Introduction to bone mechanics

1. Bone Biomechanics: Relating Mechanics Concepts to Bone Presented by: Jeff Leismer, PhD

4. Overview Mechanics (effects of forces on a body) BONE MECHANICS To Prof. Walsh BONE MECHANICS Mechanical Testing (response to loading) Statics (equilibrium of forces and moments) Dynamics (bodies in motion) Kinematics (displacements, velocities, and accelerations) Kinetics (forces responsible for motion) Failure (result of loading)

5. Bone Mechanics Overview To Jeff

7. Mechanical Lever System Mechanical Lever System d F b=fulcrum d=moment arm F~1/d To Prof. Walsh W a b c L q f

9. Overview Statics (equilibrium of forces and moments) Dynamics (bodies in motion) Statics (equilibrium of forces and moments) Mechanics (effects of forces on a body) Mechanical Testing (response to loading) Kinematics (displacements, velocities, and accelerations) Kinetics (forces responsible for motion) Failure (result of loading)

10. Statics Overview

12. Static Analysis F Static Analysis -To solve for the muscle force, remove rigid body ‘bc’ and replace the section with reaction forces at ‘b’ W c L d q f Rx Ry a b

13. F Static Analysis Resolve muscle force vector into x and y components useful angles Sum the forces and moments and set them equal to zero Use the figure to find the forces and moments in each direction Now let’s plug in some realistic values and solve for the forces What happens to F if we increase the moment arm, d? To Prof. Walsh W L d q f Rx Ry a b If W=20 lbf, L=14 in, d=0.5 in,  =70°, and  =50°:  =70° F=687 lbf Rx=-235 lbf Ry=626 lbf If d=1 in F=343 lbf Rx=-117 lbf Ry=303 lbf =Rx+F*cos(  ) Rx=-F*cos(  ) =F*sin(  )-W+Ry Ry=W-F*sin(  ) =W*L*sin(  )-d*F*sin(  ) F=W*L*sin(  )/[d*sin(  )] + ∑Fx=0 + ∑Fy=0 + ∑M b =0     =90 ° -  +  y x

14. Overview Mechanics (effects of forces on a body) Mechanical Testing (response to loading) Statics (equilibrium of forces and moments) Dynamics (bodies in motion) Mechanics (effects of forces on a body) Kinematics (displacements, velocities, and accelerations) Kinetics (forces responsible for motion) Failure (result of loading)

15. Mechanics of Materials Overview

16. Mechanics of Materials Types of stresses and their equations state of stress (can be used to show the state of stress at a point) To Jeff  xx  yy  zz  xy  xy  yz  yz  xz  xz  xz  zz  xx  yy  yz  yz  xz  xz  xy  xy F F F rigid body stress cube Normal Stresses: Bending  b =M*c/I Axial  =F/A Shear Stresses: Torsion  =T*r/J Transverse Shear  =V/A

17. Mechanics of Materials Mechanics of Materials a M -To find the stresses at point ‘e’: -Make a cut at ‘e’ -Replace the removed section with reaction forces and a moment at point ‘e’ -Remove all components to the right of the cut W q Rx Ry d L F c f b e

18. Mechanics of Materials q a M L-d Simplify analysis by rotating the coordinate system and force vectors Solve for the reaction forces and moment W Rx Ry e Wy Wx x y Wy=W*cos(  ) ; Wx=W*sin(  ) + ∑Fx=0=Rx + ∑Fy=0=-Wy+Ry Ry=Wy + ∑M=0=Wy*(L-d)-M M=Wy*(L-d) If W=20 lbf, L=14 in, d=1 in, and  =70°: M=89 in-lb

19. Mechanics of Materials q M L-d Cross-section of bone at ‘e’ Bending stress at ‘e’ due to moment ‘M’  b =M*c/I Normal stress at ‘e’ due to Rx  N =Rx/A  N =4.4 psi A=  *(ro 2 -ri 2 )=1.57 in 2 Rx=Wx=W*cos(  )=6.8 lbf Shear stress at point ‘e’ due to Ry  N =Ry/A  N =12 psi Ry=Wy=W*sin(  )=18.8 lbf c=ro I=  *(ro 4 -ri 4 )/4 For M=89 in-lb, ro=0.75 in, ri=0.25 in I=0.245 in 4  b =272 lb/in 2 = 272 psi failure strength (bending) <<  f =30,250 psi To Prof. Walsh The stresses found above were calculated for a point at the top of the cross-section. The stresses will be lower at any other point about the cross-section. W Rx Ry Wy Wx x y e a ro ri

21. Overview Mechanics (effects of forces on a body) Mechanical Testing (response to loading) Statics (equilibrium of forces and moments) Dynamics (bodies in motion) Mechanics (effects of forces on a body) Kinematics (displacements, velocities, and accelerations) Kinetics (forces responsible for motion) Failure (result of loading)

24. Overview Mechanics (effects of forces on a body) Mechanical Testing (response to loading) Statics (equilibrium of forces and moments) Dynamics (bodies in motion) Mechanics (effects of forces on a body) Kinematics (displacements, velocities, and accelerations) Kinetics (forces responsible for motion) Failure (result of loading)