61 lines
2.3 KiB
Markdown
Executable File
61 lines
2.3 KiB
Markdown
Executable File
---
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author: Akbar Rahman
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date: \today
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title: MMME2053 // Elastic Instability (Buckling)
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tags: [ elastic_instability, buckling ]
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uuid: b8b2cff7-8106-4968-bab5-f4cffcf8b5a0
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lecture_slides: [ ./lecture_slides/MMME2053-EI L1 Slides.pdf, ./lecture_slides/MMME2053-EI L2 Slides.pdf ]
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lecture_notes: [ ./lecture_notes/Elastic Instability (Buckling) Notes.pdf ]
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exercise_sheets: [ ./exercise_sheets/Elastic Instability (Buckling) Exercise Sheet.pdf, ./exercise_sheets/Elastic Instability (Buckling) Exercise Sheet Solutions.pdf ]
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worked_examples: [ ./worked_examples/MMME2053-EI WE1 Slides.pdf ]
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---
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# Notes from Lecture Slides (2)
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> In contrast to the classical cases considered here, actual compression members are seldom truly pinned or
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> completely fixed against rotation at the ends. Because of this uncertainty regarding the fixity of the ends,
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> struts or columns are often assumed to be pin-ended. This procedure is conservative.
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>
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> The above equations are not applicable in the inelastic range, i.e. for $\sigma > \sigma_y$ , and must be modified.
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>
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> The critical load formulae for struts or columns are remarkable in that they do not contain any strength
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> property of the material and yet they determine the load carrying capacity of the member. The only material
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> property required is the elastic modulus, $E$, which is a measure of the stiffness of the strut.
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# Stability of Equilibrium
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# Critical Buckling Load on a Strut
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Critical buckling load is given by:
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$$P_c = \frac{\pi^2EI}{L_\text{eff}^2}$$
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where $L_\text{eff}$ is the effective length:
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- Free-fixed -> $L_\text{eff} = 2l$
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- Hinged-hinged -> $L_\text{eff} = l$
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- Fixed-hinged -> $L_\text{eff} = 0.7l$
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- fixed-fixed -> $L_\text{eff} = 0.5l$
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where $l = 0.5L$
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Derivations detailed in lecture slides (1, pp. 8-21).
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# Compression of Rods/Columns
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Derivations detailed in lecture slides (2, pp. 3-5).
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Buckling will occur if
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$$\sigma = \frac{\pi^2E}{\frac{L^2}{K^2}}$$
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where $k$ is the radius of gyration and $\frac{L}{K}$ is the slenderness ratio.
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Plastic collapse will occur if $\sigma = \sigma_y$.
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This can be represented diagrammatically:
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