VIRTUAL WORK
The principle of virtual work is one of the most powerful tools in structural analysis. It was developed by John Bernoulli in 1717 and like other energy methods of analysis; it is based on the conservation of energy.
Definition:
Virtual work principle may be stated as follows “If a system of forces acts on a particle which is in statical equilibrium and the particle is given any virtual displacement then the net work done by the forces is zero”.
A virtual displacement is any arbitrary displacement which is mathematically conceived and does not actually have to take place, but must be geometrically possible. The forces must not only be in equilibrium but are assumed to remain constant and parallel to their original lines of action.
Although the principle of virtual work has many applications in mechanics. However we will use it to obtain the displacement and slope at various points on a deformable body.
EXPLAINATION:
Whenever a body is fixed from moving, it is necessary that the loadings satisfy the equilibrium conditions and the displacements satisfy the compatibility conditions. Specifically, equilibrium conditions require the external loads to be uniquely related to the internal loads, and the compatibility conditions require the external displacements to be uniquely related to the internal deformations.
If we consider a deformable body of any shape or size and apply a series of external loads “P” to it, these loadings will cause internal loadings “u” within the body. Here the external and internal loads are related by the equations of equilibrium. Furthermore, since the body is deformable, the external loads will be displaced “∆”, and the internal loadings will undergo displacements “δ”. In general, the material does not have to behave elastically, and so the displacements may not be related to the loads.
However, if the external displacements are known, the corresponding internal displacements are uniquely defined since the body is continuous, for this case, the conservation of energy states that
Ue = Ui
ΣP∆ = Σuδ------(1)
External work = Internal virtual work
Note:
This relation shows that for deformable bodies, virtual work can be formulated in two alternative ways.
(I)It can either be determined by multiplying real forces by virtual (imaginary) displacements,
(II)It can be obtained as a product of virtual forces & real displacements.
DETERMINATION OF DEFLECTION BY
VIRTUAL WORK PRINCIPLE
Based on the above concept, we will now develop the principle of virtual work so that it can be used to determine the displacement and slope at any point on a body. To do this, we will consider the body to be of arbitrary shape as shown in the following figures.
Fig 1: Application of Virtual Unit LoadFig 2: Application of Real Loads
This body is subjected to the real loads P1, P2 and P3. It is to be understood that these loads cause no movement of the supports; however, in general they can strain the material beyond the elastic limit. Suppose that it is necessary to determine the displacement “∆” of point “A” on the body caused by these loads. To do this, we will consider applying the conservation of energy principle as stated in equation 1.
In this case however,there is no force acting at “A”, and so the unknown displacement “∆” will not be included as an external “work term” in the equation1.
In order to get around this limitation, we will place an imaginary or “virtual” force “P′, on the body at point “A”, such that “P′ acts in the same direction as “∆”. Furthermore, this load is applied to the boy before the real loads are applied as shown in figure 1. For convenience, we will choose “P′ to have a “unit” magnitude; that is, P′=1.
It is to be emphasized that the term “virtual” is used to describe the load because it is imaginary and does not actually exist as part of the real loading. This external virtual load, however, does create an internal virtual load “u” in a representative element or fiber of the body as shown in figure1. As expected, “P′ and “u” can be related by the equations of equilibrium. Also, because of “P′ and “u”, the body and the element will each undergo a virtual displacement, although we will not be concerned with their magnitudes.
Once the virtual load is applied and then the body is subjected to the real loads P1, P2 and P3, point “A” will be displaced a real amount “∆”, which causes the element to be displaced “dL” as shown in fig2. As a result, the external virtual force “P′ and internal virtual load “u” ride along by “∆” and “dL” respectively; consequently these loads perform external virtual work of “1.∆” on the body and internal virtual work of
“u . dL” on the element. Considering only the conservation of virtual energy, the external virtual work is then equal to the internal virtual work done on all the elements of the body. Therefore, we can write the virtual work equation as.
`------(2)
Here
P′ = 1 = External virtual unit load acting in the direction of “∆”
U = Internal virtual load acting on the element
∆ = External displacement caused by the real loads
dL = Internal displacement of the element in the direction of “u” caused by the real
Loads.
By choosing “P′ = 1”, it can be seen that the solution for “∆” follows directly, since
∆ = Σ u dL.
DETERMINATION OF SLOPE BY
VIRTUAL WORK PRINCIPLE
In a similar manner, if the rotational displacement or slope of the tangent at a point on the body is to be determined, a virtual couple moment “M′, having a “unit” magnitude, is applied at that point. As a consequence, this couple moment causes a virtual load “uθ” in one of the elements of the body. Assuming that the real loads deform the element an amount “dL”, the rotation “θ” can be found from the virtual work equation given below.
------(3)
Here
M′ = 1 = External virtual unit couple moment acting in the direction of “θ”
Uθ= Internal virtual load acting on an element
θ = External rotational displacement in radians caused by the real loads.
dL = Internal displacement of the element in the direction of “Uθ”, caused by the
real loads
This method for applying the principle of virtual work is often referred to as the
“Method of virtual forces”, since a virtual force is applied, resulting in a calculation of an external real displacement. The equation of virtual work in this case represents a statement of compatibility requirements for the body.
INTERNAL VIRTUAL WORK
The terms on the right hand side of equation 2 and 3, represent the internal virtual work developed in the body. The real internal displacements “dL” in these terms can be produced in several different ways. For example geometric fabrication errors, temperature change or residual stresses.
If we assume that the material behavior is linear-elastic and the stress does not exceed the proportional limit, we can formulate the expressions for internal virtual work caused by stress using the equations of elastic strain energy. The have been listed in the center column of table 1. Recall that each of these expressions assumes that the stress resultant N, V, M or T was applied gradually from zero to its full value. As a result, the work done by the stress resultant is shown in these expressions as one-half the product of the stress resultant and its displacement.
In the case of virtual-force method, however, the full virtual loading is applied
before the real loads cause displacements, and therefore the work of the internal virtual loading is simply the product of the internal virtual load and its real displacement. Referring to these internal virtual loadings (u) by the corresponding lowercase symbols n, ν, m, and t, the virtual work due to axial load, shear, bending moment and torsional moment is listed in the right hand column of Table 1.
Table 1:
By using these results, the virtual work equation for a body subjected to a general loading can therefore be written as follows.
------(4)
METHOD OF VIRTUAL FORCES
APPLIED TO TRUSSES
Let us consider a truss shown in the figure3 below where virtual unit load is applied on a truss joint “A”. We will apply the method of virtual forces to determine the displacement of a truss joint. In order to illustrate this principle, the vertical displacement of joint “A” of the truss will be determined.
Fig 3: Application of virtual unit load
This displacement is caused by the real loads “P1” AND “P2”as shown in figure 4 as follows.
Fig 4: Application of real loads
and since these loads cause only axial force in the members, it is only necessary to consider the internal virtual work due to axial load, as shown in the table 1.
To obtain this virtual work, we will assume that each member has a constant cross-sectional area “A”. and the virtual load “n” and real load “N” are constant throughout the member’s length. As a result, the internal virtual work for a member is.
------(5)
And the virtual work equation for the entire truss is therefore given by using eq 2..
------(6)
Here
1 = External virtual unit load acting on the truss joint in the stated direction of “∆”
∆ = Joint displacement caused by the real loads on the truss
N = Internal virtual force in a truss member caused by the real loads
L = Length of a member
A = cross-sectional area of a member
E = modulus of elasticity of a member
In this case, the external virtual unit load creates internal virtual “n” forces in each of the truss, they cause the truss joint to be displaced “∆” in the same direction as the virtual unit load (fig4), and each member undergoes a displacement NL/AE, in the same direction as its respective “n” force. Consequently, the external virtual work “1.∆ “ equals the internal virtual work or the internal (virtual) strain energy stored in all the truss members as shown in above equation.
EFFECT OF TEMPERATURE
CHANGE ON TRUSS
Truss members can change their length due to a change in temperature. If “α” is the coefficient of thermal expansion for a member and ∆T is the change in temperature, the change in length of a member is ∆L = α∆TL. Hence we can determine the displacement of a selected truss joint due to this temperature change as (using eq 2)
------(7)
Here
1 = External virtual unit load acting on the truss joint in the stated direction of “∆”
∆ = External joint displacement caused by the temperature change
N = Internal virtual force in a truss member caused by the real loads
α = Co-efficient of thermal expansion of the member
∆T = Change in temperature of member
L = Length of the member
METHOD OF VIRTUAL FORCES
APPLIED TO BEAMS
Let us consider a beam shown in the figure 5 below, where virtual unit load is applied on a beam at point “A”. We will apply the method of virtual forces to determine the displacement and slope at a point on a beam.
Fig 5: Virtual loads
In order to illustrate this principle, the displacement “∆” of a point “A” on the beam shown in figure 6 below will be determined. This displacement is caused by the real distributed load “w”, and since this load causes both a shear and moment within the beam, we must actually consider the internal virtual work due to both of these loadings.
Fig 6: Beam with Real Loads
As beam deflections due to shear are negligible as compared with those caused by bending, particularly if the beam is long and slender, and this type of beam is most commonly used in practice, so we will consider only the virtual strain energy due to bending Applying the virtual work equation 2 for this case we get
------(8)
Here
1 = External virtual unit load acting on the beam in the direction of “∆”
∆ = Displacement caused by the real loads acting on the beam
M= internal virtual moment in the beam, expresses as a function of “x” and caused by the
real loads.
E = Modulus of Elasticity of the material
I = Moment of inertia of the cross-sectional area, computed about the neutral axis.
In a similar manner, if the slope “θ” of the tangent at a point on the beam’s elastic curve is to be determined, a virtual unit couple moment must be applied at the point, and the corresponding internal virtual moment “mθ” has to be determined. So applying the general equation of virtual work in terms of slope i.e’ eq 3 we get
------(9)
CASTIGLIANO’S THEOREM
This theorem is used to evaluate the deflection and slope at any point of a structural member using strain energy. It was narrated by Alberto Castigliano in 1879. It is also based on the conservation of energy.
Definition:
It states that “The displacement at any point is equal to the first partial derivative of strain energy in the body with respect to a force acting at that point and in the direction of displacement”.
In a similar manner “The slope of the tangent at a point in a body is equal to the first partial derivative of strain energy in the body with respect to the couple moment acting at a point in the direction of slope angle”.
Limitation:
This theorem can only be applied to bodies having constant temperature and material with linear elastic behavior.
EXPLAINATION:
In order to derive Castigliano’s theorem, consider a body of an arbitrary shape as shown in the fig 7.
Fig: 7
This body is subjected to a series of “n” forces P1, P2, P3…Pn. since the external work done by these forces is equal to the internal strain energy stored in the body; we can apply the conservation of energy i.e.
However, the external work is a function of the external loads, i.e. Ue = Σ ⌠P dx, so the internal work is also a function of the external loads. Thus
------(10)
Now, if any one of the external forces, say Pj, is increased by a differential amount dPj, the internal work will also be increased, such that the strain energy becomes.
------(11)
This value should not depend on the sequence in which the “n” forces are applied to the body e.g. we could apply dPj to the body first, then apply the loads P1, P2, P3…Pn. In this case, dPj would cause the body to be displaced in differential amount d∆j in the direction of dPj. As we know from strain energy that Ue = ½(Pj ∆j), the increment of strain energy would be ½ (dPj .d∆j). This quantity, however, is a second-order differential and may be neglected. Further application of the loads P1, P2, P3…Pn causes dPj to move through the displacement ∆j so that now the strain energy becomes.
------(12)
Here, the above Ui is the internal strain energy in the body caused by the loads P1, P2, P3…Pn and “dUj = dPj . ∆j” is the additional strain energy caused by dPj. Eq 11 represents the strain energy in the body determined by first applying the loads P1, P2, P3…Pn. Then dPj. Eq 12 represents the strain energy determined by first applying dPj and then the loads P1, P2, P3…Pn. Since these two equations must be equal. So we get.
------(13)
This proves the theorem.
CASTIGLIANO’S THEOREM
APPLIED TO TRUSSES
Since a truss member is subjected to an axial load, the strain energy is given by the following equation
Substituting this equation in to the general equation of castigliano’s theorem (i.e. eq 13)
for deflection and omitting subscript “i” we get
------(14)
It is generally easier to perform the differentiation prior to summation. Also, L, A and E are constant for a given member, and therefore we may write the above equation as
------(15)
Here
Joint displacement of the truss
P = External force of variable magnitude applied to the truss joint in the direction of ∆
N= Internal axial force in a member caused by both the force P and the loads on the truss
L = length of a member
A= cross sectional area of a member
E = modulus of elasticity of the material
Note:In order to determine the partial derivative ∂N/∂P, it will be necessary to treat “P” as a variable, not a specific numerical quantity. In other words, each internal axial force “N” must be expressed as a function of “P”.
By comparison, the above equation is similar to that used for the method of
virtual worki.e, “1 . ∆ = ∑ (nNL /AE)”, except that “n” is replaced by ∂N/∂P. These terms, “n” and“∂N/ ∂P”, will, however, be the same, since they represent the rate of change of the internal axial force with respect to the load “P” or, in other words, the axial force per unit load.
CASTIGLIANO’S THEOREM
APPLIED TO BEAMS
The internal strain energy for a beam is caused by both bending and shear, however, if the beam is long and slender, the strain energy due to shear can be neglected compared with that of bending. Assuming this is to be the case, the internal strain energy for a beam is given by
------(16)
Substituting this equation into the basic equation of castigliano’s theorem i.e. eq 13, and omitting the subscript “i”, we get
------(17)
Rather than squaring the expression for internal moment M, integrating, and then taking the partial derivative, it is generally easier to differentiate prior to integration. Provided “E” and “I” are constant, we have
------(18)
Where
Displacement of the point caused by the real loads acting on the beam
P = External force of variable magnitude applied to beam in the direction of ∆
N= Internal moment in the beam, expressed as a function of “x” and caused by both the force “P” and the loads on the beam
L = length of a member
E = modulus of elasticity of the material
I = Moment of inertia of cross-sectional area computed about the neutral axis
If the slope of the tangent “θ” at a point on the elastic curve is to be determined, the partial derivative of the internal moment “M” with respect to an external couple moment M’ acting at the point must be found. Then for such case.
------(19)
The above equations are similar to those used for the method of virtual work, except m and mθreplace ∂M/∂P and ∂M/∂M’ respectively.
Note:It should be mentioned that if the loading on a member causes significant strain
energy within the member due to axial load, shear, bending moment, and torsional
moment, then the effects of all these loadings should be included when applying
Castigliano’s theorem. To do with their associated partial derivatives. The result
is given below.
---- (20)