Project “The development of the didactic potential of Cracow University of Technology in the range of modern
construction” is co-financed by the European Union within the confines of the European Social Fund
and realized under surveillance of Ministry of Science and Higher Education.
1
Exercise 4.2
For a given planar force system find:
I
An equivalent coupe-force system at point A.
II.
The simplest equivalent force system. Draw the result of reduction on the figure.
q
1
=4 kN/m
17 kNm
q
2
=
2 kN/m
14 kN
10 kN
15 kN
3
2 kN
2m
4m
1m
2m
=3 kN/m2
A
x
y
The origin of the coordinate system is positioned at point A
I. An equivalent couple-force system at point A
Replacing the distributed loads by their resultants.
q
1
=4 kN/m
M
1
=17 kNm
q
2
=
2 kN/m
F
4
=14 kN
F
1
=10 kN
15 kN
3
2 kN
2m
4m
1m
2m
=3 kN/m2
A
x
y
F
6
F
5
W
1
W
2
2
3
*l
2
1
3
*l
2
W
3
F
2
F
3
1
l
2
construction” is co-financed by the European Union within the confines of the European Social Fund
and realized under surveillance of Ministry of Science and Higher Education.
2
])
[
4
],
/
[
4
(
1
1
1
1
1
m
l
m
kN
q
l
q
W
]
[
16
]
[
4
]
/
[
4
1
kN
m
m
kN
W
])
3
[
],
/
[
2
(
2
1
2
2
2
2
2
m
l
m
kN
q
l
q
W
]
[
3
]
[
3
]
/
[
2
2
1
2
kN
m
m
kN
W
])
[
2
]
[
2
]
[
1
],
/
[
3
(
2
2
3
m
m
m
area
m
kN
area
W
]
[
6
]
[
2
]
/
[
3
2
2
3
kN
m
m
kN
W
Resolving the oblique forces into components
]
[
2
3
)
45
sin(
2
kN
F
,
]
[
2
3
)
45
cos(
3
kN
F
]
[
3
3
2
kN
F
F
Where:
5
4
)
cos(
,
5
3
)
sin(
(see the figure)
]
[
12
]
[
15
)
cos(
5
kN
kN
F
]
[
9
]
[
15
)
sin(
6
kN
kN
F
Determining the sum vector
)
,
,
(
O
S
S
S
y
x
2
5
2
1
W
F
F
F
S
x
]
[
2 kN
S
x
3
1
6
4
3
W
W
F
F
F
S
y
]
[
2 kN
S
y
)
],
[
2
],
[
2
(
O
kN
kN
S
Detrmining the total moment about point A
Project “The development of the didactic potential of Cracow University of Technology in the range of modern
M
1
F
4
F
1
2m
2m
1m
2m
A
x
y
F
6
F
5
W
1
W
2
W
3
F
2
F
3
r
F1
r
F3
r
F5
r
F6
r
W3
r
W1
2m
r
W2
+
-
1m
2m
Project “The development of the didactic potential of Cracow University of Technology in the range of modern
construction” is co-financed by the European Union within the confines of the European Social Fund
and realized under surveillance of Ministry of Science and Higher Education.
3
)
,
0
,
0
(
Az
A
M
M
(for a planar force system)
1
3
3
2
2
1
1
6
6
5
5
3
3
1
1
M
r
W
r
W
r
W
r
F
r
F
r
F
r
F
M
W
W
W
F
F
F
F
Az
Remark: the moment of F
2
and F
4
about A equal zero, because their lines of action pass through that point.
]
[
4
]
[
17
]
[
1
]
[
6
]
[
1
]
[
3
]
[
2
]
[
16
]
[
4
]
[
9
]
[
2
]
[
12
]
[
2
]
[
3
]
[
1
]
[
10
kNm
kNm
m
kN
m
kN
m
kN
m
kN
m
kN
m
kN
m
kN
M
Az
])
[
4
,
0
,
0
(
kNm
M
A
Answer
The planar system of forces can be reduced at point A to a coupe-force system comprising one force
)
],
[
2
],
[
2
(
O
kN
kN
S
applied at point A, and one couple with a moment
])
[
4
,
0
,
0
(
kNm
M
A
,
II. The simplest equivalent system
)
],
[
2
],
[
2
(
O
kN
kN
S
,
])
[
4
,
0
,
0
(
kNm
M
A
hence the parameter of the system
0
A
M
S
k
0
0
k
S
the system can be reduced to a resultant force.
The equation of a central axis
0
);
,
,
(
P
M
z
y
x
P
P
A
S
M
M
A
p
)
0
,
,
(
)
0
,
,
(
int
)
0
,
0
,
0
(
int
y
x
P
A
vector
y
x
P
po
A
po
)
0
,
,
(
)
0
],
[
2
],
[
2
(
y
x
P
A
kN
kN
S
)
]
[
2
]
[
2
,
0
,
0
(
x
kN
y
kN
P
A
S
P
A
S
M
A
0
)
]
[
2
]
[
2
,
0
,
0
(
])
[
4
,
0
,
0
(
)
0
,
0
,
0
(
x
kN
y
kN
kNm
[
2
]
[
2
]
[
2
]
[
4
0
m
x
y
x
kN
y
kN
kNm
]
]
[
2 m
x
y
the central axis (or the line of action of the resultant force.)
Answer
The given planar system of forces can be reduced to a resultant force equal to the sum vector, acting along the
central
]
[
2 m
x
y
Project “The development of the didactic potential of Cracow University of Technology in the range of modern
construction” is co-financed by the European Union within the confines of the European Social Fund
and realized under surveillance of Ministry of Science and Higher Education.
4
q
1
=4 kN/m
17 kNm
q
2
=
2 kN/m
14 kN
10 kN
15 kN
3
2 kN
2m
4m
1m
2m
=3 kN/m2
A
x
y
y=
x-
2[
m
]
W=S(-2[kN ],-2[kN ])