The 500-kg engine is suspended from the jib crane at the position shown. Determine the state of stress at point B on the cross section of the boom at section a–a. Point B is just above the bottom flange.

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Support Reactions: Referring to the free-body diagram of the entire boom, Fig.a,

[latex]\curvearrowleft +\Sigma M_{C}=0 ; \quad F_{D E} \sin 30^{\circ}(6)+F_{D E} \cos 30^{\circ}(0.4)-500(9.81)(2)=0[/latex]

[latex]F_{D E}=2931.50 \mathrm{N}[/latex]

Internal Loadings: Considering the equilibrium of the free-body diagram of the boom's right cut segment, Fig. b.

[latex]\stackrel{+}{ ightarrow} \Sigma F_{x}=0 ; \quad N-2931.50 \cos 30^{\circ}=0 \quad N=2538.75 \mathrm{N}[/latex]

[latex]+\uparrow \Sigma F_{y}=0 \quad 2931.50 \sin 30^{\circ}-V=0 \quad V=1465.75 \mathrm{N}[/latex]

[latex]\curvearrowleft +\Sigma M_{O}=0 ; \quad 2931.50 \sin 30^{\circ}(2)+2931.50 \cos 30^{\circ}(0.4)-M=0[/latex]

[latex]M=3947.00 \mathrm{N} \cdot \mathrm{m}[/latex]

Section Properties 'The cross-sectional area and the moment of inertia about the centroidal axis of the boom's cross section are

[latex]A=0.15(0.3)-0.13(0.26)=0.0112 \mathrm{m}^{2}[/latex]

[latex]I=\frac{1}{12}(0.15)\left(0.3^{3} ight)-\frac{1}{12}(0.13)\left(0.26^{3} ight)=0.14709\left(10^{-3} ight) \mathrm{m}^{4}[/latex]

Referring to Fig. c, [latex]Q_{B}[/latex] is

[latex]Q_{B}=\bar{y}_{2} A_{2}^{\prime}=0.14(0.02)(0.15)=0.42\left(10^{-3} ight) \mathrm{m}^{3}[/latex]

Normal Stress: The normal stress is the combination of axial and bending stress. Thus,

[latex]\sigma=\frac{N}{A}+\frac{M y}{I}[/latex]

For point B, y=0.13 m . Then

[latex]\sigma_{B}=\frac{-2538.75}{0.0112}+\frac{3947.00(0.13)}{0.14709\left(10^{-3} ight)}=3.26 \mathrm{MPa}(T)[/latex]

Shear Stress: The shear stress is contributed by transverse shear stress only. Thus,

[latex] au_{B}=\frac{V Q_{B}}{I t}=\frac{1465.75\left[0.42\left(10^{-3} ight) ight]}{0.14709\left(10^{-3} ight)(0.02)}=0.209 \mathrm{MPa}[/latex]

The state of stress at point B is represented on the element shown in Fig. d.8.41

Question 8.41: The 500-kg engine is suspended from the jib crane at the pos...

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