Posted: September 26th, 2022

Derive a formula to describe the shear strain,

BMED 3400D&E Fall 2011HW #5Problems 7 and 8 due September 29Other problems due October 25Problem 1A human femur is mounted in a torsion testing machine, thefemur is subjected to a torsional loading until it fractures atsection a-a. The applied torque, or moment, versus total angle oftwist is shown on the graph. Assume that the bone can beapproximated as a unit cylindrical tube with cross-section asshown.a) Derive a formula to describe the shear strain, , from thegiven information.b) At the moment of fracture, what is the shear strain at the innerand outer surfaces of the bone?c) At the moment of fracture, what is the angle of twist, , at thesite of the fracture a-a relative to E?d) At the moment of fraction, what is the shear stress, , at theinner and outer surfaces of the bone?Problem 2A shaft is made of a steel alloy having an allowable shear stress of allow = 12 ksi. If the diameterof the shaft is 1.5 in, determine the maximum torque T that can be transmitted. What would bethe maximum torque ‘T’ if a 1 in diameter hole is bored though the center of the shaft? Sketchthe shear stress distribution along a radial line in each case.Problem 3Given a cylinder of radius R and length L with applied torque T and shearmodulus 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/md) 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 4You are a biomedical engineer at a small company that is developingvertebral disc implants using a new biocompatible polymer. You want to know how much torquewill be applied to the disc implant when somebody twists their spine. From an X-ray, youdetermine that the angle of twist between the points A and B, on the adjacent bony vertebralsegments is 2 degrees when the person twists as much apossible. The new polymer has Young’s modulus of 6 GPa andshear modulus 3 GPa, and fails when normal stress exceeds 15GPa, or shear stress exceeds 8 GPa.R= 1.5 cm, H =1 cma) What is the applied torque?b) Does the implant fail under this torsional load?c) Where would it fail? (show on a diagram)Problem 5An orthopedic surgeon is inserting a hip implant into the shaft of a femur.The implant can be modeled as tapered cylinder with maximum radius ofRmax = 3a, minimum radius of R=1, and length L1 connected to a solidcylinder of length L2+L3 and radius a.. The shear modulus is G.Rmax = 3aTL1L2The surgeon has pushed the bottom portion (L3) of the implant into thehollow of the bone. She twists the implant with torque T at the top of theimplant. A uniform resistive torque t N-m/m acts on the implant/boneinterface as shown.a) Derive a formula for the total angle of twist of the implant while thesurgeon twists it (in terms of the above variables). If L1=L2=L3=L, whichpart twists the most?tL3R=aProblem 6The spinal column consists of a series of vertebrae separated by intervertebral discs. Eachvertebral 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 fibrosushas 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, andradius RDi = 0.5 cm.A physical therapist applies a torque T=50 N-m to your spine as shown. Assume no slippage atthe interfaces. What is the maximum shear stress in each of the 3 parts of the spinal column?Problem 7 (Detailed Annotated Solution)The femur can be modeled as a partially hollow (BC) and partially solid (AC) cylinder. Theouter radius is ro and the inner radius is ri. End A is fixed and torques (T1 and T2) are appliedabout the long axis of the bone as shown. What is the maximum shear stress in the femur and thetotal 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-mL1/2T1should be from B-CProblem 8 (Problem Solving Journal)A gymnast is supporting his weight on the parallel bars as shown. Draw afree body diagram of one of the bars. Assume the beam has length L, thesupports are positioned at L/10 from each end, and his hands are locatedL/5 from the front support. Where would the internal shear be greatest?Sketch what the bent beam would look like.

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