16.9 Example: Uniform beam

16.9.1 With uniform loading

Cantilever beam uniformly loaded

  • Recall: \[\begin{align} \frac{d^2}{d z^2} \left[ E {I_{yy}}{u {}_{,zz}} + E {I_{xy}}{v {}_{,zz}} \right] =& \, p_x \\ \frac{d^2}{d z^2} \left[ E {I_{xy}}{u {}_{,zz}} + E {I_{xx}}{v {}_{,zz}} \right] =& \, p_y \\ \end{align}\]
  • If symmetry exists across the \(x\) or \(y\) axis, the bending equations are uncoupled. \[\begin{align} \frac{d^2}{d z^2} \left[ E {I_{yy}}{u {}_{,zz}} \right] =& \; p_x \\ \frac{d^2}{d z^2} \left[ E {I_{xx}}{v {}_{,zz}} \right] =& \; p_y \\ M_x =& \, - E{I_{xx}}{v {}_{,zz}} \\ M_y =& \, + E{I_{yy}}{u {}_{,zz}} \\ \end{align}\]

16.9.2 With uniform loading

Cantilever beam uniformly loaded

\[\begin{align} E I_{yy} \; {u {}_{,zzzz}} =& \; p_x \\ E I_{yy} \; {u {}_{,zzz}} =& \; p_x z + C_1 \\ E I_{yy} \; {u {}_{,zz}} =& \; p_x \frac{z^2}{2} + C_1 z + C_2\\ E I_{yy} \; {u {}_{,z}} =& \; p_x \frac{z^3}{6} + C_1 \frac{z^2}{2} + C_2 z +C_3\\ E I_{yy} \; u(z) =& \; p_x \frac{z^4}{24} + C_1 \frac{z^3}{6} + C_2 \frac{z^2}{2} +C_3 z +C_4\\ \end{align}\]

Cantilever beam uniformly loaded

  • What are some boundary conditions we can use? \[\begin{align} u(0) =& \; 0 \Longrightarrow C_4 = 0\\ u'(0) =& \; 0 \Longrightarrow C_3 = 0\\ \end{align}\] \[\begin{align} M(0) =& \; E I_{yy} {u {}_{,zz}} = \frac{p_x l^2}{2} \Longrightarrow C_2 = \frac{p_x l^2}{2}\\ M(l) =& \; E I_{yy} {u {}_{,zz}} = 0 \Longrightarrow C_1 = - l p_x\\ \end{align}\]

\[E I_{yy} u(z) = p_x \left(\frac{z^4}{24}-\frac{l\,z^3}{6}+\frac{l^2\,z^2}{4}\right)\]

16.9.3 Caution!!!

  • Plates and screws can often be modeled as symmetric/uniform...
  • Keep in mind bones are neither symmetric nor uniform
  • Our approach for modeling biological structures may be guided by engineering theories, but theories must not be blindly accepted as truth

16.9.4 Contact Relationship Between Components

Components Relationship Comments
Screw - Bone (i.e. screw threads) Rigid (tied contact) Fixed all DOF
Plate - Bone Contact pair Friction sliding (\(\mu=0.3\))
Fracture surfaces Contact pair Friction sliding (\(\mu=1.0\))
Plate - Screw (Conventional) Universal joint Provide a universal connection between the screw control node and nodes on the bearing surface of the plate.
Plate - Screw (Locked) Rigid Provide a rigid connection between the screw control node and nodes on the bearing surface of the plate.
Fracture type Construct Construct Construct
Neutralization plate (Conventional screw construct) Neutralization plate (Fixed angle screw construct) Lateral periarticular distal fibular plate (Fixed angle screw construct)
Comminuted fracture Model #1 image Model #3 image Model #5 image
Danis-Weber B fracture Model #2 image Model #4 image Model #6 image

: Fracture Plate Combinations \[fracture-plate-combinations\]

Displacement (\(|\Delta \vec{u}|\)) Due to Fibulotalar Reaction Load and External Moment With Danis-Weber B Fracture