EBK PHYSICS FOR SCIENTISTS AND ENGINEER
EBK PHYSICS FOR SCIENTISTS AND ENGINEER
10th Edition
ISBN: 8220106740163
Author: SERWAY
Publisher: CENGAGE L
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Chapter 12, Problem 27AP

The lintel of prestressed reinforced concrete in Figure P12.27 is 1.50 m long. The concrete encloses one steel reinforcing rod with cross-sectional area 1.50 cm2. The rod joins two strong end plates. The cross-sectional area of the concrete perpendicular to the rod is 50.0 cm2. Young’s modulus for the concrete is 30.0 × 109 N/m2. After the concrete cures and the original tension T1 in the rod is released, the concrete is to be under compressive stress 8.00 × 106 N/m2. (a) By what distance will the rod compress the concrete when the original tension in the rod is released? (b) What is the new tension T2 in the rod? (c) The rod will then be how much longer than its unstressed length? (d) When the concrete was poured, the rod should have been stretched by what extension distance from its unstressed length? (e) Find the required original tension T1 in the rod.

Figure P12.27

Chapter 12, Problem 27AP, The lintel of prestressed reinforced concrete in Figure P12.27 is 1.50 m long. The concrete encloses

(a)

Expert Solution
Check Mark
To determine

The compressed length of the rod when the original tension in the rod is released.

Answer to Problem 27AP

The compressed length of the rod when the original tension in the rod is released is 0.400mm.

Explanation of Solution

The length of the concrete is 1.50m, cross-sectional area of the rod is 1.50cm2, cross-sectional area of the concrete is 50.0cm2, Young’s modulus of the concrete is 30.0×109N/m2, and the compressive stress on concrete is 8.0×106N/m2.

Formula to calculate the compressive strain is,

    Compressivestrain=Δll

Here, Δl is the compressive length and l is the original length.

Formula to calculate the modulus of rigidity of the concrete is,

    Y=CompressivestressCompressivestrain

Substitute Δll in the above equation to find Y.

    Y=CompressivestressΔll

Substitute 30.0×109N/m2 for Y, 8.0×106N/m2 for compressive stress and 1.50m for l in the above equation to find Δl.

    (30.0×109N/m2)=(8.0×106N/m2)Δl(1.50m)Δl=(0.0004m(1000mm1m))=0.4mm

Conclusion:

Therefore, the compressed length of the rod when the original tension in the rod is released is 0.400mm.

(b)

Expert Solution
Check Mark
To determine

The magnitude of the new tension in the rod.

Answer to Problem 27AP

The magnitude of the new tension in the rod is 40.0kN.

Explanation of Solution

Formula to calculate the compressive stress is,

    Compressivestress=T2AC

Here, T2 is the new tension on the concrete and AC is the cross-sectional of the concrete.

Substitute 8.0×106N/m2 for compressive stress, and 50.0cm2 for AC in the above equation to find T2.

    (8.0×106N/m2)=T2(50.0cm2(1m2104cm2))T2=(4000N)(1kN103N)=40.0kN

Conclusion:

Therefore, the magnitude of the new tension in the rod is 40.0kN.

(c)

Expert Solution
Check Mark
To determine

The increase in length of the rod due to tension.

Answer to Problem 27AP

The increase in length of the rod due to tension is 2.00mm.

Explanation of Solution

Formula to calculate the tensile stress on the rod is,

    Tensilestress=T2Ar

Here, Ar is the cross-sectional area of the rod.

Formula to calculate the tensile strain is,

    Tensilestrain=Δltl

Here, Δlt is the increase in length.

Formula to calculate Young’s modulus is,

    Y=TensilestressTensilestrain

Substitute T2Ar for tensile stress and Δltl for tensile strain in the above equation to find Y.

    Y=T2ArΔll

Substitute 40kN for T2, 1.5cm2 for Ar, 30.0×109N/m2 for Y and 1.5m for l in the above equation to find Δl.

    (30.0×109N/m2)=(40kN)(1.5cm2(1m2104cm2))Δl1.5mΔl=(0.00133m(1000mm1m))2.00mm

Conclusion:

Therefore, the increase in length of the rod due to tension is 2.00mm.

(d)

Expert Solution
Check Mark
To determine

The required extension of the rod while concrete was poured.

Answer to Problem 27AP

The required extension of the rod while concrete was poured is 2.40mm.

Explanation of Solution

The length of the concrete is 1.50m, cross-sectional area of the rod is 1.50cm2, cross-sectional area of the concrete is 50.0cm2, Young’s modulus of the concrete is 30.0×109N/m2, and the compressive stress on concrete is 8.0×106N/m2.

Formula to calculate the required extension length of the concrete is,

    Δlrequired=Δl+Δlt

Δlrequired is the required extension length.

Substitute 0.400mm for Δl and 2.0mm for Δlt in the above equation to find Lrequired.

    Δlrequired=(0.400mm)+(2.0mm)=2.40mm

Conclusion:

Therefore, the required extension of the rod while concrete was poured is 2.40mm.

(e)

Expert Solution
Check Mark
To determine

The required original tension on the rod.

Answer to Problem 27AP

The required original tension on the rod is 48.0kN.

Explanation of Solution

Formula to calculate the stress due to original tension is,

    Tensilestress=T1Ar

Here, T1 is the original tension on the rod.

Formula to calculate the tensile strain is,

    Tensilestrain=Δlrequiredl

Formula to calculate Young’s modulus of the steel rod is,

    Ys=TensilestressTensilestrain

Here, Ys is the Young’s modulus of the steel rod.

Substitute T1Ar for tensile stress and Δlrequiredl for tensile strain in the above equation to find T1.

`    Ys=(T1Ar)(Δlrequiredl)T1=(Δlrequiredl)ArYs

Substitute 2.40mm for Δlrequired, 1.5cm2 for Ar, 20.0×1010N/m2 for Ys and 1.5m for l in the above equation to find T1.

    T1=(Δlrequiredl)ArYs=(2.40mm)(1m1000mm)1.5m(1.5cm2(1m2104cm2))(20.0×1010N/m2)=(48000N)(1kN1000N)48.0kN

Conclusion:

Therefore, the required original tension on the rod is 48.0kN.

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Chapter 12 Solutions

EBK PHYSICS FOR SCIENTISTS AND ENGINEER

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