Question

1. estimate the stress (psi) in a concrete-like material that is cast into a bar that is 50 inches long and is fully restrained at each end against axial movement. the concrete is initially cast and cured at a temperature of 100 °f and subsequently cools to a temperature of 0 °f. assume that the modulus of elasticity is 5 million psi and the thermal coefficient is 5 x10-6 in/in/°f.

Explanation:

Step1: Recall Thermal Stress Formula

The formula for thermal stress (\(\sigma\)) is \(\sigma = E \cdot \alpha \cdot \Delta T\), where \(E\) is the modulus of elasticity, \(\alpha\) is the thermal coefficient of expansion, and \(\Delta T\) is the change in temperature.

Step2: Calculate Temperature Change (\(\Delta T\))

The initial temperature \(T_1 = 100^\circ\text{F}\) and the final temperature \(T_2 = 0^\circ\text{F}\). So, \(\Delta T = T_1 - T_2 = 100 - 0 = 100^\circ\text{F}\).

Step3: Substitute Values into Formula

Given \(E = 5 \times 10^6\) psi, \(\alpha = 5 \times 10^{-6}\) in/in/\(^\circ\text{F}\), and \(\Delta T = 100^\circ\text{F}\).
Substitute into \(\sigma = E \cdot \alpha \cdot \Delta T\):
\[

$$\begin{align*} \sigma&=(5 \times 10^6)\times(5 \times 10^{-6})\times100\\ &=(5\times5\times100)\times(10^6\times10^{-6})\\ &= 2500\times1\\ &= 2500 \end{align*}$$

\]

Answer:

The stress in the concrete - like material is 2500 psi.