Question 6.9: Determine the section modulus for the tee shape used in Exam...
Determine the section modulus for the tee shape used in Example Problem 6–5. Then compute the maximum stress due to bending at the top and bottom surfaces if the bending moment is 60 N · m.
The results of Example Problem 6–5. shown in Figure 6–13, include the following:
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- Overall height of the section: 4.50 cm
- Location of the horizontal centroidal axis: 2.75 cm above the base
- Area moment of inertia: 12.125 cm^{4}
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We can now observe that the distances from the centroidal axis to the top and bottom surfaces are unequal:
c_{b} = 2.75 cm from the centroid to the bottom surface
c_{t} = 4.50 cm − 2.75 cm = 1.75 cm from the centroid to the top surface
Two values of the section modulus can be computed:
At the top surface, S_{t} = I/c_{t} = (12.125 cm^{4})/1.75 cm = 6.93 cm³
At the bottom surface, S_{b} = I/c_{b} = (12.125 cm^{4})/2.75 cm = 4.41 cm³
We can now compute the stresses at the top and bottom surfaces:
At the top surface, \sigma = M/S_{t} = (60 N·m)(100³ cm³/m³)/6.93 cm³ = 8.66 \times 10^{6} N/m² = 8.66 MPa
At the bottom surface, \sigma = M/S_{b} = (60 N·m)(100³ cm³/m³)/4.41 cm³
\sigma = 13.6 \times 10^{6} N/m² = 13.6 MPa
Note that the lower value for section modulus gives the higher value of stress and that would typically be the objective of the stress calculation. However, this will be explored in greater detail in Chapter 7 for special cases where the material of a beam has different strengths in tension and in compression.
