Calculate the load that a fibroblast cell applies to buckle a single column in a collagen tissue engineering scaffold, given that the column has a length of 100μm, is circular in cross section, with a diameter of 2μm, and has a Young's modulus of 5MPa. You can assume that the end constraint factor, n, for the column, is 0.6.
Pcr (in nN):
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answer is 1.393
To calculate the critical buckling load (Pcr) in this scenario, we can use Euler's buckling formula. Euler's formula relates the critical load to the material properties and geometric properties of the column through the relationship:
Pcr = (π^2 * E * I) / (L^2)
Where:
Pcr is the critical buckling load
E is the Young's modulus
I is the moment of inertia of the column's cross-section
L is the length of the column
To calculate Pcr, we need to find the moment of inertia (I) of the column's cross-section. For a circular cross-section, the formula for the moment of inertia is:
I = (π/4) * (d^4)
Where:
d is the diameter of the column
Let's calculate the moment of inertia first.
d = 2μm
I = (π/4) * (2μm)^4
Next, substitute the values into Euler's formula to calculate Pcr:
Pcr = (π^2 * 5MPa * I) / (100μm)^2
Now we can multiply the diameter and length in terms of meters to convert the units:
Pcr = (π^2 * 5MPa * I) / (100 * 10^-6 m)^2
Lastly, multiply the result by the end constraint factor (n) to obtain the final answer:
Pcr = n * (π^2 * 5MPa * I) / (100 * 10^-6 m)^2
Calculate Pcr using the above formula, and you will get the load in nanonewtons (nN).