Torsion of bars experiment report

Below is a visual representation of how a fixed circular rod reacts when a torque is applied. The angle of twist, φ, is greatest at the position of applied torque, T. Notice how the angle propagates out from the fixed end. In other words, the angle of twist measured at position 2 will always be greater than the angle of twist measured at position 1. 1 2 L L2 L1 T 1 2 Material Type: Average Diameter (in.): Specimen Length (in.): Length to position 1 (in.): Length to position 2 (in.): Aluminum 0,4986 52,4986 17,486 32,36 Torque (in-lb) Position 1 (deg) Material Type: Average Diameter (in.): Specimen Length (in.): Length to position 1 (in.): Length to position 2 (in.): Brass 0,5014 52,5014 16,014 33,64 Torque (in-lb) Position 1 (deg) Material Type: Average Diameter (in.): Specimen Length (in.): Length to position 1 (in.): Length to position 2 (in.): Steel 0,5007 52,5007 17,014 32,14 Torque (in-lb) Position 1 (deg) Position 2 (deg) Position 2 (deg) Position 2 (deg) 25 0,5 1 1 2 2 1,5 50 1 2 2 4 4 3,5 75 3 2,5 3,5 6 7 6 25 0,5 0,5 0,5 1,5 1,5 1,5 50 1 1 1,5 2 2 2 75 2 2 2,5 3 3 3 75 1 1 1 2 1 1,5 150 1,5 2 2 3 2,5 3 225 3,5 3,5 3 5 4 4,5 100 4 3,5 4,5 8 8 7,5 100 3 3 3 4 4 4 300 5 4,5 5 6 6,5 7 Experiment #6: Torsion Objective The purpose of this experiment is to explore the relationship between torque and angle of twist, and to determine the shear stress – shear strain relationship for various linear-elastic bars. Relevance of Experimental Design Two test setups for torsional rigidity are available for the student’s review. The first using manually applied torque from a torque wrench and where the resulting rotation is measured. The other applies a known rotation and measures the torque. The latter of the two is fraught with complexity associated with the electronic torque measurement as system compliance must be removed to correct for lost torsion shear strains. This presents a common theme for the student’s consideration wherein the quality of any measurement must be questioned. The first of the two setups is preferred despite its simplicity due to the resulting reliability. However, unless specimens are very long, angular rotations may be small and therefore difficult to read with high precision. The student is invited to entertain other means of measuring rotation for the purposes of reducing the associated errors. Report Type Worksheet / Informal with cover letter Theoretical Background According to elementary elastic torsion theory, the shear stress in a bar caused by an internal torque is given by: = Tr J 𝜏 = 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝑟 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑇 = 𝑡𝑤𝑖𝑠𝑡𝑖𝑛𝑔 𝑚𝑜𝑚𝑒𝑛𝑡 𝑎𝑡 𝑡ℎ𝑒 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑟 𝑤ℎ𝑒𝑟𝑒 𝜏 𝑖𝑠 𝑏𝑒𝑖𝑛𝑔 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝐽 = ∫ 𝑟2𝑑𝐴 = 𝐼𝑥𝑥 + 𝐼𝑦𝑦 = 𝑝𝑜𝑙𝑎𝑟 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 Assumptions made in the calculation of the shear stress, τ, are: 1. 2. 3. 4. The bar is elastic. Plane sections remain plane and parallel. Radial lines remain radial. The length of the bar does not experience any change in dimension. For a solid circular bar, 𝐽= 𝜋𝑟4 2 In the process of deriving the torsion formula for τ, a second relationship involving the angle of twist (i.e. angular deformation) evolves. This is given by: 𝜑= 𝑇𝐿 𝐽𝐺 𝜑 = 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑡𝑤𝑖𝑠𝑡 𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 𝐿 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑇 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡𝑤𝑖𝑠𝑡𝑖𝑛𝑔 𝑚𝑜𝑚𝑒𝑛𝑡 𝐺 = 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝑟𝑖𝑔𝑖𝑑𝑖𝑡𝑦 𝑜𝑟 𝑠ℎ𝑒𝑎𝑟 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 (𝐻𝑜𝑜𝑘𝑒’𝑠 𝐿𝑎𝑤 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙𝑖𝑡𝑦 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) Shear stress and shear strain are linearly related by G: 𝐺= 𝜏 𝛾 𝛾 = 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑎𝑖𝑛 From these equations you will notice that the shear strain at any radius r is related to the angle of twist by, 𝑇𝐿 𝑟 𝛾 = 𝑇𝑟 = [ ( )] 𝜑𝑟 = 𝐽𝐺 𝐽𝐺 𝐿 𝐿 NOTE: The above formulas are valid only for a solid shaft with a circular cross-section. Instructions 1. Using a caliper, measure the diameter of the provided specimen bars at three points along its length.