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GP 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 \(4.00 \mathrm{~cm}\) thick, \(1.00 \mathrm{~m}\) long, and \(1 \leq 00 \mathrm{~m}\) wide is poured for a sidewalk at an ambient temperature of \(25.0^{\circ} \mathrm{C}\) 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 \(\Delta I\). from the equation for compressive stress and strain (Eq. 9.3). (b) Use the expression found in part (a) to eliminate \(\Delta T\) from Equation \(11.3\), obtain ing a symbolic equation for thermal energy transfer \(Q\). (c) Compute the mass of the concrete slab given that its density is \(2.40 \times 10^{5} \mathrm{~kg} / \mathrm{m}^{3}\). (d) Concrete has an ultimate compressive strength of \(2.00 \times 10^{7} \mathrm{~Pa}\), specific heat of \(880 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}\), and Young's modulus of \(2.1 \times 10^{10} \mathrm{~Pa}\). 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 \(1.00 \times 10^{3} \mathrm{~W}\) 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 Lake the slab to reach the point at which it is in danger of cracking due to compressive stress?

Step by step solution

01

Compute mass from given density and volume

Using the formula for volume and given density, calculate the mass of the concrete slab.\nThe formula for volume of a rectangular cuboid is \(length \times width \times height\). So, volume would be \(1.00 m \times 1.00 m \times 0.04 m = 0.04 m^3\).\nMass is \(density \times volume\), given the density as \(2.40 \times 10^5 kg/m^3\), we can multiply with the volume to get mass of the slab, \(m = density \times volume\).

02

Substitute equations to find thermal energy transfer equation

By substituting the linear expansion equation to find the compressive strain, and then use that to substitute in place of the change in temperature in the equation for thermal energy transfer. Linear expansion equation is \(\Delta L = L \times \alpha \times \Delta T\), compressive strain is \(- \Delta L / L\), and thermal energy transfer \(Q = m \cdot c \cdot \Delta T\), where m is mass, c is specific heat, and \(\Delta T\) is change in temperature.

03

Calculate thermal energy needed and corresponding temperature change

Use the compressive stress (ultimate compressive strength) to solve for thermal energy by substituting in the obtained equation from step 2.\nSubstitute this value in the equation for thermal energy transfer to find the change in temperature.

04

Calculate the time for slab to reach dangerous temperature

Given the power delivered by the sun and the energy retention, find the actual energy absorbed in one second, which is a watt.\nThen, divide the total required thermal energy by the energy absorbed per second to find the time it takes for the slab to reach dangerous temperature.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Compressive Stress

Compressive stress is a critical concept in understanding how materials respond to force. It occurs when a material is subjected to axial load that pushes it together, causing it to shorten. This type of stress is described by how much force is applied over an area with the formula:
\[ \text{Compressive Stress} = \frac{F}{A} \]where:

• \( F \) is the force applied
• \( A \) is the area of the cross-section

In contexts like concrete slabs, if the thermal expansion of the slab is constrained, like in the exercise, compressive stress builds up, potentially leading to cracking or failure if the material's compressive strength is exceeded. This underscores the importance of accommodating thermal expansion in building design, such as by adding expansion joints to prevent the accumulation of stress.

Linear Expansion Equation

The linear expansion equation helps predict how much a material will expand or contract when the temperature changes. It is critical for materials like concrete, which can undergo significant changes due to environmental conditions. The formula is given by:
\[ \Delta L = L \times \alpha \times \Delta T \]where:

• \( \Delta L \) is the change in length
• \( L \) is the original length of the material
• \( \alpha \) is the coefficient of linear expansion
• \( \Delta T \) is the change in temperature

Using this equation, engineers can predict how structural elements like beams and slabs will behave under different temperatures, ensuring that structures remain safe and intact through various weather conditions. In the exercise, the slab's inability to expand due to lack of joints leads to increased compressive stress.

Thermal Energy Transfer

Understanding thermal energy transfer is essential for calculating how much energy is needed to change a material's temperature. This concept embodies how energy is absorbed or released as heat, affecting the temperature of the material. The formula used is:
\[ Q = m \cdot c \cdot \Delta T \]where:

• \( Q \) is the thermal energy
• \( m \) is the mass of the material
• \( c \) is the specific heat capacity
• \( \Delta T \) is the change in temperature

In the context of the exercise, it’s crucial for calculating how much energy, due to solar exposure, causes the slab to reach a point where it might crack. The absorption of too much energy without the capacity to expand leads to increased danger of stress beyond the material's limit, requiring careful calculation and preventative measures.

Young's Modulus

Young's Modulus is a measure of a material's stiffness or rigidity and is a fundamental concept in understanding material deformation. It quantifies the relationship between stress (force per unit area) and strain (deformation in the material). The formula is:
\[ E = \frac{\text{stress}}{\text{strain}} \]where:

• \( E \) is Young’s Modulus
• Stress is the force applied
• Strain is the resultant deformation

Young's Modulus provides a valuable understanding of how a material like concrete will behave when subjected to forces, affecting its ability to recover its original shape after deforming. In the exercise, a high Young's Modulus of concrete indicates it does not easily deform under stress, which, combined with squeeze from thermal expansion, can lead to cracking without adequate accommodations like expansion joints.