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. 

Homework 4 

 

Problem 1.  
 

Given the force system shown below: 
I.    Find the equivalent coupe-force system at point  A. 
II.   Find the equivalent coupe-force system at point  B.  
III.  Find the simplest equivalent force system. Draw the result of reduction on the figure. 
  
 

 

                                  
            

2m

3m

1m

2m

2m

2 kN

10 kN

5 kN

 kN

 kN

 kN

A

B

2 kNm

 
 
 
 
 
 
 
 
 
 
 
 
 
Answers: 

 

I 

 

The equivalent couple-force system at point A consists of a force 

)

],

[

2

],

[

2

(

O

kN

kN

S





 applied at point A, 

and a couple with the  moment 

])

[

4

,

0

,

0

(

kNm

M

A





 

II.  

The equivalent couple-force system at point A consists of a force 

)

],

[

2

],

[

2

(

O

kN

kN

S





 applied at point B, 

and a couple with the  moment 

])

[

6

,

0

,

0

(

kNm

M

B





 

 

III

.  (The origin of a coordinate system is at point A) 

 
The simplest equivalent force system is the resultant force equal 

)

],

[

2

],

[

2

(

O

kN

kN

S





 acting along the 

central axis 

 

]

[

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. 

 
 

2m

3m

1m

2m

2m

2 kN

10 kN

5 kN

 kN

 kN

 kN

A

B

2 kNm

2m

3m

1m

2m

2m

A

B

y=

-x-

[

2

m

]

W=S(-2[kN ],2[kN ],0)

x

 
 
 
 
 

                                                                              



 

  
 
 
 
 
 
 
 
 
 
 
 

Problem 2.  

Reduce the given force system at a point belonging to the central axis (if it’s possible). 

he origin of a coordinate system is at point O. 

 
 

13 kN

6

2 kN

10 kNm

q=4 kN/m

20 kN

4m

3m

5m

=1 kN/m2

45°

O

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 

 
 
 

(

])

[

20

,

0

,

0

(

kNm

M

O





) 

T

 

 

 

 

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. 

 
Answer: 

At an arbitrary point belonging to the central axis described  by the equation 

]

[

5

4

5

m

x

y







, the force 

)

],

[

5

],

[

4

(

O

kN

kN

S

W



system reduces to the resultant force 

13 kN

6

2 kN

10 kNm

q=4 kN/m

20 kN

4m

3m

5m

=1 kN/m2

45°

O

4m

3m

5m

O

y=

-5

/4

x+

5[m

]

W=S(4[kN ],-5[kN ],0)

x

y





. 

                                                                    

 

                                                                   

  
 
 
 
 
 
 
 

 

