Buckling

Buckling is the sudden change in shape of a structural component under a compressive load

Taken from TAM251 Lecture Notes - L13S2

The beam is still able to withstand normal loads, but buckling causes an instability. Small perturbations make the structure unstable. Failure is elastic ($ \sigma < \sigma_Y $), but if increased loads are applied, further deformation and plastic failure (yielding) / brittle failure (fracture) can occur (post-buckling failure).

Stability of Structures

Single Column

Taken from TAM251 Lecture Notes - L13S3

Column $ AB $ is supporting uniaxial compressive load $ P $. To properly design this column, the cross-section must satisfy the following:

$$ \sigma = \frac{P}{A} \le \sigma_{all}\ $$

$$ \delta = \frac{PL}{EA} \le \delta_{spec}\ $$

Increasing the load can cause the column to buckle $ \rightarrow $ instability causing failure.

Two Rods and a Torsional Spring

Taken from TAM251 Lecture Notes - L13S4

Rods $ AC $ and $ CB $ are perfectly aligned and a torsional spring connects them at point $ C $. For small perturbations, point $ C $ moves to the right.

If
If

Taken from TAM251 Lecture Notes - L13S5

The spring restoring moment is

$$ M_s = K(2\Delta\theta)= \text{restoring moment}\ $$

The moment resultant from the applied load P tends to move the rod away from the vertical position

$$ M_{load} = P\frac{L}{2}\sin\Delta\theta = P\frac{L}{2}\Delta\theta = \text{destabilizing moment}\ $$

Stable system:
Unstable system:
Equilibrium position gives:

The critical load can be found with

$$ P_{cr} = \frac{4K}{L}\ $$

**Expandable Derivation**

$$ M_s = M_{load}\ $$

$$ K(2\Delta\theta) = P_{cr}\frac{L}{2}\Delta\theta\ $$

**End Derivation**

Euler's formula

Euler's formula can be used to solve for the critical load of a uniaxially loaded column.

Pinned-end Columns

Taken from TAM251 Lecture Notes - L13S6

Rod $ AB $ is pinned on each end. Equilibrium gives

$$ M = -Py\ $$

After a small perturbation, the system reaches equilibrium

$$ M(x) = EIy $$

$$ EIy $$

$$ y $$

Linear, homogeneous differential equation of second order with constant coefficients. The general solution is

$$ y(x) = A\sin(px) + B\cos(px)\ $$

With boundary conditions

$$ y(0) = y(L) = 0\ $$

Euler's Formula

$$ P_{cr} = \frac{\pi^2EI}{L^2}\ $$

Buckling occurs at

$$ P > P_{cr}\ $$

**Expandable Derivation**

$$ y(x) = A\sin(\sqrt{\frac{P}{EI}}x) + B\cos(\sqrt{\frac{P}{EI}}x)\ $$

$$ y(x=0)=0 \rightarrow A\sin(0)+B\cos(0) = 0\ $$

$$ B=0\ $$

$$ y(x=L)=0 \rightarrow A\sin(\sqrt{\frac{P}{EI}}L)+0 = 0\ $$

$$ A\sin(\sqrt{\frac{P}{EI}}L)=0\ $$

This has two solutions

$$ A = 0 \rightarrow \text{not interesting}\ $$

$$ A = n \rightarrow n \text{any number except where} A\sin(\sqrt{\frac{P}{EI}}L) = n\pi $$

$$ \frac{P}{EI}L^2 = n^2\pi^2\ $$

$$ P_{cr} = \frac{n^2\pi^2EI}{L^2}\ $$

Buckling usually happens at the smallest non-zero value of $ P_{cr} $

$$ n=1\ $$

Higher $ n $ values can be achieved if columns are prevented from buckling at $ n=1 $

**End Derivation**

Other Boundary Conditions

Taken from TAM251 Lecture Notes - L13S8

Different boundary conditions change length used in the critical load formula to the effective length ($ L_e $).

$$ P_{cr} = \frac{\pi^2EI}{L_e^2}\ $$

Solid Mechanics Reference