Introductory Solid Mechanics

Buckling of Beams

  • Above a critical compressive axial load, a beam undergoes sudden strong deflection from straight shape (it buckles)
  • This is an instability (the straight shape is still a valid mechanical state, but becomes unstable to small perturbations)
  • The failure is elastic, i.e., upon release of the force the beam can go back to being straight
  • However, after buckling further deformation and plastic failure (yielding) or brittle failure (fracture) may happen easily (post-buckling failure)

    • Stability of Structures

    COLUMNS: member AB of length L supporting compressive loading

  • The load P is a centric axial load
  • Properly designed column - cross section is selected such that
    • \[\sigma = \frac{P}{A} \le \sigma_{all}\] \[\delta = \frac{PL}{EA} \le \delta_{spec}\]
  • As the applied load increases, the column might buckle (instead of remaining straight, the column becomes strongly curved). --> INSTABILITY CAUSING FAILURE!

  • Consider model with two rigid rods and one torsional spring
  • Both rods and load P are perfectly aligned --> system stays in position of EQUILIBRIUM
  • Point C moves slightly to the right (small perturbation)…
    • … and returns to original position – STABLE SYSTEM
    • … and then moves further away from that position – UNSTABLE SYSTEM

  • The spring restoring moment tends to bring the rod back to its original position \[M = K(2\Delta\theta) = \text{restoring moment}\]
  • 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: $M > M_{load}$
  • Unstable system: $M < M_{load}$
  • Equilibrium position gives: $M = M_{load}$
    • \[K(2\Delta\theta) = P_{cr}\frac{L}{2}\Delta\theta\] \[P_{cr} = \frac{4K}{L}\]

    Euler’s formula for Pinned-end Columns

  • Equilibrium gives $M = -Py$
  • After a small perturbation, the system reaches an equilibrium configuration such that:
    • \[EIy'' = M = -Py\] \[y'' + \frac{P}{EI}y = 0\] Linear, homogeneous differential equation of second order with constant coefficients \[y'' + p^2y = 0\]
  • The general solution is $y(x) = A\sin(px) + B\cos(px)$
  • With boundary conditions $y(0) = y(L) = 0$

    • Eulers Formula: \[P_{cr} = \frac{\pi^2EI}{L^2}\]

    Buckling occurs: $P > P_{cr}$