Mechanical Engineering Principles

THE EFFECTS OF FORCES ON MATERIALS 13

Therefore, drop in thermal strain

=

− 30 × 106 Pa

2 × 1011 Pa

=− 1. 5 × 10 −^4 =αT

from which, temperature

T=

− 1. 5 × 10 −^4

14 × 10 −^6

=− 10. 7 °C

Hence, the drop in temperatureT from 20°C

is−10.7°C

Therefore,the temperature for the prop to

be ineffective= 20 °− 10. 7 °= 9. 3 °C

Now try the following exercise

Exercise 4 Further problem on thermal

strain

1. A steel rail may assumed to be stress
   free at 5°C. If the stress required to cause
   buckling of the rail is−50 MPa, at what
   temperature will the rail buckle?. It may
   be assumed that the rail is rigidly fixed at
   its ‘ends’.
   TakeE= 2 × 1011 N/m^2 and
   α= 14 × 10 −^6 /°C. [22.86°C]

1.12 Compound bars

Compound bars are of much importance in a num-

ber of different branches of engineering, including

reinforced concrete pillars, composites, bimetallic

bars, and so on. In this section, solution of such

problems usually involve two important considera-

tions, namely

(a) compatibility

(or considerations of displacements)

(b) equilibrium

N.B. It is necessary to introduce compatibility in

this section as compound bars are, in general, stat-

ically indeterminate (see Chapter 4). The follow-

ing worked problems demonstrate the method of

solution.

Problem 19. A solid bar of cross-sectional

areaA 1 , Young’s modulusE 1 and coefficient

of linear expansionα 1 is surrounded

co-axially by a hollow tube of cross-sectional

areaA 2 , Young’s modulusE 2 and coefficient

of linear expansionα 2 , as shown in

Figure 1.12. If the two bars are secured

firmly to each other, so that no slipping takes

place with temperature change, determine the

thermal stresses due to a temperature riseT.

Both bars have an initial lengthLand

α 1 >α 2

Bar

L

1

Bar 2

Figure 1.12 Compound bar

α 1 LT

L α 2 LT

ε 1 L

ε 2 L

A

A

Bar 1

Bar 2

Figure 1.13 “Deflections” of compound bar

There are two unknown forces in these bars, namely

F 1 andF 2 ; therefore, two simultaneous equations

will be required to determine these unknown forces.

The first equation can be obtained by considering

the compatibility (i.e.‘deflections’) of the bars, with

the aid of Figure 1.13.

Free expansion of bar (1)=α 1 LT

Free expansion of bar (2)=α 2 LT

In practice, however, the final resting position of

the compound bar will be somewhere in between