Assume the maximum shear stress that bone (as a material) can withstand is tau Max = 57 MPa prior to failure. Determine the maximum torsional torque that can be carried prior to failure for each of the following hypothetical bone geometries:

a) A solid circular cylindrical bone of radius r = 2 cm

b) A hollow circular cylindrical bone with outer radius ro = 4 cm and inner radius, ri, chosen to have the same total cross-sectional area as the bone in (a).

c) A hollow circular cylindrical bone with outer radius ro = 6 cm and inner radius, ri, chosen to have the same total cross-sectional area as the bone in (a).

d) Which of these bones is strongest against torsional loads? Why?
A vertical force of F = 2500 N is applied to the femoral head as shown. This force is applied at a distance b = 65 mm laterally from the long shaft of the femur and a distance h = 250 mm above section a-a, which is at the mid shaft (half way between the hip and knee joints). Assume the long shaft of the femur is a hollow circular cylinder with inner radius ri = 200 mm and outer radius ro = 250 mm.

a) Assuming the femur is in static equilibrium, what are the reaction forces and moments that occur at section a-a?

b) What maximum axial stresses (compression and/or tension) arise at section a-a due to purely axial loading?

c) What maximum axial stresses (compression and/or tension) arise at section a-a due to purely flexural loading (i.e., bending)?

d) What then are the net axial stresses on the medial and lateral borders of the femoral shaft at section a-a? Are they different?

e) If bone (as a material) is generally stronger in compression than intension, does having an axial compressive load, in combination with bending, increase or decrease the chances of fracture in this situation? Why or why not?