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| Akbar Rahman | \today | MMME2053 // Elastic Instability (Buckling) |
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Notes from Lecture Slides (2)
In contrast to the classical cases considered here, actual compression members are seldom truly pinned or completely fixed against rotation at the ends. Because of this uncertainty regarding the fixity of the ends, struts or columns are often assumed to be pin-ended. This procedure is conservative.
The above equations are not applicable in the inelastic range, i.e. for
\sigma > \sigma_y, and must be modified.The critical load formulae for struts or columns are remarkable in that they do not contain any strength property of the material and yet they determine the load carrying capacity of the member. The only material property required is the elastic modulus,
E, which is a measure of the stiffness of the strut.
Stability of Equilibrium
Critical Buckling Load on a Strut
Critical buckling load is given by:
P_c = \frac{\pi^2EI}{L_\text{eff}^2}where L_\text{eff} is the effective length:
- Free-fixed ->
L_\text{eff} = 2l - Hinged-hinged ->
L_\text{eff} = l - Fixed-hinged ->
L_\text{eff} = 0.7l - fixed-fixed ->
L_\text{eff} = 0.5l
where l = 0.5L
Derivations detailed in lecture slides (1, pp. 8-21).
Compression of Rods/Columns
Derivations detailed in lecture slides (2, pp. 3-5).
Buckling will occur if
\sigma = \frac{\pi^2E}{\frac{L^2}{K^2}}where k is the radius of gyration and \frac{L}{K} is the slenderness ratio.
Plastic collapse will occur if \sigma = \sigma_y.
This can be represented diagrammatically:
