Crane is a mechanical device, used widely to move materials in modern production. It is reported that the energy consumptions of China are at least 5–8 times of other developing countries. Thus, energy consumption becomes an unavoidable topic. There are several reasons influencing the energy loss, and the camber of the girder is the one not to be neglected. In this paper, the problem of the deflections induced by the moving payload in the girder of overhead travelling crane is examined. The evaluation of a camber giving a counterdeflection of the girder is proposed in order to get minimum energy consumptions for trolley to move along a nonstraight support. To this aim, probabilistic payload distributions are considered instead of fixed or rated loads involved in other researches. Taking 50/10 t bridge crane as a research object, the probability loads are determined by analysis of load distribution density functions. According to load distribution, camber design under different probability loads is discussed in detail as well as energy consumptions distribution. The research results provide the design reference of reasonable camber to obtain the least energy consumption for climbing corresponding to different P0; thus energy-saving design can be achieved.
1. Introduction
Crane is a mechanical device, used to move widely materials in modern production. It plays a very important role in the national economy with greatly reduced labour intensity, improved production efficiency, and promoted social development as an indispensable auxiliary tool and process equipment [1]. Therefore, its energy consumption becomes an unavoidable topic and, in fact, energy consumption of crane is very huge [2]. There are several reasons influencing the energy loss, and the camber of the girder is the one not to be neglected [3]. When overhead travelling crane works, wheels of the trolley will press on the bridge and generate downward bend. If the deformation is too large, the phenomena of “climbing” will occur and the driving force required will increase [4]. Meanwhile, the deformation can affect the performance of crane trolley badly. The camber of girder can decrease the running resistance and ensure the crane to be safe and steady when running. Thus, it is necessary to design the camber for crane girder so as to compensate the deformation of girder by its weight and loading, which can decrease the deformation affections and energy loss, enhance the bearing capacity of crane, reduce crane climbing and slip slope, and ensure smooth running.
At present, camber curve of crane girder plate is widely adopted [5], while welding, self-weight, and the loading can cause deformation, so it is difficult to obtain an ideal camber curve. Rongbo discussed the basic principles of camber curve on the girder and introduced the camber specified by domestic and international technical standards. Furthermore, various forms of special prechange curve were proposed [6]. In literature [7], the advantages and disadvantages of popular camber curve (parabola, sinusoidal, and threefold) were discussed and a new camber was proposed. However, the above researches about camber are generally based on rated load. In fact, during the actual operation, the crane hoist/load is random varying from empty load to rated load. Under light load, climbing or downgrading of trolley is unavoidable with different energy consumptions corresponding to different loadings. Also, camber under rated load may not be the most energy-saving. As for cranes often with heavy loadings, the precamber of girder needs to be larger and for cranes often with light loadings smaller. Thus, operation under nonrated load will result in climbing and slip-on as well as increased energy loss. Professor Gao and Master Tian tested eight in-service representative bridge cranes to collect operation data (load, trolley position, working hours, etc.) and proposed the standard of hoisting load spectrum [8]. But the research was not continued deeply for energy-saving camber design. Therefore, research on camber design of crane is of great importance to accurately determine the camber curve of crane girder plate, control the girder camber, and reduce crane energy consumptions as well as to ensure crane safety. To this aim, probabilistic payload distributions are considered instead of fixed or rated loads involved in other researches.
The present work was carried out in order to seek reasonable camber to obtain the least energy consumption corresponding to different probability loads. In Section 2, load spectrum of crane is studied based on analysis of load status level and corresponding load spectrum coefficient. In Section 3, taking 50/10 t bridge crane as research object, the probability loads are determined by analysis of load distribution density functions. In Section 4, camber design for different probability loads is discussed in detail as well as energy consumptions distribution. Finally, research conclusions are summarized.
2. Probability Load Analysis of Crane
Precamber is generally determined according to rated load, while the loading at work is uncertain, varying from empty load to rated load. By analyzing the load spectrum of crane to master the crane's work conditions, corresponding camber can be determined.
2.1. Load Spectrum Coefficient
The lifting weight of crane is uncertain, varying from 0 to rated load or even overweight. Therefore, uncertain load parameters should be adopted for analysis with considerations of the camber. Load spectrum coefficient describes the randomness of crane loading and can reflect the laws of loadings statistically as well as the loading status from the view of loading changes and use frequency of each typical load. In GB/T 3811, load spectrum coefficient is defined for calculation as [9]
(1)KP=∑[CiCT(PQiPQmax)m],
where
is the load spectrum coefficient;
Ci is the number of cranes working cycles corresponding to each typical load;
CT is the total number of cranes working cycles;
PQi is the typical lifting load within expected lifetime of the cranes;
PQmax is the rated load;
m is the exponent for facilitating the level division (m=3).
Currently load status can usually be divided into four levels based on load spectrum coefficient, as shown in Table 1.
Load status level and corresponding load spectrum coefficient.
Load status levels
Load spectrum coefficient of crane, Kp
Work conditions
Q1
Kp ≤ 0.125
Seldom for rated load and often light load
Q2
0.125 < Kp ≤ 0.250
Seldom for rated load and often medium load
Q3
0.250 < Kp ≤ 0.500
Sometimes for rated load and often heavy load
Q4
0.500 < Kp ≤ 1.000
Often for rated load
2.2. Crane Hoisting Load Spectrum
It is reported in literature [8] that eight in-service overhead traveling cranes are monitored and tested for collecting operation data. Statistical analysis results of the collected data show that the loading of overhead traveling crane passes the normality test at the significant level of 0.05, suggesting that the load is normally distributed. Thus, load belonging to different load status levels will generate different normal distribution density functions as follows:
(2)fx-t,s-t2(x)=1s-t2πe-(x-x-t)2/2s-t2.
From formula (2), it can be seen that there are load mean (x-t) and standard deviation (s-t) which are two key parameters, which is closely related with rated hoisting load denoted by “x-t=μQxtmax” and “s-t=σQxtmax,” where xtmax denotes rated hoisting weight/load and μQ,σQ denote the mean and standard deviation of hoisting weight/load, respectively.
So, it is necessary to get μQ,σQ for obtaining load mean (x-t) and standard deviation (s-t). Some practical data of cranes are collected in order to fit the normal distribution and solute μQ and σQ. PQi,PQmax,Ci, and CT are recorded. μQ can be calculated by the summation of the product of loading value multiplied by probability of loading, while the probability of loading can be approximately calculated by Ci/CT, which means the portion of the number of cranes working cycles under each typical load to the total number of cranes working cycles. So,
(3)μQ=∑[CiCT(PQiPQmax)]σQ=∑CiCT(PQiPQmax-μQ)2.
Let δQ denote the variation coefficient, and
(4)δQ=σQμQ.
The hoisting/load spectrum of overhead crane can reflect the hoisting/load probability distribution under certain status level, which provide data supports for further research on the camber under different loadings.
On the other hand, the running track of trolley is also random, varying from middle to both ends or from end to middle or around the middle. In practice, the trolley generally goes through the middle of the girder. So for calculation simplification, the operation of trolley per work cycle is regarded as one climbing.
3. Probability Load Determination of Crane
Box overhead crane is a typical overhead travelling widely used at home and abroad with simple design, good manufacturing processes, structural stability, and other advantages. Take a 50/10 t overhead crane used in a practical project as the research object (shown in Figure 1).
Model of 50/10 t overhead crane.
Structural illustration of girder
Section attributes illustration
Parameters of the research object are as follows:
rated load is 50 t;
effective length of the girder is 31.5 m;
gauge of trolley is 3580 mm;
weight of total girder is 37.8 t (beam: 14.833 t and end girder: 4.067 t);
weight of trolley is 15.4 t;
H, the spacing between top and bottom plate, is 1700 mm;
B1, width of the plate, is 650 mm;
B2, the spacing between webs, is 590 mm;
T11/T12, the thickness of top/bottom plate, is 24 mm;
T21/T22, the thickness of left/right web, is 6 mm;
the spacing between big stiffening ribs is 1200 mm~2750 mm;
the spacing between small stiffening ribs is 400 mm~550 mm;
material is ordinary carbon steel Q235.
The rated load is 50 t; that is, xtmax=500KN and the parameters of x-t,s-, and confidence level corresponding to the above four work conditions are as follows:
(5)Q1:x-t=μQxtmax=229KN,s-t=σQxtmax=73KN,U1=∫05001s-t2πe-(P-x-t)2/2s-t2dP=0.99904Q2:x-t=axtmax=304.5KN,s-t=bxtmax=56KN,U2=∫05001s-t2πe-(P-x-t)2/2s-t2dP=0.9999Q3:x-t=axtmax=394KN,s-t=bxtmax=34.5KN,U3=∫05001s-t2πe-(P-x-t)2/2s-t2dP=0.998939Q4:x-t=axtmax=450KN,s-t=bxtmax=20KN,U4=∫05001s-t2πe-(P-x-t)2/2s-t2dP=0.99379.
As the crane load is impossible to be negative and cannot exceed the rated load, so all loads should occur within the interval [0, Pmax], the range of whose normal distribution function is (-∞,+∞). According to the checking by the above formulas, it can be found that the probability of loads appearing in [0, Pmax] is greater than 0.99. So [0, Pmax] can be replaced by (-∞,+∞) and the load distribution density functions, corresponding to Q1, Q2, Q3, and Q4, can be expressed as
(6)Q1:fx-t,s-t2(P)=1s-t2πe-(P-x-t)2/2s-t2=1732πe-(P-229)2/2×732=0.00546e-(P-229)2/10658Q2:fx-t,s-t2(P)=1s-t2πe-(P-x-t)2/2s-t2=1562πe-(P-304.5)2/2×562=0.00712e-(P-304.5)2/6272Q3:fx-t,s-t2(P)=1s-t2πe-(P-x-t)2/2s-t2=134.52πe-(P-394)2/2×34.52=0.01156e-(P-394)2/2380.5Q4:fx-t,s-t2(P)=1s-t2πe-(P-x-t)2/2s-t2=1202πe-(P-450)2/2×202=0.01994e-(P-450)2/800.
The distribution function curves are shown in Figure 2.
Distribution function curve corresponding to Q1, Q2, Q3, and Q4.
From Figure 2, it can be concluded that, with the load status level improved, the hoisting/load mean gradually increases and the load distribution becomes more concentrated. The loads of Q4 are distributed around the rated load with a certain probability of overloading.
Actually, some operation parameters of cranes cannot be obtained for crane design. If the load spectrum of overhead crane is not available, the load distribution cannot be obtained according to Table 2. But generally the load is considered as conformed to normal distribution [0, Pmax]. Then according to design handbook of cranes, μQ and σQ for hook hoisting of overhead crane can be basically determined. Generally μQ∈ (0.5-0.6) and σQ∈ (0.15–0.2). Due to different work conditions of different cranes with different loadings, the values of μQ and σQ are different. Here, let μQ=0.55 and σQ=0.18 by design experts. Then according to x-t=μQxtmax,s-t=σQxtmax, and xtmax=500KN, x-t=275KN, s-t=90KN.
Hanging heavy load spectrum of standard recommended by GB/T 3811[10].
Load status levels
Nominal load spectrum coefficient, Kp
MQ
σQ
Load spectrum coefficient
Q1
Kp ≤ 0.125
0.458
0.146
0.125
Q2
0.125 < Kp ≤ 0.250
0.609
0.112
0.250
Q3
0.250 < Kp ≤ 0.500
0.788
0.069
0.500
Q4
0.500 < Kp ≤ 1.000
0.900
0.04
0.9
Placing obtained x-t and s- into formula (2), the density function of load distribution can be obtained as
(7)fx-t,s-t2(P)=1s-t2πe-(P-x-t)2/2s-t2=1902πe-(P-275)2/2×902=0.004434e-(P-275)2/16200.
The distribution is illustrated as shown in Figure 3 and the probability of loads appearing in [0, Pmax] is 0.992667.
Distribution function curve corresponding to uncertain load level.
4. Matching of Camber with Probability Load/Hoisting
As shown in Figure 4, the bridge crane can be considered as simply supported beam and it is necessary to overcome the climbing resistance when trolley is moving on the bending track. The climbing resistance is closely related to the slope where trolley moves to. To eliminate the climbing resistance, the ideal camber curve should ensure that the slope remains zero (0) for trolley at any position on the girder.
Girder illustration under wheel pressure (P denotes the load, X denotes the distance between loading position and left end, S denotes the span length, and α denotes the inclination).
That is,
(8)αcamber+α=0,
where αcamber denotes the inclination with prefabricated camber and α denotes the inclination without camber which can make sure the slope remains zero (0) for trolley at any position on the girder.
At certain position X, the downwarping inclination α equals: (simply supported beam classic formula)
(9)α=-PS23EIXS(1-XS)(1-2XS)∵E=∫(f×s)ds.
Here, f is the sum of P and the weight of trolley and is (P+157), s denotes the vertical displace, and s=dx×tanα. The energy consumed equals the work done by the vertical load acting on the girder to generate a displacement (downwarping).
So, when the camber designed consumes minimum energy under the load of P0, if loading is P, then trolley running from the middle of girder to the end position will make useless power work done as much as
(10)EPP0=∫0L/2(P+157)|tgαP0-tgαP|dx,
where αP0 denotes the inclination at certain position X on the prefabricated camber, which is designed to consume minimum energy under load of P0.
If α is very small, then tanα≈α and the energy consumptions loading probability load P according to the load curve with design load P0 and with considerations of the weight of trolley (157 KN) can be expressed as
(11)EPP0=∫0L/2(P+157)|αP0-αP|dx.
The energy consumptions distribution is shown in Figure 5.
Load energy consumptions distribution.
Place formula (9) into formula (11); then
(12)EP0=∫0L/2(P+157)S26EI·xS(1-4x2S2)|P0-P|dx.
After solution of the differential equations, energy consumptions can be obtained:
(13)EP0=(P+157)|P0-P|S348EI.
Take a 50/10 t overhead crane, for example, and input the value of S, E, and I; then
(14)EP0=0.11(P+157)|P0-P|.
From Figure 5, it can be found that the minimum energy consumptions occur when P approaches P0 and the energy consumptions are less when P<P0 than when P>P0 because of heavier loading. So, heavier load can be considered for camber design to guarantee less energy consumptions when overloading.
With combinations of each load distribution density function, EP0, which denotes the energy consumptions of the crane whose camber is based on P0 and under loadings of normal distribution, can be calculated as
(15)EP0=∫0Pmax1s-t2πe-(P-x-t)2/2s-t2(P+157)|P-P0|S348EIdP.
Remove the absolute value and expand formula (15); then
(16)EP0=1s-t2πS348EI×(∫0P0e-(P-x-t)2/2s-t2(P+157)(P0-P)dP+∫P0Pmaxe-(P-x-t)2/2s-t2(P+157)(P-P0)dP).
Then, the energy consumptions under Q1, Q2, Q3, and Q4 can be obtained as well as energy consumptions distribution shown in Figures 6–9:
(17)Q1:EP0=0.0006·(∫0P0e-(P-229)2/10658·(P+157)(P0-P)dP+∫P0500e-(P-229)2/10658·(P+157)(P-P0)dP∫P0500e(-(P-229)2/10658))=-4.17+18.04e(0.0430-0.0000938P0)P0-0.0052P0+(10291.5-42.385P0)×Erf[2.22-0.00968P0]Q2:EP0=0.00078·(∫0P0e-(P-304.5)2/6272(P+157)(P0-P)dP+∫P0500e-(P-304.5)2/6272×(P+157)(P-P0)dP∫P0500e(-(P-304.5)2/6272))=-9.27+0.000896e(0.0968-0.000159P0)P0+0.018P0+(15752.2-50.60P0)Erf[3.8396-0.0126P0]Q3:EP0=0.00127(∫0P0e-(P-394)2/2380.5(P+157)(P0-P)dP+∫P0500e-(P-394)2/2380.5×(P+157)(P-P0)dP+∫P0500e(-(P-394)2/2380.5))=-40.433+7.54×10-26a~(0.33-0.00042P0)P0+0.0793P0+(23996.77-60.54P0)Erf[8.08-0.020P0]Q4:EP0=0.00219(∫0P0e-(P-450)2/800(P+157)(P0-P)dP+∫P0500e-(P-450)2/800×(P+157)(P-P0)dP+∫P0500e(-(P-450)2/800))=-229.103+1.25×10-107e(1.125-0.00125P0)P0+0.45P0+(30033.1-66.64P0)Erf[15.91-0.035P0].
Energy consumptions distribution with probability load P0 under Q1.
Energy consumptions distribution with probability load P0 under Q2.
Energy consumptions distribution with probability load P0 under Q3.
Energy consumptions distribution with probability load P0 under Q4.
From Figure 6, it can be found that when c=0, that is, without camber, energy consumptions are greatly increased. With P0 acting on the girder and increasing, EP0 decreases. When P0 approaches 242.661 KN, the minimum of EP0 occurs as 2419.86 J. If P0 continues to increase, EP0 will increase.
Similarly from Figure 7, it can be found that, when P0 approaches 311.256 KN, the minimum of EP0 occurs as 2243.65 J.
Similarly from Figure 8, it can be found that, when P0 approaches 396.3 KN, the minimum of EP0 occurs as 1654.47 J.
Similarly from Figure 9, it can be found that, when P0 approaches 450.489 KN, the minimum of EP0 occurs as 1037.59 J.
Through the above analysis, the ideal load P0 corresponding to different load status levels can be initially determined as shown in Table 3.
Ideal load P0 corresponding to different load status levels.
Load status level
P0
Q1
242 KN
Q2
311 KN
Q3
396 KN
Q4
450 KN
When the load spectrum coefficient is unknown, the load can be considered to be conformed to [0, Pmax] normal distribution. Then, the energy consumptions under the uncertain load status level Q can be obtained as well as energy consumptions distribution shown in Figure 10:
(18)Q:EP0=0.00049·(∫0P0e-(P-275)2/16200(P+157)(P0-P)dP+∫P0500e-(P-275)2/16200×(P+157)(P-P0)dP+∫P0500e(-(P-275)2/16200))=-775.587+27.4307e(0.034-0.000062P0)P0+2.32P0+(12222.6-41.62P0)Erf(2.166-0.006P0).
Energy consumptions distribution with probability load P0 under Q.
From Figure 10, energy consumption distribution law can be found when load spectrum coefficient is uncertain. When P0=0, that is, without camber, energy consumptions are greatly much. With P0 acting on the girder and increasing, EP0 decreases. When P0 approaches 396.57 KN, the minimum of EP0 occurs as 3401.02 J. If P0 continues to increase, EP0 will increase.
Different P0 determines the corresponding reasonable camber to obtain the least energy consumption for climbing. According to formula (9) and αcamber+α=0, then
(19)αcamber=-α=PS23EIXS(1-XS)(1-2XS).
Because αcamber is very small,
(20)dycamberdx=tgαcamber≈αcamber(21)dycamber=[PS23EIXS(1-XS)(1-2XS)]dx.
After integral solution of formula (21), the camber curve can be expressed as
(22)ycamber=PS36EI(XS)2(1-XS)2.
Combined with the weight of the trolley Ptrolley, the camber curve can be obtained and expressed as
(23)ycamber=(P0+Ptrolley)S36EI(XS)2(1-XS)2.
5. Conclusions
Just as expressed in formula (23), the camber curve can be designed. However, in previous researches on camber design, P0 is designed as fixed or rated loads. In fact, the hoisting weight of overhead crane is not fixed, and energy consumptions under certain camber with different hoisting/load will vary. Camber design based on the fixed or rated load may not get the optimal energy-saving. This research seeks to get ideal P0 for trolley moving with probabilistic loads to obtain the least energy consumption. Then, according to formula (23), energy-saving camber design of overhead travelling crane can be obtained. Besides, some conclusions can be drawn as below.
With the load status level improved, the hoisting/load mean gradually increases and the load distribution becomes more concentrated. The loads of Q4 are distributed around the rated load with a certain probability of overloading.
The minimum energy consumptions occur when P approaches P0 and the energy consumptions are less when P<P0 than when P>P0 because of heavier loading. So, heavier load can be considered for camber design to guarantee less energy consumptions when overloading.
The research results of this paper can also provide a great reference value to cutting girder web along a certain curve for prefabrication.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was financially supported by National Foundation of General Administration of Quality Supervision and Inspection (2012QK178), Program of Science Foundation of General Administration of Quality Supervision and Inspection of Jiangsu Province (KJ103708), and “excellence plans-zijin star” Foundation of Nanjing University of Science. Also, the work is preresearch of the National Natural Science Foundation of China (research on energy consumption modelling and methodology of energy-saving design for cranes). The supports are gratefully acknowledged.
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