Learning Goal: To use equilibrium to calculate the plane state of stress in a rotated coordinate system. In general, the three-dimensional state of stress at a point requires six stress components to be fully described: three normal stresses and three shear stresses ( Figure 1). When the external loadings are coplanar, however, the resulting internal stresses can be treated as plane stress and described using a simpler, two- dimensional analysis with just two normal stresses and one shear stress (Figure 2). The third normal stress and two other shear stresses are assumed to be zero. The normal and shear stresses for a state of stress depend on the orientation of the axes. If the stresses are given in one coordinate system (Figure 3), the equivalent stresses in a rotated coordinate system (Figure 4) can be calculated using the equations of static equilibrium. Both sets of stresses describe the same state of stress. The stresses and Ty can be found by considering the free-body diagram of a right triangular prism with the legs parallel to the original coordinate axes and a hypotenuse perpendicular to the ' axis (Figure 5). The stresses y and Try can be found using a similar right triangular prism with the hypotenuse perpendicular to the y' axis (Figure 6). The state of stress at a point can be described by = 36 MPa, = 56 MPa, and 7zy = 14 MPa. A second coordinate system is rotated by = 10 as shown (Figure 4). Part A - Calculate \sigma_{x} What is the normal stress in the direction of the '-axis? Express your answer with appropriate units to three significant figures. View Available Hint(s) 125.06 MPa Submit Previous Answers ? Incorrect; Try Again; 4 attempts remaining Figure : 1 of 6 Part B - Calculate \tau_{x'y'} What is the shear stress in the y'-direction for the faces with a normal in the '-direction? Express your answer with appropriate units to three significant figures. View Available Hint(s) Tz'y' = Value Units Submit ? Part C - Calculate \sigma_{y} What is the normal stress in the direction of the y'-axis? Express your answer with appropriate units to three significant figures. Review Part B - Calculate \tau_{x'y'} What is the shear stress in the y'-direction for the faces with a normal in the '-direction? Express your answer with appropriate units to three significant figures. View Available Hint(s) Tz'y' = Value Units Submit ? Part C - Calculate \sigma_{y} What is the normal stress in the direction of the y'-axis? Express your answer with appropriate units to three significant figures. View Available Hint(s) Submit Value Units ?