1
1 2005 Pearson Education South Asia Pte Ltd
9-4 Method of virtual work:
Trusses
External loading
Consider the vertical disp of joint B in Fig
9.7
If the applied loadings P
1
& P
2
cause a
linear elastic material response, the
element will deform
AE NL L / =
2 2005 Pearson Education South Asia Pte Ltd
9-4 Method of virtual work:
Trusses
External loading (contd)
Applying eqn 9.13, the virtual work eqn of
the truss is:
member a of elasticity modulus
member of area sectional - cross
member the of length
load real by the caused member truss a in force normal internal
truss on the loads real by the caused disp joint ext
load unit ual virt
ext by the caused member truss a in force normal l int virtua
of
direction stated in the joint truss on the acting load unit l ext virtua 1
=
=
=
=
=
=
=
E
A
L
N
n
9.15 eqn . 1
AE
nNL
=
2
3 2005 Pearson Education South Asia Pte Ltd
9-4 Method of virtual work:
Trusses
External loading (contd)
The external virtual load creates internal
virtual forces n in each of the members
The real loads caused the truss joints to be
displaced in the same direction as the
virtual unit load
Each member is disp NL/AE in the same
direction as its respective n force
Hence, ext virtual work =sum of int.
(virtual) strain energy stored in truss
members
4 2005 Pearson Education South Asia Pte Ltd
9-4 Method of virtual work:
Trusses
Temperature
In some cases, truss members may
change their length due to temperature
The disp of a selected truss joint may be
written as
TL L =
9.16 eqn . 1 TL n =
member a of ture in tempera change
member of expansion thermal of t coefficien
change ture by tempera caused disp joint ext
=
=
=
T
3
5 2005 Pearson Education South Asia Pte Ltd
9-4 Method of virtual work:
Trusses
Fabrication errors & camber
Errors in fabricating the lengths of the
members of a truss may occur
Truss members may also be made slightly
longer or shorter in order to give the truss a
camber
Camber is often built into bridge truss so
that the bottom cord will curve upward by
the same amount equivalent to the
downward deflection when subjected to the
bridges full dead weight
6 2005 Pearson Education South Asia Pte Ltd
9-4 Method of virtual work:
Trusses
Fabrication errors & camber (contd)
The disp of a truss joint from its expected
position can be written as:
A combination of right sides of eqn 9.15 to
9.17 may be necessary if both external
loads, thermal change & fabrication errors
are taking place
9.17 eqn . 1 L n =
error n fabricatio by caused as size
intended its from member the of length in difference
errors n fabricatio by caused disp joint ext
=
=
L
4
7 2005 Pearson Education South Asia Pte Ltd
Example 9.1
Determine the vertical disp of joint C of the
steel truss shown in Fig 9.8(a)
The cross-sectional area of each member =
300mm
2
E =200GPa
8 2005 Pearson Education South Asia Pte Ltd
Example 9.1
Fig 9.8
5
9 2005 Pearson Education South Asia Pte Ltd
Example 9.1 - solution
Virtual force
Only a vertical 1kN load is placed at joint C
The force in each member is calculated
using method of joints
Results are shown in Fig 9.8(b)
Real forces
The real forces are calculated using
method of joints
Results are shown in Fig 9.8(c)
10 2005 Pearson Education South Asia Pte Ltd
Example 9.1 - solution
Virtual work eqn
Arranging the data in tabular form, we have
Table
mm m
m kN m
m kN
kN
AE
m kN
AE
nNL
kN
v
v
v
c
c
c
16 . 6 00616 . 0
] / ) 10 ( 200 )[ ) 10 ( 300 (
6 . 369
. 1
6 . 369
. 1
2 6 2 6
2
2
= =
=
= =
6
11 2005 Pearson Education South Asia Pte Ltd
Example 9.2
The cross-sectional area of each member
shown in Fig 9.9(a)
A =400mm
2
E =200GPa
Determine the vertical disp of joint C if no
loads act on the truss, what would be the
vertical disp of joint C if member AB is 5mm
too short
12 2005 Pearson Education South Asia Pte Ltd
Example 9.2
Fig 9.9
7
13 2005 Pearson Education South Asia Pte Ltd
Example 9.2 - solution
Virtual forces
The support reactions at A & B are
calculated
The n force in each member is determined
using method of joints as shown in Fig
9.9(b)
Applying eqn 9.17
mm m
m kN kN
L n
v
v
c
c
33 . 3 00333 . 0
) 005 . 0 )( 667 . 0 ( . 1
. 1
= =
=
=
14 2005 Pearson Education South Asia Pte Ltd
Example 9.3
Determine the vertical disp of joint C of the
steel truss shown in Fig 9.10(a)
Due to radiant heating from the wall, member
AD is subjected to increase in temp =+60
o
C
Take =1.08(10
-5
)/
o
C and E =200GPa
The cross-sectional area of each member is
indicated in the figure
8
15 2005 Pearson Education South Asia Pte Ltd
Example 9.3
Fig 9.10
16 2005 Pearson Education South Asia Pte Ltd
Example 9.3 - solution
Virtual forces n
The forces in members are computed, Fig
9.10(b)
Real forces N
Since n forces in AB & BC are zero, N
forces need not be computed
Virtual work eqn
Both loads & temp affect the deformation
Eqn 9.15 & 9.16 are combined
9
17 2005 Pearson Education South Asia Pte Ltd
Example 9.3 - solution
Virtual work eqn (contd)
mm m
TL n
AE
nNL
v
v
c
c
93 . 1 00193 . 0
) 4 . 2 )( 60 )]( 10 ( 08 . 1 )[ 1 (
)] 10 ( 200 )[ 10 ( 900
) 3 )( 500 )( 25 . 1 (
)] 10 ( 200 )[ 10 ( 1200
) 4 . 2 )( 400 ( 1
)] 10 ( 200 )[ 10 ( 1200
) 8 . 1 )( 600 ( 75 . 0
. 1
. 1
5
6 6
6 6 6 6
= =
+
+ + =
+ =
18 2005 Pearson Education South Asia Pte Ltd
9-5 Method of virtual work: Beams
& Frames
The Principle of virtual work may be
formulated for beam & frame deflections by
considering the beam shown in Fig 9.11(a)
To compute a virtual unit load acting in the
direction of is placed on the beam at A
The internal virtual moment m is determined
by the method of sections at an arbitrary
location x from the left support, Fig 9.11(b)
When point A is displaced , the element dx
deforms or rotates d =(M/EI)dx
10
19 2005 Pearson Education South Asia Pte Ltd
9-5 Method of virtual work: Beams
& Frames
Fig 9.11
20 2005 Pearson Education South Asia Pte Ltd
9-5 Method of virtual work: Beams
& Frames
9.18 eqn . 1
0
=
L
dx
EI
mM
axis neutral about the computed area, sectional - cross of inertia of moment
material the of elasticity of modulus
loads real by the
caused & x of function a as expressed frame, or beam in the moment int
frame or beam on the acting loads real by caused point the of disp ext
load unit l ext virtua by the caused &
x of function a as expressed frame, or beam in the moment virtual internal
of direction in the frame or beam on the acting load unit virtual external
=
=
=
=
=
=
11
21 2005 Pearson Education South Asia Pte Ltd
9-5 Method of virtual work: Beams
& Frames
If the tangent rotation or slope angle at a
point on the beams elastic curve is to be
determined, a unit couple moment is applied
at the point
The corresponding int moment m
have to be
determined
9.19 eqn . 1
0
=
L
dx
EI
M m
22 2005 Pearson Education South Asia Pte Ltd
9-5 Method of virtual work: Beams
& Frames
If concentrated forces or couple moments act
on the beam or the distributed load is
discontinuous, separate x coordinates will
have to chosen within regions that have no
discontinuity of loading
12
23 2005 Pearson Education South Asia Pte Ltd
Example 9.4
Determine the disp of point B of the steel
beam shown in Fig 9.13(a)
Take
E =200GPa
I =500(10
6
) mm
4
24 2005 Pearson Education South Asia Pte Ltd
Example 9.4
Fig 9.13
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25 2005 Pearson Education South Asia Pte Ltd
Example 9.4 - solution
Virtual moment m
The vertical disp of point B is obtained by
placing a virtual unit load of 1kN at B, Fig
9.13(b)
Using method of sections, the internal
moment m is formulated as shown in Fig
9.13(b)
Real moment M
Using the same x coordinate, M is
formulated as shown in Fig 9.13(c)
26 2005 Pearson Education South Asia Pte Ltd
Example 9.4 - solution
Virtual work eqn
mm m
EI
m kN
kN
dx
EI
x x
dx
EI
mM
kN
B
B
L
B
150 150 . 0
) 10 ( 15
. 1
) 6 )( 1 (
. 1
3 2 3
10
0
2
0
= =
=
= =
14
27 2005 Pearson Education South Asia Pte Ltd
Example 9.5
Determine the tangential rotation at point A of
the steel beam shown in Fig 9.14(a)
Take
E =200GPa
I =60(10
6
) mm
4
28 2005 Pearson Education South Asia Pte Ltd
Example 9.5
Fig 9.14
15
29 2005 Pearson Education South Asia Pte Ltd
Example 9.5 - solution
Virtual moment m
The tangential rotation of point A is
obtained by placing a virtual unit couple of
1kNm at A, Fig 9.14(b)
Using method of sections, the internal
moment m
is formulated as shown in Fig
9.14(b)
Real moment, M
The internal moment is formulated as
shown in Fig 9.14(c)
30 2005 Pearson Education South Asia Pte Ltd
Example 9.5 - solution
Virtual work eqn
rad
dx x
EI
dx
EI
x
dx
EI
M m
m kN
A
L
A
000563 . 0
3
1
3
) 1 (
. . 1
3
0
3
3
0
3
0
=
=
=
=
16
31 2005 Pearson Education South Asia Pte Ltd
Example 9.9
Determine the tangential rotation at point C of
the frame shown in Fig 9.18(a)
Take
E =200GPa
I =15(10
6
) mm
4
32 2005 Pearson Education South Asia Pte Ltd
Example 9.9
Fig 9.18
17
33 2005 Pearson Education South Asia Pte Ltd
Example 9.9 - solution
Virtual moments m
A unit moment is applied at C and the int
moments m
are calculated, Fig 9.18(b)
Real moments M
In a similar manner, the real moments are
are calculated as shown in Fig 9.18(c)
34 2005 Pearson Education South Asia Pte Ltd
Example 9.9 - solution
Virtual work eqn
Using the data in Fig 9.18(b) & 9.18(c), we
have:
( ) ( )
rad
EI
kNm
EI EI
dx
EI
dx
EI
x
dx
EI
M m
C
L
C
00875 . 0
25 . 26 15 25 . 11
5 . 7 ) 1 ( 5 . 2 ) 1 (
. 1
2
2
0
2
3
0
1
1
0
= = + =
+
=
=
