Concept explainers
For each of the plane stress states listed below, draw a Mohr’s circle diagram properly labeled, find the principal normal and shear stresses, and determine the angle from the x axis to σ1. Draw stress elements as in Fig. 3–11c and d and label all details.
(a) σx = 20 kpsi, σy = 210 kpsi, τxy = 8 kpsi cw
(b) σx = 16 kpsi, σy = 9 kpsi, τxy = 5 kpsi ccw
(c) σx = 10 kpsi, σy = 24 kpsi, τxy = 6 kpsi ccw
(d) σx = 212 kpsi, σy = 22 kpsi, τxy = 12 kpsi cw
(a)
The principle normal stress.
The shear stress.
The angle from
Answer to Problem 15P
The principle normal stress
The shear stress is
The angle from
Explanation of Solution
The Figure (1) shows the state of stress on the element for
Figure (1)
Write the coordinates of the points through which the Mohr’s circle pass.
Here, the stress along x face is
Draw the
Write the formula for the center point.
Here, the center point is
Write the expression for the angle between the line joining points A and B with
Here, the angle made by the diameter with positive x-axis in the counterclockwise direction is
Write the expression of the radius of circle.
Write the expression maximum in plane normal stress.
Here, the maximum in plane normal stress are
Write the expression of maximum in plane shear stress.
Here, the maximum shear stress is
Write the expression for the angle of maximum shear plane.
Here, the angle is
Conclusion:
Substitute
Draw the Mohr’s circle diagram.
The Figure (2) shows the Mohr’s circle diagram.
Figure (2)
Substitute the value
Substitute the value
Substitute
Substitute
Thus, the principle normal stress
Substitute
Thus, the maximum shear stress is
Substitute the value
Thus, the angle from
The Figure (3) shows the maximum in plane normal stress distribution about the plane.
Figure (3)
The Figure (4) shows stress distribution at maximum shear stress plane.
Figure (4)
(b)
The principle normal stress.
The shear stress.
The angle from
Answer to Problem 15P
The principle normal stress
The shear stress is
The angle from
Explanation of Solution
The Figure (5) shows the state of stress on the element for
Figure (5)
Conclusion:
Substitute the value
Draw the Mohr’s circle diagram.
The Figure (2) shows the Mohr’s circle diagram.
Figure (6)
Substitute the value
Substitute the value
Substitute the value
Substitute the value
Thus, the principle normal stress
Substitute the value
Thus, the maximum shear stress is
Substitute
Thus, the angle from
The Figure (3) shows the maximum in plane normal stress distribution about the plane.
Figure (7)
The Figure (4) shows stress distribution at maximum shear stress plane.
Figure (8)
(c)
The principle normal stress.
The shear stress.
The angle from
Answer to Problem 15P
The principle normal stress
The shear stress is
The angle from
Explanation of Solution
The Figure (5) shows the state of stress on the element for
Figure (9)
Conclusion:
Substitute the value
Draw the Mohr’s circle diagram.
The Figure (2) shows the Mohr’s circle diagram.
Figure (10)
Substitute the value
Substitute the value
Write the value of
Substitute the value
Substitute the value
Thus, the principle normal stress
Substitute
Thus, the maximum shear stress is
Substitute the value
Thus, the angle from
The Figure (3) shows the maximum in plane normal stress distribution about the plane.
Figure (11)
The Figure (4) shows stress distribution at maximum shear stress plane.
Figure (12)
(d)
The principle normal stress.
The shear stress.
The angle from
Answer to Problem 15P
The principle normal stress
The shear stress is
The angle from
Explanation of Solution
The Figure (5) shows the state of stress on the element for
Figure (13)
Conclusion:
Substitute the value
Draw the Mohr’s circle diagram.
The Figure (2) shows the Mohr’s circle diagram.
Figure (14)
Substitute the value for
Substitute the value
Write the value of
Substitute the value
Substitute the value
Thus, the principle normal stress
Substitute the value
Thus, the shear stress is
Substitute the value
Thus, the angle from
The Figure (3) shows the maximum in plane normal stress distribution about the plane.
Figure (15)
The Figure (4) shows stress distribution at maximum shear stress plane.
Figure (16)
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Chapter 3 Solutions
Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)
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