The cantilever beam shown in Fig. P.9.11(a) has the hollow cross section shown in Fig. 9.11(b). If the 130 kN load is applied at the centroid of the end cross section calculate the maximum value of compressive stress in the cross section of the beam.

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Accepted Answer

From Fig. S.9.11 and taking moments of areas about the base of the section.

[latex](570 imes 200-500 imes 150) \bar{y}=570 imes 200 imes 285-150 imes 500 imes 300[/latex]

which gives

[latex] \bar{y}=256 mm [/latex]

The area of the beam cross section is

[latex] A=200 imes 570-150 imes 500=39000 mm ^{2} [/latex]

The second moment of area about the z axis is

[latex]I_{z}=\frac{200 imes 570^{3}}{12}+200 imes 570 imes 29^{2}-\frac{150 imes 500^{3}}{12}-150 imes 500 imes 44^{2}[/latex]

so that

[latex]I_{z}=1.47 imes 10^{9} mm ^{4}[/latex]

The beam is subjected to bending from the vertical component of the applied load together with an axial compressive force produced by the horizontal component of the applied load. The maximum compressive stress in the beam section will therefore occur at the built in end of the beam and along the bottom edge of the section. Then

[latex] \sigma_{\max }=\frac{130 imes 10^{3} \cos 15^{\circ} imes 6 imes 10^{3} imes 256}{1.47 imes 10^{9}}+\frac{130 imes 10^{3} \sin 15^{\circ}}{39000} [/latex]

which gives

[latex] \sigma_{\max }=132.1 N / mm ^{2} [/latex]

Question 9.11: The cantilever beam shown in Fig. P.9.11(a) has the hollow c...

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