Bending, Buckling, and Case Study in Stress Shielding

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Bending, Buckling, and Case Study in Stress
Shielding
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Bending

• Pure Bending
• The longitudinal strain is linearly varying
  with the distance from the neutral axis.
• ?x-y/?
• ?m-c/?

• For a linear elastic isotropic solid loaded in
  1-D ?xE?x
• ?mMc/I
• ?x-My/I
• 1/? M/EI (curvature)

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Buckling

• Eulers formula-- buckling of a centrally
  compressed bar
• Assume initially straight and loaded by centrally
  applied load
• If P is less than Pcrit then it is stable and
  experiences only axial compression
• Load is calculated from the deflection curve.

• For one fixed end
• Pcr ?2EI/4l 2
• --it is the smallest load that can keep the
  bar in a slightly bent shape.
• For a bar with built in ends
• Pcr 4?2EI/l 2
• Can occur in overload in fracture fixation
  devices

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Clinical relevance of bending

• Total Stress can be calculated by combining the
  axial and bending stresses.
• About 200,000 THR surgeries annually in the U.S.
• Main design problems
• Stress shielding
• Loosening

Circular cross section (hollow)
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Loading at Hip Joint
(courtesy Dr. Tom Andriacchi, Stanford University)
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THR Material Requirements

• Biocompatibility
• function in body without local or systemic
  rejection response
• Resistant to corrosion, degradation, and wear
• Similar mechanical properties (strength,
  stiffness, friction) to structures they replace
• High quality and low cost

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THR procedure

• Existing hip joint is completely removed and
  replaced with an artificial hip
• Diseased femoral head is removed
• Acetabulum is reamed and acetabular component is
  inserted
• Bone marrow is extracted from proximal femoral
  canal and femoral stem is inserted into cavity

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Fixating Implant

• Two methods of fixating artificial components to
  bone bone cement and bony ingrowth

(http//orthoinfo.aaos.org)
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Early Development of THR

• First THR used stainless steel femoral stem
  fixated with bone cement
• In mid 1970s started to see increase in stem
  fractures of devices implanted in the 1960s

(courtesy Dr. Tom Andriacchi, Stanford
University))
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Solution to Fracture Problem

• Material
• changed manufacturing processes
• introduced Cobalt-Chromium alloy stem (stronger
  and stiffer)

• Geometry
• made stems thicker in high stress areas

• Solved stem fracture problem

• Introduced other problems

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Bone Loss - Stress Shielding

• Wolffs Law (1869) bone adapts (remodels) in
  response to the mechanical loads placed on it
• Stiff implant changes mechanical loads on femur

Solution Make implant more flexible
(courtesy Dr. Tom Andriacchi)
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How do we make beam more flexible?

• In axial loading case

• reduce axial rigidity (EA)

• In pure bending case

• reduce flexural rigidity (EI)

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Example Problem

• Composite circular beam made of metallic core
  (stem) and outer sleave made of bone
• Axial compressive force F -3000 N (4 Body
  Weights for 168 lb person)
• Bending moment M 30 N-m
• Neglect shear force
• Bone shaft diameters
  dout 2.5 cm din 1.0 cm
• Longitudinal modulus of cortical bone 17 GPa

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Example Problem

• For simplicity, perform analysis of two loading
  cases separately
• Use three different materials for stem
• stainless steel (E 193 GPa)
• Co-Cr alloy (E 214 GPa)
• titanium alloy (E 124 GPa)
• Use two different stem diameters
• 1.1 cm
• 1.5 cm

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Composite Beam Theory

• Axial loading
• Pure Bending Moment (max stress)

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Compressive Stress Axial Loading

• Bone without implant 7.3 MPa

• 1.1 cm stainless steel stem
• Bone 2.0 MPa Stem 23.1 MPa

• 1.5 cm stainless steel stem
• Bone 1.3 MPa Stem 14.7 MPa

• 1.5 cm Co-Cr stem
• Bone 1.2 MPa Stem 14.9 MPa

• 1.5 cm Titanium stem
• Bone 1.9 MPa Stem 13.6 MPa

• 1.1 cm Titanium stem
• Bone 2.8 MPa Stem 20.1 MPa

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Compressive Stress Bending

• Bone without implant 20.0 MPa

• 1.1 cm stainless steel stem
• Bone 14.0 MPa Stem 70.4 MPa

• 1.5 cm stainless steel stem
• Bone 8.4 MPa Stem 56.8 MPa

• 1.5 cm Co-Cr stem
• Bone 7.8 MPa Stem 69.0 MPa

• 1.5 cm Titanium stem
• Bone 10.8 MPa Stem 47.2 MPa

• 1.1 cm Titanium stem
• Bone 15.8 MPa Stem 50.8 MPa

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Combined Max Compressive Stress

• Worst case for stress shielding
• 1.5 cm Co-Cr
• Best case for stress shielding
• 1.1 cm Titanium

(http//news.bbc.co.uk)
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Case Study on Bone Resorption

• Bobyn and Engh (1988) examined 411 cases of
  cementless hip replacements
• They categorized the extent of bone resorption
  for each case as none, 1st degree, 2nd degree, or
  3rd degree

• For stems gt 1.3 cm, 28 had 2nd or 3rd degree
  bone resorption
• For stems lt 1.3 cm, 6 had 2nd or 3rd degree bone
  resorption

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Summary

• THR is a great bioengineering achievement that
  has improved millions of lives
• THR design has improved greatly over the last 4
  decades through a proper understanding of the
  loading conditions and the properties of the
  materials