23.3:
Principal Stresses: Problem Solving
Two planes that are at right angles carry shearing stress along with tensile and compressive stresses. With known stress values, determine the principal planes, maximum shearing stress, and principal stresses.
The principal planes are determined using the relation where the principal plane equals the ratio of twice the shearing stress to the difference between tensile and compressive stresses.
Substitute the known values of shearing stress, normal stress, and compressive stress into the equation to calculate the principal plane values.
The average normal stress is then determined by calculating the average value of tensile and compressive normal stresses.
Next, calculate the maximum shearing stress by substituting the values of normal and shearing stresses into the equation.
Calculate the major principal stress by adding the average normal stress value to the maximum shearing stress.
The difference between the average normal stress value and the maximum shearing stress is the minimum principal stress.
23.3:
Principal Stresses: Problem Solving
When analyzing two planes intersecting at right angles under the influence of shearing, tensile, and compressive stresses, it is essential to identify principal planes, maximum shearing stress, and principal stresses. To find the principal planes, apply a formula that equates them to twice the shearing stress divided by the difference between tensile and compressive stresses.
By inserting the given shearing, tensile, and compressive stress values into this formula, one can calculate the orientations of the principal planes. Then consider the average of the tensile and compressive stresses to determine the average normal stress. This step is important for understanding the overall stress distribution across the material. The maximum shearing stress is calculated using the derived normal and shearing stress values.
This calculation highlights the peak shearing stress the material experiences, which is vital for assessing its failure risk. The major principal stress, which indicates the maximum stress the material can withstand without yielding, is computed by adding the maximum shearing stress to the average normal stress. Similarly, the minimum principal stress is found by subtracting the maximum shearing stress from the average normal stress. These computations are fundamental for engineering applications, providing insights into material behavior under complex stress conditions.