I want to use a 6061 aluminum alloy for a tie rod with a circular cross section. The yield strength of this alloy is 240 MPa, and the elastic modulus is 70 GPa. 1. If this tie rod has to hold a load of 20,000N, what is the minimum diameter of the tie rod needed to avoid yielding. Enter the answer in meters.
Engineering · College · Sun Jan 24 2021
Answered on
To calculate the minimum diameter of the tie rod needed to avoid yielding, we first need to use the yield strength of the material and the applied load to determine the area of the cross-section.
The yield strength (\( \sigma_y \)) is the maximum stress that the material can withstand before it starts to deform permanently. The formula for stress (\( \sigma \)) is given by:
\[ \sigma = \frac{F}{A} \]
where: \( F \) = applied force (load) \( A \) = cross-sectional area
Rearranging for \( A \) we get:
\[ A = \frac{F}{\sigma_y} \]
Given: \( F = 20,000 \) N (force) \( \sigma_y = 240 \) MPa (yield strength)
We need to make sure the units are consistent; hence we convert the yield strength from MPa to N/m²:
\[ \sigma_y = 240 \text{ MPa} \times 10^6 \text{ N/m}^2/\text{MPa} = 240 \times 10^6 \text{ N/m}^2 \]
Now we can calculate the required area:
\[ A = \frac{20,000 \text{ N}}{240 \times 10^6 \text{ N/m}^2} = \frac{20,000}{240 \times 10^6} \text{ m}^2 = 8.333 \times 10^{-5} \text{ m}^2 \]
Since the cross-section of the rod is circular, the area \( A \) is defined by:
\[ A = \frac{\pi d^2}{4} \]
Solving for the diameter \( d \), we get:
\[ d = \sqrt{\frac{4A}{\pi}} = \sqrt{\frac{4 \times 8.333 \times 10^{-5}}{\pi}} \]
\[ d \approx 0.0103 \text{ meters} \]
To avoid yielding, you would need a tie rod with a minimum diameter of approximately 0.0103 meters, or 10.3 millimeters.
Extra: Understanding material properties and stress-strain behavior is crucial when designing any structural component. When a material is subject to stress, the atoms in the material are displaced slightly. If the stress is removed before the yield strength is exceeded, the material will return to its original shape due to its elasticity―this is known as elastic deformation.
The yield strength of a material is an indicator of the maximum stress that can be applied without causing permanent deformation. In the case of a tie rod, it is crucial that the component is designed so the maximum stress it experiences under load is less than the yield strength to ensure that the rod retains its shape and integrity while in use.
The elastic modulus, also known as Young's modulus, is a measure of the stiffness of a material. It provides a relationship between stress and strain in the elastic region of the material's deformation behavior. In practical terms, it tells you how much a material will stretch (strain) under a certain load (stress).
In summary, when designing a structural component like a tie rod, it's essential to understand the material's yield strength and the load it must support to ensure that the component functions safely without failing or undergoing permanent deformation.