Compound Bars
THE EFFECTS OF FORCES ON MATERIALS Therefore, drop in thermal strain = −30 × 106 Pa 2 × 1011 Pa = −1.5 × 10−4 = αT f
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THE EFFECTS OF FORCES ON MATERIALS
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 temperature T from 20 ° C is −10.7 ° C
Problem 19. A solid bar of cross-sectional area A1 , Young’s modulus E1 and coefficient of linear expansion α1 is surrounded co-axially by a hollow tube of cross-sectional area A2 , Young’s modulus E2 and coefficient of linear expansion α2 , 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 rise T . Both bars have an initial length L and α1 > α2 Bar 1 Bar 2
Therefore, the temperature for the prop to be ineffective = 20° − 10.7° = 9.3 ° C Now try the following exercise
13
L
Figure 1.12 Compound bar
Exercise 4 Further problem on thermal strain A
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’. Take E = 2 × 1011 N/m2 and [22.86 ° C] α = 14 × 10−6 /° C.
Bar 1
α1LT
ε1L Bar 2
ε2L
L
α2LT
1.12 Compound bars Compound bars are of much importance in a number 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 considerations, 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, statically indeterminate (see Chapter 4). The following worked problems demonstrate the method of solution.
A
Figure 1.13
“Deflections” of compound bar
There are two unknown forces in these bars, namely F1 and F2 ; 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) = α1 LT Free expansion of bar (2) = α2 LT In practice, however, the final resting position of the compound bar will be somewhere in between
THE EFFECTS OF FORCES ON MATERIALS
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 temperature T from 20 ° C is −10.7 ° C
Problem 19. A solid bar of cross-sectional area A1 , Young’s modulus E1 and coefficient of linear expansion α1 is surrounded co-axially by a hollow tube of cross-sectional area A2 , Young’s modulus E2 and coefficient of linear expansion α2 , 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 rise T . Both bars have an initial length L and α1 > α2 Bar 1 Bar 2
Therefore, the temperature for the prop to be ineffective = 20° − 10.7° = 9.3 ° C Now try the following exercise
13
L
Figure 1.12 Compound bar
Exercise 4 Further problem on thermal strain A
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’. Take E = 2 × 1011 N/m2 and [22.86 ° C] α = 14 × 10−6 /° C.
Bar 1
α1LT
ε1L Bar 2
ε2L
L
α2LT
1.12 Compound bars Compound bars are of much importance in a number 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 considerations, 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, statically indeterminate (see Chapter 4). The following worked problems demonstrate the method of solution.
A
Figure 1.13
“Deflections” of compound bar
There are two unknown forces in these bars, namely F1 and F2 ; 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) = α1 LT Free expansion of bar (2) = α2 LT In practice, however, the final resting position of the compound bar will be somewhere in between