For Problems 10-56 through 10-83, determine the principal stresses and the maximum shear stress using Mohr’s circle. The following data sets give the stresses on the initial stress element. Perform the following operations: (a) Draw the complete Mohr’s circle, labeling critical points including σ1​,σ2​,τmax​, and σavg​. (b) On Mohr’s circle, indicate which line represents the x-axis on the initial stress element. (c) On Mohr’s circle, indicate the angles from the line representing the x-axis to the σ1​-axis and to the τmax​-axis. (d) Draw the principal stress element and the maximum shear stress element in their proper orientation relative to the initial stress element. \begin{tabular}{llcc} Problem & σx​ & σy​ & τxy​ \\ \hline 10−56 & 300MPa & −100MPa & 80MPaCW \\ 10−57 & 250MPa & −50MPa & 40MPaCW \\ 10−58 & 80MPa & −10MPa & 60MPaCW \end{tabular}

Answer

Problem 10-56

Given:

σx = 300 MPa σy = -100 MPa τxy = 80 MPa CW

a) Draw Mohr’s circle

To draw Mohr’s circle, follow these steps:

  1. Plot the point (300, 0) on the σx-axis, representing the σx stress.
  2. Plot the point (-100, 0) on the σy-axis, representing the σy stress.
  3. Draw a circle with center at the midpoint of the two points, which is (100, 0).
  4. Label the center of the circle as σavg, which is (100, 0) in this case.
  5. Draw a line passing through the center of the circle and the point (300, 0), representing the x-axis on the initial stress element.
  6. Mark the intersection of the circle and the x-axis as σ1, which is the maximum principal stress.
  7. Draw a horizontal line passing through the center of the circle, representing the τxy shear stress.
  8. Mark the intersection of the circle and the horizontal line as τmax, which is the maximum shear stress.

b) Indicate the line representing the x-axis on the initial stress element

The line passing through the center of the circle and the point (300, 0) represents the x-axis on the initial stress element.

c) Indicate the angles from the line representing the x-axis to the σ1-axis and to the τmax-axis

The angle from the line representing the x-axis to the σ1-axis is 45 degrees.

The angle from the line representing the x-axis to the τmax-axis is 67.5 degrees.

d) Draw the principal stress element and the maximum shear stress element

The principal stress element is oriented such that the σ1 stress acts along the x-axis and the σ2 stress acts along the y-axis.

The maximum shear stress element is oriented such that the τmax shear stress acts along a plane at 45 degrees to the x-axis and y-axis.

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