Ch. 5 - Structural Analysis of MSK Systems - Beam Theory

Question 1

Why using models to help us calculate the stresses in bone and bone-implant systems, why do we aim to choose the simplest model to do the job?

Answer 1

Because that way we have

• fewer assumptions
• less chance for error

Question 2

State 2 potential advantages of Beam Theory.

Answer 2

1. It can be used to describe a number of skeletal structures, as well as bone-implant systems
2. It is a simple model that can be used to validate results obtained from a more complex model (e.g. FEA)

Question 3

Give 4 examples of 1 material beams in the MSK system that could be analysed using simple beam theory.

Answer 3

1. A bone strut
2. The femoral neck component of a total hip replacement
3. The elastic behaviour of bone specimens
4. Long bone behaviour

Question 4

Elaborate on 2 examples of a 2 material beam in the MSK system that could be analysed using simple beam theory.

Answer 4

1. A fractured long bone treated using a plate attached with screws. After healing the device and bone form a composite beam. Initially the system can be thought of as a single material beam with the rod carrying all the force. As the bone heals, both materials are carrying load and now the system can be considered a composite beam.
2. Bone specimens from the longitudinal axis of a bone when tested in bending to failure. With yielding, cross-section will have sections behaving elastically and others inelastically.

Question 5

Elaborate on how both a cemented and an uncemented total hip replacement stem can be modelled as a 3 material beam.

Answer 5

• Cemented: PMMA (bone cement) is the 3rd material

- Cementless: fibrous tissue layer that forms around porous coating is the 3rd material

Question 6

The derivation of beam theory is based on 3 elements. What are they?

Answer 6

1. A strain-displacement relationship
2. Equilibrium relationships
3. Material behaviour

Question 7

What is the fundamental assumption of all beam theory?

Answer 7

That all plane sections before loading remain plane after loading.

Question 8

How is strain distribution different in response to axial loading vs. bending?

Answer 8

Axial loading - strain is constant over the cross-section

Bending - strain varies linearly from the neutral axis

Question 9

State the fundamental assumptions according to beam theory for the case of a 1-material symmetric beam under axial loading.

Answer 9

1. Plane sections remain plane after loading
2. The material is elastic with a linear stress-strain diagram (Hooke’s Law applies)
3. The only stress is the longitudinal stress in the x direction - All other stress components are assumed to be 0 and the beam is in equilibrium. (Each longitudinal fibre of the beam is subjected to only tension or compression).

Question 10

According to beam theory, what is true about the stress in a one material beam that does not hold for beam of more materials?

Answer 10

The stress in a one material beam does not depend on the material because the elastic modulus term E drops out. It depends only on load and beam geometry.

Question 11

State the fundamental assumptions according to beam theory for the case of a 1-material symmetric beam under bending.

Answer 11

1. Bending moment is applied about an axis perpendicular to the plane of symmetry (there is no out of plane bending)
2. Plane sections remain plane after loading
3. Deformations vary linearly across any cross-section
4. Reference (neutral) axis is that which neither lengthens nor shortens

Question 12

State the fundamental assumptions according to beam theory for the case of a 3-material symmetric beam.

Answer 12

• Assume the composite beam deforms as a one-material beam:
  1. Plane sections remain plane after loading
  2. Each material shows linear elastic behaviour
• Hooke’s Law applies for each component
• Piecewise linear normal stress variation sigma = E x epsilon

Question 13

Why is the neutral axis usually the geometric centroid in a symmetric 1-material beam but not in a composite beam?

Answer 13

In a composite beam, the location of the neutral axis is affected by positions and material properties of each component.

Question 14

How can you determine the location of the neutral axis for a composite beam?

Answer 14

Using the equilibrium condition for the axial force.

Question 15

How is the situation of an unsymmetrical beam fundamentally different from that of a symmetrical beam?

Answer 15

1. We can no longer assume that the member will bend in the plane of the couples
2. In general, the neutral axis of the section will not coincide with the axis of the couple

Question 16

Why do we even need to consider unsymmetrical beams for orthopaedic biomechanics?

Answer 16

1. Bone cross-sections are not symmetric

2. An implant may be symmetric, but it is not necessarily placed symmetrically wrt the bone

Question 17

Describe the main principles of Unsymmetrical Beam Theory.

Answer 17

1. We have two coordinate systems (y,z) and (t,s), where the origin of the (t,s) system is located wrt the modulus-weighted centroid (at y^,z^).
2. The cross-section is arbitrary with respect to the (y,z) coordinate system.
3. The cross-section can be defined by regions Ai, each with its own Ei.
4. Main assumption: plane sections remain plane after bending.
5. Assume that strain at any point (t,s) in the cross-section is a linear function

Question 18

State the effects of superior eccentricity of the trabecular bone in the femoral neck.

Answer 18

1. Increase internal effective bending moment on the cross-section
2. Shift the neutral axis of bending inferior to the center of the neck
3. Decrease the distance from the neutral axis to the inferior periosteal surface of the bone