Question 7.5: A beam, to be made from ASTM A36 structural steel plate, is ...

Objective    Specify the height of the rectangular cross section.

Given          Loading pattern shown in Figure 7–16; ASTM A36 structural steel

Width of the beam to be 32.0 mm; static loads

Analysis     We will use the design procedure A from this section.

Results       Step 1. Figure 7–17 shows the completed shearing force and bending moment diagrams. The maximum bending moment is 6810 N · m between the loads, in the middle of the beam span from x = 1.2 – 3.6 m.

Step 2. From Table 7–1, for static load on a ductile material,

\sigma_{d} = s_{y} /2

Step 3. From Appendix A–12, s_{y} = 248 MPa for ASTM A36 steel. For a static load, a design factor of N = 2 based on yield strength is reasonable. Then,

\sigma_{d} = \frac{ s_{y}}{N} = \frac{248  MPa}{2} = 124 MPa

Step 4. The required section modulus, S, is

S_{min} = \frac{M}{\sigma_{d}} = \frac{6810  N·m\left\lgroup \frac{1000  mm}{1  m}\right\rgroup }{124  MPa} = 54 919  mm³

Step 5. The formula for the section modulus for a rectangular section with a height h and a thickness b is

S = \frac{I}{c} = \frac{bh^{3}}{12(h/2)} = \frac{bh^{2}}{6}

For the beam in this design problem, b will be 1.25 in. Then, solving for h gives

S = \frac{bh^{3}}{12(h/2)}

h_{min} = \sqrt{\frac{6S_{min}}{b}}=\sqrt{\frac{6(54  919  mm^{3})}{32}}

h_{min} = 101.5 mm

From Appendix A–2, specify the next larger preferred size, 110 mm.

Comment   Because the beam is rather long there may be a tendency for it to deform laterally because of elastic instability. Lateral bracing may be required. Also, deflection should be checked using the methods discussed in Chapter 9.