For each section illustrated, find the second moment of area, the location of the neutral axis, and the distances from the neutral axis to the top and bottom surfaces. Consider that the section is transmitting a positive bending moment about the z axis. Mz, where Mz = 10 kip · in if the dimensions of the section are given in ips units, or Mz = 1.13 kN · m if the dimensions are in SI units. Determine the resulting stresses at the top and bottom surfaces and at every abrupt change in the cross section.
Problem 3–34
(a)
(b)
(c) Dimensions in mm
(d)
(a)
The second moment of area of the section.
The distance from the neutral axis to the top surfaces.
The distance from the neutral axis to the bottom surfaces.
The resulting stress at the top surface.
The resulting stress at the bottom surface.
The resulting stress at the abrupt change in cross-section.
Answer to Problem 34P
The second moment of area of the section is
The distance from the neutral axis to the top surface is
The distance from the neutral axis to the bottom surface is
The resulting stress at the top surface is
The resulting stress at the bottom surface is
The resulting stress at the abrupt change in cross-section is
Explanation of Solution
Figure-(1) shows two different sections divided of the in the same diagram.
Figure-(1)
Calculate the second moment of area of the section.
Here, the second moment area of the area
Write the moment of inertia of the rectangular section.
Calculate the location of the neutral axis.
Calculate the location of the neutral axis.
Here, the area of the each section is
Write the resulting bending stress on beam.
Here, the distance from the neutral axis to the top or bottom surface is
Conclusion:
Substitute
Substitute
Substitute
Thus, the second moment of area of the section is
Substitute
Since the section is symmetrical about
Thus, the distance from the neutral axis to the top surface is
Thus, the distance from the neutral axis to the bottom surface is
Substitute
Since top and bottom surfaces are at the same distance from the neutral axis so the resulting stresses are same.
Thus, the resulting stress at the top surface is
Thus, the resulting stress at the bottom surface is
Substitute
Thus the resulting stress at the abrupt change in cross-section is
(b)
The second moment of area of the section.
The distance from the neutral axis to the top surface.
The distance from the neutral axis to the bottom surface.
The resulting stress at the point A.
The resulting stress at the point D.
The resulting stress at the abrupt change in cross-section at a point B.
The resulting stress at the abrupt change in cross-section at a point C.
Answer to Problem 34P
The second moment of area of the section is
The distance from the neutral axis to the top surface is
The distance from the neutral axis to the bottom surface is
The resulting stress at the point A is
The resulting stress at the point C is
The resulting stress at the abrupt change in cross-section at a point B is
The resulting stress at the abrupt change in cross-section at a point C is
Explanation of Solution
Figure-(2) shows two different sections divided of the in the same diagram.
Figure-(2)
Write the expression for the area of section 1.
Here, the width of the section 1 is
Write the expression for the area of section 2.
Here, the width of the section 2 is
Calculate the location of the neutral axis.
Calculate the location of the neutral axis.
Here, the area of the each section is
Write the expression for the moment of inertia of section 1.
Write the expression for the moment of inertia of section 2.
Write the expression for the total moment of inertia.
Here, the distance of the neutral axis from the centroid of section 1 is
Write the resulting bending stress on beam.
Here, the distance from the neutral axis to the top or bottom surface is
Conclusion:
Substitute
Substitute
Substitute
Substitute
Substitute
Since the section is symmetrical about
Thus, the distance from the neutral axis to the top surface is
Thus, the distance from the neutral axis to the bottom surface is
Substitute
Substitute
Distance of the neutral axis from the two different centroids is
Substitute
Thus, the second moment of area of the section is
Substitute
Thus, the resulting stress at the bottom surface at a point A is
Substitute
Thus, the resulting stress at the point B is
Substitute
Thus, the resulting stress at the a point C is
Substitute
Thus, the resulting stress at the a point D is
(c)
The second moment of area of the section.
The distance from the neutral axis to the top surface.
The distance from the neutral axis to the bottom surface.
The resulting stress at the point A.
The resulting stress at the point D.
The resulting stress at the abrupt change in cross-section at a point B.
The resulting stress at the abrupt change in cross-section at a point C.
Answer to Problem 34P
The second moment of area of the section is
The distance from the neutral axis to the top surface is
The distance from the neutral axis to the bottom surface is
The resulting stress at the point A is
The resulting stress at the point D is
The resulting stress at the abrupt change in cross-section at a point B is
The resulting stress at the abrupt change in cross-section at a point C is
Explanation of Solution
Figure-(3) shows three different sections divided of the section in the same diagram.
Figure-(3)
Write the expression for the area of section 1.
Here, the width of the section 1 is
Write the expression for the area of section 2.
Here, the width of the section 2 is
Write the expression for the area of section 3.
Here, the width of the section 3 is
Calculate the location of the neutral axis.
Calculate the location of the neutral axis.
Here, the area of the each section is
Write the expression for the moment of inertia of section 1.
Write the expression for the moment of inertia of section 2.
Write the expression for the moment of inertia of section 3.
Write the expression for the total moment of inertia.
Here, the distance of the neutral axis from the centroid of section 1 is
Write the resulting bending stress on beam.
Here, the distance from the neutral axis to the top or bottom surface is
Conclusion:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Since the section is symmetrical about
Thus, the distance from the neutral axis to the top surface is
Thus, the distance from the neutral axis to the bottom surface is
Substitute
Substitute
Substitute
Distance of the neutral axis from the three different centroids of section is
Substitute
Thus, the second moment of area of the section is
Substitute
Thus, the resulting stress at the bottom surface at a point A is
Substitute
Thus, the resulting stress at the point B is
Substitute
Thus, the resulting stress at the a point C is
Substitute
Thus, the resulting stress at the a point D is
(d)
The second moment of area of the section.
The distance from the neutral axis to the top surface.
The distance from the neutral axis to the bottom surface.
The resulting stress at the point A.
The resulting stress at the point D.
The resulting stress at the abrupt change in cross-section at a point B.
Answer to Problem 34P
The second moment of area of the section is
The distance from the neutral axis to the top surface is
The distance from the neutral axis to the bottom surface is
The resulting stress at the point A is
The resulting stress at the point C is
The resulting stress at the abrupt change in cross-section at a point B is
Explanation of Solution
Figure-(4) shows two different sections divided of the section shown in the same diagram.
Figure-(4)
Write the expression for the area of section 1.
Here, the width of the section 1 is
Write the expression for the area of section 2.
Here, the width of the section 2 is
Calculate the location of the neutral axis.
Calculate the location of the neutral axis.
Here, the area of the each section is
Write the expression for the moment of inertia of section 1.
Write the expression for the moment of inertia of section 2.
Write the expression for the total moment of inertia.
Here, the distance of the neutral axis from the centroid of section 1 is
Write the resulting bending stress on beam.
Here, the distance from the neutral axis to the top or bottom surface is
Conclusion:
Substitute
Substitute
Substitute
Substitute
Substitute
Since the section is symmetrical about
Thus, the distance from the neutral axis to the top surface is
Thus, the distance from the neutral axis to the bottom surface is
Substitute
Substitute
Distance of the neutral axis from the two different centroids is
Substitute
Thus, the second moment of area of the section is
Substitute
Thus, the resulting stress at the bottom surface at a point A is
Substitute
Thus, the resulting stress at the point B is
Substitute
Thus, the resulting stress at the a point C is
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