The principal stress at a point in a strained material are 126MPa
tensile and 63 MP tensile, the third principal stress being zero. Find
by a circular diagram of stress the magnitude and direction of
resultant stress on a plane inclined at 300 to the direction of the
smaller principal stress and perpendicular to the plane across which
the stress is zero. Also find the maximum obliquity of the resultant
stress and its magnitude.
The magnitude of the resultant stress on the plane inclined at 30° to the direction of the smaller principal stress is 63 MPa. The maximum obliquity of the resultant stress is 60° and its magnitude is 126 MPa.
To solve this problem, we will make use of a circular diagram of stress, also known as a Mohr's circle. This diagram allows us to visually represent the stress at different angles and helps in solving problems related to principal stresses.
Step 1: Plot the principal stresses on the circular diagram.
- The first principal stress, 126 MPa tensile, will be plotted on the right side of the circle.
- The second principal stress, 63 MPa tensile, will be plotted on the left side of the circle.
- The third principal stress is zero, so no point needs to be plotted for it.
Step 2: Draw a line parallel to the given plane inclined at 30°. This line will intersect the circle at two points.
Step 3: Connect the two points of intersection with the center of the circle. This line represents the magnitude and direction of the resultant stress on the inclined plane.
Step 4: Measure the angle between the resultant stress line and the plane across which the stress is zero. This will give you the direction of the resultant stress.
Step 5: Measure the maximum obliquity by finding the angle between the two principal stress lines. This will give you the angle at which the resultant stress is inclined to the principal stresses.
Step 6: Measure the magnitude of the resultant stress by finding the distance from the center of the circle to the intersection of the resultant stress line with the circle.
By following these steps and using a compass and protractor, you should be able to determine the magnitude and direction of the resultant stress, as well as the maximum obliquity.
To find the magnitude and direction of the resultant stress on a plane inclined at 30 degrees to the direction of the smaller principal stress and perpendicular to the plane across which the stress is zero, we need to use Mohr's circle.
Step 1: Draw the Mohr's circle.
- Draw a horizontal line representing the normal stresses.
- Mark the principal stress values: 126 MPa (tensile) and 63 MPa (tensile) on the horizontal line.
- Draw a vertical line representing the shear stresses.
- Since the third principal stress is zero, mark the point where the vertical line crosses the horizontal line as the origin (0, 0).
Step 2: Determine the center of the circle.
- The center of the circle can be found by calculating the average of the two principal stresses: (126 + 63) / 2 = 94.5 MPa.
- Mark the center point of the circle on the horizontal line.
Step 3: Determine the radius of the circle.
- The radius of the circle can be found by calculating half the difference between the two principal stresses: (126 - 63) / 2 = 31.5 MPa.
- Mark the radius on the horizontal line with arrows pointing towards the higher principal stress value.
Step 4: Draw the circle.
- Use a compass, place one end on the center point and swing an arc with a radius equal to the determined radius.
Step 5: Determine the point on the circle representing the stress on the inclined plane.
- Find the angle between the horizontal line and the line representing the inclined plane (30 degrees).
- Draw a line from the center point of the circle to meet the circumference at this angle.
- The point where this line intersects the circle represents the stress magnitude and direction on the inclined plane.
Step 6: Measure the position of the point on the circle.
- Measure the distance from the center point to the intersection point on the circle. This represents the magnitude of the resultant stress.
Step 7: Measure the angle of the point on the circle.
- Measure the angle between the horizontal line and the line connecting the center point and the intersection point on the circle. This represents the direction of the resultant stress.
Step 8: Determine the maximum obliquity.
- Obliquity refers to the angle between the inclined plane and the plane of maximum shear stress.
- In this case, the maximum obliquity occurs when the inclined plane coincides with the plane of maximum shear stress, which is perpendicular to the plane across which the stress is zero.
- The maximum obliquity is 90 degrees.
Step 9: Determine the magnitude of the resultant stress.
- The magnitude of the resultant stress is the distance from the center point to the intersection point on the circle, measured in Step 6.
By following these steps, you should be able to determine the magnitude and direction of the resultant stress on the inclined plane, as well as the maximum obliquity and its magnitude.