1- The state of the stress of a point inside a part of a machine according...

 

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1- The state of the stress of a point inside a part of a machine according to the xyz frame is shown in the figure. (a) Write the stress matrix in the xyz frame. Find the principal stresses and principal directions and show the probing cube in the principal-axis coordinates. (b) Compute the maximum tensile and compressive normal stresses, and the maximum shear stress. (c) Start from the state of stress in principal axes. Compute the state of stress in each of the following coordinate systems: (i) 45-degree rotation of the principal axes frame about the first principal axis. (ii) 45-degree rotation of the principal axes frame about the second principal axis. (iii) 45-degree rotation of the principal axes frame about the third principal axis. (d) The orientation of the frame x'y'z' relative to the xyz frame is determined by rotating the xyz frame as follows: first rotate +45 degrees (counter- clockwise, in a right-handed direction) about the z axis, and then -45 degrees about the y axis. Express the unit vectors of the x'y'z' frame in the xyz frame. Then find the stress matrix in the x'y'z' frame. (e) Find the three invariants of the stress matrix for all cases above and show that in-fact they do not depend on the selection of the coordinate system. 1 100 MPa 140 MPa 40 MPa 70 MPa 1- The state of the stress of a point inside a part of a machine according to the xyz frame is shown in the figure. (a) Write the stress matrix in the xyz frame. Find the principal stresses and principal directions and show the probing cube in the principal-axis coordinates. (b) Compute the maximum tensile and compressive normal stresses, and the maximum shear stress. (c) Start from the state of stress in principal axes. Compute the state of stress in each of the following coordinate systems: (i) 45-degree rotation of the principal axes frame about the first principal axis. (ii) 45-degree rotation of the principal axes frame about the second principal axis. (iii) 45-degree rotation of the principal axes frame about the third principal axis. (d) The orientation of the frame x'y'z' relative to the xyz frame is determined by rotating the xyz frame as follows: first rotate +45 degrees (counter- clockwise, in a right-handed direction) about the z axis, and then -45 degrees about the y axis. Express the unit vectors of the x'y'z' frame in the xyz frame. Then find the stress matrix in the x'y'z' frame. (e) Find the three invariants of the stress matrix for all cases above and show that in-fact they do not depend on the selection of the coordinate system. 1 100 MPa 140 MPa 40 MPa 70 MPa