Replace the force-couple system shown in Fig. (a) with an equivalent force-couple system, with the force acting at point A, given that F =100 N and C =3600 N · mm.

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Accepted Answer

Moving the given force F from point B to point A requires the introduction of a couple of transfer [latex]C^{T}[/latex]. This couple is then added to the given couple-vector C, thereby obtaining the resultant couple-vector, which we label [latex]C^{R}[/latex]. The couplevector [latex]C^{R}[/latex] and the force F located at point A will then be the required force-couple system. Owing to the three-dimensional nature of this problem, it is convenient to use vector methods in the solution. Writing F in vector form, we obtain

[latex]F=100\lambda _{BE}=100\frac{\overrightarrow{BE}}{\left|\overrightarrow{BE} ight|}=100\left(\frac{−120i + 60k}{134.164} ight)=−89.44i + 44.72k N[/latex]

The position vector from A to B is [latex]r_{AB}[/latex] =120j − 60k m. The couple of transfer is equal to the moment of the given force F about point A, so we have

[latex]C^{T}=M_{A}=r_{AB} imes F=\left|\begin{matrix} i & j & k \ 0 & 120 & -60 \ −89.44 & 0 & 44.72 \end{matrix} ight|[/latex]= 5366.4i + 5366.4j + 10732.8k N · mm

Expressing the given couple-vector C shown in Fig. (a) in vector form,

[latex]C=3600\lambda _{DB}=3600\frac{\overrightarrow{DB}}{\left|\overrightarrow{DB} ight|}=3600\left(\frac{120i + 120j − 60k}{180} ight)=2400i + 2400j − 1200k N · mm[/latex]

Adding [latex]C^{T}[/latex] and C (remember that couple-vectors are free vectors), the resultant couple vector is

[latex]C^{R} =C^{T}[/latex] + C=7766.4i + 7766.4j + 9532.8k N · mm

The magnitude of [latex]C^{R}[/latex] is given by

[latex]C^{R}=\sqrt{(7766.4)^{2}+(7766.4)^{2}+(9532.8)^{2}}=14543.3 N · mm[/latex]

The equivalent force-couple system is shown in Fig. (b). Note that the force acts at point A. For convenience of representation, [latex]C^{R}[/latex] is shown at point O, but being a free vector, it could be placed anywhere.

Question 2.11: Replace the force-couple system shown in Fig. (a) with an eq...

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