Ch. 6 - Structural Analysis of MSK Systems - Advanced

Question 1

What are the major assumptions of beam theory in general?

Answer 1

1. In bending, plane cross-sections remain planar
2. Beams should be a) sufficiently long, b) approx. straight, and c) approx. prismatic
3. There are no abrupt changes in geometry, material properties, or loading (i.e. long bones)

Question 2

State which mechanical principle you would apply to model each of the following situations:

a) Load transfer from implant to bone through a cement layer
b) Torsional loading of bones
c) Contact btw articulating surfaces

Answer 2

a) Beams on elastic foundation
b) Torsion of non-circular rods
c) Contact stress analysis - Hertz Theory

Question 3

In the BOEF equation, what does the term EI represent? What does the equation tell us?

Answer 3

EI represents the structural stiffness of the beam. The equation tells us that it is equal to the sum of the structural stiffnesses of the individual components of the beam.

Question 4

State what each of the following terms of the Governing Equation for BOEF represent:

a) k
b) v(x)
c) p(x)

Answer 4

a) Foundation modulus
b) Displacement of the beam in the y direction
c) Distributed load applied to the beam (represented as force/length)

Question 5

Consider the case of an infinite length beam, loaded at center with load P. What are the system’s boundary conditions based on a BOEF analysis?

Answer 5

v(infinity) = 0
dv/dx at infinity = 0
dv/dx at 0 = 0
V(0) = -P/2

Question 6

Describe what mechanical principle you would use to model the bone-interface stem system of a fixation stem.

Answer 6

Bone-interface stem system can be represented as 2 beams (bone + stem) and an elastic foundation (cement). If no cement is used, elastic foundation is the interface layer than would form.

Question 7

What does the theory of torsion of circular sections tell us and when does it apply?

Answer 7

It is valid for both solid as well as hollow cross-sections. The theory tells us that shear stresses vary radially with the radius.

Question 8

Why do we need to use a different method to analyse circular cross section vs non-circular ones?

Answer 8

Because non-circular cross section do not remain plane under torsional loading as there will be out-of-plane displacement (“warping”).

Question 9

What are the assumptions of the theory of torsion of non-circular cross-sections?

Answer 9

1. Straight, prismatic bar
2. Homogenous and elastic material (isotropic material behaviour in cross-sectional plane - at least transversely isotropic)
3. Small deformations
4. Distortion of any transverse plane does not depend on position along the axis, but only upon the position in the cross-sectional plane

Question 10

What is torsional stiffness?

Answer 10

It describes how rigid a material is, i.e. how much resistance it offers per degree change in its angle when twisted.

Question 11

Describe the membrane analogy and how it can be useful to calculate/approximate torsion in non-circular cross-sections.

Answer 11

Torsional behaviour of a prismatic bar of an arbitrary cross-section can be predicted using the analogy of pressurized membrane of the same shape. Consider a membrane stretched across a hole that has the same shape as the cross-section and then it is inflated.

Question 12

What 3 analogies can be deduced between the membrane analogy and the torsion of a non-circular cross-section?

Answer 12

1. Steepest slope of the membrane at any point = magnitude of the max shear stress
2. Level contour line at any point (perpendicular to slope) = direction of total shear stress
3. Volume under the membrane = half of the applied torque

Question 13

What 3 basic assumptions are made when using Hertz Theory?

Answer 13

1. Contact areas are small wrt the size of the bodies in contact
2. Both surfaces deform
3. The elastic moduli of the contacting components are similar (note: a stiffer material has a higher elastic modulus)

Question 14

Why would Hertz Theory not apply to analyse contact stress between metal & polyethylene in a knee joint replacement?

Answer 14

This type of contact stress is actually better modelled by considering the metal as a rigid indenter. The deformation of the metal is actually negligible, so assumption 2 does not hold. This occurs because metal is much stiffer than PE, so assumption 3 also does not hold (elastic modulus of metal is 2 orders of magnitude greater than PE).

Question 15

In contact stress analysis, both contact area and maximum contact pressure are functions of _____, ______ and _______.

Answer 15

load, modulus and conformity

Question 16

According to contact stress anlysis, when the load increases, what happens to:

• contact area?
• stress?

Answer 16

When the load increases, both the contact area and the stress increase.

Question 17

According to Contact Stress Analysis, if the diameter of the spheres decreases, what happens to:

• contact area?
• contact pressure?

Answer 17

• contact area decreases

- contact pressue increases

Question 18

According to Contact Stress Analysis, if the elastic modulus increases, what happens to:

• contact area?
• maximum contact pressure?

Answer 18

• contact area decreases

- maximum pressure increases

Question 19

Using Contact Stress Analysis, consider the case of contact between a sphere and a concave cup. How do you adapt the model to apply to this situation?

Answer 19

For the concave object, use radius of curvature with a minus sign.

Question 20

Consider the following situation and state what model you would use to analyse them:

a) Examine transverse load transfer of an implant against a bone
b) Examine long bones and identify peak stress for irregular shapes
c) Examine the stress between to surfaces in contact

Answer 20

a) BOEF
b) Torsional analysis if non-circular cross-sections
c) Hertz Theory