A simply supported beam made of rolled steel joist (I-section: 450 mm × 200 mm) has a span of 5 m and it carries a concentrated load W. The flanges are strengthened by two 300 mm × 20 mm plates, one riveted to each flange, over the entire length of the flanges. The second moment of area of the

Question

A simply supported beam made of rolled steel joist (I-section: 450 mm × 200 mm) has a span of 5 m and it carries a concentrated load W. The flanges are strengthened by two 300 mm × 20 mm plates, one riveted to each flange, over the entire length of the flanges. The second moment of area of the joist about the principal bending axis is 35060 [latex] cm ^4 [/latex]. Calculate:
(a) the greatest central load that the beam will carry if the bending stress in the 300 mm × 20 mm plates is not to exceed 125 MPa.
(b) the minimum length of the 300 mm plates required to restrict the maximum bending stress in the flanges of the joist to 125 MPa.

Accepted Answer

(a) Let us show the beam section in Figure 6.23.

Clearly, [latex] \bar{I} [/latex] is the second moment of area about neutral axis and is found as

[latex] \bar{I}=35060(10)^4+2\left[\frac{1}{12}(300)(20)^3+(300)(20)(225+10)^2 ight]=1.0137 imes 10^9 mm ^4 [/latex]

For the beam with point load at the midspan, we know that [latex] M_{\max }=w L / 4 [/latex] Therefore, maximum bending stress, [latex] \sigma_{\max } [/latex] is

[latex] \sigma_{\max }=\frac{M_{\max }}{Z} [/latex]

so, [latex] 125=\frac{(w)(5000)}{4} imes \frac{25}{1.0137\left(10^9 ight)} [/latex]

or W = 413.76 kN

(b) Let us show the beam with plates riveted on the flanges in Figure 6.24. Let a be the minimum length of 300 mm plates riveted to the beam.

In Figure 6.24(a), the beam with two numbers of 300 mm × 20 mm plates riveted is shown; while in Figure 6.24(b), the bending moment diagram of the beam is depicted. Clearly,

[latex] M_{\max }=\sigma_{\max } Z=125 imes \frac{1.0137 imes 10^9}{245}=517.19 imes 10^6 N mm [/latex]

If the maximum bending stress in the flanges of the joist is to be limited to 125 MPa, then it corresponds to the bending moment M, which is given by

[latex] M=\sigma Z=125 imes \frac{35060 imes 10^4}{225}=194.78 imes 10^6 N mm [/latex]

Thus, in the base joist (i.e., without reinforcement by the plates), the bending moment is to be limited to 194.78 [latex] \left(10^6 ight) [/latex] N mm. Therefore, if a is the minimum length of two numbers of 300 mm × 20 mm plates, then from the bending moment diagram, we have

[latex] \frac{a / 2}{2.5}=\frac{517.19-194.78}{517.19} [/latex]

or a = 1.56 m

Thus, the minimum length of the plates is 1.56 m.

Question 6.8: A simply supported beam made of rolled steel joist (I-sectio...

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