Mechanical Engineering Principles

62 MECHANICAL ENGINEERING PRINCIPLES

(b) For the beam to be in equilibrium, the forces
acting upwards must be equal to the forces
acting downwards, thus

RA+RB=( 2 + 7 + 3 )kN

=12 kN

RB= 6 .3kN,

thus RA= 12 − 6. 3 = 5 .7kN

Problem 7. For the beam shown in

Figure 5.12 calculate (a) the force acting on

supportA, (b) distanced, neglecting any

forces arising from the mass of the beam.

10 N 15 N 30 N

40 N

0.5 m

1.0 m

2.0 m

2.5 m

d

A

RA B

Figure 5.12

(a) From Section 5.2, (the forces acting in an up-

ward direction) = (the forces acting in a

downward direction)

Hence (RA+ 40 )N=( 10 + 15 + 30 )N

RA= 10 + 15 + 30 − 40

=15 N

(b) Taking moments about the left-hand end of the
beam and applying the principle of moments
gives:

clockwise moments=anticlockwise moments

( 10 × 0. 5 )+( 15 × 2. 0 )Nm+30 N×d

=( 15 × 1. 0 )+( 40 × 2. 5 )Nm

i.e. 35 N m+30 N×d=115 N m

from which,distance,

d=

( 115 − 35 )Nm

30 N

= 2 .67 m

Problem 8. A metal barABis 4.0 m long

and is supported at each end in a horizontal

position. It carries loads of 2.5 kN and

5.5 kN at distances of 2.0 m and 3.0 m,

respectively, fromA. Neglecting the mass of

the beam, determine the reactions of the

supports when the beam is in equilibrium.

The beam and its loads are shown in Figure 5.13.

At equilibrium,

RA+RB= 2. 5 + 5. 5 = 8 .0kN ( 1 )

AB

RA RB

2.0 m

2.5 kN 5.5 kN

1.0 m

Figure 5.13

Taking moments aboutA,

clockwise moments=anticlockwise moment,

i.e. ( 2. 5 × 2. 0 )+( 5. 5 × 3. 0 )= 4. 0 RB

or 5. 0 + 16. 5 = 4. 0 RB

from which, RB=

21. 5

4. 0

= 5 .375 kN

From equation (1),RA= 8. 0 − 5. 375 = 2 .625 kN

Thus the reactions at the supports at equilibrium

are 2.625 kN atAand 5.375 kN atB

Problem 9. A beamPQis 5.0 m long and

is supported at its ends in a horizontal

position as shown in Figure 5.14. Its mass is

equivalent to a force of 400 N acting at its

centre as shown. Point loads of 12 kN and

20 kN act on the beam in the positions

shown. When the beam is in equilibrium,

determine (a) the reactions of the supports,

RPandRQ, and (b) the position to which

the 12 kN load must be moved for the force

on the supports to be equal.