Lecture 9

.title[
# Lecture 9
]
.subtitle[
## Coarse Structures
]
.author[
### Dr. Christopher Kenaley
]
.institute[
### Boston College
]
.date[
### 2024/02/13
]

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<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.14.0/css/all.min.css">

- Consider safety factors

- Consider how structures fail

- Consider how structures resist bending

]

.pull-right[
![](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTXL4A1lCI63p7w9qJwAPua5vytkoEB5x8rjw&usqp=CAU)

]

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# What if we have both?

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Voigt model

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]

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# Why this in a tendon?

.center[
<img src="https://www.physio-pedia.com/images/thumb/b/be/Rep_loading-unloading_curve_intechopen.jpeg/357px-Rep_loading-unloading_curve_intechopen.jpeg" width="450" />

]

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# A comparison of SF

Compression safety factor during locomotion

<table class="table" style="margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> species </th>
   <th style="text-align:left;"> SF </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Dog jumping </td>
   <td style="text-align:left;"> 2.8 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Kangaroo hopping </td>
   <td style="text-align:left;"> 3.2 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Possum </td>
   <td style="text-align:left;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Elephant running </td>
   <td style="text-align:left;"> 1-1.7 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Man weightlifting </td>
   <td style="text-align:left;"> 6 </td>
  </tr>
</tbody>
</table>
.center[

<br>
<br>
<br>
<br>
Alexander, R.McN. 1981. Sci. Prog. Oxf. 67:109-120

]
 ]

Mean bending safety factor during running

<table class="table" style="margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> species </th>
   <th style="text-align:right;"> SF </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Bullfrog </td>
   <td style="text-align:right;"> 8.0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Salamanders </td>
   <td style="text-align:right;"> 10.5 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Possum </td>
   <td style="text-align:right;"> 5.1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Eutherians </td>
   <td style="text-align:right;"> 2.5 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Iguana </td>
   <td style="text-align:right;"> 10.8 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Turtles </td>
   <td style="text-align:right;"> 4.4 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Crocs </td>
   <td style="text-align:right;"> 6.3 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Birds </td>
   <td style="text-align:right;"> 3.3 </td>
  </tr>
</tbody>
</table>
.center[

Blob, R.W. et al. 2014. Int. & Comp. Biol., 54 (6):1058–1071

]
 ]
 
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# A comparison of SF

.body[
These are SF from experiments: the real world is different. Loads (direction, scale) vary and so does the capacity to withstand them. Geometry and structural arrangement is important!]

![](Lecture09_files/figure-html/unnamed-chunk-6-1.svg)

<br>
.pull-left[
.footnote[
`\(\textrm{0.0004 yr}^{-1} \textrm{ human humerus}\)`

`\(\textrm{0.0006 yr}^{-1} \textrm{ human femur}\)`
]
]

`\(\textrm{0.5 life}^{-1} \textrm{ deer antler/spider webs}\)`
]
]

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# Shape matters

<img src="img/buckling.png" width="250" />
 ]

local buckling

`\(\sigma_{cr}  = kEt/D\)`

(k ~ 0.7) 
]
]

Euler buckling

`\(\sigma_{cr}={\frac {\pi ^{2}EI}{(kL)^{2}}}\)`

(k ~ 0.5) 
]
]

k = column effective length factor
]

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# Shape matters

EI: Flexural stiffness of a structure

I = second moment of area: a measure of how shape affects stress distribution in a structure

]

Euler buckling

`\(\sigma_{cr}={\frac {\pi ^{2}EI}{(kL)^{2}}}\)`

]
]

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# Shape matters

EI: Flexural stiffness of a structure

I = second moment of area: a measure of how shape affects stress distribution in a structure

]

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# Shape matters

Where are hip fractures most common? Where will hip most likely fail?

]

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# Thanks!

Slides created via the R package [**xaringan**](https://github.com/yihui/xaringan).