Given an aluminum shell perfectly bonded to a steel core, what is the thermal stress in the aluminum shell when the system is heated from the unstressed state (20∘𝐶) to 180∘𝐶? Consider only deformations in the axial directions. Give your answer in 𝑀𝑃𝑎

Question

Given an aluminum shell perfectly bonded to a steel core, what is the thermal stress in the aluminum shell when the system is heated from the unstressed state (20∘𝐶) to 180∘𝐶? Consider only deformations in the axial directions. Give your answer in 𝑀𝑃𝑎

𝛼𝐴𝑙=23.6×10−6𝐾−1

𝐸𝐴𝑙=70𝐺𝑃𝑎

𝛼𝑆𝑡𝑒𝑒𝑙=11.7×10−6𝐾−1

𝐸𝑆𝑡𝑒𝑒𝑙=200𝐺𝑃𝑎

Cilinder dimensios= 200mm, 50 mm and 20mm

Thermal stress in Al shell (in 𝑀𝑃𝑎):

Answer 1

To calculate the thermal stress in the aluminum shell, we need to use the following formula:

𝜎 = 𝛼 * 𝐸 * Δ𝑇

Where:
𝜎 is the thermal stress,
𝛼 is the coefficient of thermal expansion,
𝐸 is the Young's modulus,
Δ𝑇 is the change in temperature.

First, let's calculate the change in temperature (Δ𝑇) by subtracting the initial temperature (20∘𝐶) from the final temperature (180∘𝐶):

Δ𝑇 = 180 - 20 = 160 degrees Celsius

Next, we can calculate the thermal stress in the aluminum shell. Given the values provided, we have:
𝛼𝐴𝑙 = 23.6 × 10^(-6) 𝐾^(-1)
𝐸𝐴𝑙 = 70 𝐺𝑃𝑎

Substituting these values into the formula, we have:

𝜎 = 𝛼𝐴𝑙 * 𝐸𝐴𝑙 * Δ𝑇

𝜎 = (23.6 × 10^(-6) 𝐾^(-1)) * (70 𝐺𝑃𝑎) * (160 degrees Celsius)

To convert GPa to MPa, we divide by 1000:

𝜎 = (23.6 × 10^(-6) 𝐾^(-1)) * (70 𝐺𝑃𝑎) * (160 degrees Celsius) / 1000

Simplifying the calculation:

𝜎 = 2.3472 𝑀𝑃𝑎

Therefore, the thermal stress in the aluminum shell when heated from 20∘𝐶 to 180∘𝐶 is 2.3472 MPa.