The following problems are to be solved using Mohr's circle. Neatly and carefully, hand-generate Mohr's circle solutions so that the final solution is easy to identify and grade. Draw Mohr's circles to scale, and completely label with all pertinent information. Be sure to label the reference points (coordinate pairs of normal and shear stress/strain components) as plotted from the faces of the given element on Mohr's Circle. In your final solution, show a neatly drawn sketch of the point of interest on the component (if no component is given, draw one) with original reference axes (like "x" and "y" for theoretical stress analysis, or "A" and "C" for strain gage rosette reduction). Principal axis orientations should be shown relative to an original reference axis of your choice. Scan your hand solutions to PDF, combine any PDF files, and upload as a single PDF file (be sure your last name is in the filename as always) to this assignment link on Canvas. Retain your hard copy to use as study materials. 1) A state of stress that occurs at a point on the free surface of a solid body is: ox= 60 MPa; oy = 0 MPa; Txy = 30 MPa. Evaluate the two principal normal stresses and the one principal shear stress that can be found by coordinate system rotations in the x-y plane, and show the coordinate system rotations to these principal stresses with sketches of the elements for both principal normal stresses and principal shear stresses, correctly oriented relative to x & y axes. Determine the maximum normal stress and the maximum shear stress at this point. Indicate which planes upon which these maxima are acting. That is, be sure the resulting maxima are labeled with the correct and appropriate subscripts. 2) A state of stress that occurs at a point on the free surface of a solid body is: ox=-100 MPa; oy = 40 MPa; Txy = -50 MPa. Evaluate the two principal normal stresses and the one principal shear stress that can be found by coordinate system rotations in the x-y plane, and show the coordinate system rotations to these principal stresses with sketches of the elements for both principal normal stresses and principal shear stresses, correctly oriented relative to x & y axes. Determine the maximum normal stress and the maximum shear stress at this point. Indicate which planes upon which these maxima are acting. That is, be sure the resulting maxima are labeled with the correct and appropriate subscripts. 3) An element of material is subjected to the following state of stress: ox= 100 MPa; oy= 40 MPa; oz= -60 MPa; Txy = 30 MPa; Tyz = Txz=0 MPa. Evaluate the principal normal stresses and the principal shear stresses that can be found in all three orthogonal planes. Show the orientations of the principal stresses on sketches. 4) Strains are measured on the surface of a titanium alloy part as follows: &x= 3400 , &y= 1300 , and yxy=2100 . Estimate the maximum normal stress and the maximum shear stress, and show the orientation of the principal element on a sketch. (Assume that the gages were bonded to the metal when there was no load on the part, that there has been no yielding, and that no loading is applied directly to the surface). Find properties for titanium in the Dowling text. 5) A pipe with closed ends has an outer diameter of 80 mm and wall thickness of 2.0 mm. It is subjected to an internal pressure of 10 MPa and a bending moment of 2.0 kN.m. Determine the maximum normal stress and the maximum shear stress, and show the orientation of the principal normal element on a sketch of the pipe. Neglect localized effects of the end closure. 6) A tube has an outer diameter of 60 mm and wall thickness of 3.0 mm. It is subjected to a bending moment of 1.8 kN-m and a torque of 2.5 kN-m. Determine the maximum normal stress and the maximum shear stress, and show the orientation of the principal normal element on a sketch of the tube. 7) A rectangular strain gage rosette is employed to measure strains on the free surface of a titanium alloy part, with the result being &A=2200 , = 1500 , and c = 1800 . Determine the maximum normal stress and the maximum shear stress, and show the orientation of the principal normal element on a sketch of the part. Assume that no yielding has occurred. 8) A rectangular strain gage rosette is employed to measure strains on the free surface of an aluminum alloy part, with the result being &a=-1550 , &b= -3800 , and &c=720 . Determine the maximum normal stress and the maximum shear stress, and determine the orientation of the principal normal element relative to reference axes. Assume that no yielding has occurred.