2. BME 315 – Biomechanics Chapter 4. Mechanical Properties of the Body Professor: Darryl Thelen University of Wisconsin-Madison Fall 2009

3. TORSION  refers to the twisting of a straight shaft  loading is due to moment about the long axis (a.k.a. torque)  moment axis is parallel to the shaft axis Units of torque?

4. Torsion Demo  For a circular cross-section, assume that plane sections remain plane and undistorted  Square elements on surface undergo shear strain and likewise shear stress

5. STRESS AND STRAIN IN A CIRCULAR SHAFT Main Concepts:  Torsional loading induces shearing strains   Shearing strains vary linearly with the distance from the central axis of the shaft.  Shear stresses must also vary linearly with distance from shaft axis.  Max shear strain and shear stress occur at the outer shaft radius  Shear stress is present both on a cross-section and along a planar cut through shaft

6. SHEAR STRESSES (ELASTIC RANGE) To relate shear stress to the internal torque, T, use equilibrium: The result is: r  T – internal torsional moment at a cross-section  r – radial distance from the central shaft axis to point of interest  J – polar moment of inertia about the central shaft axis r

7. Polar Moment of Inertia

8. Example Problem A femur is potted in cement, and two loads f are applied at a distance d from the centerline. The bone has inner and outer radii a and c.  What is the maximum shear stress to be observed in the bone?  Where is the maximum shear stress located?

9. Example Problem A solid circular cylinder has a c=2 cm outer radius. If the twisting torque T applied at the free end is 1000 Nm, Determine the maximum shear stress If the cylinder is hollowed out by drilling a a=1.5cm radius hole down the center Sketch the shear stress distribution for the 2 cases

10. Stresses in Elastic Range Recall that the sum of the moments from the internal stress distribution is equal to the torque on the shaft at the section, The results are known as the elastic torsion formulas, From Hooke’s Law for shear,, so The shearing stress varies linearly with the radial position in the section.

11. 10 Define:  - the angle of twist of the shaft due to torsion. Angle of Twist This equation is valid for calculating the angle of twist for shafts of length L, circular cross-section and a shear modulus G under a constant torque T r

12. Example Problem  A solid cylinder (G=10GPa) 10 cm long and 2 cm in outer radius is subjected to a torque T=3kN-m. Determine the angle of twist.

13. Angle of Twist in Composite Shaft  Shaft has a shear modulus G, determine the angle of twist for: Segment AB Segment BC If L 1 =L 2 =L and d 1 =2d 2 =d, what is the relative angle of twist across the two segments?

14. Example Given L 1 = L 2 = 50 in, d 1 = 2.0 in, d 2 = 1.5 in, G = 10x10 6 psi:  What is the allowable torque, T, if the angle of twist is not to exceed 0.02 radians ©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license.

15. Problem 4.2  Determine an expression for the torque over each region of the shaft.

16. Rate of twist, and Torsion Formula

17. 13916 Stresses in Elastic Range Recall that the sum of the moments from the internal stress distribution is equal to the torque on the shaft at the section, The results are known as the elastic torsion formulas, Multiplying the previous equation by the shear modulus, From Hooke’s Law for shear,, so The shearing stress varies linearly with the radial position in the section.

18. Example T  let’s use “Best Mechanics of Materials”  http://web.umr.edu/~mecmovie/index.html Notation issues: –they use I P for J –they call the constitutive relations “Torque-twist relations” –they call the compatibility relations “Geometry of deformation” –when the constitutive reln’s are subbed into the compatibility reln’s, they refer to this new equation as the “compatibility” relation. I just call it “algebra”.

19. EXAMPLE PROBLEM The torques shown are exerted on pulleys A, B, C. Knowing that both shafts are solid, determine the maximum shearing stress in (a) shaft AB, (b) shaft BC.

20. Concept Questions  Consider a shaft with a hollow center of What would be an expression for J? How would shear stress vary as a function of radius r?  We now know that

21. These photos represent a torsional force that all may be familiar with. Photo 1 shows a rolled leaf of paper into cylindrical form. Photo 2 is shown after I applied equal and opposite torsional forces to the ends of the paper. Photo by Sean Schutten

22. 13621 From observation, the angle of twist  of the shaft is proportional to the applied torque T and to the shaft length L. Shaft Deformations When subjected to torsion, every cross-section of a circular shaft remains plane and undistorted. Cross-sections of noncircular (non- axisymmetric) shafts are distorted when subjected to torsion. Cross-sections for hollow and solid circular shafts remain plain and undistorted because a circular shaft is axisymmetric.

23. Recall the orthogonal nature of shear stresses At what orientation of the element would you observe principal stresses?

24. Example  Represent the state of stress in terms of a coordinate system corresponding w/ max shear stress

25. EXAMPLE PROBLEM A suturing thread is to be subjected to a 2.5-lb tensile load. Knowing that E=0.5(10 6 ) psi, that the maximum allowable normal stress is 6 ksi, and that the length of the thread must not increase by more than 1%, determine the required diameter of the thread.