You have a stainless steel hollow rod (E=193GPa,ν=0.29) that has an inner radius r of 1.2cm, a wall thickness t of 0.7mm, and a length L of 5cm. Calculate the critical loads, in kN, for local buckling and Euler bucking (Pcr,local and Pcr,Euler, respectively). Which would happen first? Assume n=1. For a thin-walled cylinder, A=2πrt and I=πr3t.
To calculate the critical loads for local buckling and Euler buckling, we need to determine the moment of inertia (I) of the hollow rod and use the appropriate formulas.
First, let's calculate the outer radius of the hollow rod:
Outer radius = Inner radius + Wall thickness
= 1.2 cm + 0.7 mm
Convert wall thickness to cm:
Wall thickness = 0.7 mm = 0.07 cm
Outer radius = 1.2 cm + 0.07 cm
= 1.27 cm
Now, let's calculate the moment of inertia (I) for the hollow rod:
I = πr^3t
Substituting the values:
I = π*(1.27 cm)^3 * 0.07 cm
Now, let's calculate the critical load for local buckling (Pcr,local):
Pcr,local = π^2 * E * I / (L / r)^2
Substituting the values:
Pcr,local = π^2 * 193 GPa * I / (5 cm / 1.27 cm)^2
Finally, let's calculate the critical load for Euler buckling (Pcr,Euler):
Pcr,Euler = π^2 * E * I / L^2
Substituting the values:
Pcr,Euler = π^2 * 193 GPa * I / (5 cm)^2
To determine which type of buckling would occur first, compare the values of Pcr,local and Pcr,Euler. The lower value corresponds to the critical load that would occur first.
Let me do the calculations for you and provide the results.
To calculate the critical loads for local buckling and Euler buckling, we need to use the formulas:
For local buckling:
P_cr,local = (π^2 * E * I) / (4 * L^2)
For Euler buckling:
P_cr,Euler = (π^2 * E * A) / (L^2)
Let's start by calculating the values required for the formulas:
Inner radius, r = 1.2 cm = 0.012 m
Wall thickness, t = 0.7 mm = 0.0007 m
Length, L = 5 cm = 0.05 m
Now we can calculate the area and moment of inertia:
Area, A = 2 * π * r * t
= 2 * π * 0.012 * 0.0007
≈ 0.0000528 m²
Moment of inertia, I = π * r^3 * t
= π * (0.012^3) * 0.0007
≈ 1.61328e-8 m^4
Next, we need to substitute these values into the formulas to calculate the critical loads:
For local buckling:
P_cr,local = (π^2 * E * I) / (4 * L^2)
= (π^2 * 193e9 * 1.61328e-8) / (4 * 0.05^2)
≈ 620.462 kN
For Euler buckling:
P_cr,Euler = (π^2 * E * A) / (L^2)
= (π^2 * 193e9 * 0.0000528) / (0.05^2)
≈ 661.776 kN
So the critical loads for local buckling and Euler buckling are approximately 620.462 kN and 661.776 kN, respectively.
To determine which would happen first, we compare the two critical loads. Since P_cr,local < P_cr,Euler (620.462 kN < 661.776 kN), local buckling would happen first.