 23 3
L
INTxW u EA u dx
L L Lδ δ⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞= ⋅ − ⋅⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦∫
Carrying out the integration gives,
2 2INTAEW u uL
δ δ= ⋅ ⋅ (6e)
Setting equations (5e) and (6e) equal to one another imparts the principle of
virtual work and after admitting that the virtual displacement at 2 is arbitrary,
but not zero we have;
2 2 232
AE u u P uL
δ δ⎛ ⎞⋅ ⋅ = ⋅⎜ ⎟⎝ ⎠
232
PLuAE
=
This is again the exact or actual displacement at node 2.
Case C:
The real and virtual displacements are now assumed to take the following form;
2
21 43
x xu uL L
⎡ ⎤⎛ ⎞ ⎛ ⎞= − ⋅⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
2sin2
xu uL
πδ δ⎛ ⎞= ⎜ ⎟⎝ ⎠
As before, the choice for the virtual displacement is admissible in that it
satisfies the boundary conditions for the problem.
The virtual work done by real external forces is computed as follows;
2 20
sin2
L
EXTP xW P u u dxL L
πδ δ δ⎡ ⎤⎛ ⎞= + ⋅ + ⋅ ⋅⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∫
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
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37
( ) 21.637EXTW P uδ δ= ⋅ (7e)
The virtual work done by real internal forces through application of equation
(25) after taking appropriate derivatives;
22
4 23 3
du x udx L L
⎡ ⎤= − ⋅⎢ ⎥⎣ ⎦ ( )
2cos2 2
d u x udx L Lδ π π δ⎛ ⎞= ⎜ ⎟
⎝ ⎠
2 220
4 2cos2 2 3 3
L
INTx xW u EA u dx
L L L Lπ πδ δ⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞= ⋅ − ⋅⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦∫
Integrating gives;
2 21.091INTAEW u uL
δ δ= ⋅ ⋅ ⋅ (8e)
Setting equation (7e) equal to equation (8e) and admitting that the virtual
displacement at 2 is arbitrary, but not zero gives,
( )2 2 21.091 1.637AE u u P uL
δ δ⋅ ⋅ ⋅ = ⋅
2 1.50 PLuAE
= ⋅
Once again we have obtained the actual or exact displacement at node 2.
Therefore, Case B and Case C have shown that the choice of the virtual
displacement form is arbitrary as long as the form chosen is admissible.
No matter what choice of admissible virtual displacement form, as long as
the real displacements are exact, you will get the exact solution to the
problem using virtual work.
Case D:
The results of Case A should be expected. If you know the answer to a problem
and apply it in the principle of virtual work, you would certainly hope to get the
exact solution. The results in Cases B and C may not be expected, but they
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
38
should be confidence building. However, what happens if we don’t know the
answer to the problem before we solve it? Isn’t this always the case? Case D
illustrates what happens when we make an admissible assumption for both the
real and virtual displacement forms. In other words, we do not know the
solution to the problem now, but we have some idea how the system should
behave in terms of displacements.
The real and virtual displacements now take assumed forms that are known to
not be the exact solution. These forms are shown below,
2sin2
xu uL
π⎛ ⎞= ⎜ ⎟⎝ ⎠
2sin2
xu uL
πδ δ⎛ ⎞= ⎜ ⎟⎝ ⎠
These forms are admissible because they satisfy the boundary conditions to the
problem.
The virtual work done by real external forces is the same as it was in Case C
because the form of the virtual displacement has not changed;
( ) 21.637EXTW P uδ δ= ⋅ (9e)
The virtual work done by real internal forces is again computed using equation
(25) after appropriate derivatives are taken;
( )2cos
2 2d u x u
dx L Lδ π π δ⎛ ⎞= ⎜ ⎟
⎝ ⎠ 2cos
2 2du x udx L L
π π⎛ ⎞= ⎜ ⎟⎝ ⎠
2 20
cos cos2 2 2 2
L
INTx xW u EA u dx
L L L Lπ π π πδ δ⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞= ⋅ ⋅⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦∫
Carrying out the integration gives,
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
39
2 21.234INTAEW u uL
δ δ= ⋅ ⋅ ⋅ (10e)
Setting equations (9e) and (10e) equal to one another imparts the principle and
we have,
( )2 2 21.234 1.637AE u u P uL
δ δ⋅ ⋅ ⋅ = ⋅
2 1.327 PLuAE
= ⋅
We now have not obtained the exact or actual solution to the problem, but we
are very close (approximately 11% difference).
Case D illustrates the following very important point related to displacements
computed using the principle of virtual work. As long as we assume
admissible forms for the real and virtual displacements in our solutions, we
will get good solutions that will always be less than the exact or actual
solution. In other words, our models will be too stiff.
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
40
Example 5 – Choice of Virtual Displacement Form
Consider the nonprismatic axial rod with loading shown below.
,y v
,x u
L
2xF1 2
1 12xA AL
⎛ ⎞= −⎜ ⎟⎝ ⎠
Assume that the displacements along the axial rod can be written as a linear
function that varies with distance along the word using the displacement at
point 2 on the rod as a control point. Therefore, assume that the real and virtual
displacements take the following forms;
2xu uL
δ δ⎛ ⎞= ⎜ ⎟⎝ ⎠
2xu uL
⎛ ⎞= ⎜ ⎟⎝ ⎠
Use the principle of virtual work to compute the displacement at the tip of the
rod.
Solution:
By inspection, we should appreciate that the real and virtual displacement forms
are admissible. In other words, when x is 0, the real and virtual displacements
are ________ and when x = L, the real and virtual displacements are
_________ and __________, respectively.
However, our knowledge of mechanics should tell us that the stress along the
rod will not be constant when there is a single loading applied at the tip as
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
41
shown in the problem statement. A simple FBD of a segment of axial rod will
bear this out. Our assumed displacement forms imply that strain is
___________________. Therefore, when we assume the linear forms for
displacement, we are also making the implicit assumption that strain is constant
and as a result, our current displacement form assumptions will result in a
violation of equilibrium. At this point, however, let’s continue our solution in
ignorant bliss and see what happens.
The virtual work done by real internal forces can be computed using the
following derivatives (pretty simple derivatives at that);
( )2
1d uu
dx Lδ
δ⎛ ⎞= ⎜ ⎟⎝ ⎠
21du u
dx L⎛ ⎞= ⎜ ⎟⎝ ⎠
Substituting into equation (25) gives,
2 1 20
1 112
L
INTxW u E A u dx
L L Lδ δ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ⋅ ⋅ ⋅ − ⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ ⎣ ⎦∫
Carrying out the integration yields,
12 2
34INT
EAW u uL
δ δ= ⋅ ⋅ ⋅
The virtual work done by real external forces is;
2 2EXT xW F uδ δ= + ⋅
Applying the principle of virtual work gives,
12 2 2 2
34 x
EA u u F uL
δ δ⋅ ⋅ ⋅ = ⋅
Admitting to the arbitrary nature of the virtual displacements gives,
( ) ( )
2 22
1 13 4 0.75x xF L F Lu
E A E A= =
⋅ ⋅
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
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42
The exact or actual solution to the problem is;
( ) ( )2
210.7213
xexact
F LuE A⋅
Therefore, even though our displacement forms violate equilibrium (in strict
theory), we still get very good solutions. When we apply the principle of
virtual work we are implicitly requiring equilibrium. Therefore, enforcing the
principle demands that we satisfy equilibrium in an average sense along the
body. As a result, the principle of virtual work as we’ll apply it in this course is
considered a weak form solution to the differential equations of equilibrium.
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
43
K.2 Twisting Deformations The next deformation type that we will consider is twisting (or torsional)
deformations. As with axial loading, we will begin by considering a differential
length that twists.
The rotations that are shown are positive using the right-hand-rule (about the x-
axis). The real shear stress and shear strain at the outer surface of the shaft is a
maximum and its magnitude is given by,
Using the geometry of the twisted length, the shear strain can be related to
the angle of twist over the segment length,
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
44
Setting the expressions for shear strain equal to one another gives,
xx
dM GJdxθ
=
The internal virtual work done within the small segment can be written as,
( ) ( )x xINT x x x x x
d dW M dx M M dx
dx dxδθ δθ
δ δθ δθ⎡ ⎤
= − + + =⎢ ⎥⎣ ⎦
(26)
All that is left is to integrate the internal virtual work over the length of the
entire member giving,
( )
0
Lx x
INTd dW GJ dx
dx dxδθ θδ
⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦∫ (27)
K.3 Flexural (Bending) Deformations
We can now consider flexural deformations and the associated internal virtual
work. Consider the infinitely small segment of beam subjected to bending
about the z-axis at both ends.
Positive rotations are shown in the figure above. That is, a positive rotation
about the z-axis occurs with positive change in x- and positive change in y-
directions. The virtual work done by the real internal bending moments is
computed as,
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
45
( )
( )
zINT z z z z
zz
dW M dx M
dx
dM dx
dx
δθδ δθ δθ
δθ
⎡ ⎤= − + +⎢ ⎥
⎣ ⎦
=
The change in slope over the short length dx should be pointed out. The
curvature over a short length dx is the change in slope over that short length. If
we define the transverse displacement in the y-direction as v we can use
mechanics of materials to define the curvature. Therefore, the virtual curvature
over the length, dx, is,
( ) ( )2
2z
zd d v
dx dxδθ δ
δκ = =
The internal virtual work can now be restated as,
( )2
2INT zd v
W M dxdx
δδ =
We should be able to recall the definition of curvature for a linear elastic
member from mechanics of materials. The real bending moment can be written
as,
Plugging the expression for real moment above into the internal virtual work
expression for bending about the z-axis previously derived gives,
( )2 2
2 20
L
INT zd v d vW EI dx
dx dxδ
δ⎡ ⎤ ⎡ ⎤
= ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦
∫ (28)
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
46
L. External Virtual Work We have covered the virtual work done by internal forces (axial force, torque,
and bending moment). We can now turn our attention to the virtual work done
by the external forces. The simplest virtual work computation for external
forces is that due to concentrated forces,
( )1
s
EXT i ii
W Pδ δ=
= Δ ⋅∑ (29)
where:
iδΔ is the virtual displacement in the direction of the applied loading at
the point of loading application;
iP is the applied loading;
s is the number of applied loadings.
It should be noted that one may have to compute the virtual displacement at
several loading application points to completely and correctly compute the
external virtual work.
In the case of distributed loading (uniform or otherwise), the work done by
external loading must be taken care of in an integral sense,
0
L
EXTW q dxδ δ′
= Δ∫ (30)
where:
L′ is the length over which the distributed loading is applied;
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
47
q is the expression describing the loading magnitude and variation
along the length of the member;
δΔ is the virtual displacement expression.
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
48
Example 6 – Application of Virtual Work to Bending Problem
Consider the three-span continuous beam shown below.
,y v
,x u
1 2 3 4
P
A
2L
2L
L L L
Compute the vertical displacement at point A using the principle of virtual
displacements (virtual work). Assume that the real and virtual displacements
take the following forms,
sin Axv v
Lπ⎛ ⎞= ⋅⎜ ⎟
⎝ ⎠ sin A
xv vL
πδ δ⎛ ⎞= ⋅⎜ ⎟⎝ ⎠
The only kinematic degree of freedom for the problem is therefore, the
displacement at point A.
Solution:
By inspection, we should be able to verify that the real and virtual displacement
assumptions satisfy the boundary conditions for the problem as a result of,
sin 0xL
π= when 0, ,2 ,3x L L L=
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
49
The virtual work done by real internal forces is computed using equation (28).
It should be re-emphasized that the body in this problem is bent and no
stretching or twisting anywhere in the system occurs. Therefore, the
appropriate derivatives are;
cos Adv x vdx L L
π π⎛ ⎞= ⋅ ⋅⎜ ⎟⎝ ⎠
( ) cos Ad v x v
dx L Lδ π π δ⎛ ⎞= ⋅ ⋅⎜ ⎟
⎝ ⎠
22
2 sin Ad v x vdx L L
π π⎛ ⎞ ⎛ ⎞= − ⋅ ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
( ) 22
2 sin Ad v x v
dx L Lδ π π δ⎛ ⎞ ⎛ ⎞= − ⋅ ⋅⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
Plugging into equation (28) gives,
3 2 2
0
sin sinL
INT A z Ax xW v EI v dx
L L L Lπ π π πδ δ
⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − ⋅ ⋅ − ⋅ ⋅⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
∫
Carrying out the integration gives,
4
33
2z
INT A AEIW v vL
πδ δ= ⋅ ⋅ ⋅
The virtual work done by real external forces is;
EXT AW P vδ δ= − ⋅
The virtual displacement at point A is;
( )3 2
3sin 12A A Ax L
v v v vπδ δ δ δ=
⎛ ⎞= = ⋅ = −⎜ ⎟⎝ ⎠
Plugging into the previous expression gives,
( )1EXT A AW P v P vδ δ δ= − ⋅ − = ⋅⎡ ⎤⎣ ⎦ (work done is positive)
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
50
Substituting into the principle of virtual work gives,
4
33
2z
A A AEI v v P vL
π δ δ⋅ ⋅ ⋅ = ⋅
3
42
3Az
PLvEIπ
= + ⋅
The (+) sign means that the deflection at point A is in the same direction as the
assumed virtual displacement (i.e. downward).
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
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51
Let’s solve the problem again, but this time considering that the loading is
moved to point B in the left-most span as indicated below. The kinematic
degree of freedom describing the deformation remains vertical displacement at
point A.
,y v
,x u
1 2 3 4
P
B
2L
2L
L L L
A
2L
2L
The virtual work done by real internal forces does not change;
4
33
2z
INT A AEIW v vL
πδ δ= ⋅ ⋅ ⋅
The virtual work done by real external forces does change because the point
where the loading is applied has moved. The virtual work done by external
forces is therefore,
( ) ( )2
sin2EXT Ax L
A
W P v P v
P v
πδ δ δ
δ
=
⎡ ⎤⎛ ⎞= − ⋅ = − ⋅ ⋅⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦= − ⋅
The virtual work done by the external force P is now _________________
because the assumed virtual displacement is actually vertically
__________________ at the point where the load is applied.
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
52
Applying the virtual work principle gives,
4
33
2z
A A AEI v v P vL
π δ δ⋅ ⋅ ⋅ = − ⋅
3
42
3Az
PLvEIπ
= − ⋅
The negative sign means the displacement is opposite to the assumed virtual
displacement.
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
53
Example 7 – Application of Virtual Work to Bending Problem
Consider the cantilever beam shown below with linearly varying distributed
loading.
,y v
,x u
L
( ) 1oxq x qL
⎛ ⎞= −⎜ ⎟⎝ ⎠
1 32
2L
Compute the displacement at the tip of the cantilever using the coordinate
system shown. Solve the problems using two approaches.
In the first approach, assume that the real and virtual displacements follow the
polynomial form shown below,
2 31 2v a x a x= +
Fit the polynomial to the problem at hand by defining the generalized
coordinates as displacements 3v and 3zθ . Apply the principle of virtual work to
determine the displacements at the tip of the cantilever.
In the second approach, assume that the real and virtual displacements take the
following form,
2 31 2v a x a x= + 2 3
1 2v a x a xδ δ δ= +
Use the principle of virtual work to define the generalized coordinates and then
compute the displacements at the tip of the cantilever.
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
54
Solution:
This problem is best solved using MathCAD.
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
Lecture Notes C.M. Foley – Marquette University
55
CEEN 111 - Matrix Structural AnalysisPrinciple of Virtual WorkExample 7 - Solution Methodology 1
The assumed form of the displacements has been given;
v x( ) a1 x2⋅ a2 x3
⋅+:= a1
The "fitting" process follows below. One should note the sign convention used.
x L:= We want our expression for "v" to work for displacements possible at "3"
v x( ) a2 L3⋅ a1 L2
⋅+→ (displacement at point 3)
xv x( )d
d3 a2⋅ L2
⋅ 2 a1⋅ L⋅+→ (slope at point 3)
The equations above form two equations in terms of the unknown generalized coordinates.These two equations can be written in matrix form as,
L2
2 L⋅
L3
3 L2⋅
⎛⎜⎜⎝
⎞⎟⎟⎠
a1
a2
⎛⎜⎜⎝
⎞⎟⎟⎠
⋅v3
θ3
⎛⎜⎜⎝
⎞⎟⎟⎠
:=2
Inverting the coefficient matrix yields,
L2
2 L⋅
L3
3 L2⋅
⎛⎜⎜⎝
⎞⎟⎟⎠
1−3
L2
2
L3−
1L
−
1
L2
⎛⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎠
→
Solving for the vector of generalized coefficients gives;
a1
L4
3 L2⋅
2− L⋅
L3−
L2
⎛⎜⎜⎝
⎞⎟⎟⎠
⋅v3
θ3
⎛⎜⎜⎝
⎞⎟⎟⎠
⋅
3 v3⋅
L2
θ3L
−
θ3
L2
2 v3⋅
L3−
⎛⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎠
→:=L
a1 a0
3 v3⋅
L2
θ3L
−→:=a1 a2 a1
θ3
L2
2 v3⋅
L3−→:=a2
Quickly change "x" back to an unknown (i.e. definable quantity):
x x:=
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The real displacement expression in terms of the displacements at the end of the cantilever isgiven by,
v a1 x2⋅ a2 x3⋅+⎛⎝
⎞⎠ x3 θ3
L2
2 v3⋅
L3−
⎛⎜⎜⎝
⎞⎟⎟⎠
⋅ x2 θ3L
3 v3⋅
L2−
⎛⎜⎜⎝
⎞⎟⎟⎠
⋅−→:= x2
A little manual algebraic reorganization gives an expression for the real displacement:
v3 x2⋅
L2
2 x3⋅
L3−
⎛
⎝
⎞
⎠v3⋅
x3
L3
x2
L2−
⎛
⎝
⎞
⎠L⋅ θ3⋅+:=
L
We will use the same form for the virtual expression and therefore,
δv3 x2⋅
L2
2 x3⋅
L3−
⎛
⎝
⎞
⎠δv3⋅
x3
L3
x2
L2−
⎛
⎝
⎞
⎠L⋅ δθ3⋅+:=
LL
The internal virtual work can then be computed;
δWint
0
L
x2x
3 x2⋅
L2
2 x3⋅
L3−
⎛
⎝
⎞
⎠δv3⋅
x3
L3
x2
L2−
⎛
⎝
⎞
⎠L⋅ δθ3⋅+
⎡
⎣
⎤
⎦
d
d
2EIz⋅ 2x
3 x2⋅
L2
2 x3⋅
L3−
⎛
⎝
⎞
⎠v3⋅
x3
L3
x2
L2−
⎛
⎝
⎞
⎠L⋅ θ3⋅+
⎡
⎣
⎤
⎦
d
d
2⋅
⌠⎮⎮⎮⌡
d:=L
δWint2 EIz⋅ 6 v3⋅ δv3⋅ 3 L⋅ v3⋅ δθ3⋅− 3 L⋅ θ3⋅ δv3⋅− 2 L2⋅ θ3⋅ δθ3⋅+⋅
L3→
δWint collect δv3, δθ3, 2 EIz⋅ 6 v3⋅ 3 L⋅ θ3⋅−( )⋅
L3
⎡
⎣
⎤
⎦δv3⋅
2 EIz⋅ 2 L2⋅ θ3⋅ 3 L⋅ v3⋅−⋅
L3
⎡
⎣
⎤
⎦δθ3⋅+→
Given the expression for the applied loading and virtual displacement expression derived, the virtual work doneby the external loading is,
δWext
0
L
x3 x2
⋅
L2
2 x3⋅
L3−
⎛
⎝
⎞
⎠δv3⋅
x3
L3
x2
L2−
⎛
⎝
⎞
⎠L⋅ δθ3⋅+
⎡
⎣
⎤
⎦q0⋅ 1
xL
−⎛⎝
⎞⎠
⋅
⌠⎮⎮⎮⌡
d:=L
δWext collect δv3, δθ3, 3 L⋅ q0⋅
20
⎛⎝
⎞⎠
δv3⋅L2 q0⋅
30−
⎛
⎝
⎞
⎠δθ3⋅+→
Solution can be finished by hand relatively easily. Applying the principle of
virtual work and recognizing that the virtual displacements are indeed non-zero
and arbitrary results in a system of two equations in two unknowns shown on
the following sheet.
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Solving the two equations gives the desired displacements;
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CEEN 111 - Matrix Structural AnalysisPrinciple of Virtual WorkExample 7 - Solution Methodology 2
In this approach, we will determine the generalized coefficients using the principle of virtual work.
The real and virtual forms of the transverse displacement are written as:
v x( ) a1 x2⋅ a2 x3
⋅+:= a1
δv x( ) δa1 x2⋅ δa2 x3
⋅+:= δa1
The virtual work done by real internal forces is:
δWint
0
L
x2xδv x( )d
d
2EI⋅ 2x
v x( )d
d
2⋅
⌠⎮⎮⎮⌡
d:= δv ( NOTE how EI is defined !!!! )
δWint 2 EI⋅ L⋅ 2 a1⋅ δa1⋅ 3 L⋅ a1⋅ δa2⋅+ 3 L⋅ a2⋅ δa1⋅+ 6 L2⋅ a2⋅ δa2⋅+⎛
⎝⎞⎠⋅→
δWint collect δa1, δa2, 2 EI⋅ L⋅ 2 a1⋅ 3 L⋅ a2⋅+( )⋅ δa1⋅ 2 EI⋅ L⋅ 6 a2⋅ L2⋅ 3 a1⋅ L⋅+⎛
⎝⎞⎠⋅ δa2⋅+→
The virtual work done by real external forces is:
δWext0
L
xqo 1xL
−⎛⎜⎝
⎞⎟⎠
⋅⎡⎢⎣
⎤⎥⎦
δv x( )⋅⌠⎮⎮⌡
d:= δv ( NOTE the coordinate system chosen !!! )
δWextL3 qo⋅ 5 δa1⋅ 3 L⋅ δa2⋅+( )⋅
60→
We can now use principle of virtual work to solve for a's:
As4 EI⋅ L⋅
6 EI⋅ L2⋅
6 EI⋅ L2⋅
12 EI⋅ L3⋅
⎛⎜⎜⎝
⎞⎟⎟⎠
1− 112
L3⋅ qo⋅
120
L4⋅ qo⋅
⎛⎜⎜⎜⎝
⎞⎟⎟⎟⎠
⋅:=EI
As
7 L2⋅ qo⋅
120 EI⋅
L qo⋅
40 EI⋅−
⎛⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎠
→
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Plug back into polynomial for "v":
v x( ) As0 0, x2⋅ As1 0, x3
⋅+:= As
v x( )7 L2
⋅ qo⋅ x2⋅
120 EI⋅
L qo⋅ x3⋅
40 EI⋅−→
θz x( )x
v x( )dd
7 L2⋅ qo⋅ x⋅
60 EI⋅
3 L⋅ qo⋅ x2⋅
40 EI⋅−→:=x
vL v x( ) substitute x L=, L4 qo⋅
30 EI⋅→:= L
θzL θz x( ) substitute x L=, L3 qo⋅
24 EI⋅→:= L
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Example 8 – Application of Virtual Work to General Problem
Consider the structural system shown below;
,y v
,x u1 2
34
40 kN
3 m
5 m
3 210 10A mm= ×
2 m3 m
6 440 10I mm= ×
The system now involves both members that stretch and compress as well as
members that bend. Compute the displacement in the system at point 3
assuming that the vertical displacement along the beam member can be written
as;
3sin10
xv vπ⎛ ⎞= − ⋅⎜ ⎟⎝ ⎠
Use the principle of virtual work (principle of virtual displacements). Assume
that the material modulus can be taken as; 200,000E MPa= .
It should be noted that the only kinematic degree of freedom used to solve the
problem is the vertical displacement at point 3 in the system.
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Solution:
The solution to this problem can be nearly completely carried out in MathCAD.
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CEEN 111 - Matrix Structural AnalysisPrinciple of Virtual WorkExample 8
The assumed form of the displacements has been given;
v x( ) sinπ x⋅10
⎛⎜⎝
⎞⎟⎠
⎛⎜⎝
⎞⎟⎠
− v3⋅:= v3 δv x( ) sinπ x⋅10
⎛⎜⎝
⎞⎟⎠
⎛⎜⎝
⎞⎟⎠
− δv3⋅:= δv3
The virtual work done by real internal forces comes from two parts:
δWINT δWINTbeam δWINTcol+:= δWINTbeam
Computing the virtual work done by real internal forces demands that we know what the real and virtualstrain is in the vertical column member. (Note: the column member does not bend and the beam memberis free to get longer or shorter.)
The virtual work done by the real internal (bending) forces in the beam is computed using
δWINTbeam
0
10
x2xδv x( )d
d
2EI⋅ 2x
v x( )d
d
2⋅
⌠⎮⎮⎮⌡
d:= δv ( NOTE how EI is defined and the integrationlimits - the beam is 10 meters long !!!! )
Computing the virtual work done by the real internal forces in the column member is a little bit tricky.The real and virtual displacement of the top-end of the column can be written in terms of the displacementalong the beam as;
ucol sinπ 3⋅10
⎛⎜⎝
⎞⎟⎠
⎛⎜⎝
⎞⎟⎠
− v3⋅ float 4, 0.809− v3⋅→:= v3 (NOTE: deflection at the column topis the same as the vertical deflectionin the beam at x = 3 m.)
δucol sinπ 3⋅10
⎛⎜⎝
⎞⎟⎠
⎛⎜⎝
⎞⎟⎠
− δv3⋅ float 4, 0.809− δv3⋅→:= δv3
The real and virtual strains in the column are therefore,
Lcol 3:=
εcolucolLcol
float 4, 0.2697− v3⋅→:=εcol δεcolδucolLcol
float 4, 0.2697− δv3⋅→:=δεcol
The virtual work done by real internal forces in the column member is therefore,
δWINTcol0
3xδεcol AE⋅ εcol⋅
⌠⎮⌡
d:= δεcol
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The total virtual work done by real internal forces in the structural system is therefore,
AE 2.0000 109⋅:=
( NOTE: all units are N and m )EI 8.0 106
⋅:=
δWINT δWINTbeam δWINTcol+collect δv3,
float 5, δv3 0.21821 AE⋅ v3⋅ 0.048705 EI⋅ v3⋅+( )⋅→:=
δv3
δWINT .48705e-1 EI v3⋅⋅ .21821 AE v3⋅⋅+( ) δv3⋅ float 5, 4.3681e8 v3⋅ δv3⋅→:= v3
The virtual work done by real external forces is computed as follows;
δWEXT 40000.0−( ) sinπ 5⋅10
⎛⎝
⎞⎠
⎛⎝
⎞⎠
− δv3⋅⎡⎣
⎤⎦
⋅ float 5, 40000.0 δv3⋅→:= δv3 ( NOTE the coordinate system and all units of N !!! )
The solution for the displacement at point 3 then follows directly from the
principle of virtual work;
( )( ) ( )
( )3 3 3
3
0.43681 9 40000
9.157 5 0.092
e v v v
v e m mm
δ δ=
= − = ↓
The deformation is in the same direction as the assumed virtual displacement at
point 3.
The 0.92 mm deflection should seem a little strange. The reason is that the
model we used for the real and virtual displacement is not reflective of the
displacement likely to occur at point 3 in the system.
We can alter the problem and model slightly to see the impact on the resulting
solution.
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CEEN 111 - Matrix Structural AnalysisPrinciple of Virtual WorkExample 9 - Additional Example Problem
Assume that the structural system has changed from that of Example 8 to the following;
,y v
,x u1 2
34
40 kN
3 m
1 m
3 210 10A mm= ×
2 m3 m
6 440 10I mm= ×,y v
,x u1 2
34
40 kN
3 m
1 m
3 210 10A mm= ×
2 m3 m
6 440 10I mm= ×
Compute the vertical displacement at point 3 in the system using the following real andvirtual displacement form;
v x( ) sinπ x⋅3
⎛⎜⎝
⎞⎟⎠
⎛⎜⎝
⎞⎟⎠
− v3⋅:= v3 δv x( ) sinπ x⋅3
⎛⎜⎝
⎞⎟⎠
⎛⎜⎝
⎞⎟⎠
− δv3⋅:= δv3
Solution:
The virtual work done by real internal forces comes from two parts:
δWINT δWINTbeam δWINTcol+:= δWINTbeam
The virtual work done by the real internal (bending) forces in the beam is computed using
( NOTE how EI is defined and the integrationlimits - the beam is 6 meters long !!!! )δWINTbeam
0
6
x2xδv x( )d
d
2EI⋅ 2x
v x( )d
d
2⋅
⌠⎮⎮⎮⌡
d:= δv
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Computing the virtual work done by the real internal forces in the column member is a little bit tricky.The real and virtual displacement of the top-end of the column can be written in terms of the displacementalong the beam as;
ucol v 3( ) float 4, 0.0→:= (NOTE: deflection at the column topis the same as the vertical deflectionin the beam at x = 3 m.)
δucol δv 3( ) float 4, 0.0→:=
The real and virtual strains in the column are therefore,
Lcol 3:=
εcolucolLcol
float 4, 0.0→:= δεcolδucolLcol
float 4, 0.0→:=
The virtual work done by real internal forces in the column member is therefore,
δWINTcol0
3xδεcol AE⋅ εcol⋅
⌠⎮⌡
d:= AE
The total virtual work done by real internal forces in the structural system is therefore,
AE 2.0000 109⋅:=
( NOTE: all units are N and m )EI 8.0 106
⋅:=
δWINT δWINTbeam δWINTcol+collect δv3,
float 6, 3.60774 EI⋅ v3⋅ δv3⋅→:=δWINT
The virtual work done by real external forces is computed as follows;
( NOTE the coordinate system and all units of N !!! )δWEXT 40000.0− δv 5( )⋅( ) float 6, 34641.0− δv3⋅→:= 40000.0−
Application of the principle of virtual work results in the deflection at point 3
being equal to 1.2 mm downward. A negative sign appears because the
displacement is in a direction opposite to the assumed virtual displacement.
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M. Internal Virtual Work – General Form
A generalized form for the internal virtual work would allow application to a
variety of problems. To see how generalized we can get, let’s consider a 3D
state of stress at some hypothetical point.
We will start by considering the x-direction. The real force applied in the x-
direction can be computed using,
The virtual displacement in the x-direction is computed using the virtual strain
and length,
The internal virtual work done by the real force acting through the virtual strain
is therefore, computed as,
CEEN 111 – Matrix Structural Analysis Principle of Virtual Work:
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We have only considered one component of virtual work. In reality, all
stress components will do virtual work as the little “cube” deforms. Therefore,
the internal virtual work is a summation of the virtual work done by ALL real
stresses acting through ALL virtual strains. In general terms, the internal virtual
work can be written as,
[ ]int x x y y z z xy xy yz yz zx zxW dVδ δε σ δε σ δε σ δγ τ δγ τ δγ τ= + + + + +
If the differential volume is part of a larger volume, we must integrate over the
volume to get the total internal virtual work,
[ ]int x x y y z z xy xy yz yz zx zxV
W dVδ δε σ δε σ δε σ δγ τ δγ τ δγ τ= + + + + +∫ (31)
Our course entitled “Matrix Structural Analysis” would be misleading if
we didn’t include some matrices in the virtual work formulation. It is often
very useful to write equation (31) in matrix-vector form,
{ }intV
W dVδ δε σ= ⋅⎢ ⎥⎣ ⎦∫ (32)
in which,
{ }
x y z xy yz zx
Tx y z xy yz zx
δε δε δε δε δγ δγ δγ
σ σ σ σ τ τ τ
⎢ ⎥=⎢ ⎥⎣ ⎦ ⎣ ⎦
⎢ ⎥= ⎣ ⎦
It should be noted that we are integrating the virtual strain energy density (in
three dimensions) over the volume.
It is instructive to check that our general three-dimensional formulation
remains valid when we consider “stick” or “line” elements. Therefore, let’s
only consider the bending deformations and check that the general form given
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by equation (32) gives the same result as our previous equation (28). Let’s
begin with the general form,
{ }intV
W dVδ δε σ= ⋅⎢ ⎥⎣ ⎦∫
If we only consider x-directional stress (and normal stress only for that matter),
our virtual strain and real stress for bending can be incorporated into the general
form,
( )2
2z
intzV
d v M yW y dVdx I
δδ
⎡ ⎤ ⎡ ⎤= − −⎢ ⎥ ⎢ ⎥
⎣ ⎦⎣ ⎦∫
The real moment can be written in terms of real curvature giving,
( )2 2
2 2int zzV
d v y d vW y EI dVdx I dx
δδ
⎡ ⎤ ⎡ ⎤= − − ⋅⎢ ⎥ ⎢ ⎥
⎣ ⎦⎣ ⎦∫
Simplifying a little gives,
( )2 22
2 2intV
d v d vW E y dVdx dx
δδ
⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥
⎣ ⎦⎣ ⎦∫
The volume integration can be rewritten as integration over length and cross-
sectional area. Rewriting the volume integration in this manner gives,
( )2 22
2 2intL A
d v d vW E y dA dxdx dx
δδ
⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥
⎣ ⎦⎣ ⎦∫ ∫
The area integration reveals an integral form of the moment of inertia,
2z
A
I y dA= ∫
The internal virtual work can then be written as,
( )2 2
2 2int zL
d v d vW EI dxdx dx
δδ
⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥
⎣ ⎦⎣ ⎦∫
which is the same expression previously derived.
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There may be situations where three-dimensional consideration will be
needed. In this case, a material matrix will be needed. The internal virtual
work for general three-dimensional problems can then be written as,
[ ] { }
[ ] { } generalized Hooke's Law
intV
W E dV
E
δ δε ε
ε
= ⋅ ⋅⎢ ⎥⎣ ⎦
⋅ =
∫ (33)
N. References
McGuire, W., Gallagher, R.H., Ziemian, R.D. (2000) Matrix Structural
Analysis, 2nd Edition, John Wiley and Sons, Inc.
Sack, R.L. (1989) Matrix Structural Analysis, Waveland Press, Inc.