BMES 345 CH08 Problem Set 20240402 (1)

BMES 345 CHAPTER 8 PROBLEMS: NON-UNIFORM CONDITIONS AND STATICALLY INDETERMINATE SYSTEMS Contents Problem 1: Non-Uniform Axial Loading ...................................................................................................................... 3 Problem 2: Non-Uniform Axial Loading ...................................................................................................................... 4 Problem 3: Non-Uniform Axial Loading ...................................................................................................................... 5 Problem 4: Non-Uniform Axial Loading ...................................................................................................................... 6 Problem 5: Non-Uniform Axial Loading ...................................................................................................................... 7 Problem 6: Non-Uniform Axial Loading ...................................................................................................................... 8 Problem 7: Non-Uniform Axial Loading ...................................................................................................................... 9 Problem 8: Non-Uniform Axial Loading .................................................................................................................... 10 Problem 9: Statically Indeterminate Axially Loaded Systems ............................................................................. 11 Problem 10: Statically Indeterminate Axially Loaded Systems .......................................................................... 12 Problem 11: Statically Indeterminate Axially Loaded Systems .......................................................................... 13 Problem 12: Statically Indeterminate Axially Loaded Systems .......................................................................... 14 Problem 13: Statically Indeterminate Axially Loaded Systems .......................................................................... 15 Problem 14: Axial Loading ........................................................................................................................................... 16 Problem 15: Non-Uniform Axial Loading .................................................................................................................. 17 Problem 16: Non-Uniform Axial Loading .................................................................................................................. 18 Problem 17: Non-Uniform Axial Loading .................................................................................................................. 19 Problem 18: Non-Uniform Axial Loading .................................................................................................................. 20 Problem 19: Non-Uniform Torsion ............................................................................................................................. 21 Problem 20: Non-Uniform Torsion ............................................................................................................................. 22 Problem 21: Torsion ...................................................................................................................................................... 23 Problem 22: Statically Indeterminate Systems in Torsion ................................................................................... 24 Problem 23: Statically Indeterminate Systems in Torsion ................................................................................... 25 Problem 24: Statically Indeterminate Systems in Torsion ................................................................................... 26 Problem 25: Statically Indeterminate Systems in Torsion ................................................................................... 27 Problem 26: Torsion ...................................................................................................................................................... 28 [Solution] Problem 1 ...................................................................................................................................................... 29 [Solution] Problem 2 ...................................................................................................................................................... 31 [Solution] Problem 3 ...................................................................................................................................................... 33 1

[Solution] Problem 4 ...................................................................................................................................................... 34 [Solution] Problem 5 ...................................................................................................................................................... 36 [Solution] Problem 6 ...................................................................................................................................................... 38 [Solution] Problem 7 ...................................................................................................................................................... 39 [Solution] Problem 8 ...................................................................................................................................................... 41 [Solution] Problem 9 ...................................................................................................................................................... 43 [Solution] Problem 10 .................................................................................................................................................... 46 [Solution] Problem 11 .................................................................................................................................................... 48 [Solution] Problem 12 .................................................................................................................................................... 51 [Solution] Problem 13 .................................................................................................................................................... 54 [Solution] Problem 14 .................................................................................................................................................... 56 [Solution] Problem 15 .................................................................................................................................................... 59 [Solution] Problem 16 .................................................................................................................................................... 61 [Solution] Problem 17 .................................................................................................................................................... 62 [Solution] Problem 18 .................................................................................................................................................... 64 [Solution] Problem 19 .................................................................................................................................................... 67 [Solution] Problem 20 .................................................................................................................................................... 69 [Solution] Problem 21 .................................................................................................................................................... 71 [Solution] Problem 22 .................................................................................................................................................... 73 [Solution] Problem 23 .................................................................................................................................................... 75 [Solution] Problem 24 .................................................................................................................................................... 78 [Solution] Problem 25 .................................................................................................................................................... 81 [Solution] Problem 26 .................................................................................................................................................... 84 2

Problem 1: Non-Uniform Axial Loading Consider the portion of the cervical spine known as C3- C6, consisting of four cervical vertebrae and three interposing intervertebral discs (figure on the right). Assume you can model each vertebral body and intervertebral disc as prismatic bars made of linear elastic materials. Each vertebral body is made of cortical bone ( E = 15 GPa) and has a height of 15 mm. Each intervertebral disc has a height of 9 mm and E = 980 kPa. Both vertebrae and intervertebral discs have cross- sectional areas of 750 mm 2 . Under a compressive load F = 50 N (roughly equivalent to the weight of a human head), what is the total deformation (shortening) of the C3-C6 segment of the cervical spine? 3

Problem 2: Non-Uniform Axial Loading Synthetic biomaterial scaffolds are being developed to induce bone regeneration in critical size defects (defects that are too large for normal healing to occur). In this example, the scaffold fills a 1.5 cm gap in the tibia. In the diagram, in addition to the axial compressive loads, there is an intermediate tensile load applied by the patellar ligament (transmitting the quadriceps muscle force): Assume the bone has a circular cross-sectional area of A = 480 mm 2 . The elastic modulus of the tibia is E = 15 GPa, and the elastic modulus of the space filler is E = 3.5 GPa. The force applied by the ankle joint is 1.5 kN. The force applied by the knee is 2.2 kN. Determine the total change in length of the tibia. 4

Problem 3: Non-Uniform Axial Loading After fracture of the tibia, a callus forms around the fracture site, consisting of a mixture of woven bone and cartilage. Over time, the callus is replaced with normal, organized bone tissue (called lamellar bone) that is much stronger than the temporary callus. Let us consider a tibia at the early stages of the callus. The callus has a larger radius than the normal tibia, but is composed of weaker woven bone: Let us assume that both the cortical bone of the normal tibia and the woven bone of the callus are linear elastic materials with E cortical = 18 GPa and E woven = 0.5 GPa. The proximal and distal tibia are prismatic bars with cross-sectional areas A = 350 mm 2 . The callus is also a prismatic bar, with cross- sectional area A = 720 mm 2 . The entire tibia is subjected to end-applied compressive forces F = 500 N, as shown above. Based on these assumptions and parameters, answer the following questions: a) What is the internal force in the callus? b) What is the change in length of the entire tibia? 5