17.4 Advanced Structural Analysis of Musculoskeletal Systems

17.4.0.1 Beam on elastic foundation

Many engineered structures and orthopaedic structures behave as a beam on an elastic foundation

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17.4.0.1.1 Examples

17.4.0.1.2 Free body diagram

Sum forces in \[\begin{split} V(x+dx) - V(x) +p(x) dx - q(x) dx =0 \\ {\frac{d V}{d x}} = q - p \end{split}\]

Elastic force from the foundation: \(q(x) = k \cdot v(x)\)

Sum moments about \(x+dx\) \[\begin{split} M(x+dx)-M(x) + V(x) dx + q dx \frac{dx}{2} + p dx \frac{dx}{2} =0 \\ {\frac{d M}{d x}} = p - q \end{split}\]

17.4.0.1.3 Governing differential equation

Moment equation \[M = EI \frac{d^2 v}{x^2}\]
Thus \[\frac{d^2}{x^2} \left(EI \frac{d^2 v}{x^2}\right) + k v(x) = p(x)\]
This is an ordinary differential equation that must be solved with boundary conditions
“Relatively easy” for distributed loads p(x) and infinite beams
Challenge for point loads and/or finite beam lengths as in the diagram... multiple equations needed for each “section”
The easiest of these problems don’t exist in orthopaedics

17.4.0.1.4

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17.4.0.1.5

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17.4.0.1.6

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17.4.0.1.7

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17.4.1 Torsion

17.4.1.0.1

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17.4.1.0.2

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17.4.1.0.3

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17.4.1.0.4

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17.4.2 Contact stress analysis

17.4.2.0.1 Contact stress analysis

Joints and joint replacement put surfaces into contact
Most human and engineering joints have interfaces with similar material properties (modulus)
- Bone-to-bone with cartilage bearing surfaces
- Ceramic-to-ceramic for some implant interfaces
Orthopaedic implants are often composed of materials with dissimilar material properties
- Metal-to-polyethylene
When dissimilar, one material may be “rigid” relative to the other

17.4.2.0.2 Conforming vs non-conforming surfaces

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Non-conforming surfaces has surface contours which are not coincident
Conforming surfaces have coincident surfaces
The degree of conformity clearly influences the stress for a given force that is passed
The elastic modulus influences the amount of conformity through deformation of the surfaces

17.4.2.0.3 Contact stress

Wikipedia

The contact stress is not uniform over the contact surface (in most situations–perhaps all joints)
For spherical (or toroidal) surfaces, the contact pressure is largest at the center (or centerline)
For other surfaces, max pressure is often at or near the edge
All contact produces normal and shear stresses below the contact surface (even frictionless)
Appropriate friction and non-linear material properties are necessary for accurate models of joint-replacement failure
Fatigue due to cyclic loading may drive failure (ie gait)

17.4.2.0.4 Hertz theory for contact between elastic bodies

17.4.2.0.4.1 with similar moduli

Good approximate solution method for materials with similar moduli
Provides insights into other contact problems (rigid-compliant)

17.4.2.0.5 Hertz theory for contact between elastic bodies

17.4.2.0.5.1 with similar moduli

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Maximum pressure \[\pmax = \frac{3 P}{2 \pi a^2}\]

Contact radius \[a = 0.721 \left(P C_G C_M\right)^{1/3}\]

Shortening of the distance between spheres \[\delta = 1.04 \left(\frac{P^2 C_M^2}{C_G}\right)^{1/3}\]

\(C_G, C_M\) are geometric and material parameters

Material parameter \[C_M = \frac{1-\nu_1^2}{E_1} + \frac{1-\nu_2^2}{E_2}\]

Geometric parameter for this problem \[C_G = \frac{D_1 D_2}{D_1+D_2}\]

This is a non-linear response, as are all contact problems

Non-linearity in

Deformation
Stress
Material
Geometry

The equations can be used to make general statements about the response when material or geometry changes

Must be aware of the limitations (assumptions)

Contact area is small relative to body
Both surfaces deform
Similar moduli, isotropic material

17.4.2.0.6 Hertz theory for contact between elastic bodies

17.4.2.0.6.1 Maximum normal stress as a function of load

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Warning: other (non-normal) stresses are of interest as well!

17.4.2.0.7

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17.4.2.0.8 Sub-surface stresses in Hertzian contact

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Tensile stress in radial direction around edges
Nearly hydrostatic at the center of the contact area
Maximum shear stress below the center of the contact area (\(z/a=0.51\)) with magnitude \(\tau_\mathrm{max} = 0.31 \pmax\)

17.4.2.0.9 Summary and conclusion of contact stress discussion

Analytical solutions offer insights that numerical solutions cannot
- ie, the non-linear dependence on material properties, geometry, etc
- Analytical solutions guide the thought process for “real” contact problems
Numerical solutions are likely to be used for “accurate solutions to real problems”
Contact problems are non-linear
Contact involves shear and normal stresses, fatigue is an issue

Potentially drop from here to the next section