Question 6.17: Review Example Analysis 16. Consider now a bracket with a su......
Review Example Analysis 16. Consider now a bracket with a supporting web as in Figure a. To explore the usefulness of the web in reducing cantilever support bending stress, let the dimensions of the web be 28 inches long and 12 inches high as in Figure b. Let the web thickness be 1/8 inch.
a. Compute the centroidal second moment of area of the vertical end of the web against the wall portion of the bracket.
b. Compute the corresponding centroidal second moment of area of the end of the horizontal portion of the bracket (the “shelf”) using the dimensions of Example Analysis 16.
c. Compare the results of a) and b). Discuss the implication in bracket design and in mechanical design in general.
a. Recall from elementary beam theory that the second moment of area I relative to the centroidal bending axis of a rectangular cross-section is simply:
I=\frac{bh^{3}}{12} (a)
where here b is the width of the base of the cross-section and h is the height as in Figure c.
By using the web dimensions of Figure b we have:
I_{web}=(1/12)(1/8)(12)^{3}=18 in^{4} (b)
b. Recall from Example Analysis 15 the bracket shelf base and height of the bracket shelf are 1 inch and 3/16 inch as in Figure d.
Using Eq. (a) we have:
I_{shelf}=(1/12)(1)(3/16)^{3}=5.495(10^{-4}) in^{4} (c)
c. Comparing the results of a) and b) we see that the second moment of area of the wall end of the web, I_{web,} is more than 30,000 larger than that of the shelf cross-section.
Since flexural (bending) stress is inversely proportional to the second moment of area, the presence of the web greatly reduces the bending stress.
To be more specific, consider the familiar bending stress formula:
\sigma =\frac{Mc}{I} (d)
where c is now the distance from the bending axis to the extreme point of the cross-section. For a rectangular cross-section, c is simply half the height of the section. That is:
c=\frac{h}{2} (e)
By substituting from Eqs. (a) and (e) into (d), the stress becomes:
\sigma =\frac{6M}{bh^{2}} (f)
By substituting the given data values for the cross-section dimensions into Eq. (f) we have:
\sigma_{shelf}=170.7 M and \sigma_{web}=0.333 M (g)
The ratio: \sigma_{shelf}/\sigma_{web} exceeds: 500. That is the web reduces the stresses by more than a factor of 500.
Finally, observe in Eq. (f), the cross-section h appears squared in the denominator of the stress expression. This observation, and the results of Eqs. (b), (c), and (f), demonstrate the fundamental design principle:
Whenever a dimension variable in an expression is raised to a power, the value of the expression is changed exponentially by changes in the dimension variable.
