Question 17.1: Select a wire rope for a hoist required to lift 800 kg. The ...
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.
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 |
d_{r} / 9 | d_{r} / 13 to d_{r} / 16 | d_{r} / 22 | d_{r} / 15 to d_{r} / 19 |
| 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 |
0.384 d_{r}^{2} | 0.404 d_{r}^{2} s^{4} | 0.427 d_{r}^{2}( FW )^{4} | 0.366 d_{r}^{2}( W )^{4} |
| Approx metallic crosssection of rope. Core: IWRC |
0.451 d_{r}^{2} | 0.470 d_{r}^{2} s^{4} | 0.493 d_{r}^{2}( FW )^{4} | 0.497 d_{r}^{2}( W )^{4} |
| Standard nominal | ¼ \text { to } 5 / 8 \times 1 / 16^{\text {th }} s | ¼\text { to } 5 / 8 \times 1 / 16^{\text {th }} s | ¼ \text { to } 5 / 8 \times 1 / 16^{\text {th }} s | ¼ \text { to } 5 / 8 \times 1 / 16^{\text {th }} s |
| rope diameters | ¾ \text { to } 1 ½ \times 1 / 8^{\text {th }} s | ¾ \text { to } 2 ¾\times 1 / 8^{\text {th }} s | ¾ \text { to } 3¼ \times 1 / 8^{\text {th }} s | ¾ \text { to } 1½ \times 1 / 8^{\text {th }} s |
| Unit weight of rope (lb/ft) |
1.5 d_{r}^{2} | 1.6 d_{r}^{2} | 1.55 d_{r}^{2} | 1.45 d_{r}^{2} |
| Approx modulus of elasticity of the rope (psi) 0% to 20% of S_u |
13 \times 10^{6}( FC ) | 12 \times 10^{6}( FC ) | 11.6 \times 10^{6}(F C) | 9 \times 10^{6} |
| 15 \times 10^{6}(\text { IWRC }) | 14 \times 10^{6}(\text { IWRC }) | |||
| Recommended minimum sheave or drum diameter |
42 d_{r} | 34 d_{r} | 18 d_{r} | 26 d_{r} |
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,
From Eqn (17.1),
\sigma_{t}=\frac{T}{A_{\text {rope }}} (17.1)
\sigma_{\text {max suddenly applied }}=2 \sigma_{\text {max static }}=2 \frac{T}{A_{\text {rope }}}
Each wire rope carries 3924/2 = 1962 N. From Table 17.3, A_{\text {rope }}=0.427 d_{r}^{2}
\sigma_{\text {max suddenly applied }}=2 \frac{T}{A_{\text {rope }}}=2 \frac{1962}{0.427 d_{\text {rope }}^{2}}
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
\sigma_{d}=1379 / 5=275.8 MPaEquating the suddenly applied stress and the design stress, \sigma_{\text{max suddenly applied}}=\sigma_{\operatorname{design}}, and solving for the rope diameter gives
d_{\text {rope }}=\sqrt{\frac{3924}{275.8 \times 10^{6} \times 0.427}}=8.164 mmFrom 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, d_{r \text { static }}=9.525 mm.
From Table 17.3 the minimum recommended sheave diameter is given by
d_{\text {sheave }}=18 d_{\text {rope }}=18 \times 9.525=171.5 mmThe bending stress in the outer wires can be estimated from
\sigma_{b}=\frac{d_{\text {wire }}}{d_{\text {sheave }}} E_{\text {rope }}The approximate diameter of the outer wires, in this case, from Table 17.3 is d_{\text {wire }}=d_{\text {rope }} / 22.
So,
\sigma_{b}=\frac{d_{\text {wire }}}{d_{\text {sheave }}} E_{\text {rope }}=\frac{8.164 / 22}{171.5} 75.8 \times 10^{9}=191.5 MPaThe number of cycles is
N_{d}=20 \times 7.5 \times 220 \times 2=66,000From Figure 17.5, the fatigue strength parameter, R_N = 0.0055.
The pressure corresponding to failure in 66,000 cycles is
p_{N_{f}}=R_{N} \sigma_{ uts }=0.0055 \times 1379=7.584 MPaThe fatigue safety facture is 1.5, so
p_{d} \text { fatigue }=\frac{p_{N_{f}}}{1.5}=\frac{7.584}{1.5}=5.056 MPaThe wire rope diameter based on fatigue is given by
d_{\text {rope fatigue }}=\frac{2 T}{p_{d \text { fatigue }} d_{\text {sheave }}}=\frac{2 \times 1962}{5.056 \times 10^{6} \times 0.1715}=9.054 mmFrom 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
d_{\text {rope fatigue }}=\frac{2 T}{p_{d \text { fatigue }} d_{\text {sheave }}}=\frac{2 \times 1962}{5.056 \times 10^{6} \times 0.1715}=9.054 mmIn 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.
