• Author / Uploaded
• Sandeep Petwal

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PROBLEM 2.50 The concrete post ( Ec 3.6 106 psi and c 5.5 10 6 / F) is reinforced with six steel bars, each of 78 -in. diameter ( Es 29 106 psi and s 6.5 10 6 / F). Determine the normal stresses induced in the steel and in the concrete by a temperature rise of 65°F. 6 ft

10 in.

10 in.

SOLUTION

Let Pc

As

6

Ac

102

4

d

7 6 4 8

2

102

As

2

3.6079 in 2 3.6079 96.392 in 2

tensile force developed in the concrete.

For equilibrium with zero total force, the compressive force in the six steel rods equals Pc . Strains:

s

Matching:

c

Pc Es As

T)

s(

Pc Ec Ac

s

c

c(

1 Ec Ac

T)

Pc Ec Ac Pc Es As

1 Pc Es As

1 (3.6 10 )(96.392) 6

Pc c

s

Pc Ac

5.2254 103 96.392

Pc As

c(

(

T) T)

s(

s

c )(

T)

1 Pc (29 10 )(3.6079) 6

(1.0 10 6 )(65)

5.2254 103 lb

54.210 psi

5.2254 103 3.6079

c

1448.32 psi

s

54.2 psi

1.448 ksi

PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 148

PROBLEM 2.51

A 30-mm diameter

250 mm

A rod consisting of two cylindrical portions AB and BC is restrained at both ends. Portion AB is made of steel ( Es 200 GPa, s 11.7 10 6 / C) and portion BC is made of brass ( Eb 105 GPa, b 20.9 10 6 / C). Knowing that the rod is initially unstressed, determine the compressive force induced in ABC when there is a temperature rise of 50 C.

B 50-mm diameter 300 mm

C

SOLUTION AAB ABC

4 4

2 d AB 2 d BC

4 4

(30) 2

706.86 mm 2

706.86 10 6 m 2

(50)2

1.9635 103 mm 2

1.9635 10 3 m 2

Free thermal expansion: T

LAB

s(

T ) LBC

b(

T)

(0.250)(11.7 10 6 )(50) (0.300)(20.9 10 6 )(50) 459.75 10 6 m Shortening due to induced compressive force P: P

PL Es AAB

PL Eb ABC

0.250 P (200 10 )(706.86 10 6 ) 9

0.300 P (105 10 )(1.9635 10 3 ) 9

3.2235 10 9 P For zero net deflection,

P

T

3.2235 10 9 P

459.75 10

6

P 142.624 103 N

P 142.6 kN

PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 149

24 in. A

PROBLEM 2.52

32 in. B

A rod consisting of two cylindrical portions AB and BC is restrained at both ends. Portion AB is made of steel ( Es 29 106 psi, s 6.5 10 6 / F) and portion BC is made of aluminum ( Ea 10.4 106 psi, a 13.3 10 6 /°F). Knowing that the rod is initially unstressed, determine (a) the normal stresses induced in portions AB and BC by a temperature rise of 70°F, (b) the corresponding deflection of point B.

C

1 2 14 -in. diameter 1 2 -in. diameter

SOLUTION AAB T

Free thermal expansion.

4

(2.25) 2

3.9761 in 2 ABC

4

(1.5)2

1.76715 in 2

70 F

(

T ) AB

LAB

s(

T)

(24)(6.5 10 6 )(70) 10.92 10 3 in.

(

T ) BC

LBC

a(

T)

(32)(13.3 10 6 )(70)

Total:

T

(

T ) AB

(

T ) BC

29.792 10 3 in.

40.712 10 3 in.

Shortening due to induced compressive force P. (

P ) AB

PLAB Es AAB

24 P (29 106 )(3.9761)

(

P ) BC

PLBC Ea ABC

32 P (10.4 106 )(1.76715)

Total:

P

For zero net deflection,

T

P ) AB

(

P ) BC

1949.32 10 9 P

AB

P AAB

20.885 103 3.9761

5.25 103 psi

BC

P ABC

20.885 103 1.76715

11.82 103 psi

(a)

(b)

P

(

(

P ) AB B

(208.14 10 9 )(20.885 103 ) (

T ) AB

(

P ) AB

208.14 10 9 P 1741.18 10 9 P

1949.32 10 9 P 40.712 10

3

20.885 103 lb

P

5.25 ksi

AB

BC

11.82 ksi

4.3470 10 3 in.

10.92 10

3

4.3470 10

3

B

6.57 10

3

in.

or (

P ) BC B

(1741.18 10 9 )(20.885 103 ) 36.365 10 3 in. (

T ) BC

(

P ) BC

29.792 10

3

36.365 10

3

6.57 10 3 in.

(checks)

PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 150