How does the first Euler Case look like and what are the Boundary Conditions?
Single clamping in the bottom.
How does the second Euler Case look like and what are the Boundary Conditions?
Moment free supports on both ends. Deflection and Moment (EIw’’) = 0
How does the third Euler Case look like and what are the Boundary Conditions?
Clamping at one end, Moment free suport at the loaded end
How does the foruth Euler Case look like and what are the Boundary Conditions?
Clamping at both ends
What is the general approach to beam buckling (without an additional load)
We utilize the differential equation already derrived when concerned with beams under bending and compression/ tension:
(lambda = F/EI)
And the general Ansatz:
How do we generally solve in buckling cases, after the Ansatz is chosen?
With the given Boundary conditions we derrive a Coefficient Matrix.
To avoid the trivial solution (coefficients = 0 —> F = 0) the determinant of the coefficient matrix is set to zero.
Give the buckling condition and the buckling load for Euler case I
Buckling condition
Buckling load
Give the buckling condition and the buckling load for Euler case II
Give the buckling condition and the buckling load for Euler case III
Give the buckling load for Euler case IV
How does the buckling load scale over the different Euler Cases?
What can support nummerical calculations of unkown buckling cases?
Taking in the nearest Euler Cases as upper and lower bound.
Name two easy numerical methods to numerically solve e.g. a buckling problem.
Full enumeration (Calculate every point between bounds, if sign change detected go back and decrease increment)
Newton method (use the derrivative to find the new x value to calculate the function for, it will be on the other side of the x-axis crossing. Redo until condition is sufficiently met)
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