Mechanical Engineering Principles

46 MECHANICAL ENGINEERING PRINCIPLES

Now try the following exercise

Exercise 20 Further problems on a graph-

ical method

Determine the internal forces in the following

pin-jointed trusses using a graphical method:

1.

12

3

60 ° 30 °

R 1 R 2

4 kN

Figure 4.23

⎡

⎢

⎢

⎣

R 1 = 3 .0kN, R 2 = 1 .0kN,

1–2, 1.7 kN, 1–3,− 3 .5kN,

2–3,− 2 .0kN

⎤

⎥

⎥

⎦

2.

R (^1) R
2
60 ° 30 ° H 2
3
2
6 kN
1
Figure 4.24
⎡
⎢
⎢
⎣
R 1 =− 2 .6kN, R 2 = 2 .6kN,
H 2 = 6 .0 kN, 1–2,− 1 .5kN,
1–3, 3.0 kN, 2–3,− 5 .2kN
⎤
⎥
⎥
⎦
3.
R 1 R 2
H 11 45 °
3
2
4 kN
6 kN
45 °
Figure 4.25
⎡
⎢
⎢
⎣
R 1 = 5 .0kN, R 2 = 1 .0kN,
H 1 = 4 .0kN 1–2,1.0kN,
1–3,−7.1 kN, 2–3,−1.4 kN
⎤
⎥
⎥
⎦
4.
2 kN
4 kN
6 kN
3 5
2
4
(^130) ° 30 ° 30 ° 30 °
6
12 m
R 1 R 2
Figure 4.26
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
R 1 = 5 .0kN, R 2 = 7 .0kN,
1–3,− 10 .0kN, 1–6,− 8 .7kN,
3–4,− 8 .0kN, 3–6,− 2 .0kN,
4–6, 4.0kN, 4–5,8.0kN,
5–6,− 6 .0kN, 5–2,− 14 .0kN,
6–2, 12.1kN
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

4.4 Method of joints (a mathematical

method)

In this method, all unknown internal member forces

are initially assumed to be in tension. Next, an

imaginary cut is made around a joint that has two

or less unknown forces, so that a free body diagram

is obtained for this joint. Next, by resolving forces

in respective vertical and horizontal directions at

this joint, the unknown forces can be calculated.

To continue the analysis, another joint is selected

with two or less unknowns and the process repeated,

remembering that this may only be possible because

some of the unknown member forces have been

previously calculated. By selecting, in turn, other

joints where there are two or less unknown forces,

the entire framework can be analysed.

It must be remembered that if the calculated force

in a member isnegative, then that member is in

compression. Vice-versa is true for a member in

tension.

To demonstrate the method, some pin-jointed

trusses will now be analysed in Problems 8 to 10.