SAS206B: STRUCTURAL ANALYSIS AND STRENGTH OF

MATERIALS

UNIT 10: VIRTUAL WORK METHOD

1
References
1 Structural Analysis, Aslam Kassimali SI Edition, 5th Edition
CENGAGE Learning, Published 2015, ISBN 9781285051505
7 Deflection of Trusses, Beams and Frames: Work – Energy
Methods
2 Structural and Stress Analysis, 3rd Edition, T.H.G. Megson,
Butterworth-Heinemann, Published 2014, ISBN
9780080999364
15 Virtual Work and Energy Methods
3 Class Notes 2019 prepared by Dr Chris Ackerman

2
Learning Outcomes:
1. Demonstrate knowledge and understanding of the principle
of the Virtual Work method in analysing structures

3
Assessment criteria
 The basic principle of the virtual work method is formulated
 Primary and redundancy structures are constructed
 Equations that describe the bending moment diagrams / axial
forces in trusses are developed
 Integrate the bending moment equations
 Free body, shear force and bending moment diagrams of static
indeterminate structures are drawn
 Tabulated Integrals as alternative solution to integration of
bending moment equations are applied.

4
 Dimensioned drawing of the free body, shear force and
bending moment diagrams are produced.
 Position of the maximum bending moment is located
 Primary and redundancy structure deflections in laboratory is
measured
 PROKON software is operated to calculate deflections and
end-moments
 Measured and calculated deflections are compared
 Results are reflected upon

5
10.1 INTRODUCTION
 The Virtual work method is a method that can be used to
determine:
o Deflections and axial forces of static indeterminate pin
jointed frames and
o Calculate and then draw the bending moment and shear
force diagrams of static indeterminate flexural structures.
 In the case of solving indeterminate structures, the principle of
superposition is used.

6
 With static indeterminate flexural structures, this method
follows the same basic algorithm of the moment area method
except that the areas of bending moments are not determined
but the equations that describe the bending moments are
integrated.

7
10.2 THEORETICAL CONSIDERATIONS
Basic Equations
10.2.1 For Axial Load carrying Structures
 From the Engesser and Castigliano energy theorems a
relationship exists between the applied force in a member and
the extension of the member due to the applied force.

Relationship between force and extension due to applied

force
8
 From the figure, V is the strain energy under the curve which
is the work done by P stretching the member therefore,

 While C is the complementary energy

 Where is an initial extension caused by lack of fit

(thermal effects etc).
9
 For a constant force, express e as:

 And integrating,

 Thus

10
 For a structure containing n number of redundancies, partial
differentiation for the complementary energy C is

 and for the strain energy V

 R is a redundancy in a hyperstatic pin jointed frame which

may be a reaction or member force.

11
 Apply influence coefficients i.e. unit loads for R say r1. With
partial differentiation and constant force in the complementary
energy:

 Where

 Po is the member forces (Primary Structure) when r1 (Unit

Load Structure) is equal to zero

12
 Substitute P in

 Then

R = The redundant
r1 = Virtual Force in the member 1
L = Length of member 1
A = Cross sectional area of member 1
E = Young Modulus of member 1
Po = Real Force in the member 1

13
 For deflection differentiate complementary energy:

 And due to lack of fit and thermal strain:

14
10.2.2 For hyper-static flexural structures
 Equation 2.11 is written as

 The deflection Equation 2.12 can be written as:

 Where Mo is the equation(s) describing the bending moment

diagram of the Primary structure and M describes the final
bending moment diagram. m1 is the equation(s) describing
the bending moment diagram of the redundancy.
 In the case of deflection calculations, a unit load is placed at
the point of enquiry and the equation (s) describing the
bending moment diagram is m1.
15
10.3 STATIC INDETERMINATE PIN JOINTED FRAMES
 Steps for the method to solve statically indeterminate pin
jointed frames is as follows:
1) Determine the stability and the degree of static
indeterminacy (n=m+r-2j)
2) Remove n number of redundancies (stability
requirements must still be met i.e. m = 2 j – 3).
3) Calculate the member forces of the static determinate
frame (Primary structure).
4) Remove the real loading and apply a unit virtual load at
the redundancy (unit load structure).
5) Calculate the member forces of the frame with the unit
load.

16
6) Calculate R using Equation 2.11

7) To obtain the final member forces of the frame, determine

P for each member using Equation 2.9.

17
10.3.1 Examples
Example 10.1
Calculate the member forces of the truss shown in the Figure
E10.1 below using the principle of virtual work. The truss consists
of 50 x 228 timber beams with an elastic modulus of 7 GPa.

Figure E10.1
18
Solution E10.1
1) Determine the stability and the degree of static indeterminacy
(n=m+r-2j)

o The degree of static indeterminacy is n = m + r - 2 j.

o This is n = 5 + 4 - 2*4.
o Therefore, the truss is statically indeterminate to the first
degree.

19
2) Remove n = 1 redundancy (stability requirements must
still be met i.e. m = 2 j – 3).

 Remove for instance support B i.e. the reaction Bv.

20
 Check stability: m = 2 j – 3.
o This is 5 = 8 – 3.
o Therefore, the truss is stable.

21
3) Calculate the member forces (Po) of the statically
determinate frame (Primary structure).
 The Primary structure is shown below.

 The member forces (Po) can be calculated using

several methods such as the method of:
o Sections,
o Nodes and
o Force diagrams.

22
 The member forces are normally summarized in a
table as shown below.

Primary structure member forces and reactions

23
4) Remove the real loading and apply a unit virtual load at the
redundancy (unit load structure).
 The next step is to remove the real loading and apply a
unit virtual load at the redundancy B.

Unit load at the redundancy

24
5) Calculate the member forces of the frame with the unit
virtual load.

Unit load at the redundancy, member forces and reactions

 The member virtual forces are calculated and
summarized in the table below.

25
 Table showing the above calculated member axial
forces Po due to real load acting on the primary
structure and r1 the member virtual forces due to unit
virtual load acting at the redundancy.

26
 The value -323,9 x10-6 m is the vertical deflection at B
due to the real loads with support B removed (shown
below). Note that the unit load was chosen up and
therefore the negative deflection.
 If the unit load were chosen down, the deflection
would have been positive.

27
28
 The value 169,2 x10-6 is the vertical deflection at B due
to a virtual load of 1 at B with support B removed.

29
6) Calculate the redundant R = Bv using Equation 2.11
 The value of Bv is calculated using Equation 2.11 (see
table below for data).

30
 It is a fact that the actual deflection at B is zero
therefore the 169,2 x10-6 deflection need to be
multiplied by 1,195 to equate it to 323,9 x10-6.
 This 1,195 is the reaction at B, resisting the loads that
try to push down node B by 323,9 x10-6 m.

31
7) To obtain the final member forces of the frame, determine P
for each member using Equation 2.9.

 To obtain the final member forces of the frame,

determine P for each member using superposition
o P = Po + r1 * Bv
o The forces are summarized in the table below.

32