Final answer:
The combined stress on the cylindrical surface of the pin is 56 MPa. The principal stresses are 28 MPa and -28 MPa for tensile and compressive stresses, respectively. The maximum normal stress theory predicts a factor of safety of 1, indicating the pin is safe. Without the values for ultimate shear stress and the load line, we are unable to determine the factor of safety predicted by Mohr's failure criterion or draw the stress envelope.
Step-by-step explanation:
(a) To calculate the combined stress state on a cylindrical surface of the pin on the standard plane stress element, we need to consider both the tensile and compressive stresses. Since the pin is loaded in a way that causes both tensile and compressive stresses on the surface of the pin, we can calculate the combined stress using the formula:
Combined stress = Tensile stress + Compressive stress
Given the ultimate tensile and compressive stress of the pin is 28 MPa, the combined stress on the cylindrical surface of the pin on the standard plane stress element is 56 MPa.
(b) To determine the principal stresses, we need to consider the maximum and minimum values of the stress. Since the pin is loaded in a way that causes both tensile and compressive stresses, the maximum principal stress is the tensile stress and the minimum principal stress is the compressive stress. Thus, the principal stresses are:
Maximum Principal Stress: Tensile stress = 28 MPa
Minimum Principal Stress: Compressive stress = -28 MPa
(c) To determine the factors of safety predicted by the maximum normal-stress theory of failure, we compare the maximum normal (tensile) stress with the ultimate tensile stress of the pin. If the maximum normal stress is less than the ultimate tensile stress, the pin is safe and the factor of safety is given by:
Factor of Safety = Ultimate Tensile Stress / Maximum Normal (Tensile) Stress
In this case, the maximum normal stress is 28 MPa, which is less than the ultimate tensile stress of 28 MPa. Therefore, the factor of safety is 1, indicating that the pin is safe.
(d) To determine the factor of safety predicted by Mohr's failure criterion, we need to calculate the maximum shear stress and compare it with the ultimate shear stress of the pin. Since the pin is loaded in a way that causes both tensile and compressive stresses, the maximum shear stress can be calculated using the formula:
Maximum Shear Stress = (Maximum Principal Stress - Minimum Principal Stress) / 2
Substituting the values, we have:
Maximum Shear Stress = (28 MPa - (-28 MPa)) / 2 = 56 MPa
Comparing the maximum shear stress with the ultimate shear stress of the pin, we can determine the factor of safety:
Factor of Safety = Ultimate Shear Stress / Maximum Shear Stress
Since the ultimate shear stress is not given, we cannot determine the factor of safety predicted by Mohr's failure criterion.
(e) To draw the stress envelope for both theories, we plot the maximum shear stress and maximum normal stress on a stress diagram. The load line is a plot of the maximum normal stress versus the maximum shear stress for each case. However, without the values for the ultimate shear stress and the load line, we are unable to draw the stress envelope or determine if the pin fails.