Select a wire rope for a hoist required to lift 800 kg. The hoist design uses two lines to support the load attached to a vertically moving sheave, on which the swivel hook is mounted. The desired design life is 2 years, and the anticipated maximum use is 20 lifts per hour,7.5 h a day for 220 days

Question

Select a wire rope for a hoist required to lift 800 kg. The hoist design uses two lines to support the load attached to a vertically moving sheave, on which the swivel hook is mounted. The desired design life is 2 years, and the anticipated maximum use is 20 lifts per hour, 7.5 h a day for 220 days per year. A safety factor of 5 is required based on static ultimate strength and 1.5 based on fatigue.

Accepted Answer

Load = 400 × 9.81 = 3924 N.

From Figure 17.4, going for a balance of resistance to abrasion and fatigue, a 6 × 31 WS or a 6 × 25 FW wire rope construction might be appropriate.

From Table 17.3,

Table 17.3: Material and construction data for selected wire rope classes.

Classification	6 × 7	6 × 19	6 × 37	8 × 19
Number of outer strands	6	6	6	8
Number of wires per strand	3 ̶ 14	15 ̶ 26	27 ̶ 49	15 ̶ 26
Maximum number of outer wires	9	12	18	12
Approx diameter of outer wires	[latex]d_{r} / 9[/latex]	[latex]d_{r} / 13[/latex] to [latex]d_{r} / 16[/latex]	[latex]d_{r} / 22[/latex]	[latex]d_{r} / 15[/latex] to [latex]d_{r} / 19[/latex]
Material available (typically). Core: FC	IPS (200)	I (80)	IPS (200)	I (80)
		T (130)		T (130)
		IPS (200)		IPS (200)
Material available (typically). Core: IWRC	IPS (190)	I (190)	EIPS (220)	IPS (190)
		T (220)	EIPS (255)	EIPS (200)
		IPS (255)
Approx metallic crosssection of rope. Core: FC	[latex]0.384 d_{r}^{2}[/latex]	[latex]0.404 d_{r}^{2} s^{4}[/latex]	[latex]0.427 d_{r}^{2}( FW )^{4}[/latex]	[latex]0.366 d_{r}^{2}( W )^{4}[/latex]
Approx metallic crosssection of rope. Core: IWRC	[latex]0.451 d_{r}^{2}[/latex]	[latex]0.470 d_{r}^{2} s^{4}[/latex]	[latex]0.493 d_{r}^{2}( FW )^{4}[/latex]	[latex]0.497 d_{r}^{2}( W )^{4}[/latex]
Standard nominal	[latex]¼ ext { to } 5 / 8 imes 1 / 16^{ ext {th }} s[/latex]	[latex]¼ ext { to } 5 / 8 imes 1 / 16^{ ext {th }} s[/latex]	[latex]¼ ext { to } 5 / 8 imes 1 / 16^{ ext {th }} s[/latex]	[latex]¼ ext { to } 5 / 8 imes 1 / 16^{ ext {th }} s[/latex]
rope diameters	[latex]¾ ext { to } 1 ½ imes 1 / 8^{ ext {th }} s[/latex]	[latex]¾ ext { to } 2 ¾ imes 1 / 8^{ ext {th }} s[/latex]	[latex]¾ ext { to } 3¼ imes 1 / 8^{ ext {th }} s[/latex]	[latex]¾ ext { to } 1½ imes 1 / 8^{ ext {th }} s[/latex]
Unit weight of rope (lb/ft)	[latex]1.5 d_{r}^{2}[/latex]	[latex]1.6 d_{r}^{2}[/latex]	[latex]1.55 d_{r}^{2}[/latex]	[latex]1.45 d_{r}^{2}[/latex]
Approx modulus of elasticity of the rope (psi) 0% to 20% of [latex]S_u[/latex]	[latex]13 imes 10^{6}( FC )[/latex]	[latex]12 imes 10^{6}( FC )[/latex]	[latex]11.6 imes 10^{6}(F C)[/latex]	[latex]9 imes 10^{6}[/latex]
		[latex]15 imes 10^{6}( ext { IWRC })[/latex]	[latex]14 imes 10^{6}( ext { IWRC })[/latex]
Recommended minimum sheave or drum diameter	[latex]42 d_{r}[/latex]	[latex]34 d_{r}[/latex]	[latex]18 d_{r}[/latex]	[latex]26 d_{r}[/latex]

the nearest classifications compatible with 6 × 31 WS and 6 × 25 FW are 6 × 37 and 6 × 19, respectively. As a first iteration for the design, a 6 × 37 WS wire rope construction will be explored.

A safety factor of 5 based on ultimate tensile strength has been defined.
For a suddenly applied load,

[latex]\sigma_{ ext {max suddenly applied }}=2 \sigma_{ ext {max static }}[/latex]

From Eqn (17.1),

[latex]\sigma_{t}=\frac{T}{A_{ ext {rope }}}[/latex] (17.1)

[latex]\sigma_{ ext {max suddenly applied }}=2 \sigma_{ ext {max static }}=2 \frac{T}{A_{ ext {rope }}}[/latex]

Each wire rope carries 3924/2 = 1962 N. From Table 17.3, [latex]A_{ ext {rope }}=0.427 d_{r}^{2}[/latex]

[latex]\sigma_{ ext {max suddenly applied }}=2 \frac{T}{A_{ ext {rope }}}=2 \frac{1962}{0.427 d_{ ext {rope }}^{2}}[/latex]

From Table 17.3, the static ultimate tensile strength of improved plow steel (IPS) is 200,000 psi ≈ 1379 MPa.

With a static safety factor of 5, the design stress is given by

[latex]\sigma_{d}=1379 / 5=275.8 MPa[/latex]

Equating the suddenly applied stress and the design stress, [latex]\sigma_{ ext{max suddenly applied}}=\sigma_{\operatorname{design}}[/latex], and solving for the rope diameter gives

[latex]d_{ ext {rope }}=\sqrt{\frac{3924}{275.8 imes 10^{6} imes 0.427}}=8.164 mm[/latex]

From Table 17.3, the nearest larger standard rope diameter is 11/32 in or 12/32 in. Going for the larger of these, 12/32 in, [latex]d_{r ext { static }}=9.525 mm[/latex].

From Table 17.3 the minimum recommended sheave diameter is given by

[latex]d_{ ext {sheave }}=18 d_{ ext {rope }}=18 imes 9.525=171.5 mm[/latex]

The bending stress in the outer wires can be estimated from

[latex]\sigma_{b}=\frac{d_{ ext {wire }}}{d_{ ext {sheave }}} E_{ ext {rope }}[/latex]

The approximate diameter of the outer wires, in this case, from Table 17.3 is [latex]d_{ ext {wire }}=d_{ ext {rope }} / 22.[/latex]

So,

[latex]\sigma_{b}=\frac{d_{ ext {wire }}}{d_{ ext {sheave }}} E_{ ext {rope }}=\frac{8.164 / 22}{171.5} 75.8 imes 10^{9}=191.5 MPa[/latex]

The number of cycles is

[latex]N_{d}=20 imes 7.5 imes 220 imes 2=66,000[/latex]

From Figure 17.5, the fatigue strength parameter, [latex]R_N = 0.0055[/latex].

The pressure corresponding to failure in 66,000 cycles is

[latex]p_{N_{f}}=R_{N} \sigma_{ uts }=0.0055 imes 1379=7.584 MPa[/latex]

The fatigue safety facture is 1.5, so

[latex]p_{d} ext { fatigue }=\frac{p_{N_{f}}}{1.5}=\frac{7.584}{1.5}=5.056 MPa[/latex]

The wire rope diameter based on fatigue is given by

[latex]d_{ ext {rope fatigue }}=\frac{2 T}{p_{d ext { fatigue }} d_{ ext {sheave }}}=\frac{2 imes 1962}{5.056 imes 10^{6} imes 0.1715}=9.054 mm[/latex]

From Table 17.4,

Table 17.4: Maximum allowable bearing pressure between rope and sheave.

	Sheave material
Rope	Wood	Cast iron	Cast steel	Manganese steel
Regular lay
6 × 7	1.0	2.1	3.8	10.1
6 × 19	1.7	3.3	6.2	16.6
6 × 37	2.1	4.0	7.4	20.7
8 × 19	2.4	4.7	8.7	24.1
Lang lay
6 × 7	1.1	2.4	4.1	11.4
6 × 19	1.9	3.8	6.9	19.0
6 × 37	2.3	4.6	8.1	22.8
8 × 19	2.7	5.5	10.0	27.6

for a 6 × 37 wire rope on a cast-carbon steel sheave (BHN 160), the
allowable bearing pressure load based on wear is 8.1 MPa.

The wire rope diameter based on wear is given by

[latex]d_{ ext {rope fatigue }}=\frac{2 T}{p_{d ext { fatigue }} d_{ ext {sheave }}}=\frac{2 imes 1962}{5.056 imes 10^{6} imes 0.1715}=9.054 mm[/latex]

In this case, examining the wire diameters based on static loading, fatigue and wear, the largest diameter is that for static loading.

In summary, the wire rope required is 3/8 in 6 × 37 WS IPS fiber core. The sheave is carbon steel (BHN-160) with a diameter of 171.45 mm.

Question 17.1: Select a wire rope for a hoist required to lift 800 kg. The ...