Question 7.1: For the beam shown in Figure 7–6, compute the maximum stress...
For the beam shown in Figure 7–6, compute the maximum stress due to bending. The cross section of the beam is a rectangle 100 mm high and 25 mm wide. The load at the middle of the beam is 1500 N, and the length of the beam is 3.40 m.
Objective Compute the maximum stress due to bending.
Given Beam and loading shown in Figure 7–6
Analysis The guidelines defined in this section will be used.
Results Step 1. The shearing force and bending moment diagrams have been drawn and included in Figure 7–6. The maximum bending moment is 1275 N · m at the middle of the beam.
Step 2. The centroid of the rectangular cross section is at the intersection of the two axes of symmetry, 50 mm from either the top or the bottom surface of the beam.
Step 3. The moment of inertia of the area for the rectangular shape with respect to the cen- troidal axis is
I = \frac{bh^{3}}{12} = \frac{25(100)^{3}}{12} = 2.08 \times 10^{6} mm^{4}
Step 4. The distance c is 50 mm from the centroidal axis to either the top or the bottom surface.
Step 5. The maximum stress due to bending occurs at both the top and the bottom of the beam at the point of maximum bending moment. Applying Equation (7–1) gives
\sigma_{max} = \frac{Mc}{I} = \frac{(1275 N·m)(50 mm)}{2.08 \times 10^{6} mm^{4}} \times \frac{10^{3} mm}{m}
\sigma_{max} = = 30.6 N/mm² = 30.6 MPa
Note that this stress level is compressive on the top surface of the rectangular beam and tensile at the bottom surface.