From 572d86e60f30688dc26aa2b64a4c317f9b77ad76 Mon Sep 17 00:00:00 2001 From: Alvie Rahman Date: Fri, 24 Mar 2023 15:37:43 +0000 Subject: [PATCH] mmme2053 note on elastic instability (buckling) --- .../elastic_instability.md | 49 ++++++++++++++++++ .../images/bucking_vs_plastic_collapse.png | Bin 0 -> 9061 bytes .../images/perturbed.png | Bin 0 -> 6268 bytes .../images/stable_unstable_equilibria.png | Bin 0 -> 76182 bytes ...ot-2023-03-24T15:00:01,985512777+00:00.png | Bin 0 -> 7342 bytes 5 files changed, 49 insertions(+) create mode 100644 uni/mmme/2053_mechanics_of_solids/images/bucking_vs_plastic_collapse.png create mode 100644 uni/mmme/2053_mechanics_of_solids/images/perturbed.png create mode 100644 uni/mmme/2053_mechanics_of_solids/images/stable_unstable_equilibria.png create mode 100644 uni/mmme/2053_mechanics_of_solids/images/vimscrot-2023-03-24T15:00:01,985512777+00:00.png diff --git a/uni/mmme/2053_mechanics_of_solids/elastic_instability.md b/uni/mmme/2053_mechanics_of_solids/elastic_instability.md index 8945542..8ec5d24 100755 --- a/uni/mmme/2053_mechanics_of_solids/elastic_instability.md +++ b/uni/mmme/2053_mechanics_of_solids/elastic_instability.md @@ -9,3 +9,52 @@ lecture_notes: [ ./lecture_notes/Elastic Instability (Buckling) Notes.pdf ] exercise_sheets: [ ./exercise_sheets/Elastic Instability (Buckling) Exercise Sheet.pdf, ./exercise_sheets/Elastic Instability (Buckling) Exercise Sheet Solutions.pdf ] worked_examples: [ ./worked_examples/MMME2053-EI WE1 Slides.pdf ] --- + +# 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 + +![(a) is a stable equilibrium (it will return to equilibrium if it deviates) whereas (b) is an unstable equilibrium (it will not return to equilibrium if it deviates)](./images/stable_unstable_equilibria.png) + +# 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: + +![](./images/bucking_vs_plastic_collapse.png) diff --git a/uni/mmme/2053_mechanics_of_solids/images/bucking_vs_plastic_collapse.png b/uni/mmme/2053_mechanics_of_solids/images/bucking_vs_plastic_collapse.png new file mode 100644 index 0000000000000000000000000000000000000000..06afc272577850df72390a38c505ce0dbcf6ced2 GIT binary patch literal 9061 zcmch7cT`hfx9y=LT?v9zDQWg^_WRu=hT@tToqMYo1SVHAQj~dJ+Ht$d#4kGywo77<_9I z6N0}m(Y4Hh7ee!AigEz<`Y*E~Hy!})0?KkS+Hb#X%@{ivt|C@!d!PVxW!DrnN}5s8*#i?zc@0WX4jtX!2QJ63DlbKbUK-aohIkT3Ip zN-^UiirC6|D+Ne<4eu*WH$NI(tZ>^yY;;@7 zz?E{V{u248wN$TR{j502~(~l^v_XDhJhqr>+Ppa~S49N2nOGQ4NEbuJmE!f%h zoJ=y>R+@8x3a0gKmT%cqO*ojh2mpVKSjAH9j~>&0zy%T%T6tY@06roZ4iF;^1%O+` z1b{Sd762&Sg#f?}u*uyZH~?@^kpO}C|6dwq<-ym)#@!S&IX06Q`yb^FO}s1!!3BPy zF)_o8*vInVgU zYJckuZLd@^0GeDcJOBFf*Uz5AkX%DiZ%lscW_uh2uf*G|q<53+ 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