1. By: Anthony Beeman

2. P L P v v P N A M x y Euler’s Fundamental Buckling Problem Assumptions: Straight Column Homogeneous Material Boundary Conditions: Pinned-Pinned Governing Equations: n=mode L= Original Column Length E= Young’s Modulus I= Moment of Inertia

3. P L e =L x y P L e =2L L P P Other End Conditions Modified Euler Buckling Formula: L= Original Column Length L e = Effective Column Length E= Young’s Modulus I= Moment of Inertia

4. P L=10 M AA Cross Section A-A r=0.5 M VariableValueDescription ρ [kg/m 3 ]7800Density υ [Dim]0.3Poisson's ratio E [Pa]2e11Young's Modulus Mechanical Properties Calculated Critical Load Problem Analyzed [N]

5. Case Number Analytical Critical Load [N] Theoretical Critical Load [N] Percent Error [%] Case 12.422e8 0.00% Case 22.414e82.422e80.33% Case 32.410e82.422e80.49% Case 1 2,904 DOF Case 2 12,723 DOF Case 3 73,623 DOF

6. Case 1 285 DOF Case 2 490 DOF Case 3 48,145 DOF Case Number Analytical Critical Load [N] Theoretical Critical Load [N] Percent Error [%] Case 13.102e82.422e82.86 Case 22.851e82.422e81.82 Case 32.447e82.422e81.52

8. Case 1 2,904 DOF Case 2 12,723 DOF Case 3 73,623 DOF Case Number Analytical Critical Load [N] Theoretical Critical Load [N] Percent Error [%] Case 12.406e82.422e80.16 Case 22.406e82.422e80.16 Case 32.406e82.422e80.16