What is the critical buckling force that will buckle a 9 foot tall 2 x 4 (K = 1.0)?
Please show how to set up the problem and the steps to solve. I do not understand. Thank you.
To find the critical buckling force for a 9 foot tall 2 x 4 with a given K value, we need to use Euler's formula for buckling.
Euler's formula states that the critical buckling force (Pc) is equal to the product of the Euler's critical buckling load (Pcr) and the slenderness ratio (λ).
The slenderness ratio (λ) is defined as the ratio of the effective length (L) of the column to the least radius of gyration (r).
In this case, we have a 2 x 4, which means the actual dimensions of the lumber are 1.5 inches by 3.5 inches. Let's assume the effective length (L) of the column is 9 feet or 108 inches.
The least radius of gyration (r) can be calculated as follows:
r = sqrt[(I_min) / (A)]
Where:
I_min = min[(b * h^3) / 12]
A = b * h
b = 1.5 inches (thickness of the lumber)
h = 3.5 inches (height of the lumber)
Substituting the values, we can calculate r.
Next, we need to calculate the slenderness ratio (λ) using the formula:
λ = L / r
With the given K value of 1.0, we can use the table provided to find the Euler's critical buckling load (Pcr) corresponding to the value of λ.
Finally, we can calculate the critical buckling force (Pc) using the formula:
Pc = Pcr * λ
By following these steps and performing the necessary calculations, you can find the critical buckling force for the given 9 foot tall 2 x 4 with a K value of 1.0.