Structure Analysis I
Chapter
9
Deflection
Energy Method
Structural Analysis
I
Dr. Mohammed Arafa
Energy Method
P
U
dx
F
U
e
x
e
2
1
0
When a force F undergoes a displacement
dx
in the same direction as the force, the work
done is
If the total displacement is
x
the work become
dx
F
dU
e
External Work
The force applied gradually
Energy Method
'
'
e
U P
If P is already applied to the bar and that
another force F` is now applied
The work done by P when the bar undergoes
the further deflection
` is then
External Work
The work of a moment is defined by the
product of the magnitude of the moment
M
and the angle then
If the total angle of rotation is the work
become:
d
M
dU
e
d
M
U
d
M
U
e
e
2
1
0
The moment applied gradually
Structural Analysis
I
Dr. Mohammed Arafa
Energy Method
Strain Energy
–
Axial Force
L
A
N
E
AE
NL
AE
L
N
U
P
U
N
P
i
2
2
2
1
N =
internal normal force in a truss member caused
by the real load
L =
length of member
A =
cross

sectional area of a member
E =
modulus of elasticity of a member
Energy Method
Strain Energy
–
Bending
M
U
dx
EI
M
d
2
1
L
i
i
EI
dx
M
U
EI
dx
M
dU
0
2
2
2
2
Principle of Virtual Work
u
P
Work of
External
Loads
Work of
Internal
Loads
dL
u
.
.
1
Virtual Load
Real displacement
Method of Virtual Work: Trusses
External Loading
AE
NL
n
.
.
1
dL
u
.
.
1
1
=
external virtual unit load acting on the truss joint in the stated direction of
n =
internal virtual normal force in a truss member caused by the external
virtual unit load
= external joint displacement caused by the real load on the truss
N =
internal normal force in a truss member caused by the real load
L =
length of member
A =
cross

sectional area of a member
E =
modulus of elasticity of a member
Method of Virtual Work: Trusses
Temperature
1. T L
n
1
=
external virtual unit load acting on the truss joint in the stated direction of
n =
internal virtual normal force in a truss member caused by the external virtual
unit load
=
external joint displacement caused by the temperature change.
=
coefficient of thermal expansion of member
T
=
change in temperature of member
L =
length of member
Method of Virtual Work: Trusses
Fabrication Errors and Camber
1. L
n
1
=
external virtual unit load acting on the truss joint in the stated direction of
n =
internal virtual normal force in a truss member caused by the external virtual
unit load
=
external joint displacement caused by fabrication errors
L
=
difference in length of the member from its intended size as caused by a
fabrication error.
Structural Analysis
I
Dr. Mohammed Arafa
Example
1
The cross sectional area of each member of the truss show, is A =
400
mm
2
& E =
200
GPa.
a) Determine the vertical displacement of joint C if a
4

kN force is applied
to the truss at C
A virtual force of
1
kN is applied at C in the vertical direction
AE
NL
n
.
.
1
Solution
Member
n (KN)
N (KN)
L (m)
nNL
AB
0.667
2
8
10.67
AC

0.833
2.5
5

10.41
CB

0.833

2.5
5
10.41
Sum
10.67
mm
m
AE
AE
nNL
m
kN
m
kN
133
.
0
000133
.
0
10
200
10
400
)
67
.
10
67
.
10
.
1
)
/
(
6
)
(
6
(
2
2
Structural Analysis
I
Dr. Mohammed Arafa
Text book Example
8

14
Determine vertical displacement at C
A =
0.5
in
2
E =
29
(
10
)
3
ksi
Example
2
Example
3
Method of Virtual Work: Beam
dx
EI
M
d
dL
u
.
.
1
dx
EI
M
m
L
0
.
1
1
=
external virtual unit load acting on the beam in the stated direction of
Δ
m =
internal virtual moment in a beam caused by the external virtual unit load
Δ
=
external joint displacement caused by the real load on the truss
M =
internal moment in a beam caused by the real load
L =
length of beam
I =
moment of inertia of cross

sectional
E =
modulus of elasticity of the beam
Method of Virtual Work: Beam
dx
EI
M
m
L
m
KN
0
)
.
(
.
1
Similarly the rotation angle at any point on the beam can be
determine, a unit couple moment is applied at the point and the
corresponding internal moment have to be determine
m
Example
4
Determine the displacement at point B of a steel beam
E =
200
Gpa , I =
500
(
10
6
) mm
4
Structural Analysis
I
Dr. Mohammed Arafa
Solution
m
EI
EI
x
EI
dx
x
EI
dx
x
x
dx
EI
M
m
L
15
.
0
)
10
)(
10
(
500
)
10
(
200
)
10
(
15
)
10
(
15
4
6
6
)
6
(
)
1
(
.
1
12
6
6
3
3
10
0
4
10
0
3
10
0
2
0
m
B
B
15
.
0
5
.
7
)
10
)(
10
(
500
)
10
(
200
2000
12
6
6
Another Solution
Real Load
Virtual Load
Example
5
Example
6
Determine the Slope
at
point B of a steel beam
E =
200
Gpa
, I =
60
(
10
6
) mm
4
Solution
Virtual Load
rad
0094
.
0
)
10
(
60
)
10
(
200
2
)
5
(
3
)
10
(
3
2
3
3
)
3
(
)
1
(
)
3
(
)
0
(
.
1
6
6
2
2
10
5
2
10
5
10
5
5
0
0
B
L
EI
x
EI
dx
x
EI
dx
x
EI
dx
x
dx
EI
M
m
Real Load
rad
EI
EI
B
0094
.
0
112
1
112
6
9
10
60
)
10
(
200
112
Another Solution
Real Load
Virtual Load
Example
7
Example
8
Example
9
Determine both the horizontal deflection at
A
Example
4
Solution
Real Load
Virtual Load
m
EI
EI
EI
A
031
.
0
1250
5
.
2
500
0
100
6
9
10
200
)
10
(
200
1250
Structural Analysis
I
Dr. Mohammed Arafa
Virtual Strain Energy
Caused by: Axial Load
n
nNL
U
AE
n =
internal virtual axial load caused by the external virtual unit load
N =
internal normal force in the member caused by the real load
L =
length of member
A =
cross

sectional area of the member
E =
modulus of elasticity of the material
Virtual Strain Energy
Caused by: Shear
0
L
S
V
U K dx
GA
/
/
/
/
S
dy dx
G
dy G dx
K V A
V
dy K dx
GA
dU dy K V A dx
0
L
S
V
U K dx
GA
=
internal virtual shear in the member caused by the external virtual unit load
V =
internal shear in the member caused by the real load
G=
shear modulus of elasticity for the material
A =
cross

sectional area of the member
K =
form factor for the cross

sectional area
K=
1.2
for rectangular cross sections
K=
10
/
9
for circular cross sections
K=
1.0
for wide

flange and I

beams where A is the area of the web.
Virtual Strain Energy
Caused by: Shear
Virtual Strain Energy
Caused by: Torsion
t
tTL
U
GJ
/
/
t
cd dx
G
Tc
J
T
d dx dx dx
c Gc GJ
tT
dU td dx
GJ
Virtual Strain Energy
Caused by: Torsion
t
=
internal virtual torque caused by the external virtual unit load
T=
internal shear in the member caused by the real load
G=
shear modulus of elasticity for the material
J =
polar moment of inertia of the cross

sectional
L =
member length
t
tTL
U
GJ
Structural Analysis
I
Dr. Mohammed Arafa
Virtual Strain Energy
Caused by:
Temperature
0
m
L
m
temp
T
d dx
c
m T
U dx
c
Virtual Strain Energy
Caused by: Temperature
0
L
m
temp
m T
U dx
c
m
=
internal virtual moment in the beam caused by the external virtual unit load or
unit moment
=
coefficient of thermal expansion
Tm
=
change in temperature between the mean temperature and the temperature
at the top or the bottom of the beam
c =
mid

depth of the beam
Example
10
Example
11
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