METHOD OF VIRTUAL WORK
Group 3
Chirarapasi Ashington S N02220287P
Chirinda Tinotenda N02213152Y
Gatsi Blessing N02220764T
Gwini Rutendo N02211683N
Makumbe Adrian N02220790K
Musvaire Malvern N02213188P
Mwenje Jeremiah N02220301W
Ngoni Anthony Tinarwo N02220411W
EXTERNAL WORK AND STRAIN ENERGY
• In order to find displacement/deflection at points in complicated structures such as
trusses, frames and beams subjected to complicated loading, we use energy
methods.
• Most energy methods are based on the conservation of energy principle ,which
states that the work done by all the external forces acting on a structure, 𝑈𝑒 , is
transformed into internal work or strain energy, 𝑈𝑖 .
𝑈𝑒 = 𝑈𝑖
• To develop any energy method based on this principle, we need to determine the
external force and strain energy caused by a force.
WORK OF A FORCE:
• A force does work when it undergoes a displacement dx in the same direction as the force.
• Work done is a scalar, defined as 𝑑𝑈𝑒 = 𝐹𝑑𝑥.
𝑥
• If total displacement is 𝑥, work becomes 𝑈𝑒 = 0 𝐹𝑑𝑥
• Now consider the effect caused by an axial force applied at the end of a bar shown below. As
magnitude of F is gradually increased from zero to limiting value 𝐹 = 𝑃, final displacement of
the end of bar becomes Δ.
• For linear-elastic behavior of material, 𝐹 = 𝑃 Δ 𝑥. Substitute into the equation above, we
get the equation below, which represents the shaded triangular area in the image above.
1
2
𝑈𝑒 = 𝑃Δ
• Suppose that 𝑃 is already applied to the bar and another force 𝐹′ is now
applied, so end of bar is further displaced by an amount Δ′.
• Work done by 𝑃 (not 𝐹′) when the bar goes a further deflection is then 𝑈𝑒′ =
𝑃Δ′ , represented by the shaded rectangular area in the image below.
Triangular area BCD represents work by the force 𝐹′ since this force causes
displacement Δ′.
• When a force 𝑃 is applied to the bar, followed by the force 𝐹′, total work done by
both forces is represented by the area of the entire triangle in graph shown.
STRAIN ENERGY (AXIAL FORCE):
• Given a bar of uniform cross-sectional area 𝐴 and length 𝐿. If the bar is subjected
to a gradual force 𝑁, work is going to be done on the bar and is converted into
strain energy in the bar.
• Provided the material is linearly elastic, Hooke’s Law is valid, 𝜎 = 𝐸𝜖. The normal
stress, 𝜎 = 𝑁 𝐴 and the final strain 𝜖 = Δ 𝐿.
• Therefore, 𝑁 𝐴 = 𝐸 Δ 𝐿, and the final deflection is
𝑁𝐿
Δ=
𝐴𝐸
1
• Substituting this equation into 𝑈𝑒 = 𝑃Δ, with 𝑃 = 𝑁, the strain energy in the bar
2
becomes
𝑈𝑖 =
𝑁2 𝐿
2𝐴𝐸
PRINCIPLE OF VIRTUAL WORK:
 It provides a general means of obtaining displacement/deflection at a specific
point on a structure, be it a beam, frame, or truss.
 Before developing the principle of virtual work, it is necessary to make some
general statements regarding work and energy we discussed in the previous
section.
 If we take a deformable body of any shape or size and apply a series of external loads 𝑃
to it, it will cause internal loads 𝑢 at points throughout the structure.
 As a consequence of these loading, external displacement ∆ will occur at 𝑃 loads and
internal displacement 𝛿 will occur at each point in internal load 𝑢.
 These are related by the principle of work and energy as:
 Based on this concept, we will now develop the principle of virtual work.
 We will consider the structure (or body) to be of arbitrary shape
 Suppose it is necessary to determine the displacement Δ at point A on the body caused
by the ‘real loads’ 𝑃1 , 𝑃2 and 𝑃3 ;
 Since no external loads act on the body at A and in the direction Δ, the displacement Δ
can be determined by first placing on the body a virtual load 𝑃′ such that this force
acts in the same direction Δ.
 For convenience we choose 𝑃′ to have a ‘unit’ magnitude, that is, 𝑃′ = 1.
 So, the term ‘virtual’ is used to describe the load, since it is imaginary and does not
actually exist as part of the real loading.
 The unit load (𝑃′) does, however, create an internal load 𝑢 in a representative element,
or fibre of the body.
• Once the virtual loadings are applied, then the body is subjected to the real loads 𝑃1 ,
𝑃2 and 𝑃3 . Point A will be displaced an amount Δ, causing the element to deform an
amount 𝑑𝐿.
• As a result, the external virtual force 𝑃′ and internal virtual load 𝑢 “ride along” by Δ
and 𝑑𝐿. As a result, by the principle of work and energy:
Where,
𝑃′ = 1 = external virtual load acting in the direction Δ.
𝑢 = internal virtual load acting on the element in the direction 𝑑𝐿.
Δ = external displacement caused by the real loads.
𝑑𝐿 = internal deformation of the element caused by the real loads
METHOD OF VIRTUAL WORK: TRUSSES
1. Due to External Loading:
• Consider the vertical displacement Δ at joint B of the truss above.
• A typical element of the truss would be one of it’s member having length L. If the applied
loadings 𝑃1 and 𝑃2 cause linear elastic material response, then this element deforms an
amount Δ𝐿 = 𝑁𝐿 𝐴𝐸, where 𝑁 is the normal or axial force in the member, caused by the
loads.
• Applying the equation 1 ∙ Δ =
𝑢 ∙ 𝑑𝐿, the virtual-work equation of the truss is therefore
1∙Δ=
𝑛𝑁𝐿
𝐴𝐸
Where
1 = external virtual load acting on the truss joint in the stated direction of Δ.
n = internal virtual normal force in a truss member caused by the external virtual unit load.
Δ = external joint displacement caused by the real loads on the truss.
𝑁 = internal normal 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
• The external virtual unit load creates internal forces 𝑛 in each of the truss members. The real
loads then cause the truss joint to displace Δ in the same direction as the virtual unit load,
and each member is displaced 𝑁𝐿 𝐴𝐸 in the same direction as it’s respective 𝑛 force.
• Consequently, the external virtual force 1 ∙ Δ equals the internal virtual work or internal
(virtual) strain energy stored in all the truss members, that is, 𝑛𝑁𝐿 𝐴𝐸.
2. Due to Temperature:
• In some cases where truss members may change length due to temperature.
• If 𝛼 is the coefficient of thermal expansion for a member and Δ𝑇 is the change in
temperature, the change in length of a member is
Δ𝐿 = 𝛼 Δ𝑇 L.
• Hence, we can determine the displacement of a selected truss joint due to this
temperature change from this equation
1∙Δ=
𝑛𝛼 Δ𝑇 𝐿
PROCEDURES OF ANALYSES:
The following procedure may be used to determine a specific displacement of any joint on a
truss using the method of virtual work
1. Calculation of virtual forces, n
 Place the unit load on the truss at the joint where the desired displacement is to be
determined. The load should be in the same direction as the specified displacement
(horizontal and vertical).
 With the unit load so placed, and all the real loads removed from the truss, use the
method of joints or the method of sections and calculate the internal n force in each truss
member.
 Assume that tensile forces are positive and compressive forces are negative.
2. Calculation of real forces, N
 Use the method of sections or the method of joints to determine the N force in each member.
 These forces are caused only by the real loads acting on the truss. Again, assume tensile
forces are positive and compressive forces are negative.
3. Virtual-work equation
 Apply the equation of virtual work, to determine the desired displacement. It is important to
retain the algebraic sign for each of the corresponding n and N forces when substituting
these terms into the equation.
𝑛𝑁𝐿
 If the resultant sum
is positive, the displacement is in the same direction as the unit
𝐴𝐸
load. If a negative value results, Δ is opposite to the unit load.
 When applying any formula, attention should be paid to the units of each numerical quantity.
In particular, the virtual unit load can be assigned any arbitrary unit (𝑙𝑏, 𝑘𝑖𝑝, 𝑁, etc.), since
the n forces will have these same units, and as a result the units for both the virtual unit load
and the n forces will cancel from both sides of the equation.
EXAMPLE 1.
EXAMPLE 2
EXAMPLE 3