The thin-walled box section beam ABCD shown in Fig. P.11.13 is attached at each end to supports which allow rotation of the ends of the beam in the longitudinal vertical plane of symmetry but prevent rotation of the ends in vertical planes perpendicular to the longitudinal axis of the beam.

Question

The thin-walled box section beam ABCD shown in Fig. P.11.13 is attached at each end to supports which allow rotation of the ends of the beam in the longitudinal vertical plane of symmetry but prevent rotation of the ends in vertical planes perpendicular to the longitudinal axis of the beam. The beam is subjected to a uniform torque loading of 20 Nm/mm over the portion BC of its span. Calculate the maximum shear stress in the cross section of the beam and the distribution of angle of twist along its length; G = 70 000 N/mm².

Accepted Answer

The total torque applied to the beam is [latex] 20 imes 4 imes 10^{3} Nm [/latex]. From symmetry the reactive torques at A and D will be equal and are [latex] 40 imes 10^{3} Nm [/latex]. Therefore,

[latex]\begin{gathered}T_{ AB }=40000 Nm \T_{ BC }=40000-20(x-1000)=60000-20 x Nm (x ext { in mm })\end{gathered}[/latex]

Note that the torque distribution is antisymmetrical about the centre of the beam. The maximum torque in the beam is therefore 40 000 Nm so that, from Eq. (11.22)

[latex] au_{\max }=\frac{T_{\max }}{2 A t_{\min }} [/latex] (11.22)

[latex] au_{\max }=\frac{40000 imes 10^{3}}{2 imes 200 imes 350 imes 4}=71.4 N / mm ^{2} [/latex]

The rate of twist along the length of the beam is given by Eq. (11.25) in which

[latex] \frac{ d heta}{ d x}=\frac{T}{4 A^{2} G} \oint \frac{ d s}{t} [/latex] (11.25)

[latex] \oint \frac{ d s}{t}=\frac{2 imes 200}{4}+\frac{2 imes 350}{6}=216.7 [/latex]

Then

[latex] \frac{ d heta}{ d x}=\left[\frac{216.7}{4 imes(200 imes 350)^{2} imes 70000} ight] T=15.79 imes 10^{-14} T [/latex]

In AB, [latex] T_{ AB }=40000 Nm [/latex] so that

[latex] heta_{ AB }=6.32 imes 10^{-6} x+ B [/latex]

When [latex] x=0, heta_{ AB }=0 ext { so that } B =0 ext { and when } x=1000 mm , heta_{ AB }=0.0063 rad \left(0.361^{\circ} ight) [/latex] In BC, [latex] T_{ BC }=60000-20 x Nm [/latex]. Then, from Eq. (11.25)

[latex] heta_{ BC }=15.79 imes 10^{-14}\left(60000 x-10 x^{2} ight) imes 10^{3}+ C [/latex]

When [latex] x=1000 mm , heta_{ BC }=0.0063 ext { so that } C =-0.0016 [/latex]. Then

[latex] heta_{ BC }=1.579 imes 10^{-10}\left(60000 x-10 x^{2} ight)-0.0016 [/latex]

At mid-span where [latex] x=3000 mm , heta_{ BC }=0.0126 rad \left(0.722^{\circ} ight) [/latex].

Question 11.13: The thin-walled box section beam ABCD shown in Fig. P.11.13 ...

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