STEEL PLATE LIFTING METHOD WITH USE OF LIFTING MAGNET, LIFTING MAGNET, AND METHOD FOR MANUFACTURING STEEL PLATE BY USING LIFTING MAGNET
20240101396 ยท 2024-03-28
Assignee
Inventors
- Yuki TAKAKI (Tokyo, JP)
- Yusaku Takemura (Tokyo, JP)
- Atsushi KURIMOTO (Tokyo, JP)
- Ayaka KAWAI (Tokyo, JP)
Cpc classification
International classification
Abstract
A method for using a lifting magnet and a lifting magnet. The lifting magnet includes a plurality of electromagnet coils that are each independently ON/OFF-controllable and voltage-controllable, and a magnetic pole that is excited by application of a voltage to the electromagnet coils. An electromagnet coil to be used for lifting steel plates is determined based on a total thickness of the steel plates to be lifted. An amount of passing magnetic flux ?.sub.r in the magnetic pole in a case where magnetic flux passes through only the steel plates to be lifted when the electromagnet coil is used is calculated. An application voltage to be applied to the electromagnet coil used for lifting the steel plates is determined based on the amount of passing magnetic flux ?.sub.r. The application voltage is applied to the electromagnet coil.
Claims
1. A steel plate lifting method for using a lifting magnet configured to lift at least one steel plate to be lifted from among a plurality of stacked steel plates, the steel plate lifting method comprising: providing the lifting magnet including a plurality of electromagnet coils that are each independently ON/OFF-controllable and voltage-controllable, and a magnetic pole that is excited by application of a voltage to the electromagnet coils; determining, based on a total thickness of the at least one steel plate to be lifted, an electromagnet coil to be used for lifting the at least one steel plate; calculating an amount of passing magnetic flux ?.sub.r in the magnetic pole in a case where magnetic flux flowing out of the magnetic pole passes through only the at least one steel plate to be lifted when the electromagnet coil is used; determining an application voltage to be applied to the electromagnet coil used for lifting the at least one steel plate, based on the amount of passing magnetic flux ?.sub.r; and applying the application voltage to the electromagnet coil to lift only the at least one steel plate to be lifted from among the plurality of stacked steel plates.
2. The steel plate lifting method for using a lifting magnet according to claim 1, wherein the lifting magnet further includes a magnetic flux sensor that measures an amount of passing magnetic flux ?.sub.a in the magnetic pole, and the steel plate lifting method further comprises, when applying the application voltage to the electromagnet coil, adjusting the application voltage for the electromagnet coil such that a difference between the calculated amount of passing magnetic flux ?.sub.r in the magnetic pole and an amount of passing magnetic flux ?.sub.a in the magnetic pole is equal to or less than a threshold, the amount of passing magnetic flux ?.sub.a in the magnetic pole being measured by the magnetic flux sensor.
3. The steel plate lifting method for using a lifting magnet according to claim 1, wherein the amount of passing magnetic flux ?.sub.r in the magnetic pole is calculated based on a thickness and a saturation magnetic flux density of each steel plate to be lifted and a size of the magnetic pole excited by application of the application voltage to the electromagnet coil.
4. The steel plate lifting method for using a lifting magnet according to claim 1, further comprising: after starting lifting of the at least one steel plate with the lifting magnet, performing, before moving the lifting magnet with which the at least one steel plate is lifted, at least one of: (I) increasing the application voltage for the electromagnet coil being used for lifting the at least one steel plate, and (II) applying a voltage to one or more other electromagnet coils in addition to the electromagnet coil being used for lifting the at least one steel plate.
5. The steel plate lifting method using a lifting magnet according to claim 1, wherein the plurality of electromagnet coils include a plurality of electromagnet coils that are arranged at least one of concentrically and vertically in layers.
6. A lifting magnet comprising: a plurality of electromagnet coils that are each independently ON/OFF-controllable and voltage-controllable; a magnetic pole that is excited by application of a voltage to the electromagnet coils; and a controller configured to execute the steps of: determining, when at least one steel plate to be lifted is to be lifted from among a plurality of stacked steel plates, an electromagnet coil to be used for lifting the at least one steel plate, based on a total thickness of the at least one steel plate to be lifted, calculating an amount of passing magnetic flux ?.sub.r in the magnetic pole in a case where magnetic flux flowing out of the magnetic pole passes through only the at least one steel plate to be lifted when the electromagnet coil is used, determining an application voltage to be applied to the electromagnet coil used for lifting the at least one steel plate, based on the amount of passing magnetic flux ?.sub.r, and applying the application voltage to the electromagnet coil.
7. The lifting magnet according to claim 6, further comprising a magnetic flux sensor that measures an amount of passing magnetic flux ?.sub.a in the magnetic pole, wherein the controller is further configured to, when applying the application voltage to the electromagnet coil, execute the step of adjusting the application voltage for the electromagnet coil such that a difference between the calculated amount of passing magnetic flux ?.sub.r in the magnetic pole and an amount of passing magnetic flux ?.sub.a in the magnetic pole is equal to or less than a threshold, the amount of passing magnetic flux ?.sub.a in the magnetic pole being measured by the magnetic flux sensor.
8. The lifting magnet according to claim 6, wherein the control device is further configured to execute the step of calculating the amount of passing magnetic flux ?.sub.r in the magnetic pole, based on a thickness and a saturation magnetic flux density of each steel plate to be lifted and a size of the magnetic pole excited by application of the application voltage to the electromagnet coil to be used.
9. The lifting magnet according to claim 6, wherein the plurality of electromagnet coils include a plurality of electromagnet coils that are arranged at least one of concentrically and vertically in layers.
10. A method for manufacturing a steel plate by using the lifting magnet according to claim 6 to form a steel plate.
Description
BRIEF DESCRIPTION OF THE DRAWINGS
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[0041]
DETAILED DESCRIPTION
[0042] The disclosed embodiments are directed to a method for lifting only at least one steel plate to be lifted (including a plurality of steel plates; the same applies to the following description) from among a plurality of stacked steel plates by using a lifting magnet. The disclosed embodiments are based on the use of a novel lifting magnet having a special configuration. That is, the lifting magnet of the disclosed embodiments includes a plurality of electromagnet coils 2 that are each independently ON/OFF-controllable and voltage-controllable, and a magnetic pole 3 that is excited by application of a voltage to these electromagnet coils 2 (i.e., a magnetic pole through which the magnetic flux generated by application of a voltage passes). As will be described below, according to a lifting magnet 1 having the configuration described above, when a large magnetic flux penetration depth (holding force) is required, the required magnetic flux penetration depth can be secured by simultaneously using the plurality of electromagnet coils 2. Further, at least one of the individual electromagnet coils 2 having a relatively small number of coil turns is selectively used to control the magnetic flux penetration depth with high accuracy.
[0043] The lifting magnet 1 used in the disclosed embodiments desirably includes the plurality of electromagnet coils 2, and the electromagnet coils 2 are not arranged in any special manner. However, the lifting magnet 1 particularly preferably includes a plurality of electromagnet coils 2 that are arranged concentrically or/and arranged vertically in layers, as will be described below.
[0044] An embodiment in which a lifting magnet including a plurality of electromagnet coils that are arranged concentrically is used will be described hereinafter.
[0045]
[0046] The lifting magnet 1 of the present embodiment includes two electromagnet coils 2 that are arranged concentrically, that is, a first electromagnet coil 2a on the inner side and a second electromagnet coil 2b on the outer side (hereinafter, electromagnet coil is simply referred to as coil, for convenience of description).
[0047] The first coil 2a and the second coil 2b are, for example, insulated ring-shaped excitation coils that are formed by winding enameled copper wires a large number of turns, like coils included in an existing lifting magnet. The two coils 2a and 2b are arranged concentrically (in a nest structure) with an outer pole (outer pole iron core) interposed therebetween. Thus, the two coils 2a and 2b have different ring diameters.
[0048] In the disclosed embodiments, the expression the plurality of coils 2 that are arranged concentrically means the plurality of coils 2 that are arranged in a nest structure, and the plurality of coils 2 need not be exactly concentric.
[0049] An inner pole 3x (inner pole iron core) is arranged inside the first coil 2a on the inner side. A first outer pole 3a (ring-shaped outer pole iron core) is arranged outside the first coil 2a, that is, between the first coil 2a and the second coil 2b. A second outer pole 3b (ring-shaped outer pole iron core) is arranged outside the second coil 2b. Furthermore, a yoke 6 is arranged in contact with the respective upper ends of the inner pole 3x and the first and second outer poles 3a and 3b. The yoke 6 is secured to the respective upper ends of the inner pole 3x and the first and second outer poles 3a and 3b.
[0050] Although not illustrated, gaps between the coils 2 and the magnetic pole 3 and between the coils 2 and the yoke 6 are usually filled with a non-magnetic material (such as a resin, for example) to secure the coils 2 in place. The inner pole 3x, the first outer pole 3a, the second outer pole 3b, and the yoke 6 are typically made of a soft magnetic material such as mild steel. Accordingly, at least one or all of them may have an integral structure (may be configured as an integral member).
[0051] The lifting magnet 1 in which the plurality of electromagnet coils 2 are arranged concentrically, which is used in the disclosed embodiments, may include three or more coils that are arranged concentrically. Also in this case, the inner pole 3x is arranged inside the coil on the innermost side, and the outer poles 3a, 3b, etc. are sequentially arranged outside the respective coils. In this manner, three or more coils that are arranged concentrically are included, which advantageously achieves a large voltage control range in each case, for example, when the number of steel plates to be lifted is more finely specified, such as one, two to three, four to five, or six to seven.
[0052] The lifting magnet 1 used in the disclosed embodiments includes a plurality of coils that are arranged concentrically. For example, in the embodiment in
[0053] Consideration will be given of the lifting of steel plates with the lifting magnet as illustrated in
M=??R.sub.I??.sub.k=1-n(t.sub.k)?B.sub.s(i)
[0054] When the cross-sectional area of the inner pole is S (mm.sup.2) and the average magnetic flux density of the inner pole is B (T), the amount of magnetic flux M is expressed by multiplying the cross-sectional area S by the average magnetic flux density B (S?B). Thus, formula (i) above is expressed by formula (ii) as follows.
S?B=??R.sub.I??.sub.k=1-n(t.sub.k)?B.sub.s(ii)
[0055] Since the average magnetic flux density B is proportional to the product of the number of turns N of the coil and a current I in the coil, formula (ii) above is expressed by formula (iii) as follows (?: constant of proportionality).
N?I???S=??R.sub.I??.sub.k=1-n(t.sub.k)?B.sub.s(iii)
[0056] Here, decreasing the number of turns N of the coil decreases the amount of change in the value of the left side with respect to the amount of error of the current I. Accordingly, control for satisfying formula (iii), that is, control of the magnetic flux penetration depth, can be performed with high accuracy, and the number of thin steel plates to be lifted can be controlled.
[0057]
[0058] In the disclosed embodiments, an example will be described in which a lifting magnet 1 including m coils 2 (coils 2.sub.1 to 2.sub.m) as illustrated in
[Math. 1]
t=?.sub.k=1.sup.nt.sub.k(1)
[0059] For example, in the embodiment using the lifting magnet 1 in
[0060] Subsequently, the amount of passing magnetic flux ?.sub.r in the magnetic pole 3 in a case where the magnetic flux flowing out of the magnetic pole 3 passes through only the n steel plates to be lifted when the selected coil 2 is used (excited) is calculated. Here, the amount of passing magnetic flux ?.sub.r in the magnetic pole 3 is calculated based on the thickness of each steel plate to be lifted, the saturation magnetic flux density of each steel plate to be lifted, and the size (outer diameter) of the magnetic pole 3 inscribed in the outermost coil 2 among the coils to be used (excited). That is, when the outer diameter of the magnetic pole 3.sub.i inscribed in the outermost coil 2.sub.i (1?i?m) among the coils 2 selected in the way described above is R.sub.i (mm), the thickness of each steel plate to be lifted is t.sub.k (mm), and the saturation magnetic flux density of each steel plate to be lifted is Bs.sub.k (T), the amount of passing magnetic flux ?.sub.r (T.Math.mm.sup.2) is calculated by formula (2) below. For example, in
[Math. 2]
?.sub.r=?R.sub.i?.sub.k=1.sup.n(Bs.sub.k.Math.t.sub.k)(2)
[0061] The theoretical basis of the amount of passing magnetic flux ?.sub.r will be described with reference to
[0062] Subsequently, an application voltage to be applied to the coil 2 to be used for lifting the steel plates is determined based on the calculated amount of passing magnetic flux ?.sub.r, and the determined voltage is applied to the coil 2. Since the relationship between the application voltage and the amount of passing magnetic flux ?.sub.r is determined in advance, the voltage is applied based on the relationship. This leads to a state in which the magnetic flux flowing out of the magnetic pole 3 passes through only the n steel plates to be lifted, making it possible to lift only the n steel plates to be lifted from among the plurality of stacked steel plates.
[0063] In the disclosed embodiments, preferably, after the lifting of the steel plates with the lifting magnet 1 is started, (iv) and/or (v) below is performed before the lifting magnet 1 with which the steel plates are lifted is moved, to prevent falling of the lifted steel plates.
[0064] (iv) Increase of the application voltage for the coil 2 being used for lifting the steel plates.
[0065] (v) Application of a voltage to one or more other coils 2 in addition to the coil 2 being used for lifting the steel plates.
[0066] The matters described in (iv) above correspond to the matters described in (I) increasing the application voltage for the electromagnet coil being used for lifting the at least one steel plate, and the matters described in (v) correspond to the matters described in (II) applying a voltage to one or more other electromagnet coils in addition to the electromagnet coil being used for lifting the at least one steel plate.
[0067]
[0068] In a preferred embodiment, the lifting magnet 1 may include a magnetic flux sensor 4 that measures the amount of passing magnetic flux ?.sub.a in the magnetic pole 3. When a voltage is to be applied to the coils 2, an application voltage is adjusted (controlled) so that the difference between the amount of passing magnetic flux ?.sub.a in the magnetic pole 3 (actual measurement value), which is measured by the magnetic flux sensor 4, and the amount of passing magnetic flux ?.sub.r (target value), which is calculated in the way described above, is equal to or less than a threshold. The adjustment (control) of the application voltage is preferably performed by feedback control.
[0069] Accordingly, the lifting magnet of the embodiment in
[0070] The level of the threshold is not particularly limited. However, typically, it is preferable that the threshold is set to a value equal to or less than 10% of the amount of passing magnetic flux ?.sub.r (target value).
[0071] Examples of the magnetic flux sensor 4 include a search coil and a Hall element. In the present embodiment, the magnetic flux sensor 4 is constituted by a search coil.
[0072] The magnetic flux sensor 4 may be attached at any position where the amount of passing magnetic flux in the magnetic pole can be measured. In the embodiment in
[0073] As in the embodiment in
[0074] When the magnetic flux sensor 4 is constituted by a Hall element, the magnetic flux sensor 4 is typically attached so as to be embedded in a lower end of the magnetic pole.
[0075]
[0076] First, the total thickness t of the steel plates to be lifted is determined from the number n of steel plates to be lifted from among a plurality of stacked steel plates (the number n of steel plates to be lifted) and thicknesses t.sub.1, t.sub.2, t.sub.3, . . . , and t.sub.n of the steel plates. A coil 2 to be used for lifting the steel plates is determined in accordance with the total thickness t. Accordingly, the coils 2 to be used are determined in advance in accordance with the range of the total thickness t. For example, the number of coils is m. In this case, a plurality of different thresholds 1 to m-1 (e.g., the threshold 1: 10 mm, the threshold 2: 20 mm, . . . the threshold m-1: 50 mm) are set in a stepwise manner. If the total thickness t is less than the threshold 1 (total thickness t<threshold 1), only the first coil 2.sub.1 is used. If the total thickness t is equal to or greater than the threshold 1 and less than the threshold 2 (threshold 1?total thickness t<threshold 2), the first coil 2.sub.1 and the second coil 2.sub.2 are used. Likewise, if the total thickness t is greater than the threshold m-1 (threshold m-1<total thickness t), the first coil 2.sub.1 to the m-th coil 2.sub.m are used. In this way, the coil or coils 2 to be used for lifting the steel plates are determined. When the number of coils is two as illustrated in
[0077]
[0078] Subsequently, the amount of passing magnetic flux ?.sub.r in the magnetic pole 3 (target value) in a case where the magnetic flux flowing out of the magnetic pole 3 passes through only the n steel plates to be lifted when the coils 2 are used is calculated from formula (2) above. Since the application voltage value for obtaining the predetermined amount of passing magnetic flux ?.sub.r is determined in advance, an application voltage to be applied to the coils 2 is determined based on the calculated amount of passing magnetic flux ?.sub.r, and the determined voltage is applied to the coils 2.
[0079] Since the application (excitation) of the voltage to the coils 2 generates magnetic flux, the amount of passing magnetic flux ?.sub.a in the magnetic pole 3 is measured by the magnetic flux sensor 4. The difference between the amount of passing magnetic flux ?.sub.a (actual measurement value), which is measured by the magnetic flux sensor 4, and the amount of passing magnetic flux ?.sub.r (target value), which is calculated in the way described above, is compared with a threshold. If the difference is equal to or less than the threshold (difference threshold), it is determined that the magnetic flux passes through only the n steel plates to be lifted. Accordingly, the lifting magnet 1, which is held in place by the crane, is raised to start lifting the steel plates. On the other hand, if the difference is greater than the threshold (difference>threshold), the application voltage is adjusted until the difference becomes equal to or less than the threshold (difference?threshold). If the difference becomes equal to or less than the threshold (difference?threshold), lifting of the steel plates is started. Such adjustment (control) of the application voltage for the coils 2 is preferably performed by feedback control by a control device 5 as described below.
[0080] The lifting magnet 1, which is held in place by the crane, is raised, and the steel plates to be lifted are lifted with the lifting magnet 1. In this state, preferably, the number of lifted steel plates is checked again by measurement of the amount of passing magnetic flux by the magnetic flux sensor 4, weight measurement by a load cell, and the like. In addition, the application voltage is increased or any other coil 2 is additionally excited to prevent falling of the steel plates. As a result, the amount of magnetic flux passing through the steel plates (magnetic flux penetration depth) is increased. Thereafter, the crane is traversed to transport the lifted steel plates.
[0081]
[0082] The lifting magnet 1 may include a magnetic flux sensor 4 that measures the amount of passing magnetic flux in the magnetic pole 3. In a case where the lifting magnet 1 includes the magnetic flux sensor 4, when applying a voltage to the coil 2, the control device 5 further adjusts (controls) the application voltage for the coil 2 so that the difference between the calculated amount of passing magnetic flux ?.sub.r in the magnetic pole 3 (target value) and the amount of passing magnetic flux ?.sub.a in the magnetic pole 3 (actual measurement value), which is measured by the magnetic flux sensor 4, is equal to or less than a threshold. Preferably, the control device 5 is configured to adjust the application voltage by using feedback control.
[0083] Accordingly, the control device 5 in
[0084] The control device 5 having the configuration described above to automatically control the lifting of steel plates is included, which allows lifting control to be performed with particularly high accuracy and a more efficient lifting and transporting operation of steel plates.
[0085]
[0086] The amount of passing magnetic flux ?.sub.a (actual measurement value) is measured again by the magnetic flux sensor 4 (S9) to check again the number of lifted steel plates in the state where the steel plates are lifted in the manner described above, that is, in the state before the steel plates are moved with the steel plates lifted. The number of attracted steel plates is determined based on whether the difference between the amount of passing magnetic flux ?.sub.a (actual measurement value) and the amount of passing magnetic flux ?.sub.r (target value) is equal to or less than a threshold (S10). If the difference exceeds the threshold, that is, if the number of steel plates is unacceptable (if the number of steel plates does not match the designated number of steel plates), the process returns to S3 described above, and the steel plates are lowered to the original position and grounded. On the other hand, if the difference is equal to or less than the threshold in S10, that is, if the number of steel plates is acceptable (if the number of steel plates matches the designated number of steel plates), the weight of the lifted steel plates is further measured by a weight measurement means or the like attached to a means for suspending the lifting magnet 1 (S11). The number of attracted steel plates is determined based on the measured weight (S12). If the number of steel plates is unacceptable (if the number of steel plates does not match the designated number of steel plates), the process returns to S3 described above, and the steel plates are lowered to the original position and grounded. On the other hand, if the number of steel plates is acceptable in S12 (if the number of steel plates matches the designated number of steel plates), the application voltage for the first coil 2a is increased to prevent falling of the lifted steel plates. Alternatively, the voltage is also applied to the second coil 2b in addition to the first coil 2a (S13). Thereafter, the movement (transportation of the lifted steel plates) by the crane is started (S14).
[0087] In the embodiment described above, the lifting magnet 1 including a plurality of coils 2 that are arranged concentrically is used. In an alternative embodiment, for example, (vi) a lifting magnet 1 including a plurality of coils 2 that are arranged vertically in layers or (vii) a lifting magnet 1 including a plurality of coils 2 that are arranged concentrically and arranged vertically in layers may be used.
[0088]
[0089]
[0090] In the disclosed embodiments, even in a case where the lifting magnet 1 as illustrated in
Example
[0091] To evaluate the controllability of the number of steel plates to be lifted in the disclosed embodiments, the following test was performed. A lifting magnet as illustrated in
[0092] In this Example, only the first coil 2a was used (excited) if the total thickness of the steel plates to be lifted was less than 20 mm, and the first coil 2a and the second coil 2b were used (excited) if the total thickness was not less than 20 mm. A threshold was set to 10% of the calculated amount of passing magnetic flux ?.sub.r in the magnetic pole (target value). Feedback control was performed so that the difference between the calculated amount of passing magnetic flux ?.sub.r in the magnetic pole 3 (target value) and the amount of passing magnetic flux ?.sub.a (actual measurement value), which was measured by the magnetic flux sensor, was equal to or less than the threshold, and the application voltage for the coil or coils 2 was adjusted.
[0093] The results of this Example are shown in Table 1. According to the results, the first coil 2a was excited when the number of steel plates to be lifted was 1 to 4. When the number of steel plates to be lifted was 5 or 6, the first and second coils 2a and 2b were excited. In this way, the application voltage is controlled based on the amount of passing magnetic flux ?.sub.a in the magnetic pole 3 (actual measurement value), which is measured by the magnetic flux sensor 4, with respect to the amount of passing magnetic flux ?.sub.r in the magnetic pole (target value), which is calculated in accordance with the outer diameter sizes of the inner pole 3x and the first outer pole 3a. As a result, the number of steel plates to be lifted can be controlled under the condition in which the number of steel plates to be lifted is any of 1 to 6.
TABLE-US-00001 TABLE 1 Amount of Amount of Evaluation of Number R.sub.i passing passing magnetic controllability of steel (mm) in magnetic flux ?.sub.r Application flux ?.sub.a [actual of number of plates to formula [target value] voltage measurement steel plates No. be lifted Excitation coil (2) (T .Math. mm.sup.2) (V) value] (T .Math. mm.sup.2) to be lifted 1 1 First coil 2a 100 2121 14.2 2028 ? 2 2 First coil 2a 100 4241 22.7 4605 ? 3 3 First coil 2a 100 6362 31.8 6688 ? 4 4 First coil 2a 100 8482 41.2 8657 ? 5 5 First coil 2a, 180 19085 27.5 19439 ? Second coil 2b 6 6 First coil 2a, 180 22902 32.2 23297 ? Second coil 2b
Comparative Example
[0094] A similar test was conducted by using a lifting magnet (single-layer structure), as illustrated in
[0095] In this Comparative Example, the amount of passing magnetic flux ?.sub.r in the magnetic pole (target value) in a case where the magnetic flux flowing out of the magnetic pole (the inner pole 101) passed through only the steel plates to be lifted when the coil 100 was excited was calculated, and a voltage was applied to the coil 100 based on the amount of magnetic flux ?.sub.r. At this time, the amount of passing magnetic flux ?.sub.a in the magnetic pole (actual measurement value) was measured by a magnetic flux sensor attached to a lower end of the outer circumference of the coil 100.
[0096] The results of this Comparative Example are shown in Table 2. The size of the magnetic pole of the lifting magnet used in this Comparative Example is larger than that of the inner pole 3x in the Example. Thus, under the condition in which the number of steel plates to be lifted is one, the amount of passing magnetic flux ?.sub.a in the magnetic pole (actual measurement value), which is measured by the magnetic flux sensor, greatly exceeds the amount of passing magnetic flux ?.sub.r (target value) even if the application voltage is equal to or less than 10 V, and the number of steel plates to be lifted is not controllable.
TABLE-US-00002 TABLE 2 Amount of Amount of Evaluation of Number R.sub.i passing passing magnetic controllability of steel (mm) in magnetic flux ?.sub.r Application flux ?.sub.a [actual of number of plates to formula [target value] voltage measurement steel plates to No. be lifted (2) (T .Math. mm.sup.2) (V) value] (T .Math. mm.sup.2) be lifted 1 1 150 3181 8.9 4053 x 2 2 150 6362 14.1 5813 ? 3 3 150 9543 20.5 8905 ? 4 4 150 12723 27.8 12670 ? 5 5 150 15904 36.5 15792 ? 6 6 150 19085 43.3 18678 ?