Problem 1:
A human femur is mounted in a torsion testing machine, the femur is subjected to a torsional loading until it fractures at section a-a. The applied torque, or moment, versus total angle of twist is shown on the graph. Assume that the bone can be approximated as a unit cylindrical tube with cross-section as shown.
a) Derive a formula to describe the shear strain, , from the given information.
b) At the moment of fracture, what is the shear strain at the inner and outer surfaces of the bone?
c) At the moment of fracture, what is the angle of twist, , at the site of the fracture a-a relative to E?
d) At the moment of fraction, what is the shear stress, , at the inner and outer surfaces of the bone?
Problem 2:
A shaft is made of a steel alloy having an allowable shear stress of allow = 12 ksi. If the diameter of the shaft is 1.5 in, determine the maximum torque T that can be transmitted. What would be the maximum torque ‘T' if a 1 in diameter hole is bored though the center of the shaft? Sketch the shear stress distribution along a radial line in each case.
Problem 3:
Given a cylinder of radius R and length L with applied torque T and shear modulus G:
a) What is the shear stress in the cylinder as a function of the radius ρ?
b) What is the shear strain in the cylinder as a function of ρ?
c) What is the angle of twist of the cylinder?
Given the same cylinder with a distributed torque of t N-m/m
d) What is the shear stress as a function of x and ρ in the cylinder?
e) What is the shear strain as a function of x and ρ in the cylinder?
f) What is the total angle of twist of the cylinder?
Problem 4:
You are a biomedical engineer at a small company that is developing vertebral disc implants using a new biocompatible polymer. You want to know how much torque will be applied to the disc implant when somebody twists their spine. From an X-ray, you determine that the angle of twist between the points A and B, on the adjacent bony vertebral segments is 2 degrees when the person twists as much a possible. The new polymer has Young's modulus of 6 GPa and shear modulus 3 GPa, and fails when normal stress exceeds 15 GPa, or shear stress exceeds 8 GPa.
R= 1.5 cm, H =1 cm
a) What is the applied torque?
b) Does the implant fail under this torsional load?
c) Where would it fail? (show on a diagram)
Problem 5:
An orthopedic surgeon is inserting a hip implant into the shaft of a femur. The implant can be modeled as tapered cylinder with maximum radius of Rmax = 3a, minimum radius of R=1, and length L1 connected to a solid cylinder of length L2+L3 and radius a.. The shear modulus is G. Rmax = 3a
T
L1
L2
The surgeon has pushed the bottom portion (L3) of the implant into the hollow of the bone. She twists the implant with torque T at the top of the implant. A uniform resistive torque t N-m/m acts on the implant/bone interface as shown.
a) Derive a formula for the total angle of twist of the implant while the surgeon twists it (in terms of the above variables). If L1=L2=L3=L, which part twists the most?
t
L3
R=a
Problem 6:
The spinal column consists of a series of vertebrae separated by intervertebral discs. Each vertebral body can be modeled as a solid cylinder with height HB=1.5 cm, radius RB =1.5 cm, and shear modulus GB= 5 GPa. Each intervertebral disc has two parts. The outer annulus fibrosus has shear modulus GA= 1 GPa, height HD=0.5 cm, inner radius RDi = 0.5 cm and outer radius RDo = 1.5 cm. The inner nucleus pulposus has shear modulus GN= 0.5 GPa, height HD=0.5 cm, and radius RDi = 0.5 cm.
A physical therapist applies a torque T=50 N-m to your spine as shown. Assume no slippage at the interfaces. What is the maximum shear stress in each of the 3 parts of the spinal column?
Problem 7:
The femur can be modeled as a partially hollow (BC) and partially solid (AC) cylinder. The outer radius is ro and the inner radius is ri. End A is fixed and torques (T1 and T2) are applied about the long axis of the bone as shown. What is the maximum shear stress in the femur and the total angle of twist? The shear modulus of the femur, G=4.6 GPa, L1=L2=10cm, ro=2cm, ri=1cm, T1=50 N-m, T2=100 N-m
L1/2T1
Problem 8: (Problem Solving Journal)
A gymnast is supporting his weight on the parallel bars as shown. Draw a free body diagram of one of the bars. Assume the beam has length L, the supports are positioned at L/10 from each end, and his hands are located L/5 from the front support. Where would the internal shear be greatest? Sketch what the bent beam would look like.