Mechanical Engineering Principles

52 MECHANICAL ENGINEERING PRINCIPLES

4.5 The method of sections

(a mathematical method)

In this method, an imaginary cut is made through

the framework and the equilibrium of this part of

the structure is considered through a free body

diagram. No more than three unknown forces can

be determined through any cut section, as only three

equilibrium considerations can be made, namely

(a) resolve forces horizontally

(b) resolve forces vertically

(c) take moments about a convenient point.

Worked Problem 11 demonstrates the method of

sections.

Problem 11. Determine the unknown

member forcesF 2 ,F 5 andF 6 of the truss of

Figure 4.34, Problem 10, by the method of

sections (whereF 2 ,F 5 andF 6 are defined in

the solution on pages 49 to 51).

Firstly, all members will be assumed to be in tension

and an imaginary cut will be made through the

framework, as shown by Figure 4.40.

B

A 30 ° 30 ° 30 ° 30 °

60 ° 60 °

4 kN

3 kN

5 kN

R 1 = 5.75 kN R 2 = 6.25 kN

Imaginary

cut

Figure 4.40

Taking moments about B;see Figure 4.41.

Clockwise moments=anti-clockwise moments

Hence, 5.75 kN×2m=F 2 × 1 .155 m

where 2 tan 30°= 1 .155 m (from Figure 4.41)

i.e. F 2 =

5. 75 × 2

1. 155

= 9 .96 kN(tensile) ( 4. 28 )

1.155 m

4 kN

B

F 6

F 5

F 2

R 1 = 5.75 kN

A

2 m

30 °

30 °

30 °

Figure 4.41

Resolving forces vertically:

5 .75 kN+F 6 sin 30°=F 5 sin 30°+4kN

i.e. F 5 =F 6 +

5. 75

0. 5

−

4

0. 5

i.e. F 5 =F 6 + 3. 5 ( 4. 29 )

Resolving forces horizontally:

0 =F 2 +F 5 cos 30°+F 6 cos 30°

from which, F 5 cos 30°=−F 2 −F 6 cos 30°

and F 5 =−

F 2

cos 30°

−F 6 ( 4. 30 )

Substituting equation (4.28) into equation (4.30)

gives:

F 5 =−

9. 96

0. 866

−F 6

or F 5 =− 11. 5 −F 6 ( 4. 31 )

Equating equation (4.29) to equation (4.31) gives:

F 6 + 3. 5 =− 11. 5 −F 6

from which, 2F 6 =− 11. 5 − 3. 5 =− 15

and F 6 =−

15

2

=− 7 .5kN(compressive)

( 4. 32 )

Substituting equation (4.32) into equation (4.31)

gives:

F 5 =− 11. 5 −(− 7. 5 )

=−4kN(compressive) ( 4. 33 )