Given a 6 × 19 monitor steel (Su = 240 kpsi) wire rope. (a) Develop the expressions for rope tension Ft , fatigue tension Ff , equivalent bending tensions Fb, and fatigue factor of safety nf for a 531.5-ft, 1-ton cage-and-load mine hoist with a starting acceleration of 2 ft/s2 as depicted in Fig. 17–23. The sheave diameter is 72 in.

Question

Given a 6 × 19 monitor steel ([latex]S_{u}=240[/latex] kpsi) wire rope.

(a) Develop the expressions for rope tension [latex]F_{t}[/latex] , fatigue tension [latex]F_{f}[/latex] , equivalent bending tensions [latex]F_{b}[/latex], and fatigue factor of safety [latex]n_{f}[/latex] for a 531.5-ft, 1-ton cage-and-load mine hoist with a starting acceleration of 2 ft/s2 as depicted in Fig. 17–23. The sheave diameter is 72 in.

(b) Using the expressions developed in part (a), examine the variation in factor of safety [latex]n_{f}[/latex] for various wire rope diameters d and number of supporting ropes m.

Accepted Answer

(a) Rope tension [latex]F_{t}[/latex] from Eq. (17–46) is given by

[latex]F_{t}=\left(\frac{W}{m}+w l ight)\left(1+\frac{a}{g} ight)[/latex] (17–46)

[latex]\begin{aligned}F_{t} &=\left(\frac{W}{m}+w l ight)\left(1+\frac{a}{g} ight)=\left[\frac{2000}{m}+1.60 d^{2}(531.5) ight]\left(1+\frac{2}{32.2} ight) \&=\frac{2124}{m}+903 d^{2} ext { lbf }\end{aligned}[/latex]

From Fig. 17–21, use [latex]p / S_{u}=0.0014[/latex]. Fatigue tension [latex]F_{f}[/latex] from Eq. (17–47) is given by

[latex]F_{f}=\frac{\left(p / S_{u} ight) S_{u} D d}{2}[/latex] (17–47)

[latex]F_{f}=\frac{\left(p / S_{u} ight) S_{u} D d}{2}=\frac{0.0014(240000) 72 d}{2}=12096 d lbf[/latex]

Equivalent bending tension [latex]F_{b}[/latex] from Eq. (17–48) and Table 17–27 is given by

[latex]F_{b}=\frac{E_{r} d_{u} A_{m}}{D}[/latex] (17–48)

Table 17–27 Some Useful Properties of 6 × 7, 6 × 19, and 6 × 37 Wire Ropes
Wire Rope	Weight per Foot w, lbf/ft	Weight per Foot Including Core w, lbf/ft	Minimum Sheave Diameter D, in	Better Sheave Diameter D, in	Diameter of Wires [latex]d_{w r} ext { in }[/latex]	Area of Metal [latex]A_{m r} ext { in }^{2}[/latex]	Rope Young’s Modulus [latex]E_{r,} ext { psi }[/latex]
6 × 7	[latex]1.50 d^{2}[/latex]		42d	72d	0.111d	[latex]0.38 d^{2}[/latex]	[latex]13 imes 10^{6}[/latex]
6 × 19	[latex]1.60 d^{2}[/latex]	[latex]1.76 d^{2}[/latex]	30d	45d	0.067d	[latex]0.40 d^{2}[/latex]	[latex]12 imes 10^{6}[/latex]
6 × 37	[latex]1.55 d^{2}[/latex]	[latex]1.71 d^{2}[/latex]	18d	27d	0.048d	[latex]0.40 d^{2}[/latex]	[latex]12 imes 10^{6}[/latex]

[latex]F_{b}=\frac{E_{r} d_{w} A_{m}}{D}=\frac{12\left(10^{6} ight) 0.067 d\left(0.40 d^{2} ight)}{72}=4467 d^{3} lbf[/latex]

Factor of safety [latex]n_{f}[/latex] in fatigue from Eq. (17–50) is given by

[latex]n_{f}=\frac{F_{f}-F_{b}}{F}[/latex] (17–50)

[latex]n_{f}=\frac{F_{f}-F_{b}}{F_{t}}=\frac{12096 d-4467 d^{3}}{2124 / m+903 d^{2}}[/latex]

(b) Form a table as follows:

[latex]n_{f}[/latex]
d	m = 1	m = 2	m = 3	m = 4
0.25	1.355	2.641	3.865	5.029
0.375	1.910	3.617	5.150	6.536
0.500	2.336	4.263	5.879	7.254
0.625	2.612	4.573	6.099	7.331
0.750	2.731	4.578	5.911	6.918
0.875	2.696	4.33	5.425	6.210
1.000	2.520	3.882	4.735	5.320

Wire rope sizes are discrete, as is the number of supporting ropes. Note that for each m the factor of safety exhibits a maximum. Predictably the largest factor of safety increases with m. If the required factor of safety were to be 6, only three or four ropes could meet the requirement. The sizes are different: [latex]\frac{5}{8}[/latex] -in ropes with three ropes or [latex]\frac{3}{8}[/latex] -in ropes with four ropes. The costs include not only the wires, but the grooved winch drums.

Question 17.6: Given a 6 × 19 monitor steel (Su = 240 kpsi) wire rope. (a) ...

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