   Chapter 11, Problem 16P

Chapter
Section
Textbook Problem

In the summer of 1958 in St. Petersburg, Florida, a new sidewalk was poured near the childhood home of one of the authors. No expansion joints were supplied, and by mid-July, the sidewalk had been completely destroyed by thermal expansion and had to be replaced, this time with the important addition of expansion joints! This event is modeled here. A slab of concrete 4.00 cm thick, 1.00 m long, and 1.00 m wide is poured for a sidewalk at an ambient temperature of 25.0°C and allowed to set. The slab is exposed to direct sunlight and placed in a series of such slabs without proper expansion joints, so linear expansion is prevented. (a) Using the linear expansion equation (Eq. 10.4), eliminate ΔL from the equation for compressive stress and strain (Eq. 9.3). (b) Use the expression found in pan (a) to eliminate ΔT from Equation 11.3, obtaining a symbolic equation for thermal energy transfer Q. (c) Compute the mass of the concrete slab given that its density is 2.40 × 103 kg/m3. (d) Concrete has an ultimate compressive strength of 2.00 × 107 Pa, specific heat of 880J/kg · °C, and Young’s modulus of 2.1 × 1010 Pa. How much thermal energy must be transferred to the slab to reach this compressive stress? (e) What temperature change is required? (f) If the Sun delivers 1.00 × 103 W of power to the top surface of the slab and if half the energy, on the average, is absorbed and retained, how long does it take the slab to reach the point at which it is in danger of cracking due to compressive stress?

(a)

To determine
The expression for the compressive stress and strain.

Explanation

Given Info:

Expression for the compressive stress and strain is,

FA=Y(ΔLL0)

• F is the force
• A is the area,
• Y is the Young’s modulus,
• ΔL is change in length,
• L0 is the original length,

Expression for change in length due to linear expansion is,

ΔLL0=α(ΔT)

• α is the coefficient of linear expansion,
• ΔT is the change in temperature,

(b)

To determine
Obtain symbolic equation for thermal energy transfer.

(c)

To determine
The mass of the concrete slab.

(d)

To determine
How much thermal energy must be transferred to the slab to reach the compressive stress.

(e)

To determine
The change in temperature.

(f)

To determine
The time required for the slab to reach the point of danger of cracking due to compressive stress.

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