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_1:  Reduce a force system depicted below at a point belonging to the central axis. 
 

 

 
 
Solution: 
 
1. Finding the coordinates of the forces 
 

























kN

F

kN

F

kN

m

m

kN

F

kN

m

m

kN

F

0

,

14

,

0

0

,

0

,

16

4

,

4

,

8

6

2

2

,

2

,

4

6

4

6

,

10

,

8

50

3

,

5

,

4

50

2

4

3

2

1



























 

 
2. Calculating the sum vector

. 

 





kN

S

2

,

0

,

0



 

3. Computing the total moment about a chosen point (point L)

. 





























kNm

kNm

kNm

m

kN

m

kN

KL

F

EL

F

M

L

96

,

40

,

12

80

,

0

,

0

16

,

40

,

12

0

,

5

,

0

0

,

0

,

16

3

,

0

,

4

4

,

4

,

8

3

2



























 

4. Determining the parameter of the system. 
 



 



0

192

96

,

40

,

12

2

,

0

,

0

2

2









m

kN

m

kN

M

S

k

L





 

Parameter of the system is different from zero hence at the points belonging to the central axis the system is 
reduced to the wrench. 
 
5. Finding the moment of the wrench 
 









kNm

kN

kN

m

kN

S

S

k

M

P

96

,

0

,

0

2

,

0

,

0

4

192

2

2

2







 

 

 

 

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. 

2

 

 
6. Determining the equation of the central axis  
 

LP

S

M

M

L

P







 







5

,

5

,

0

,

,

,





L

z

y

x

P



 

















m

z

y

x

kN

kNm

kNm

5

,

5

,

2

,

0

,

0

96

,

40

,

12

96

,

0

,

0











 

 

























0

96

96

2

40

0

10

2

12

0

x

y

 

 
The equation of the central axis. 













20

11

x

y

    

 
7. Choosing an arbitrary point  belonging to the central axis. 
 





0

,

11

,

20



P

 

 
8. Answer 
 
At the point P belonging to the central axis the system is reduced to the wrench comprising one force, equal 
to the sum, applied at this point and a couple with the moment 

P

M , parallel to the sum vector. 

 



























m

P

kN

S

F

0

,

11

,

20

2

,

0

,

0

   and a couple  





kNm

M

P

96

,

0

,

0



.