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.
Pcr,local (in kN):
Pcr,Euler (in MN):
Which occurs first?
Local Buckling
Euler Buckling
local occurs first
To calculate the critical loads for local buckling and Euler buckling, we need to determine the cross-sectional properties of the hollow rod. Given that the rod is thin-walled, we can use the formulas:
A = 2πrt (cross-sectional area)
I = πr^3t (moment of inertia)
First, let's convert the given dimensions to SI units:
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, let's calculate the cross-sectional properties:
A = 2π(0.012)(0.012 + 0.0007) = 0.000957 m^2
I = π(0.012)^3(0.0007) = 1.27e-8 m^4
Next, we can calculate the critical loads using the following formulas:
For local buckling: Pcr,local = (nπ^2EI) / (L^2)
For Euler buckling: Pcr,Euler = (nπ^2EI) / (L^2)
Given n = 1, E = 193 GPa = 193e9 Pa, and the values we calculated for A and I, we can substitute the values into the formulas:
Pcr,local = (1π^2(193e9)(1.27e-8)) / (0.05^2) = 766.36 kN
Pcr,Euler = (1π^2(193e9)(1.27e-8)) / (0.05^2) = 766.36 kN
As we can see, the critical loads for local buckling and Euler buckling are the same.
Therefore, both local buckling and Euler buckling will occur simultaneously. Neither phenomenon will occur first; instead, they will occur at the same critical load.