Structure Analysis I Chapter 9

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Nov 15, 2013 (3 years and 11 months ago)

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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