Research Article Research on Energy-Saving Design of...
10
Research Article Research on Energy-Saving Design of Overhead Travelling Crane Camber Based on Probability Load Distribution Tong Yifei, 1 Tang Zhaohui, 1 Mei Song, 2 Shen Guomin, 1 and Gu Feng 3 1 School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China 2 Nanjing Research Institute for Agricultural Mechanization, Ministry of Agriculture, Nanjing 210014, China 3 Nantong Vocational University, Nantong 226007, China Correspondence should be addressed to Tong Yifei; [email protected] Received 20 November 2013; Revised 11 March 2014; Accepted 14 March 2014; Published 16 April 2014 Academic Editor: Massimo Scalia Copyright © 2014 Tong Yifei et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. us, energy consumption becomes an unavoidable topic. ere 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. e 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 50/10t 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. e research results provide the design reference of reasonable camber to obtain the least energy consumption for climbing corresponding to different 0 ; 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 inten- sity, improved production efficiency, and promoted social development as an indispensable auxiliary tool and process equipment [1]. erefore, its energy consumption becomes an unavoidable topic and, in fact, energy consumption of crane is very huge [2]. ere 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. e camber of girder can decrease the running resistance and ensure the crane to be safe and steady when running. us, 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 disad- vantages 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 Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 484635, 9 pages http://dx.doi.org/10.1155/2014/484635
Transcript of Research Article Research on Energy-Saving Design of...
Research ArticleResearch on Energy-Saving Design of Overhead Travelling CraneCamber Based on Probability Load Distribution
Tong Yifei1 Tang Zhaohui1 Mei Song2 Shen Guomin1 and Gu Feng3
1 School of Mechanical Engineering Nanjing University of Science and Technology Nanjing 210094 China2Nanjing Research Institute for Agricultural Mechanization Ministry of Agriculture Nanjing 210014 China3Nantong Vocational University Nantong 226007 China
Correspondence should be addressed to Tong Yifei tyf51129aliyuncom
Received 20 November 2013 Revised 11 March 2014 Accepted 14 March 2014 Published 16 April 2014
Academic Editor Massimo Scalia
Copyright copy 2014 Tong Yifei et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Crane is a mechanical device used widely to move materials in modern production It is reported that the energy consumptionsof China are at least 5ndash8 times of other developing countries Thus energy consumption becomes an unavoidable topic There areseveral reasons influencing the energy loss and the camber of the girder is the one not to be neglected In this paper the problem ofthe deflections induced by the moving payload in the girder of overhead travelling crane is examined The evaluation of a cambergiving a counterdeflection of the girder is proposed in order to get minimum energy consumptions for trolley to move along anonstraight support To this aim probabilistic payload distributions are considered instead of fixed or rated loads involved in otherresearches Taking 5010 t bridge crane as a research object the probability loads are determined by analysis of load distributiondensity functions According to load distribution camber design under different probability loads is discussed in detail as well asenergy consumptions distributionThe research results provide the design reference of reasonable camber to obtain the least energyconsumption for climbing corresponding to different 119875
0 thus energy-saving design can be achieved
1 Introduction
Crane is a mechanical device used to move widely materialsin modern production It plays a very important role inthe national economy with greatly reduced labour inten-sity improved production efficiency and promoted socialdevelopment as an indispensable auxiliary tool and processequipment [1] Therefore its energy consumption becomesan unavoidable topic and in fact energy consumption ofcrane is very huge [2] There are several reasons influencingthe energy loss and the camber of the girder is the onenot to be neglected [3] When overhead travelling craneworks wheels of the trolley will press on the bridge andgenerate downward bend If the deformation is too large thephenomena of ldquoclimbingrdquo will occur and the driving forcerequired will increase [4] Meanwhile the deformation canaffect the performance of crane trolley badly The camber ofgirder can decrease the running resistance and ensure thecrane to be safe and steadywhen runningThus it is necessary
to design the camber for crane girder so as to compensate thedeformation of girder by its weight and loading which candecrease the deformation affections and energy loss enhancethe bearing capacity of crane reduce crane climbing and slipslope and ensure smooth running
At present camber curve of crane girder plate is widelyadopted [5] while welding self-weight and the loadingcan cause deformation so it is difficult to obtain an idealcamber curve Rongbo discussed the basic principles ofcamber curve on the girder and introduced the camberspecified by domestic and international technical standardsFurthermore various forms of special prechange curve wereproposed [6] In literature [7] the advantages and disad-vantages of popular camber curve (parabola sinusoidal andthreefold) were discussed and a new camber was proposedHowever the above researches about camber are generallybased on rated load In fact during the actual operation thecrane hoistload is random varying from empty load to ratedload Under light load climbing or downgrading of trolley is
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 484635 9 pageshttpdxdoiorg1011552014484635
2 Mathematical Problems in Engineering
unavoidablewith different energy consumptions correspond-ing to different loadings Also camber under rated load maynot be themost energy-saving As for cranes oftenwith heavyloadings the precamber of girder needs to be larger andfor cranes often with light loadings smaller Thus operationunder nonrated load will result in climbing and slip-on aswell as increased energy loss Professor Gao and Master Tiantested eight in-service representative bridge cranes to collectoperation 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-savingcamber designTherefore research on camber design of craneis of great importance to accurately determine the cambercurve of crane girder plate control the girder camber andreduce crane energy consumptions as well as to ensure cranesafety To this aim probabilistic payload distributions areconsidered instead of fixed or rated loads involved in otherresearches
The present work was carried out in order to seekreasonable camber to obtain the least energy consumptioncorresponding to different probability loads In Section 2load spectrum of crane is studied based on analysis of loadstatus level and corresponding load spectrum coefficientIn Section 3 taking 5010 t bridge crane as research objectthe probability loads are determined by analysis of loaddistribution density functions In Section 4 camber designfor different probability loads is discussed in detail as well asenergy consumptions distribution Finally research conclu-sions are summarized
2 Probability Load Analysis of Crane
Precamber is generally determined according to rated loadwhile the loading at work is uncertain varying from emptyload to rated load By analyzing the load spectrum of craneto master the cranersquos work conditions corresponding cambercan be determined
21 Load Spectrum Coefficient The lifting weight of crane isuncertain varying from 0 to rated load or even overweightTherefore uncertain load parameters should be adopted foranalysis with considerations of the camber Load spectrumcoefficient describes the randomness of crane loading and canreflect the laws of loadings statistically as well as the loadingstatus from the view of loading changes and use frequency ofeach typical load In GBT 3811 load spectrum coefficient isdefined for calculation as [9]
119870119875= sum[
119862119894
119862119879
(
119875119876119894
119875119876max)
119898
] (1)
where119870119875is the load spectrum coefficient119862119894is the number of cranes working cycles corre-
sponding to each typical load119862119879is the total number of cranes working cycles
119875119876119894is the typical lifting load within expected lifetime
of the cranes
Table 1 Load status level and corresponding load spectrum coeffi-cient
Load statuslevels
Load spectrumcoefficient of crane119870
119901
Work conditions
1198761 119870119901le 0125 Seldom for rated load
and often light load
1198762 0125 lt 119870119901le 0250 Seldom for rated load
and often medium load
1198763 0250 lt 119870119901le 0500 Sometimes for rated load
and often heavy load1198764 0500 lt 119870
119901le 1000 Often for rated load
119875119876max is the rated load
119898 is the exponent for facilitating the level division(119898 = 3)
Currently load status can usually be divided into fourlevels based on load spectrum coefficient as shown in Table 1
22 Crane Hoisting Load Spectrum It is reported in litera-ture [8] that eight in-service overhead traveling cranes aremonitored and tested for collecting operation data Statisticalanalysis results of the collected data show that the loadingof overhead traveling crane passes the normality test at thesignificant level of 005 suggesting that the load is normallydistributedThus load belonging to different load status levelswill generate different normal distribution density functionsas follows
119891119909119905 1199041199052(119909)=1
119904119905radic2120587
119890minus(119909minus119909119905)
221199041199052
(2)
From formula (2) it can be seen that there are loadmean (119909
119905) and standard deviation (119904
119905) which are two key
parameters which is closely related with rated hoisting loaddenoted by ldquo119909
119905= 120583119876119909119905maxrdquo and ldquo119904
119905= 120590119876119909119905maxrdquo where
119909119905max denotes rated hoisting weightload and 120583
119876 120590119876denote
the mean and standard deviation of hoisting weightloadrespectively
So it is necessary to get 120583119876 120590119876for obtaining load mean
(119909119905) and standard deviation (119904
119905) Some practical data of cranes
are collected in order to fit the normal distribution and solute120583119876and 120590
119876 119875119876119894 119875119876max 119862119894 and 119862119879 are recorded 120583
119876can be
calculated by the summation of the product of loading valuemultiplied by probability of loading while the probabilityof loading can be approximately calculated by 119862
119894119862119879 which
means the portion of the number of cranes working cyclesunder each typical load to the total number of cranes workingcycles So
120583119876= sum[
119862119894
119862119879
(
119875119876119894
119875119876max)]
120590119876= radicsum
119862119894
119862119879
(
119875119876119894
119875119876max
minus 120583119876)
2
(3)
Mathematical Problems in Engineering 3
Let 120575119876denote the variation coefficient and
120575119876=120590119876
120583119876
(4)
The hoistingload spectrum of overhead crane can reflectthe hoistingload probability distribution under certain statuslevel which provide data supports for further research on thecamber under different loadings
On the other hand the running track of trolley is alsorandom varying from middle to both ends or from end tomiddle or around themiddle In practice the trolley generallygoes through the middle of the girder So for calculationsimplification the operation of trolley per work cycle isregarded as one climbing
3 Probability Load Determination of Crane
Box overhead crane is a typical overhead travelling widelyused at home and abroad with simple design good manu-facturing processes structural stability and other advantagesTake a 5010 t overhead crane used in a practical project as theresearch object (shown in Figure 1)
Parameters of the research object are as follows
(i) rated load is 50 t
(ii) effective length of the girder is 315m
(iii) gauge of trolley is 3580mm
(iv) weight of total girder is 378 t (beam 14833 t and endgirder 4067 t)
(v) weight of trolley is 154 t
(vi) 119867 the spacing between top and bottom plate is1700mm
(vii) 1198611 width of the plate is 650mm
(viii) 1198612 the spacing between webs is 590mm
(ix) 1198791111987912 the thickness of topbottom plate is 24mm
(x) 1198792111987922 the thickness of leftright web is 6mm
(xi) the spacing between big stiffening ribs is 1200mmsim2750mm
(xii) the spacing between small stiffening ribs is400mmsim550mm
(xiii) material is ordinary carbon steel 119876235
The rated load is 50 t that is 119909119905max = 500KN and the
parameters of 119909119905 119904 and confidence level corresponding to the
above four work conditions are as follows
1198761 119909119905= 120583119876119909119905max = 229KN 119904
119905= 120590119876119909119905max = 73KN
1198801= int
500
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
119889119875 = 099904
1198762 119909119905= 119886119909119905max = 3045KN 119904
119905= 119887119909119905max = 56KN
1198802= int
500
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
119889119875 = 09999
1198763 119909119905= 119886119909119905max = 394KN 119904
119905= 119887119909119905max = 345KN
1198803= int
500
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
119889119875 = 0998939
1198764 119909119905= 119886119909119905max = 450KN 119904
119905= 119887119909119905max = 20KN
1198804= int
500
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
119889119875 = 099379
(5)
As the crane load is impossible to be negative and cannotexceed the rated load so all loads should occur within theinterval [0 119875max] the range of whose normal distributionfunction is (minusinfin +infin) According to the checking by theabove formulas it can be found that the probability of loadsappearing in [0 119875max] is greater than 099 So [0 119875max] canbe replaced by (minusinfin +infin) and the load distribution densityfunctions corresponding to 1198761 1198762 1198763 and 1198764 can beexpressed as
1198761 1198911199091199051199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
73radic2120587
119890minus(119875minus229)
22times73
2
= 000546119890minus(119875minus229)
210658
1198762 1198911199091199051199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
56radic2120587
119890minus(119875minus3045)
22times56
2
= 000712119890minus(119875minus3045)
26272
1198763 1198911199091199051199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
345radic2120587
119890minus(119875minus394)
22times345
2
4 Mathematical Problems in Engineering
1 2 3 4
5 6
(1) Top plate(2) Big stiffening ribs(3) Small stiffening ribs
(4) Horizontal angle iron(5) Bottom plate(6) Webs
(a) Structural illustration of girder
H
x
y
T11
T12
T21 T22
B1
B2
(b) Section attributes illustration
Figure 1 Model of 5010 t overhead crane
= 001156119890minus(119875minus394)
223805
1198764 119891119909119905 1199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
20radic2120587
119890minus(119875minus450)
22times20
2
= 001994119890minus(119875minus450)
2800
(6)
The distribution function curves are shown in Figure 2From Figure 2 it can be concluded that with the load
status level improved the hoistingload mean graduallyincreases and the load distribution becomes more concen-trated The loads of 1198764 are distributed around the rated loadwith a certain probability of overloading
Actually some operation parameters of cranes cannotbe obtained for crane design If the load spectrum ofoverhead crane is not available the load distribution cannotbe obtained according to Table 2 But generally the load isconsidered as conformed to normal distribution [0 119875max]Then according to design handbook of cranes 120583
119876and 120590
119876for
hook hoisting of overhead crane can be basically determinedGenerally 120583
119876isin (05-06) and 120590
119876isin (015ndash02) Due to different
work conditions of different cranes with different loadingsthe values of 120583
119876and 120590
119876are different Here let 120583
119876= 055
and 120590119876= 018 by design experts Then according to 119909
119905=
120583119876119909119905max 119904119905 = 120590119876119909119905max and 119909119905max = 500KN 119909119905 = 275KN
119904119905= 90KN
100 200 300 400 500
Prob
abili
ty
0020
0015
0010
0005
Hoisting load (kN)
Q1
Q2
Q3
Q4
Figure 2 Distribution function curve corresponding to119876111987621198763and 1198764
Placing obtained 119909119905and 119904 into formula (2) the density
function of load distribution can be obtained as
119891119909119905 1199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
90radic2120587
119890minus(119875minus275)
22 times 90
2
= 0004434119890minus(119875minus275)
216200
(7)
Thedistribution is illustrated as shown in Figure 3 and theprobability of loads appearing in [0 119875max] is 0992667
Mathematical Problems in Engineering 5
Table 2 Hanging heavy load spectrum of standard recommended by GBT 3811 [10]
Load status levels Nominal load spectrum coefficient 119870119901
119872119876
120590119876
Load spectrum coefficient1198761 119870
119901le 0125 0458 0146 0125
1198762 0125 lt 119870119901le 0250 0609 0112 0250
1198763 0250 lt 119870119901le 0500 0788 0069 0500
1198764 0500 lt 119870119901le 1000 0900 004 09
100 200 300 400 500Hoisting load (kN)
Prob
abili
ty
0004
0003
0002
0001
Q
Figure 3 Distribution function curve corresponding to uncertainload level
S
X a
P
Figure 4 Girder illustration under wheel pressure (119875 denotes theload 119883 denotes the distance between loading position and left end119878 denotes the span length and 120572 denotes the inclination)
4 Matching of Camber with ProbabilityLoadHoisting
As shown in Figure 4 the bridge crane can be considered assimply supported beam and it is necessary to overcome theclimbing resistance when trolley is moving on the bendingtrack The climbing resistance is closely related to the slopewhere trolley moves to To eliminate the climbing resistancethe ideal camber curve should ensure that the slope remainszero (0) for trolley at any position on the girder
That is
120572camber + 120572 = 0 (8)
where 120572camber denotes the inclination with prefabricatedcamber and 120572 denotes the inclination without camber whichcan make sure the slope remains zero (0) for trolley at anyposition on the girder
At certain position 119883 the downwarping inclination 120572equals (simply supported beam classic formula)
E(P
0)
P0 P
30000
20000
10000
0
00
400 400
200 200
Figure 5 Load energy consumptions distribution
120572 = minus1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878)
∵ 119864 = int (119891 times 119904) 119889119904
(9)
Here 119891 is the sum of 119875 and the weight of trolley andis (119875 + 157) 119904 denotes the vertical displace and 119904 = 119889119909 timestan120572 The energy consumed equals the work done by thevertical load acting on the girder to generate a displacement(downwarping)
So when the camber designed consumes minimumenergy under the load of 119875
0 if loading is 119875 then trolley
running from the middle of girder to the end position willmake useless power work done as much as
1198641198750
119875= int
1198712
0
(119875 + 157)100381610038161003816100381610038161199051198921205721198750minus 119905119892120572119875
10038161003816100381610038161003816119889119909 (10)
where 1205721198750
denotes the inclination at certain position 119883 onthe prefabricated camber which is designed to consumeminimum energy under load of 119875
0
If 120572 is very small then tan120572 asymp 120572 and the energyconsumptions loading probability load 119875 according to theload curve with design load 119875
0and with considerations of the
weight of trolley (157 KN) can be expressed as
1198641198750
119875= int
1198712
0
(119875 + 157)100381610038161003816100381610038161205721198750minus 120572119875
10038161003816100381610038161003816119889119909 (11)
The energy consumptions distribution is shown inFigure 5
Place formula (9) into formula (11) then
1198641198750 = int
1198712
0
(119875 + 157)1198782
6119864119868sdot119909
119878(1 minus
41199092
1198782)10038161003816100381610038161198750 minus 119875
1003816100381610038161003816 119889119909 (12)
6 Mathematical Problems in Engineering
After solution of the differential equations energy con-sumptions can be obtained
1198641198750 = (119875 + 157)
10038161003816100381610038161198750 minus 1198751003816100381610038161003816
1198783
48119864119868 (13)
Take a 5010 t overhead crane for example and input thevalue of 119878 119864 and 119868 then
1198641198750 = 011 (119875 + 157)
10038161003816100381610038161198750 minus 1198751003816100381610038161003816
(14)
From Figure 5 it can be found that the minimum energyconsumptions occur when 119875 approaches 119875
0and the energy
consumptions are less when 119875 lt 1198750than when 119875 gt 119875
0
because of heavier loading So heavier load can be consideredfor camber design to guarantee less energy consumptionswhen overloading
With combinations of each load distribution densityfunction 1198641198750 which denotes the energy consumptions of thecrane whose camber is based on 119875
0and under loadings of
normal distribution can be calculated as
1198641198750 = int
119875max
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157)1003816100381610038161003816119875 minus 1198750
1003816100381610038161003816
1198783
48119864119868119889119875
(15)
Remove the absolute value and expand formula (15) then
1198641198750 =
1
119904119905radic2120587
1198783
48119864119868
times (int
1198750
0
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157) (1198750 minus 119875) 119889119875
+int
119875max
1198750
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157) (119875 minus 1198750) 119889119875)
(16)
Then the energy consumptions under 1198761 1198762 1198763and 1198764 can be obtained as well as energy consumptionsdistribution shown in Figures 6ndash9
1198761
1198641198750 = 00006 sdot (int
1198750
0
119890minus(119875minus229)
210658
sdot (119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus229)
210658
sdot (119875 + 157) (119875 minus 1198750) 119889119875)
= minus417 + 1804119890(00430minus000009381198750)1198750
minus 000521198750+ (102915 minus 42385119875
0)
times Erf [222 minus 0009681198750]
1198762
1198641198750 = 000078 sdot (int
1198750
0
119890minus(119875minus3045)
26272(119875 + 157) (1198750 minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus3045)
26272
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus927 + 0000896119890(00968minus00001591198750)1198750 + 0018119875
0
+ (157522 minus 50601198750)Erf [38396 minus 00126119875
0]
1198763
1198641198750 = 000127 (int
1198750
0
119890minus(119875minus394)
223805
(119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus394)
223805
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus40433 + 754 times 10minus26119886(033minus0000421198750)1198750 + 00793119875
0
+ (2399677 minus 60541198750)Erf [808 minus 0020119875
0]
1198764
1198641198750 = 000219 (int
1198750
0
119890minus(119875minus450)
2800(119875 + 157) (119875
0minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus450)
2800
Mathematical Problems in Engineering 7
2000
4000
6000
8000
10000
100 200 300 400 500P0
242661 241986
E(P
0)
Figure 6 Energy consumptions distribution with probability load1198750under 1198761
5000
10000
15000
100 200 300 400 500P0
311256 224365
E(P
0)
Figure 7 Energy consumptions distribution with probability load1198750under 1198762
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus229103 + 125 times 10minus107119890(1125minus0001251198750)1198750 + 045119875
0
+ (300331 minus 66641198750)Erf [1591 minus 0035119875
0]
(17)
From Figure 6 it can be found that when 119888 = 0 that iswithout camber energy consumptions are greatly increasedWith 119875
0acting on the girder and increasing 1198641198750 decreases
When1198750approaches 242661 KN theminimumof1198641198750 occurs
as 241986 J If 1198750continues to increase 1198641198750 will increase
Similarly from Figure 7 it can be found that when1198750approaches 311256KN the minimum of 1198641198750 occurs as
224365 JSimilarly from Figure 8 it can be found that when
1198750approaches 3963 KN the minimum of 1198641198750 occurs as
165447 JSimilarly from Figure 9 it can be found that when
1198750approaches 450489KN the minimum of 1198641198750 occurs as
103759 J
5000
10000
15000
20000
100 200 300 400 500P0
3963 165447
E(P
0)
Figure 8 Energy consumptions distribution with probability load1198750under 1198763
5000
10000
15000
20000
25000
30000
100 200 300 400 500P0
450489 103759
E(P
0)
Figure 9 Energy consumptions distribution with probability load1198750under 1198764
Table 3 Ideal load 1198750corresponding to different load status levels
Load status level 1198750
1198761 242KN1198762 311 KN1198763 396KN1198764 450KN
Through the above analysis the ideal load1198750correspond-
ing to different load status levels can be initially determinedas shown in Table 3
When the load spectrum coefficient is unknown theload can be considered to be conformed to [0 119875max] nor-mal distribution Then the energy consumptions under the
8 Mathematical Problems in Engineering
5000
10000
15000
20000
200 400 600 800 1000P0
39657 340102
E(P
0)
Figure 10 Energy consumptions distribution with probability load1198750under 119876
uncertain load status level119876 can be obtained as well as energyconsumptions distribution shown in Figure 10
119876
1198641198750 = 000049 sdot (int
1198750
0
119890minus(119875minus275)
216200
(119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus275)
216200
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus775587 + 274307119890(0034minus00000621198750)1198750 + 232119875
0
+ (122226 minus 41621198750)Erf (2166 minus 0006119875
0)
(18)
FromFigure 10 energy consumption distribution law canbe found when load spectrum coefficient is uncertain When1198750= 0 that is without camber energy consumptions are
greatly much With 1198750acting on the girder and increasing
1198641198750 decreasesWhen119875
0approaches 39657 KN theminimum
of 1198641198750 occurs as 340102 J If 1198750continues to increase 1198641198750 will
increaseDifferent 119875
0determines the corresponding reasonable
camber to obtain the least energy consumption for climbingAccording to formula (9) and 120572camber + 120572 = 0 then
120572camber = minus120572 =1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878) (19)
Because 120572camber is very small
119889119910camber119889119909
= 119905119892120572camber asymp 120572camber (20)
119889119910camber = [1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878)]119889119909
(21)
After integral solution of formula (21) the camber curvecan be expressed as
119910camber =1198751198783
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(22)
Combined with the weight of the trolley 119875trolley thecamber curve can be obtained and expressed as
119910camber =(1198750+ 119875trolley) 119878
3
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(23)
5 Conclusions
Just as expressed in formula (23) the camber curve can bedesigned However in previous researches on camber design1198750is designed as fixed or rated loads In fact the hoisting
weight of overhead crane is not fixed and energy consump-tions under certain camber with different hoistingload willvary Camber design based on the fixed or rated loadmay notget the optimal energy-savingThis research seeks to get ideal1198750for trolley moving with probabilistic loads to obtain the
least energy consumption Then according to formula (23)energy-saving camber design of overhead travelling cranecan be obtained Besides some conclusions can be drawn asbelow
(1) With the load status level improved the hoistingloadmean gradually increases and the load distributionbecomes more concentrated The loads of 1198764 aredistributed around the rated load with a certainprobability of overloading
(2) The minimum energy consumptions occur when 119875approaches 119875
0and the energy consumptions are less
when 119875 lt 1198750than when 119875 gt 119875
0because of
heavier loading So heavier load can be considered forcamber design to guarantee less energy consumptionswhen overloading
The research results of this paper can also provide a greatreference value to cutting girder web along a certain curve forprefabrication
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was financially supported by National Founda-tion of General Administration of Quality Supervision andInspection (2012QK178) Program of Science Foundation ofGeneral Administration of Quality Supervision and Inspec-tion of Jiangsu Province (KJ103708) and ldquoexcellence plans-zijin starrdquo Foundation of Nanjing University of Science Alsothe work is preresearch of theNational Natural Science Foun-dation of China (research on energy consumption modellingand methodology of energy-saving design for cranes) Thesupports are gratefully acknowledged
Mathematical Problems in Engineering 9
References
[1] P R Patel and V K Patel ldquoA review on structural analysisof overhead crane girder using FEA techniquerdquo InternationalJournal of Engineering Science and Innovative Technology vol 2no 4 pp 41ndash44 2013
[2] T Yifei Y Wei Y Zhen L Dongbo and L XiangdongldquoResearch on multidisciplinary optimization design of bridgecranerdquoMathematical Problems in Engineering vol 2013 ArticleID 763545 10 pages 2013
[3] S G Lee and N Q Hoang ldquoEnergy-based approach forcontroller design of overhead cranes a comparative studyrdquoApplied Mechanics and Materials vol 365-366 pp 784ndash7872010
[4] C Li-Feng ldquoMethods and analysis of bridge camber anddeformation of main girder in gantry cranerdquo Equipment Manu-facturing Technology no 5 pp 124ndash125 2010
[5] C R Bradlee ldquoMethod of measuring camberrdquo Us Patent4794773 1989
[6] F Rongbo ldquoCamber amp pre arch curve of cranesrdquo Hoisting andConveying Machinery vol 2 pp 16ndash18 1990
[7] G Shen X Li D Li and C Zhou ldquoReaseach on energy-savingdesign of arch curve of bridge crane girderrdquo in Proceedings ofthe International Conference on Remote Sensing Environmentand Transportation Engineering (RSETE rsquo11) pp 1448ndash1450Nanjing China June 2011
[8] T Jiantao Load Statistical Analisys and Reliable IndicatorResearch of Bridge Crane Taiyuan University of Science andTechnology Taiyuan China 2011
[9] S Zivanovic R P Johnson H V Dang and J Dobric ldquoDesignand construction of a very lively bridgerdquo in Topics in Dynamicsof Civil Structures vol 4 pp 371ndash380 2013
[10] X Gelin GBT3811-2008 Design Rules for Cranes ChineseStandard Press Beijing China 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
unavoidablewith different energy consumptions correspond-ing to different loadings Also camber under rated load maynot be themost energy-saving As for cranes oftenwith heavyloadings the precamber of girder needs to be larger andfor cranes often with light loadings smaller Thus operationunder nonrated load will result in climbing and slip-on aswell as increased energy loss Professor Gao and Master Tiantested eight in-service representative bridge cranes to collectoperation 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-savingcamber designTherefore research on camber design of craneis of great importance to accurately determine the cambercurve of crane girder plate control the girder camber andreduce crane energy consumptions as well as to ensure cranesafety To this aim probabilistic payload distributions areconsidered instead of fixed or rated loads involved in otherresearches
The present work was carried out in order to seekreasonable camber to obtain the least energy consumptioncorresponding to different probability loads In Section 2load spectrum of crane is studied based on analysis of loadstatus level and corresponding load spectrum coefficientIn Section 3 taking 5010 t bridge crane as research objectthe probability loads are determined by analysis of loaddistribution density functions In Section 4 camber designfor different probability loads is discussed in detail as well asenergy consumptions distribution Finally research conclu-sions are summarized
2 Probability Load Analysis of Crane
Precamber is generally determined according to rated loadwhile the loading at work is uncertain varying from emptyload to rated load By analyzing the load spectrum of craneto master the cranersquos work conditions corresponding cambercan be determined
21 Load Spectrum Coefficient The lifting weight of crane isuncertain varying from 0 to rated load or even overweightTherefore uncertain load parameters should be adopted foranalysis with considerations of the camber Load spectrumcoefficient describes the randomness of crane loading and canreflect the laws of loadings statistically as well as the loadingstatus from the view of loading changes and use frequency ofeach typical load In GBT 3811 load spectrum coefficient isdefined for calculation as [9]
119870119875= sum[
119862119894
119862119879
(
119875119876119894
119875119876max)
119898
] (1)
where119870119875is the load spectrum coefficient119862119894is the number of cranes working cycles corre-
sponding to each typical load119862119879is the total number of cranes working cycles
119875119876119894is the typical lifting load within expected lifetime
of the cranes
Table 1 Load status level and corresponding load spectrum coeffi-cient
Load statuslevels
Load spectrumcoefficient of crane119870
119901
Work conditions
1198761 119870119901le 0125 Seldom for rated load
and often light load
1198762 0125 lt 119870119901le 0250 Seldom for rated load
and often medium load
1198763 0250 lt 119870119901le 0500 Sometimes for rated load
and often heavy load1198764 0500 lt 119870
119901le 1000 Often for rated load
119875119876max is the rated load
119898 is the exponent for facilitating the level division(119898 = 3)
Currently load status can usually be divided into fourlevels based on load spectrum coefficient as shown in Table 1
22 Crane Hoisting Load Spectrum It is reported in litera-ture [8] that eight in-service overhead traveling cranes aremonitored and tested for collecting operation data Statisticalanalysis results of the collected data show that the loadingof overhead traveling crane passes the normality test at thesignificant level of 005 suggesting that the load is normallydistributedThus load belonging to different load status levelswill generate different normal distribution density functionsas follows
119891119909119905 1199041199052(119909)=1
119904119905radic2120587
119890minus(119909minus119909119905)
221199041199052
(2)
From formula (2) it can be seen that there are loadmean (119909
119905) and standard deviation (119904
119905) which are two key
parameters which is closely related with rated hoisting loaddenoted by ldquo119909
119905= 120583119876119909119905maxrdquo and ldquo119904
119905= 120590119876119909119905maxrdquo where
119909119905max denotes rated hoisting weightload and 120583
119876 120590119876denote
the mean and standard deviation of hoisting weightloadrespectively
So it is necessary to get 120583119876 120590119876for obtaining load mean
(119909119905) and standard deviation (119904
119905) Some practical data of cranes
are collected in order to fit the normal distribution and solute120583119876and 120590
119876 119875119876119894 119875119876max 119862119894 and 119862119879 are recorded 120583
119876can be
calculated by the summation of the product of loading valuemultiplied by probability of loading while the probabilityof loading can be approximately calculated by 119862
119894119862119879 which
means the portion of the number of cranes working cyclesunder each typical load to the total number of cranes workingcycles So
120583119876= sum[
119862119894
119862119879
(
119875119876119894
119875119876max)]
120590119876= radicsum
119862119894
119862119879
(
119875119876119894
119875119876max
minus 120583119876)
2
(3)
Mathematical Problems in Engineering 3
Let 120575119876denote the variation coefficient and
120575119876=120590119876
120583119876
(4)
The hoistingload spectrum of overhead crane can reflectthe hoistingload probability distribution under certain statuslevel which provide data supports for further research on thecamber under different loadings
On the other hand the running track of trolley is alsorandom varying from middle to both ends or from end tomiddle or around themiddle In practice the trolley generallygoes through the middle of the girder So for calculationsimplification the operation of trolley per work cycle isregarded as one climbing
3 Probability Load Determination of Crane
Box overhead crane is a typical overhead travelling widelyused at home and abroad with simple design good manu-facturing processes structural stability and other advantagesTake a 5010 t overhead crane used in a practical project as theresearch object (shown in Figure 1)
Parameters of the research object are as follows
(i) rated load is 50 t
(ii) effective length of the girder is 315m
(iii) gauge of trolley is 3580mm
(iv) weight of total girder is 378 t (beam 14833 t and endgirder 4067 t)
(v) weight of trolley is 154 t
(vi) 119867 the spacing between top and bottom plate is1700mm
(vii) 1198611 width of the plate is 650mm
(viii) 1198612 the spacing between webs is 590mm
(ix) 1198791111987912 the thickness of topbottom plate is 24mm
(x) 1198792111987922 the thickness of leftright web is 6mm
(xi) the spacing between big stiffening ribs is 1200mmsim2750mm
(xii) the spacing between small stiffening ribs is400mmsim550mm
(xiii) material is ordinary carbon steel 119876235
The rated load is 50 t that is 119909119905max = 500KN and the
parameters of 119909119905 119904 and confidence level corresponding to the
above four work conditions are as follows
1198761 119909119905= 120583119876119909119905max = 229KN 119904
119905= 120590119876119909119905max = 73KN
1198801= int
500
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
119889119875 = 099904
1198762 119909119905= 119886119909119905max = 3045KN 119904
119905= 119887119909119905max = 56KN
1198802= int
500
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
119889119875 = 09999
1198763 119909119905= 119886119909119905max = 394KN 119904
119905= 119887119909119905max = 345KN
1198803= int
500
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
119889119875 = 0998939
1198764 119909119905= 119886119909119905max = 450KN 119904
119905= 119887119909119905max = 20KN
1198804= int
500
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
119889119875 = 099379
(5)
As the crane load is impossible to be negative and cannotexceed the rated load so all loads should occur within theinterval [0 119875max] the range of whose normal distributionfunction is (minusinfin +infin) According to the checking by theabove formulas it can be found that the probability of loadsappearing in [0 119875max] is greater than 099 So [0 119875max] canbe replaced by (minusinfin +infin) and the load distribution densityfunctions corresponding to 1198761 1198762 1198763 and 1198764 can beexpressed as
1198761 1198911199091199051199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
73radic2120587
119890minus(119875minus229)
22times73
2
= 000546119890minus(119875minus229)
210658
1198762 1198911199091199051199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
56radic2120587
119890minus(119875minus3045)
22times56
2
= 000712119890minus(119875minus3045)
26272
1198763 1198911199091199051199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
345radic2120587
119890minus(119875minus394)
22times345
2
4 Mathematical Problems in Engineering
1 2 3 4
5 6
(1) Top plate(2) Big stiffening ribs(3) Small stiffening ribs
(4) Horizontal angle iron(5) Bottom plate(6) Webs
(a) Structural illustration of girder
H
x
y
T11
T12
T21 T22
B1
B2
(b) Section attributes illustration
Figure 1 Model of 5010 t overhead crane
= 001156119890minus(119875minus394)
223805
1198764 119891119909119905 1199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
20radic2120587
119890minus(119875minus450)
22times20
2
= 001994119890minus(119875minus450)
2800
(6)
The distribution function curves are shown in Figure 2From Figure 2 it can be concluded that with the load
status level improved the hoistingload mean graduallyincreases and the load distribution becomes more concen-trated The loads of 1198764 are distributed around the rated loadwith a certain probability of overloading
Actually some operation parameters of cranes cannotbe obtained for crane design If the load spectrum ofoverhead crane is not available the load distribution cannotbe obtained according to Table 2 But generally the load isconsidered as conformed to normal distribution [0 119875max]Then according to design handbook of cranes 120583
119876and 120590
119876for
hook hoisting of overhead crane can be basically determinedGenerally 120583
119876isin (05-06) and 120590
119876isin (015ndash02) Due to different
work conditions of different cranes with different loadingsthe values of 120583
119876and 120590
119876are different Here let 120583
119876= 055
and 120590119876= 018 by design experts Then according to 119909
119905=
120583119876119909119905max 119904119905 = 120590119876119909119905max and 119909119905max = 500KN 119909119905 = 275KN
119904119905= 90KN
100 200 300 400 500
Prob
abili
ty
0020
0015
0010
0005
Hoisting load (kN)
Q1
Q2
Q3
Q4
Figure 2 Distribution function curve corresponding to119876111987621198763and 1198764
Placing obtained 119909119905and 119904 into formula (2) the density
function of load distribution can be obtained as
119891119909119905 1199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
90radic2120587
119890minus(119875minus275)
22 times 90
2
= 0004434119890minus(119875minus275)
216200
(7)
Thedistribution is illustrated as shown in Figure 3 and theprobability of loads appearing in [0 119875max] is 0992667
Mathematical Problems in Engineering 5
Table 2 Hanging heavy load spectrum of standard recommended by GBT 3811 [10]
Load status levels Nominal load spectrum coefficient 119870119901
119872119876
120590119876
Load spectrum coefficient1198761 119870
119901le 0125 0458 0146 0125
1198762 0125 lt 119870119901le 0250 0609 0112 0250
1198763 0250 lt 119870119901le 0500 0788 0069 0500
1198764 0500 lt 119870119901le 1000 0900 004 09
100 200 300 400 500Hoisting load (kN)
Prob
abili
ty
0004
0003
0002
0001
Q
Figure 3 Distribution function curve corresponding to uncertainload level
S
X a
P
Figure 4 Girder illustration under wheel pressure (119875 denotes theload 119883 denotes the distance between loading position and left end119878 denotes the span length and 120572 denotes the inclination)
4 Matching of Camber with ProbabilityLoadHoisting
As shown in Figure 4 the bridge crane can be considered assimply supported beam and it is necessary to overcome theclimbing resistance when trolley is moving on the bendingtrack The climbing resistance is closely related to the slopewhere trolley moves to To eliminate the climbing resistancethe ideal camber curve should ensure that the slope remainszero (0) for trolley at any position on the girder
That is
120572camber + 120572 = 0 (8)
where 120572camber denotes the inclination with prefabricatedcamber and 120572 denotes the inclination without camber whichcan make sure the slope remains zero (0) for trolley at anyposition on the girder
At certain position 119883 the downwarping inclination 120572equals (simply supported beam classic formula)
E(P
0)
P0 P
30000
20000
10000
0
00
400 400
200 200
Figure 5 Load energy consumptions distribution
120572 = minus1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878)
∵ 119864 = int (119891 times 119904) 119889119904
(9)
Here 119891 is the sum of 119875 and the weight of trolley andis (119875 + 157) 119904 denotes the vertical displace and 119904 = 119889119909 timestan120572 The energy consumed equals the work done by thevertical load acting on the girder to generate a displacement(downwarping)
So when the camber designed consumes minimumenergy under the load of 119875
0 if loading is 119875 then trolley
running from the middle of girder to the end position willmake useless power work done as much as
1198641198750
119875= int
1198712
0
(119875 + 157)100381610038161003816100381610038161199051198921205721198750minus 119905119892120572119875
10038161003816100381610038161003816119889119909 (10)
where 1205721198750
denotes the inclination at certain position 119883 onthe prefabricated camber which is designed to consumeminimum energy under load of 119875
0
If 120572 is very small then tan120572 asymp 120572 and the energyconsumptions loading probability load 119875 according to theload curve with design load 119875
0and with considerations of the
weight of trolley (157 KN) can be expressed as
1198641198750
119875= int
1198712
0
(119875 + 157)100381610038161003816100381610038161205721198750minus 120572119875
10038161003816100381610038161003816119889119909 (11)
The energy consumptions distribution is shown inFigure 5
Place formula (9) into formula (11) then
1198641198750 = int
1198712
0
(119875 + 157)1198782
6119864119868sdot119909
119878(1 minus
41199092
1198782)10038161003816100381610038161198750 minus 119875
1003816100381610038161003816 119889119909 (12)
6 Mathematical Problems in Engineering
After solution of the differential equations energy con-sumptions can be obtained
1198641198750 = (119875 + 157)
10038161003816100381610038161198750 minus 1198751003816100381610038161003816
1198783
48119864119868 (13)
Take a 5010 t overhead crane for example and input thevalue of 119878 119864 and 119868 then
1198641198750 = 011 (119875 + 157)
10038161003816100381610038161198750 minus 1198751003816100381610038161003816
(14)
From Figure 5 it can be found that the minimum energyconsumptions occur when 119875 approaches 119875
0and the energy
consumptions are less when 119875 lt 1198750than when 119875 gt 119875
0
because of heavier loading So heavier load can be consideredfor camber design to guarantee less energy consumptionswhen overloading
With combinations of each load distribution densityfunction 1198641198750 which denotes the energy consumptions of thecrane whose camber is based on 119875
0and under loadings of
normal distribution can be calculated as
1198641198750 = int
119875max
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157)1003816100381610038161003816119875 minus 1198750
1003816100381610038161003816
1198783
48119864119868119889119875
(15)
Remove the absolute value and expand formula (15) then
1198641198750 =
1
119904119905radic2120587
1198783
48119864119868
times (int
1198750
0
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157) (1198750 minus 119875) 119889119875
+int
119875max
1198750
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157) (119875 minus 1198750) 119889119875)
(16)
Then the energy consumptions under 1198761 1198762 1198763and 1198764 can be obtained as well as energy consumptionsdistribution shown in Figures 6ndash9
1198761
1198641198750 = 00006 sdot (int
1198750
0
119890minus(119875minus229)
210658
sdot (119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus229)
210658
sdot (119875 + 157) (119875 minus 1198750) 119889119875)
= minus417 + 1804119890(00430minus000009381198750)1198750
minus 000521198750+ (102915 minus 42385119875
0)
times Erf [222 minus 0009681198750]
1198762
1198641198750 = 000078 sdot (int
1198750
0
119890minus(119875minus3045)
26272(119875 + 157) (1198750 minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus3045)
26272
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus927 + 0000896119890(00968minus00001591198750)1198750 + 0018119875
0
+ (157522 minus 50601198750)Erf [38396 minus 00126119875
0]
1198763
1198641198750 = 000127 (int
1198750
0
119890minus(119875minus394)
223805
(119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus394)
223805
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus40433 + 754 times 10minus26119886(033minus0000421198750)1198750 + 00793119875
0
+ (2399677 minus 60541198750)Erf [808 minus 0020119875
0]
1198764
1198641198750 = 000219 (int
1198750
0
119890minus(119875minus450)
2800(119875 + 157) (119875
0minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus450)
2800
Mathematical Problems in Engineering 7
2000
4000
6000
8000
10000
100 200 300 400 500P0
242661 241986
E(P
0)
Figure 6 Energy consumptions distribution with probability load1198750under 1198761
5000
10000
15000
100 200 300 400 500P0
311256 224365
E(P
0)
Figure 7 Energy consumptions distribution with probability load1198750under 1198762
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus229103 + 125 times 10minus107119890(1125minus0001251198750)1198750 + 045119875
0
+ (300331 minus 66641198750)Erf [1591 minus 0035119875
0]
(17)
From Figure 6 it can be found that when 119888 = 0 that iswithout camber energy consumptions are greatly increasedWith 119875
0acting on the girder and increasing 1198641198750 decreases
When1198750approaches 242661 KN theminimumof1198641198750 occurs
as 241986 J If 1198750continues to increase 1198641198750 will increase
Similarly from Figure 7 it can be found that when1198750approaches 311256KN the minimum of 1198641198750 occurs as
224365 JSimilarly from Figure 8 it can be found that when
1198750approaches 3963 KN the minimum of 1198641198750 occurs as
165447 JSimilarly from Figure 9 it can be found that when
1198750approaches 450489KN the minimum of 1198641198750 occurs as
103759 J
5000
10000
15000
20000
100 200 300 400 500P0
3963 165447
E(P
0)
Figure 8 Energy consumptions distribution with probability load1198750under 1198763
5000
10000
15000
20000
25000
30000
100 200 300 400 500P0
450489 103759
E(P
0)
Figure 9 Energy consumptions distribution with probability load1198750under 1198764
Table 3 Ideal load 1198750corresponding to different load status levels
Load status level 1198750
1198761 242KN1198762 311 KN1198763 396KN1198764 450KN
Through the above analysis the ideal load1198750correspond-
ing to different load status levels can be initially determinedas shown in Table 3
When the load spectrum coefficient is unknown theload can be considered to be conformed to [0 119875max] nor-mal distribution Then the energy consumptions under the
8 Mathematical Problems in Engineering
5000
10000
15000
20000
200 400 600 800 1000P0
39657 340102
E(P
0)
Figure 10 Energy consumptions distribution with probability load1198750under 119876
uncertain load status level119876 can be obtained as well as energyconsumptions distribution shown in Figure 10
119876
1198641198750 = 000049 sdot (int
1198750
0
119890minus(119875minus275)
216200
(119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus275)
216200
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus775587 + 274307119890(0034minus00000621198750)1198750 + 232119875
0
+ (122226 minus 41621198750)Erf (2166 minus 0006119875
0)
(18)
FromFigure 10 energy consumption distribution law canbe found when load spectrum coefficient is uncertain When1198750= 0 that is without camber energy consumptions are
greatly much With 1198750acting on the girder and increasing
1198641198750 decreasesWhen119875
0approaches 39657 KN theminimum
of 1198641198750 occurs as 340102 J If 1198750continues to increase 1198641198750 will
increaseDifferent 119875
0determines the corresponding reasonable
camber to obtain the least energy consumption for climbingAccording to formula (9) and 120572camber + 120572 = 0 then
120572camber = minus120572 =1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878) (19)
Because 120572camber is very small
119889119910camber119889119909
= 119905119892120572camber asymp 120572camber (20)
119889119910camber = [1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878)]119889119909
(21)
After integral solution of formula (21) the camber curvecan be expressed as
119910camber =1198751198783
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(22)
Combined with the weight of the trolley 119875trolley thecamber curve can be obtained and expressed as
119910camber =(1198750+ 119875trolley) 119878
3
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(23)
5 Conclusions
Just as expressed in formula (23) the camber curve can bedesigned However in previous researches on camber design1198750is designed as fixed or rated loads In fact the hoisting
weight of overhead crane is not fixed and energy consump-tions under certain camber with different hoistingload willvary Camber design based on the fixed or rated loadmay notget the optimal energy-savingThis research seeks to get ideal1198750for trolley moving with probabilistic loads to obtain the
least energy consumption Then according to formula (23)energy-saving camber design of overhead travelling cranecan be obtained Besides some conclusions can be drawn asbelow
(1) With the load status level improved the hoistingloadmean gradually increases and the load distributionbecomes more concentrated The loads of 1198764 aredistributed around the rated load with a certainprobability of overloading
(2) The minimum energy consumptions occur when 119875approaches 119875
0and the energy consumptions are less
when 119875 lt 1198750than when 119875 gt 119875
0because of
heavier loading So heavier load can be considered forcamber design to guarantee less energy consumptionswhen overloading
The research results of this paper can also provide a greatreference value to cutting girder web along a certain curve forprefabrication
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was financially supported by National Founda-tion of General Administration of Quality Supervision andInspection (2012QK178) Program of Science Foundation ofGeneral Administration of Quality Supervision and Inspec-tion of Jiangsu Province (KJ103708) and ldquoexcellence plans-zijin starrdquo Foundation of Nanjing University of Science Alsothe work is preresearch of theNational Natural Science Foun-dation of China (research on energy consumption modellingand methodology of energy-saving design for cranes) Thesupports are gratefully acknowledged
Mathematical Problems in Engineering 9
References
[1] P R Patel and V K Patel ldquoA review on structural analysisof overhead crane girder using FEA techniquerdquo InternationalJournal of Engineering Science and Innovative Technology vol 2no 4 pp 41ndash44 2013
[2] T Yifei Y Wei Y Zhen L Dongbo and L XiangdongldquoResearch on multidisciplinary optimization design of bridgecranerdquoMathematical Problems in Engineering vol 2013 ArticleID 763545 10 pages 2013
[3] S G Lee and N Q Hoang ldquoEnergy-based approach forcontroller design of overhead cranes a comparative studyrdquoApplied Mechanics and Materials vol 365-366 pp 784ndash7872010
[4] C Li-Feng ldquoMethods and analysis of bridge camber anddeformation of main girder in gantry cranerdquo Equipment Manu-facturing Technology no 5 pp 124ndash125 2010
[5] C R Bradlee ldquoMethod of measuring camberrdquo Us Patent4794773 1989
[6] F Rongbo ldquoCamber amp pre arch curve of cranesrdquo Hoisting andConveying Machinery vol 2 pp 16ndash18 1990
[7] G Shen X Li D Li and C Zhou ldquoReaseach on energy-savingdesign of arch curve of bridge crane girderrdquo in Proceedings ofthe International Conference on Remote Sensing Environmentand Transportation Engineering (RSETE rsquo11) pp 1448ndash1450Nanjing China June 2011
[8] T Jiantao Load Statistical Analisys and Reliable IndicatorResearch of Bridge Crane Taiyuan University of Science andTechnology Taiyuan China 2011
[9] S Zivanovic R P Johnson H V Dang and J Dobric ldquoDesignand construction of a very lively bridgerdquo in Topics in Dynamicsof Civil Structures vol 4 pp 371ndash380 2013
[10] X Gelin GBT3811-2008 Design Rules for Cranes ChineseStandard Press Beijing China 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Let 120575119876denote the variation coefficient and
120575119876=120590119876
120583119876
(4)
The hoistingload spectrum of overhead crane can reflectthe hoistingload probability distribution under certain statuslevel which provide data supports for further research on thecamber under different loadings
On the other hand the running track of trolley is alsorandom varying from middle to both ends or from end tomiddle or around themiddle In practice the trolley generallygoes through the middle of the girder So for calculationsimplification the operation of trolley per work cycle isregarded as one climbing
3 Probability Load Determination of Crane
Box overhead crane is a typical overhead travelling widelyused at home and abroad with simple design good manu-facturing processes structural stability and other advantagesTake a 5010 t overhead crane used in a practical project as theresearch object (shown in Figure 1)
Parameters of the research object are as follows
(i) rated load is 50 t
(ii) effective length of the girder is 315m
(iii) gauge of trolley is 3580mm
(iv) weight of total girder is 378 t (beam 14833 t and endgirder 4067 t)
(v) weight of trolley is 154 t
(vi) 119867 the spacing between top and bottom plate is1700mm
(vii) 1198611 width of the plate is 650mm
(viii) 1198612 the spacing between webs is 590mm
(ix) 1198791111987912 the thickness of topbottom plate is 24mm
(x) 1198792111987922 the thickness of leftright web is 6mm
(xi) the spacing between big stiffening ribs is 1200mmsim2750mm
(xii) the spacing between small stiffening ribs is400mmsim550mm
(xiii) material is ordinary carbon steel 119876235
The rated load is 50 t that is 119909119905max = 500KN and the
parameters of 119909119905 119904 and confidence level corresponding to the
above four work conditions are as follows
1198761 119909119905= 120583119876119909119905max = 229KN 119904
119905= 120590119876119909119905max = 73KN
1198801= int
500
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
119889119875 = 099904
1198762 119909119905= 119886119909119905max = 3045KN 119904
119905= 119887119909119905max = 56KN
1198802= int
500
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
119889119875 = 09999
1198763 119909119905= 119886119909119905max = 394KN 119904
119905= 119887119909119905max = 345KN
1198803= int
500
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
119889119875 = 0998939
1198764 119909119905= 119886119909119905max = 450KN 119904
119905= 119887119909119905max = 20KN
1198804= int
500
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
119889119875 = 099379
(5)
As the crane load is impossible to be negative and cannotexceed the rated load so all loads should occur within theinterval [0 119875max] the range of whose normal distributionfunction is (minusinfin +infin) According to the checking by theabove formulas it can be found that the probability of loadsappearing in [0 119875max] is greater than 099 So [0 119875max] canbe replaced by (minusinfin +infin) and the load distribution densityfunctions corresponding to 1198761 1198762 1198763 and 1198764 can beexpressed as
1198761 1198911199091199051199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
73radic2120587
119890minus(119875minus229)
22times73
2
= 000546119890minus(119875minus229)
210658
1198762 1198911199091199051199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
56radic2120587
119890minus(119875minus3045)
22times56
2
= 000712119890minus(119875minus3045)
26272
1198763 1198911199091199051199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
345radic2120587
119890minus(119875minus394)
22times345
2
4 Mathematical Problems in Engineering
1 2 3 4
5 6
(1) Top plate(2) Big stiffening ribs(3) Small stiffening ribs
(4) Horizontal angle iron(5) Bottom plate(6) Webs
(a) Structural illustration of girder
H
x
y
T11
T12
T21 T22
B1
B2
(b) Section attributes illustration
Figure 1 Model of 5010 t overhead crane
= 001156119890minus(119875minus394)
223805
1198764 119891119909119905 1199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
20radic2120587
119890minus(119875minus450)
22times20
2
= 001994119890minus(119875minus450)
2800
(6)
The distribution function curves are shown in Figure 2From Figure 2 it can be concluded that with the load
status level improved the hoistingload mean graduallyincreases and the load distribution becomes more concen-trated The loads of 1198764 are distributed around the rated loadwith a certain probability of overloading
Actually some operation parameters of cranes cannotbe obtained for crane design If the load spectrum ofoverhead crane is not available the load distribution cannotbe obtained according to Table 2 But generally the load isconsidered as conformed to normal distribution [0 119875max]Then according to design handbook of cranes 120583
119876and 120590
119876for
hook hoisting of overhead crane can be basically determinedGenerally 120583
119876isin (05-06) and 120590
119876isin (015ndash02) Due to different
work conditions of different cranes with different loadingsthe values of 120583
119876and 120590
119876are different Here let 120583
119876= 055
and 120590119876= 018 by design experts Then according to 119909
119905=
120583119876119909119905max 119904119905 = 120590119876119909119905max and 119909119905max = 500KN 119909119905 = 275KN
119904119905= 90KN
100 200 300 400 500
Prob
abili
ty
0020
0015
0010
0005
Hoisting load (kN)
Q1
Q2
Q3
Q4
Figure 2 Distribution function curve corresponding to119876111987621198763and 1198764
Placing obtained 119909119905and 119904 into formula (2) the density
function of load distribution can be obtained as
119891119909119905 1199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
90radic2120587
119890minus(119875minus275)
22 times 90
2
= 0004434119890minus(119875minus275)
216200
(7)
Thedistribution is illustrated as shown in Figure 3 and theprobability of loads appearing in [0 119875max] is 0992667
Mathematical Problems in Engineering 5
Table 2 Hanging heavy load spectrum of standard recommended by GBT 3811 [10]
Load status levels Nominal load spectrum coefficient 119870119901
119872119876
120590119876
Load spectrum coefficient1198761 119870
119901le 0125 0458 0146 0125
1198762 0125 lt 119870119901le 0250 0609 0112 0250
1198763 0250 lt 119870119901le 0500 0788 0069 0500
1198764 0500 lt 119870119901le 1000 0900 004 09
100 200 300 400 500Hoisting load (kN)
Prob
abili
ty
0004
0003
0002
0001
Q
Figure 3 Distribution function curve corresponding to uncertainload level
S
X a
P
Figure 4 Girder illustration under wheel pressure (119875 denotes theload 119883 denotes the distance between loading position and left end119878 denotes the span length and 120572 denotes the inclination)
4 Matching of Camber with ProbabilityLoadHoisting
As shown in Figure 4 the bridge crane can be considered assimply supported beam and it is necessary to overcome theclimbing resistance when trolley is moving on the bendingtrack The climbing resistance is closely related to the slopewhere trolley moves to To eliminate the climbing resistancethe ideal camber curve should ensure that the slope remainszero (0) for trolley at any position on the girder
That is
120572camber + 120572 = 0 (8)
where 120572camber denotes the inclination with prefabricatedcamber and 120572 denotes the inclination without camber whichcan make sure the slope remains zero (0) for trolley at anyposition on the girder
At certain position 119883 the downwarping inclination 120572equals (simply supported beam classic formula)
E(P
0)
P0 P
30000
20000
10000
0
00
400 400
200 200
Figure 5 Load energy consumptions distribution
120572 = minus1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878)
∵ 119864 = int (119891 times 119904) 119889119904
(9)
Here 119891 is the sum of 119875 and the weight of trolley andis (119875 + 157) 119904 denotes the vertical displace and 119904 = 119889119909 timestan120572 The energy consumed equals the work done by thevertical load acting on the girder to generate a displacement(downwarping)
So when the camber designed consumes minimumenergy under the load of 119875
0 if loading is 119875 then trolley
running from the middle of girder to the end position willmake useless power work done as much as
1198641198750
119875= int
1198712
0
(119875 + 157)100381610038161003816100381610038161199051198921205721198750minus 119905119892120572119875
10038161003816100381610038161003816119889119909 (10)
where 1205721198750
denotes the inclination at certain position 119883 onthe prefabricated camber which is designed to consumeminimum energy under load of 119875
0
If 120572 is very small then tan120572 asymp 120572 and the energyconsumptions loading probability load 119875 according to theload curve with design load 119875
0and with considerations of the
weight of trolley (157 KN) can be expressed as
1198641198750
119875= int
1198712
0
(119875 + 157)100381610038161003816100381610038161205721198750minus 120572119875
10038161003816100381610038161003816119889119909 (11)
The energy consumptions distribution is shown inFigure 5
Place formula (9) into formula (11) then
1198641198750 = int
1198712
0
(119875 + 157)1198782
6119864119868sdot119909
119878(1 minus
41199092
1198782)10038161003816100381610038161198750 minus 119875
1003816100381610038161003816 119889119909 (12)
6 Mathematical Problems in Engineering
After solution of the differential equations energy con-sumptions can be obtained
1198641198750 = (119875 + 157)
10038161003816100381610038161198750 minus 1198751003816100381610038161003816
1198783
48119864119868 (13)
Take a 5010 t overhead crane for example and input thevalue of 119878 119864 and 119868 then
1198641198750 = 011 (119875 + 157)
10038161003816100381610038161198750 minus 1198751003816100381610038161003816
(14)
From Figure 5 it can be found that the minimum energyconsumptions occur when 119875 approaches 119875
0and the energy
consumptions are less when 119875 lt 1198750than when 119875 gt 119875
0
because of heavier loading So heavier load can be consideredfor camber design to guarantee less energy consumptionswhen overloading
With combinations of each load distribution densityfunction 1198641198750 which denotes the energy consumptions of thecrane whose camber is based on 119875
0and under loadings of
normal distribution can be calculated as
1198641198750 = int
119875max
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157)1003816100381610038161003816119875 minus 1198750
1003816100381610038161003816
1198783
48119864119868119889119875
(15)
Remove the absolute value and expand formula (15) then
1198641198750 =
1
119904119905radic2120587
1198783
48119864119868
times (int
1198750
0
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157) (1198750 minus 119875) 119889119875
+int
119875max
1198750
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157) (119875 minus 1198750) 119889119875)
(16)
Then the energy consumptions under 1198761 1198762 1198763and 1198764 can be obtained as well as energy consumptionsdistribution shown in Figures 6ndash9
1198761
1198641198750 = 00006 sdot (int
1198750
0
119890minus(119875minus229)
210658
sdot (119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus229)
210658
sdot (119875 + 157) (119875 minus 1198750) 119889119875)
= minus417 + 1804119890(00430minus000009381198750)1198750
minus 000521198750+ (102915 minus 42385119875
0)
times Erf [222 minus 0009681198750]
1198762
1198641198750 = 000078 sdot (int
1198750
0
119890minus(119875minus3045)
26272(119875 + 157) (1198750 minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus3045)
26272
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus927 + 0000896119890(00968minus00001591198750)1198750 + 0018119875
0
+ (157522 minus 50601198750)Erf [38396 minus 00126119875
0]
1198763
1198641198750 = 000127 (int
1198750
0
119890minus(119875minus394)
223805
(119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus394)
223805
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus40433 + 754 times 10minus26119886(033minus0000421198750)1198750 + 00793119875
0
+ (2399677 minus 60541198750)Erf [808 minus 0020119875
0]
1198764
1198641198750 = 000219 (int
1198750
0
119890minus(119875minus450)
2800(119875 + 157) (119875
0minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus450)
2800
Mathematical Problems in Engineering 7
2000
4000
6000
8000
10000
100 200 300 400 500P0
242661 241986
E(P
0)
Figure 6 Energy consumptions distribution with probability load1198750under 1198761
5000
10000
15000
100 200 300 400 500P0
311256 224365
E(P
0)
Figure 7 Energy consumptions distribution with probability load1198750under 1198762
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus229103 + 125 times 10minus107119890(1125minus0001251198750)1198750 + 045119875
0
+ (300331 minus 66641198750)Erf [1591 minus 0035119875
0]
(17)
From Figure 6 it can be found that when 119888 = 0 that iswithout camber energy consumptions are greatly increasedWith 119875
0acting on the girder and increasing 1198641198750 decreases
When1198750approaches 242661 KN theminimumof1198641198750 occurs
as 241986 J If 1198750continues to increase 1198641198750 will increase
Similarly from Figure 7 it can be found that when1198750approaches 311256KN the minimum of 1198641198750 occurs as
224365 JSimilarly from Figure 8 it can be found that when
1198750approaches 3963 KN the minimum of 1198641198750 occurs as
165447 JSimilarly from Figure 9 it can be found that when
1198750approaches 450489KN the minimum of 1198641198750 occurs as
103759 J
5000
10000
15000
20000
100 200 300 400 500P0
3963 165447
E(P
0)
Figure 8 Energy consumptions distribution with probability load1198750under 1198763
5000
10000
15000
20000
25000
30000
100 200 300 400 500P0
450489 103759
E(P
0)
Figure 9 Energy consumptions distribution with probability load1198750under 1198764
Table 3 Ideal load 1198750corresponding to different load status levels
Load status level 1198750
1198761 242KN1198762 311 KN1198763 396KN1198764 450KN
Through the above analysis the ideal load1198750correspond-
ing to different load status levels can be initially determinedas shown in Table 3
When the load spectrum coefficient is unknown theload can be considered to be conformed to [0 119875max] nor-mal distribution Then the energy consumptions under the
8 Mathematical Problems in Engineering
5000
10000
15000
20000
200 400 600 800 1000P0
39657 340102
E(P
0)
Figure 10 Energy consumptions distribution with probability load1198750under 119876
uncertain load status level119876 can be obtained as well as energyconsumptions distribution shown in Figure 10
119876
1198641198750 = 000049 sdot (int
1198750
0
119890minus(119875minus275)
216200
(119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus275)
216200
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus775587 + 274307119890(0034minus00000621198750)1198750 + 232119875
0
+ (122226 minus 41621198750)Erf (2166 minus 0006119875
0)
(18)
FromFigure 10 energy consumption distribution law canbe found when load spectrum coefficient is uncertain When1198750= 0 that is without camber energy consumptions are
greatly much With 1198750acting on the girder and increasing
1198641198750 decreasesWhen119875
0approaches 39657 KN theminimum
of 1198641198750 occurs as 340102 J If 1198750continues to increase 1198641198750 will
increaseDifferent 119875
0determines the corresponding reasonable
camber to obtain the least energy consumption for climbingAccording to formula (9) and 120572camber + 120572 = 0 then
120572camber = minus120572 =1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878) (19)
Because 120572camber is very small
119889119910camber119889119909
= 119905119892120572camber asymp 120572camber (20)
119889119910camber = [1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878)]119889119909
(21)
After integral solution of formula (21) the camber curvecan be expressed as
119910camber =1198751198783
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(22)
Combined with the weight of the trolley 119875trolley thecamber curve can be obtained and expressed as
119910camber =(1198750+ 119875trolley) 119878
3
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(23)
5 Conclusions
Just as expressed in formula (23) the camber curve can bedesigned However in previous researches on camber design1198750is designed as fixed or rated loads In fact the hoisting
weight of overhead crane is not fixed and energy consump-tions under certain camber with different hoistingload willvary Camber design based on the fixed or rated loadmay notget the optimal energy-savingThis research seeks to get ideal1198750for trolley moving with probabilistic loads to obtain the
least energy consumption Then according to formula (23)energy-saving camber design of overhead travelling cranecan be obtained Besides some conclusions can be drawn asbelow
(1) With the load status level improved the hoistingloadmean gradually increases and the load distributionbecomes more concentrated The loads of 1198764 aredistributed around the rated load with a certainprobability of overloading
(2) The minimum energy consumptions occur when 119875approaches 119875
0and the energy consumptions are less
when 119875 lt 1198750than when 119875 gt 119875
0because of
heavier loading So heavier load can be considered forcamber design to guarantee less energy consumptionswhen overloading
The research results of this paper can also provide a greatreference value to cutting girder web along a certain curve forprefabrication
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was financially supported by National Founda-tion of General Administration of Quality Supervision andInspection (2012QK178) Program of Science Foundation ofGeneral Administration of Quality Supervision and Inspec-tion of Jiangsu Province (KJ103708) and ldquoexcellence plans-zijin starrdquo Foundation of Nanjing University of Science Alsothe work is preresearch of theNational Natural Science Foun-dation of China (research on energy consumption modellingand methodology of energy-saving design for cranes) Thesupports are gratefully acknowledged
Mathematical Problems in Engineering 9
References
[1] P R Patel and V K Patel ldquoA review on structural analysisof overhead crane girder using FEA techniquerdquo InternationalJournal of Engineering Science and Innovative Technology vol 2no 4 pp 41ndash44 2013
[2] T Yifei Y Wei Y Zhen L Dongbo and L XiangdongldquoResearch on multidisciplinary optimization design of bridgecranerdquoMathematical Problems in Engineering vol 2013 ArticleID 763545 10 pages 2013
[3] S G Lee and N Q Hoang ldquoEnergy-based approach forcontroller design of overhead cranes a comparative studyrdquoApplied Mechanics and Materials vol 365-366 pp 784ndash7872010
[4] C Li-Feng ldquoMethods and analysis of bridge camber anddeformation of main girder in gantry cranerdquo Equipment Manu-facturing Technology no 5 pp 124ndash125 2010
[5] C R Bradlee ldquoMethod of measuring camberrdquo Us Patent4794773 1989
[6] F Rongbo ldquoCamber amp pre arch curve of cranesrdquo Hoisting andConveying Machinery vol 2 pp 16ndash18 1990
[7] G Shen X Li D Li and C Zhou ldquoReaseach on energy-savingdesign of arch curve of bridge crane girderrdquo in Proceedings ofthe International Conference on Remote Sensing Environmentand Transportation Engineering (RSETE rsquo11) pp 1448ndash1450Nanjing China June 2011
[8] T Jiantao Load Statistical Analisys and Reliable IndicatorResearch of Bridge Crane Taiyuan University of Science andTechnology Taiyuan China 2011
[9] S Zivanovic R P Johnson H V Dang and J Dobric ldquoDesignand construction of a very lively bridgerdquo in Topics in Dynamicsof Civil Structures vol 4 pp 371ndash380 2013
[10] X Gelin GBT3811-2008 Design Rules for Cranes ChineseStandard Press Beijing China 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
1 2 3 4
5 6
(1) Top plate(2) Big stiffening ribs(3) Small stiffening ribs
(4) Horizontal angle iron(5) Bottom plate(6) Webs
(a) Structural illustration of girder
H
x
y
T11
T12
T21 T22
B1
B2
(b) Section attributes illustration
Figure 1 Model of 5010 t overhead crane
= 001156119890minus(119875minus394)
223805
1198764 119891119909119905 1199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
20radic2120587
119890minus(119875minus450)
22times20
2
= 001994119890minus(119875minus450)
2800
(6)
The distribution function curves are shown in Figure 2From Figure 2 it can be concluded that with the load
status level improved the hoistingload mean graduallyincreases and the load distribution becomes more concen-trated The loads of 1198764 are distributed around the rated loadwith a certain probability of overloading
Actually some operation parameters of cranes cannotbe obtained for crane design If the load spectrum ofoverhead crane is not available the load distribution cannotbe obtained according to Table 2 But generally the load isconsidered as conformed to normal distribution [0 119875max]Then according to design handbook of cranes 120583
119876and 120590
119876for
hook hoisting of overhead crane can be basically determinedGenerally 120583
119876isin (05-06) and 120590
119876isin (015ndash02) Due to different
work conditions of different cranes with different loadingsthe values of 120583
119876and 120590
119876are different Here let 120583
119876= 055
and 120590119876= 018 by design experts Then according to 119909
119905=
120583119876119909119905max 119904119905 = 120590119876119909119905max and 119909119905max = 500KN 119909119905 = 275KN
119904119905= 90KN
100 200 300 400 500
Prob
abili
ty
0020
0015
0010
0005
Hoisting load (kN)
Q1
Q2
Q3
Q4
Figure 2 Distribution function curve corresponding to119876111987621198763and 1198764
Placing obtained 119909119905and 119904 into formula (2) the density
function of load distribution can be obtained as
119891119909119905 1199041199052(119875)=1
119904119905radic2120587
119890minus(119875minus119909119905)
221199041199052
=1
90radic2120587
119890minus(119875minus275)
22 times 90
2
= 0004434119890minus(119875minus275)
216200
(7)
Thedistribution is illustrated as shown in Figure 3 and theprobability of loads appearing in [0 119875max] is 0992667
Mathematical Problems in Engineering 5
Table 2 Hanging heavy load spectrum of standard recommended by GBT 3811 [10]
Load status levels Nominal load spectrum coefficient 119870119901
119872119876
120590119876
Load spectrum coefficient1198761 119870
119901le 0125 0458 0146 0125
1198762 0125 lt 119870119901le 0250 0609 0112 0250
1198763 0250 lt 119870119901le 0500 0788 0069 0500
1198764 0500 lt 119870119901le 1000 0900 004 09
100 200 300 400 500Hoisting load (kN)
Prob
abili
ty
0004
0003
0002
0001
Q
Figure 3 Distribution function curve corresponding to uncertainload level
S
X a
P
Figure 4 Girder illustration under wheel pressure (119875 denotes theload 119883 denotes the distance between loading position and left end119878 denotes the span length and 120572 denotes the inclination)
4 Matching of Camber with ProbabilityLoadHoisting
As shown in Figure 4 the bridge crane can be considered assimply supported beam and it is necessary to overcome theclimbing resistance when trolley is moving on the bendingtrack The climbing resistance is closely related to the slopewhere trolley moves to To eliminate the climbing resistancethe ideal camber curve should ensure that the slope remainszero (0) for trolley at any position on the girder
That is
120572camber + 120572 = 0 (8)
where 120572camber denotes the inclination with prefabricatedcamber and 120572 denotes the inclination without camber whichcan make sure the slope remains zero (0) for trolley at anyposition on the girder
At certain position 119883 the downwarping inclination 120572equals (simply supported beam classic formula)
E(P
0)
P0 P
30000
20000
10000
0
00
400 400
200 200
Figure 5 Load energy consumptions distribution
120572 = minus1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878)
∵ 119864 = int (119891 times 119904) 119889119904
(9)
Here 119891 is the sum of 119875 and the weight of trolley andis (119875 + 157) 119904 denotes the vertical displace and 119904 = 119889119909 timestan120572 The energy consumed equals the work done by thevertical load acting on the girder to generate a displacement(downwarping)
So when the camber designed consumes minimumenergy under the load of 119875
0 if loading is 119875 then trolley
running from the middle of girder to the end position willmake useless power work done as much as
1198641198750
119875= int
1198712
0
(119875 + 157)100381610038161003816100381610038161199051198921205721198750minus 119905119892120572119875
10038161003816100381610038161003816119889119909 (10)
where 1205721198750
denotes the inclination at certain position 119883 onthe prefabricated camber which is designed to consumeminimum energy under load of 119875
0
If 120572 is very small then tan120572 asymp 120572 and the energyconsumptions loading probability load 119875 according to theload curve with design load 119875
0and with considerations of the
weight of trolley (157 KN) can be expressed as
1198641198750
119875= int
1198712
0
(119875 + 157)100381610038161003816100381610038161205721198750minus 120572119875
10038161003816100381610038161003816119889119909 (11)
The energy consumptions distribution is shown inFigure 5
Place formula (9) into formula (11) then
1198641198750 = int
1198712
0
(119875 + 157)1198782
6119864119868sdot119909
119878(1 minus
41199092
1198782)10038161003816100381610038161198750 minus 119875
1003816100381610038161003816 119889119909 (12)
6 Mathematical Problems in Engineering
After solution of the differential equations energy con-sumptions can be obtained
1198641198750 = (119875 + 157)
10038161003816100381610038161198750 minus 1198751003816100381610038161003816
1198783
48119864119868 (13)
Take a 5010 t overhead crane for example and input thevalue of 119878 119864 and 119868 then
1198641198750 = 011 (119875 + 157)
10038161003816100381610038161198750 minus 1198751003816100381610038161003816
(14)
From Figure 5 it can be found that the minimum energyconsumptions occur when 119875 approaches 119875
0and the energy
consumptions are less when 119875 lt 1198750than when 119875 gt 119875
0
because of heavier loading So heavier load can be consideredfor camber design to guarantee less energy consumptionswhen overloading
With combinations of each load distribution densityfunction 1198641198750 which denotes the energy consumptions of thecrane whose camber is based on 119875
0and under loadings of
normal distribution can be calculated as
1198641198750 = int
119875max
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157)1003816100381610038161003816119875 minus 1198750
1003816100381610038161003816
1198783
48119864119868119889119875
(15)
Remove the absolute value and expand formula (15) then
1198641198750 =
1
119904119905radic2120587
1198783
48119864119868
times (int
1198750
0
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157) (1198750 minus 119875) 119889119875
+int
119875max
1198750
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157) (119875 minus 1198750) 119889119875)
(16)
Then the energy consumptions under 1198761 1198762 1198763and 1198764 can be obtained as well as energy consumptionsdistribution shown in Figures 6ndash9
1198761
1198641198750 = 00006 sdot (int
1198750
0
119890minus(119875minus229)
210658
sdot (119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus229)
210658
sdot (119875 + 157) (119875 minus 1198750) 119889119875)
= minus417 + 1804119890(00430minus000009381198750)1198750
minus 000521198750+ (102915 minus 42385119875
0)
times Erf [222 minus 0009681198750]
1198762
1198641198750 = 000078 sdot (int
1198750
0
119890minus(119875minus3045)
26272(119875 + 157) (1198750 minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus3045)
26272
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus927 + 0000896119890(00968minus00001591198750)1198750 + 0018119875
0
+ (157522 minus 50601198750)Erf [38396 minus 00126119875
0]
1198763
1198641198750 = 000127 (int
1198750
0
119890minus(119875minus394)
223805
(119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus394)
223805
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus40433 + 754 times 10minus26119886(033minus0000421198750)1198750 + 00793119875
0
+ (2399677 minus 60541198750)Erf [808 minus 0020119875
0]
1198764
1198641198750 = 000219 (int
1198750
0
119890minus(119875minus450)
2800(119875 + 157) (119875
0minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus450)
2800
Mathematical Problems in Engineering 7
2000
4000
6000
8000
10000
100 200 300 400 500P0
242661 241986
E(P
0)
Figure 6 Energy consumptions distribution with probability load1198750under 1198761
5000
10000
15000
100 200 300 400 500P0
311256 224365
E(P
0)
Figure 7 Energy consumptions distribution with probability load1198750under 1198762
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus229103 + 125 times 10minus107119890(1125minus0001251198750)1198750 + 045119875
0
+ (300331 minus 66641198750)Erf [1591 minus 0035119875
0]
(17)
From Figure 6 it can be found that when 119888 = 0 that iswithout camber energy consumptions are greatly increasedWith 119875
0acting on the girder and increasing 1198641198750 decreases
When1198750approaches 242661 KN theminimumof1198641198750 occurs
as 241986 J If 1198750continues to increase 1198641198750 will increase
Similarly from Figure 7 it can be found that when1198750approaches 311256KN the minimum of 1198641198750 occurs as
224365 JSimilarly from Figure 8 it can be found that when
1198750approaches 3963 KN the minimum of 1198641198750 occurs as
165447 JSimilarly from Figure 9 it can be found that when
1198750approaches 450489KN the minimum of 1198641198750 occurs as
103759 J
5000
10000
15000
20000
100 200 300 400 500P0
3963 165447
E(P
0)
Figure 8 Energy consumptions distribution with probability load1198750under 1198763
5000
10000
15000
20000
25000
30000
100 200 300 400 500P0
450489 103759
E(P
0)
Figure 9 Energy consumptions distribution with probability load1198750under 1198764
Table 3 Ideal load 1198750corresponding to different load status levels
Load status level 1198750
1198761 242KN1198762 311 KN1198763 396KN1198764 450KN
Through the above analysis the ideal load1198750correspond-
ing to different load status levels can be initially determinedas shown in Table 3
When the load spectrum coefficient is unknown theload can be considered to be conformed to [0 119875max] nor-mal distribution Then the energy consumptions under the
8 Mathematical Problems in Engineering
5000
10000
15000
20000
200 400 600 800 1000P0
39657 340102
E(P
0)
Figure 10 Energy consumptions distribution with probability load1198750under 119876
uncertain load status level119876 can be obtained as well as energyconsumptions distribution shown in Figure 10
119876
1198641198750 = 000049 sdot (int
1198750
0
119890minus(119875minus275)
216200
(119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus275)
216200
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus775587 + 274307119890(0034minus00000621198750)1198750 + 232119875
0
+ (122226 minus 41621198750)Erf (2166 minus 0006119875
0)
(18)
FromFigure 10 energy consumption distribution law canbe found when load spectrum coefficient is uncertain When1198750= 0 that is without camber energy consumptions are
greatly much With 1198750acting on the girder and increasing
1198641198750 decreasesWhen119875
0approaches 39657 KN theminimum
of 1198641198750 occurs as 340102 J If 1198750continues to increase 1198641198750 will
increaseDifferent 119875
0determines the corresponding reasonable
camber to obtain the least energy consumption for climbingAccording to formula (9) and 120572camber + 120572 = 0 then
120572camber = minus120572 =1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878) (19)
Because 120572camber is very small
119889119910camber119889119909
= 119905119892120572camber asymp 120572camber (20)
119889119910camber = [1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878)]119889119909
(21)
After integral solution of formula (21) the camber curvecan be expressed as
119910camber =1198751198783
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(22)
Combined with the weight of the trolley 119875trolley thecamber curve can be obtained and expressed as
119910camber =(1198750+ 119875trolley) 119878
3
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(23)
5 Conclusions
Just as expressed in formula (23) the camber curve can bedesigned However in previous researches on camber design1198750is designed as fixed or rated loads In fact the hoisting
weight of overhead crane is not fixed and energy consump-tions under certain camber with different hoistingload willvary Camber design based on the fixed or rated loadmay notget the optimal energy-savingThis research seeks to get ideal1198750for trolley moving with probabilistic loads to obtain the
least energy consumption Then according to formula (23)energy-saving camber design of overhead travelling cranecan be obtained Besides some conclusions can be drawn asbelow
(1) With the load status level improved the hoistingloadmean gradually increases and the load distributionbecomes more concentrated The loads of 1198764 aredistributed around the rated load with a certainprobability of overloading
(2) The minimum energy consumptions occur when 119875approaches 119875
0and the energy consumptions are less
when 119875 lt 1198750than when 119875 gt 119875
0because of
heavier loading So heavier load can be considered forcamber design to guarantee less energy consumptionswhen overloading
The research results of this paper can also provide a greatreference value to cutting girder web along a certain curve forprefabrication
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was financially supported by National Founda-tion of General Administration of Quality Supervision andInspection (2012QK178) Program of Science Foundation ofGeneral Administration of Quality Supervision and Inspec-tion of Jiangsu Province (KJ103708) and ldquoexcellence plans-zijin starrdquo Foundation of Nanjing University of Science Alsothe work is preresearch of theNational Natural Science Foun-dation of China (research on energy consumption modellingand methodology of energy-saving design for cranes) Thesupports are gratefully acknowledged
Mathematical Problems in Engineering 9
References
[1] P R Patel and V K Patel ldquoA review on structural analysisof overhead crane girder using FEA techniquerdquo InternationalJournal of Engineering Science and Innovative Technology vol 2no 4 pp 41ndash44 2013
[2] T Yifei Y Wei Y Zhen L Dongbo and L XiangdongldquoResearch on multidisciplinary optimization design of bridgecranerdquoMathematical Problems in Engineering vol 2013 ArticleID 763545 10 pages 2013
[3] S G Lee and N Q Hoang ldquoEnergy-based approach forcontroller design of overhead cranes a comparative studyrdquoApplied Mechanics and Materials vol 365-366 pp 784ndash7872010
[4] C Li-Feng ldquoMethods and analysis of bridge camber anddeformation of main girder in gantry cranerdquo Equipment Manu-facturing Technology no 5 pp 124ndash125 2010
[5] C R Bradlee ldquoMethod of measuring camberrdquo Us Patent4794773 1989
[6] F Rongbo ldquoCamber amp pre arch curve of cranesrdquo Hoisting andConveying Machinery vol 2 pp 16ndash18 1990
[7] G Shen X Li D Li and C Zhou ldquoReaseach on energy-savingdesign of arch curve of bridge crane girderrdquo in Proceedings ofthe International Conference on Remote Sensing Environmentand Transportation Engineering (RSETE rsquo11) pp 1448ndash1450Nanjing China June 2011
[8] T Jiantao Load Statistical Analisys and Reliable IndicatorResearch of Bridge Crane Taiyuan University of Science andTechnology Taiyuan China 2011
[9] S Zivanovic R P Johnson H V Dang and J Dobric ldquoDesignand construction of a very lively bridgerdquo in Topics in Dynamicsof Civil Structures vol 4 pp 371ndash380 2013
[10] X Gelin GBT3811-2008 Design Rules for Cranes ChineseStandard Press Beijing China 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Table 2 Hanging heavy load spectrum of standard recommended by GBT 3811 [10]
Load status levels Nominal load spectrum coefficient 119870119901
119872119876
120590119876
Load spectrum coefficient1198761 119870
119901le 0125 0458 0146 0125
1198762 0125 lt 119870119901le 0250 0609 0112 0250
1198763 0250 lt 119870119901le 0500 0788 0069 0500
1198764 0500 lt 119870119901le 1000 0900 004 09
100 200 300 400 500Hoisting load (kN)
Prob
abili
ty
0004
0003
0002
0001
Q
Figure 3 Distribution function curve corresponding to uncertainload level
S
X a
P
Figure 4 Girder illustration under wheel pressure (119875 denotes theload 119883 denotes the distance between loading position and left end119878 denotes the span length and 120572 denotes the inclination)
4 Matching of Camber with ProbabilityLoadHoisting
As shown in Figure 4 the bridge crane can be considered assimply supported beam and it is necessary to overcome theclimbing resistance when trolley is moving on the bendingtrack The climbing resistance is closely related to the slopewhere trolley moves to To eliminate the climbing resistancethe ideal camber curve should ensure that the slope remainszero (0) for trolley at any position on the girder
That is
120572camber + 120572 = 0 (8)
where 120572camber denotes the inclination with prefabricatedcamber and 120572 denotes the inclination without camber whichcan make sure the slope remains zero (0) for trolley at anyposition on the girder
At certain position 119883 the downwarping inclination 120572equals (simply supported beam classic formula)
E(P
0)
P0 P
30000
20000
10000
0
00
400 400
200 200
Figure 5 Load energy consumptions distribution
120572 = minus1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878)
∵ 119864 = int (119891 times 119904) 119889119904
(9)
Here 119891 is the sum of 119875 and the weight of trolley andis (119875 + 157) 119904 denotes the vertical displace and 119904 = 119889119909 timestan120572 The energy consumed equals the work done by thevertical load acting on the girder to generate a displacement(downwarping)
So when the camber designed consumes minimumenergy under the load of 119875
0 if loading is 119875 then trolley
running from the middle of girder to the end position willmake useless power work done as much as
1198641198750
119875= int
1198712
0
(119875 + 157)100381610038161003816100381610038161199051198921205721198750minus 119905119892120572119875
10038161003816100381610038161003816119889119909 (10)
where 1205721198750
denotes the inclination at certain position 119883 onthe prefabricated camber which is designed to consumeminimum energy under load of 119875
0
If 120572 is very small then tan120572 asymp 120572 and the energyconsumptions loading probability load 119875 according to theload curve with design load 119875
0and with considerations of the
weight of trolley (157 KN) can be expressed as
1198641198750
119875= int
1198712
0
(119875 + 157)100381610038161003816100381610038161205721198750minus 120572119875
10038161003816100381610038161003816119889119909 (11)
The energy consumptions distribution is shown inFigure 5
Place formula (9) into formula (11) then
1198641198750 = int
1198712
0
(119875 + 157)1198782
6119864119868sdot119909
119878(1 minus
41199092
1198782)10038161003816100381610038161198750 minus 119875
1003816100381610038161003816 119889119909 (12)
6 Mathematical Problems in Engineering
After solution of the differential equations energy con-sumptions can be obtained
1198641198750 = (119875 + 157)
10038161003816100381610038161198750 minus 1198751003816100381610038161003816
1198783
48119864119868 (13)
Take a 5010 t overhead crane for example and input thevalue of 119878 119864 and 119868 then
1198641198750 = 011 (119875 + 157)
10038161003816100381610038161198750 minus 1198751003816100381610038161003816
(14)
From Figure 5 it can be found that the minimum energyconsumptions occur when 119875 approaches 119875
0and the energy
consumptions are less when 119875 lt 1198750than when 119875 gt 119875
0
because of heavier loading So heavier load can be consideredfor camber design to guarantee less energy consumptionswhen overloading
With combinations of each load distribution densityfunction 1198641198750 which denotes the energy consumptions of thecrane whose camber is based on 119875
0and under loadings of
normal distribution can be calculated as
1198641198750 = int
119875max
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157)1003816100381610038161003816119875 minus 1198750
1003816100381610038161003816
1198783
48119864119868119889119875
(15)
Remove the absolute value and expand formula (15) then
1198641198750 =
1
119904119905radic2120587
1198783
48119864119868
times (int
1198750
0
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157) (1198750 minus 119875) 119889119875
+int
119875max
1198750
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157) (119875 minus 1198750) 119889119875)
(16)
Then the energy consumptions under 1198761 1198762 1198763and 1198764 can be obtained as well as energy consumptionsdistribution shown in Figures 6ndash9
1198761
1198641198750 = 00006 sdot (int
1198750
0
119890minus(119875minus229)
210658
sdot (119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus229)
210658
sdot (119875 + 157) (119875 minus 1198750) 119889119875)
= minus417 + 1804119890(00430minus000009381198750)1198750
minus 000521198750+ (102915 minus 42385119875
0)
times Erf [222 minus 0009681198750]
1198762
1198641198750 = 000078 sdot (int
1198750
0
119890minus(119875minus3045)
26272(119875 + 157) (1198750 minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus3045)
26272
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus927 + 0000896119890(00968minus00001591198750)1198750 + 0018119875
0
+ (157522 minus 50601198750)Erf [38396 minus 00126119875
0]
1198763
1198641198750 = 000127 (int
1198750
0
119890minus(119875minus394)
223805
(119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus394)
223805
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus40433 + 754 times 10minus26119886(033minus0000421198750)1198750 + 00793119875
0
+ (2399677 minus 60541198750)Erf [808 minus 0020119875
0]
1198764
1198641198750 = 000219 (int
1198750
0
119890minus(119875minus450)
2800(119875 + 157) (119875
0minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus450)
2800
Mathematical Problems in Engineering 7
2000
4000
6000
8000
10000
100 200 300 400 500P0
242661 241986
E(P
0)
Figure 6 Energy consumptions distribution with probability load1198750under 1198761
5000
10000
15000
100 200 300 400 500P0
311256 224365
E(P
0)
Figure 7 Energy consumptions distribution with probability load1198750under 1198762
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus229103 + 125 times 10minus107119890(1125minus0001251198750)1198750 + 045119875
0
+ (300331 minus 66641198750)Erf [1591 minus 0035119875
0]
(17)
From Figure 6 it can be found that when 119888 = 0 that iswithout camber energy consumptions are greatly increasedWith 119875
0acting on the girder and increasing 1198641198750 decreases
When1198750approaches 242661 KN theminimumof1198641198750 occurs
as 241986 J If 1198750continues to increase 1198641198750 will increase
Similarly from Figure 7 it can be found that when1198750approaches 311256KN the minimum of 1198641198750 occurs as
224365 JSimilarly from Figure 8 it can be found that when
1198750approaches 3963 KN the minimum of 1198641198750 occurs as
165447 JSimilarly from Figure 9 it can be found that when
1198750approaches 450489KN the minimum of 1198641198750 occurs as
103759 J
5000
10000
15000
20000
100 200 300 400 500P0
3963 165447
E(P
0)
Figure 8 Energy consumptions distribution with probability load1198750under 1198763
5000
10000
15000
20000
25000
30000
100 200 300 400 500P0
450489 103759
E(P
0)
Figure 9 Energy consumptions distribution with probability load1198750under 1198764
Table 3 Ideal load 1198750corresponding to different load status levels
Load status level 1198750
1198761 242KN1198762 311 KN1198763 396KN1198764 450KN
Through the above analysis the ideal load1198750correspond-
ing to different load status levels can be initially determinedas shown in Table 3
When the load spectrum coefficient is unknown theload can be considered to be conformed to [0 119875max] nor-mal distribution Then the energy consumptions under the
8 Mathematical Problems in Engineering
5000
10000
15000
20000
200 400 600 800 1000P0
39657 340102
E(P
0)
Figure 10 Energy consumptions distribution with probability load1198750under 119876
uncertain load status level119876 can be obtained as well as energyconsumptions distribution shown in Figure 10
119876
1198641198750 = 000049 sdot (int
1198750
0
119890minus(119875minus275)
216200
(119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus275)
216200
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus775587 + 274307119890(0034minus00000621198750)1198750 + 232119875
0
+ (122226 minus 41621198750)Erf (2166 minus 0006119875
0)
(18)
FromFigure 10 energy consumption distribution law canbe found when load spectrum coefficient is uncertain When1198750= 0 that is without camber energy consumptions are
greatly much With 1198750acting on the girder and increasing
1198641198750 decreasesWhen119875
0approaches 39657 KN theminimum
of 1198641198750 occurs as 340102 J If 1198750continues to increase 1198641198750 will
increaseDifferent 119875
0determines the corresponding reasonable
camber to obtain the least energy consumption for climbingAccording to formula (9) and 120572camber + 120572 = 0 then
120572camber = minus120572 =1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878) (19)
Because 120572camber is very small
119889119910camber119889119909
= 119905119892120572camber asymp 120572camber (20)
119889119910camber = [1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878)]119889119909
(21)
After integral solution of formula (21) the camber curvecan be expressed as
119910camber =1198751198783
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(22)
Combined with the weight of the trolley 119875trolley thecamber curve can be obtained and expressed as
119910camber =(1198750+ 119875trolley) 119878
3
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(23)
5 Conclusions
Just as expressed in formula (23) the camber curve can bedesigned However in previous researches on camber design1198750is designed as fixed or rated loads In fact the hoisting
weight of overhead crane is not fixed and energy consump-tions under certain camber with different hoistingload willvary Camber design based on the fixed or rated loadmay notget the optimal energy-savingThis research seeks to get ideal1198750for trolley moving with probabilistic loads to obtain the
least energy consumption Then according to formula (23)energy-saving camber design of overhead travelling cranecan be obtained Besides some conclusions can be drawn asbelow
(1) With the load status level improved the hoistingloadmean gradually increases and the load distributionbecomes more concentrated The loads of 1198764 aredistributed around the rated load with a certainprobability of overloading
(2) The minimum energy consumptions occur when 119875approaches 119875
0and the energy consumptions are less
when 119875 lt 1198750than when 119875 gt 119875
0because of
heavier loading So heavier load can be considered forcamber design to guarantee less energy consumptionswhen overloading
The research results of this paper can also provide a greatreference value to cutting girder web along a certain curve forprefabrication
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was financially supported by National Founda-tion of General Administration of Quality Supervision andInspection (2012QK178) Program of Science Foundation ofGeneral Administration of Quality Supervision and Inspec-tion of Jiangsu Province (KJ103708) and ldquoexcellence plans-zijin starrdquo Foundation of Nanjing University of Science Alsothe work is preresearch of theNational Natural Science Foun-dation of China (research on energy consumption modellingand methodology of energy-saving design for cranes) Thesupports are gratefully acknowledged
Mathematical Problems in Engineering 9
References
[1] P R Patel and V K Patel ldquoA review on structural analysisof overhead crane girder using FEA techniquerdquo InternationalJournal of Engineering Science and Innovative Technology vol 2no 4 pp 41ndash44 2013
[2] T Yifei Y Wei Y Zhen L Dongbo and L XiangdongldquoResearch on multidisciplinary optimization design of bridgecranerdquoMathematical Problems in Engineering vol 2013 ArticleID 763545 10 pages 2013
[3] S G Lee and N Q Hoang ldquoEnergy-based approach forcontroller design of overhead cranes a comparative studyrdquoApplied Mechanics and Materials vol 365-366 pp 784ndash7872010
[4] C Li-Feng ldquoMethods and analysis of bridge camber anddeformation of main girder in gantry cranerdquo Equipment Manu-facturing Technology no 5 pp 124ndash125 2010
[5] C R Bradlee ldquoMethod of measuring camberrdquo Us Patent4794773 1989
[6] F Rongbo ldquoCamber amp pre arch curve of cranesrdquo Hoisting andConveying Machinery vol 2 pp 16ndash18 1990
[7] G Shen X Li D Li and C Zhou ldquoReaseach on energy-savingdesign of arch curve of bridge crane girderrdquo in Proceedings ofthe International Conference on Remote Sensing Environmentand Transportation Engineering (RSETE rsquo11) pp 1448ndash1450Nanjing China June 2011
[8] T Jiantao Load Statistical Analisys and Reliable IndicatorResearch of Bridge Crane Taiyuan University of Science andTechnology Taiyuan China 2011
[9] S Zivanovic R P Johnson H V Dang and J Dobric ldquoDesignand construction of a very lively bridgerdquo in Topics in Dynamicsof Civil Structures vol 4 pp 371ndash380 2013
[10] X Gelin GBT3811-2008 Design Rules for Cranes ChineseStandard Press Beijing China 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
After solution of the differential equations energy con-sumptions can be obtained
1198641198750 = (119875 + 157)
10038161003816100381610038161198750 minus 1198751003816100381610038161003816
1198783
48119864119868 (13)
Take a 5010 t overhead crane for example and input thevalue of 119878 119864 and 119868 then
1198641198750 = 011 (119875 + 157)
10038161003816100381610038161198750 minus 1198751003816100381610038161003816
(14)
From Figure 5 it can be found that the minimum energyconsumptions occur when 119875 approaches 119875
0and the energy
consumptions are less when 119875 lt 1198750than when 119875 gt 119875
0
because of heavier loading So heavier load can be consideredfor camber design to guarantee less energy consumptionswhen overloading
With combinations of each load distribution densityfunction 1198641198750 which denotes the energy consumptions of thecrane whose camber is based on 119875
0and under loadings of
normal distribution can be calculated as
1198641198750 = int
119875max
0
1
119904119905radic2120587
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157)1003816100381610038161003816119875 minus 1198750
1003816100381610038161003816
1198783
48119864119868119889119875
(15)
Remove the absolute value and expand formula (15) then
1198641198750 =
1
119904119905radic2120587
1198783
48119864119868
times (int
1198750
0
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157) (1198750 minus 119875) 119889119875
+int
119875max
1198750
119890minus(119875minus119909119905)
221199042
119905 (119875 + 157) (119875 minus 1198750) 119889119875)
(16)
Then the energy consumptions under 1198761 1198762 1198763and 1198764 can be obtained as well as energy consumptionsdistribution shown in Figures 6ndash9
1198761
1198641198750 = 00006 sdot (int
1198750
0
119890minus(119875minus229)
210658
sdot (119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus229)
210658
sdot (119875 + 157) (119875 minus 1198750) 119889119875)
= minus417 + 1804119890(00430minus000009381198750)1198750
minus 000521198750+ (102915 minus 42385119875
0)
times Erf [222 minus 0009681198750]
1198762
1198641198750 = 000078 sdot (int
1198750
0
119890minus(119875minus3045)
26272(119875 + 157) (1198750 minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus3045)
26272
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus927 + 0000896119890(00968minus00001591198750)1198750 + 0018119875
0
+ (157522 minus 50601198750)Erf [38396 minus 00126119875
0]
1198763
1198641198750 = 000127 (int
1198750
0
119890minus(119875minus394)
223805
(119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus394)
223805
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus40433 + 754 times 10minus26119886(033minus0000421198750)1198750 + 00793119875
0
+ (2399677 minus 60541198750)Erf [808 minus 0020119875
0]
1198764
1198641198750 = 000219 (int
1198750
0
119890minus(119875minus450)
2800(119875 + 157) (119875
0minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus450)
2800
Mathematical Problems in Engineering 7
2000
4000
6000
8000
10000
100 200 300 400 500P0
242661 241986
E(P
0)
Figure 6 Energy consumptions distribution with probability load1198750under 1198761
5000
10000
15000
100 200 300 400 500P0
311256 224365
E(P
0)
Figure 7 Energy consumptions distribution with probability load1198750under 1198762
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus229103 + 125 times 10minus107119890(1125minus0001251198750)1198750 + 045119875
0
+ (300331 minus 66641198750)Erf [1591 minus 0035119875
0]
(17)
From Figure 6 it can be found that when 119888 = 0 that iswithout camber energy consumptions are greatly increasedWith 119875
0acting on the girder and increasing 1198641198750 decreases
When1198750approaches 242661 KN theminimumof1198641198750 occurs
as 241986 J If 1198750continues to increase 1198641198750 will increase
Similarly from Figure 7 it can be found that when1198750approaches 311256KN the minimum of 1198641198750 occurs as
224365 JSimilarly from Figure 8 it can be found that when
1198750approaches 3963 KN the minimum of 1198641198750 occurs as
165447 JSimilarly from Figure 9 it can be found that when
1198750approaches 450489KN the minimum of 1198641198750 occurs as
103759 J
5000
10000
15000
20000
100 200 300 400 500P0
3963 165447
E(P
0)
Figure 8 Energy consumptions distribution with probability load1198750under 1198763
5000
10000
15000
20000
25000
30000
100 200 300 400 500P0
450489 103759
E(P
0)
Figure 9 Energy consumptions distribution with probability load1198750under 1198764
Table 3 Ideal load 1198750corresponding to different load status levels
Load status level 1198750
1198761 242KN1198762 311 KN1198763 396KN1198764 450KN
Through the above analysis the ideal load1198750correspond-
ing to different load status levels can be initially determinedas shown in Table 3
When the load spectrum coefficient is unknown theload can be considered to be conformed to [0 119875max] nor-mal distribution Then the energy consumptions under the
8 Mathematical Problems in Engineering
5000
10000
15000
20000
200 400 600 800 1000P0
39657 340102
E(P
0)
Figure 10 Energy consumptions distribution with probability load1198750under 119876
uncertain load status level119876 can be obtained as well as energyconsumptions distribution shown in Figure 10
119876
1198641198750 = 000049 sdot (int
1198750
0
119890minus(119875minus275)
216200
(119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus275)
216200
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus775587 + 274307119890(0034minus00000621198750)1198750 + 232119875
0
+ (122226 minus 41621198750)Erf (2166 minus 0006119875
0)
(18)
FromFigure 10 energy consumption distribution law canbe found when load spectrum coefficient is uncertain When1198750= 0 that is without camber energy consumptions are
greatly much With 1198750acting on the girder and increasing
1198641198750 decreasesWhen119875
0approaches 39657 KN theminimum
of 1198641198750 occurs as 340102 J If 1198750continues to increase 1198641198750 will
increaseDifferent 119875
0determines the corresponding reasonable
camber to obtain the least energy consumption for climbingAccording to formula (9) and 120572camber + 120572 = 0 then
120572camber = minus120572 =1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878) (19)
Because 120572camber is very small
119889119910camber119889119909
= 119905119892120572camber asymp 120572camber (20)
119889119910camber = [1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878)]119889119909
(21)
After integral solution of formula (21) the camber curvecan be expressed as
119910camber =1198751198783
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(22)
Combined with the weight of the trolley 119875trolley thecamber curve can be obtained and expressed as
119910camber =(1198750+ 119875trolley) 119878
3
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(23)
5 Conclusions
Just as expressed in formula (23) the camber curve can bedesigned However in previous researches on camber design1198750is designed as fixed or rated loads In fact the hoisting
weight of overhead crane is not fixed and energy consump-tions under certain camber with different hoistingload willvary Camber design based on the fixed or rated loadmay notget the optimal energy-savingThis research seeks to get ideal1198750for trolley moving with probabilistic loads to obtain the
least energy consumption Then according to formula (23)energy-saving camber design of overhead travelling cranecan be obtained Besides some conclusions can be drawn asbelow
(1) With the load status level improved the hoistingloadmean gradually increases and the load distributionbecomes more concentrated The loads of 1198764 aredistributed around the rated load with a certainprobability of overloading
(2) The minimum energy consumptions occur when 119875approaches 119875
0and the energy consumptions are less
when 119875 lt 1198750than when 119875 gt 119875
0because of
heavier loading So heavier load can be considered forcamber design to guarantee less energy consumptionswhen overloading
The research results of this paper can also provide a greatreference value to cutting girder web along a certain curve forprefabrication
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was financially supported by National Founda-tion of General Administration of Quality Supervision andInspection (2012QK178) Program of Science Foundation ofGeneral Administration of Quality Supervision and Inspec-tion of Jiangsu Province (KJ103708) and ldquoexcellence plans-zijin starrdquo Foundation of Nanjing University of Science Alsothe work is preresearch of theNational Natural Science Foun-dation of China (research on energy consumption modellingand methodology of energy-saving design for cranes) Thesupports are gratefully acknowledged
Mathematical Problems in Engineering 9
References
[1] P R Patel and V K Patel ldquoA review on structural analysisof overhead crane girder using FEA techniquerdquo InternationalJournal of Engineering Science and Innovative Technology vol 2no 4 pp 41ndash44 2013
[2] T Yifei Y Wei Y Zhen L Dongbo and L XiangdongldquoResearch on multidisciplinary optimization design of bridgecranerdquoMathematical Problems in Engineering vol 2013 ArticleID 763545 10 pages 2013
[3] S G Lee and N Q Hoang ldquoEnergy-based approach forcontroller design of overhead cranes a comparative studyrdquoApplied Mechanics and Materials vol 365-366 pp 784ndash7872010
[4] C Li-Feng ldquoMethods and analysis of bridge camber anddeformation of main girder in gantry cranerdquo Equipment Manu-facturing Technology no 5 pp 124ndash125 2010
[5] C R Bradlee ldquoMethod of measuring camberrdquo Us Patent4794773 1989
[6] F Rongbo ldquoCamber amp pre arch curve of cranesrdquo Hoisting andConveying Machinery vol 2 pp 16ndash18 1990
[7] G Shen X Li D Li and C Zhou ldquoReaseach on energy-savingdesign of arch curve of bridge crane girderrdquo in Proceedings ofthe International Conference on Remote Sensing Environmentand Transportation Engineering (RSETE rsquo11) pp 1448ndash1450Nanjing China June 2011
[8] T Jiantao Load Statistical Analisys and Reliable IndicatorResearch of Bridge Crane Taiyuan University of Science andTechnology Taiyuan China 2011
[9] S Zivanovic R P Johnson H V Dang and J Dobric ldquoDesignand construction of a very lively bridgerdquo in Topics in Dynamicsof Civil Structures vol 4 pp 371ndash380 2013
[10] X Gelin GBT3811-2008 Design Rules for Cranes ChineseStandard Press Beijing China 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
2000
4000
6000
8000
10000
100 200 300 400 500P0
242661 241986
E(P
0)
Figure 6 Energy consumptions distribution with probability load1198750under 1198761
5000
10000
15000
100 200 300 400 500P0
311256 224365
E(P
0)
Figure 7 Energy consumptions distribution with probability load1198750under 1198762
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus229103 + 125 times 10minus107119890(1125minus0001251198750)1198750 + 045119875
0
+ (300331 minus 66641198750)Erf [1591 minus 0035119875
0]
(17)
From Figure 6 it can be found that when 119888 = 0 that iswithout camber energy consumptions are greatly increasedWith 119875
0acting on the girder and increasing 1198641198750 decreases
When1198750approaches 242661 KN theminimumof1198641198750 occurs
as 241986 J If 1198750continues to increase 1198641198750 will increase
Similarly from Figure 7 it can be found that when1198750approaches 311256KN the minimum of 1198641198750 occurs as
224365 JSimilarly from Figure 8 it can be found that when
1198750approaches 3963 KN the minimum of 1198641198750 occurs as
165447 JSimilarly from Figure 9 it can be found that when
1198750approaches 450489KN the minimum of 1198641198750 occurs as
103759 J
5000
10000
15000
20000
100 200 300 400 500P0
3963 165447
E(P
0)
Figure 8 Energy consumptions distribution with probability load1198750under 1198763
5000
10000
15000
20000
25000
30000
100 200 300 400 500P0
450489 103759
E(P
0)
Figure 9 Energy consumptions distribution with probability load1198750under 1198764
Table 3 Ideal load 1198750corresponding to different load status levels
Load status level 1198750
1198761 242KN1198762 311 KN1198763 396KN1198764 450KN
Through the above analysis the ideal load1198750correspond-
ing to different load status levels can be initially determinedas shown in Table 3
When the load spectrum coefficient is unknown theload can be considered to be conformed to [0 119875max] nor-mal distribution Then the energy consumptions under the
8 Mathematical Problems in Engineering
5000
10000
15000
20000
200 400 600 800 1000P0
39657 340102
E(P
0)
Figure 10 Energy consumptions distribution with probability load1198750under 119876
uncertain load status level119876 can be obtained as well as energyconsumptions distribution shown in Figure 10
119876
1198641198750 = 000049 sdot (int
1198750
0
119890minus(119875minus275)
216200
(119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus275)
216200
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus775587 + 274307119890(0034minus00000621198750)1198750 + 232119875
0
+ (122226 minus 41621198750)Erf (2166 minus 0006119875
0)
(18)
FromFigure 10 energy consumption distribution law canbe found when load spectrum coefficient is uncertain When1198750= 0 that is without camber energy consumptions are
greatly much With 1198750acting on the girder and increasing
1198641198750 decreasesWhen119875
0approaches 39657 KN theminimum
of 1198641198750 occurs as 340102 J If 1198750continues to increase 1198641198750 will
increaseDifferent 119875
0determines the corresponding reasonable
camber to obtain the least energy consumption for climbingAccording to formula (9) and 120572camber + 120572 = 0 then
120572camber = minus120572 =1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878) (19)
Because 120572camber is very small
119889119910camber119889119909
= 119905119892120572camber asymp 120572camber (20)
119889119910camber = [1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878)]119889119909
(21)
After integral solution of formula (21) the camber curvecan be expressed as
119910camber =1198751198783
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(22)
Combined with the weight of the trolley 119875trolley thecamber curve can be obtained and expressed as
119910camber =(1198750+ 119875trolley) 119878
3
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(23)
5 Conclusions
Just as expressed in formula (23) the camber curve can bedesigned However in previous researches on camber design1198750is designed as fixed or rated loads In fact the hoisting
weight of overhead crane is not fixed and energy consump-tions under certain camber with different hoistingload willvary Camber design based on the fixed or rated loadmay notget the optimal energy-savingThis research seeks to get ideal1198750for trolley moving with probabilistic loads to obtain the
least energy consumption Then according to formula (23)energy-saving camber design of overhead travelling cranecan be obtained Besides some conclusions can be drawn asbelow
(1) With the load status level improved the hoistingloadmean gradually increases and the load distributionbecomes more concentrated The loads of 1198764 aredistributed around the rated load with a certainprobability of overloading
(2) The minimum energy consumptions occur when 119875approaches 119875
0and the energy consumptions are less
when 119875 lt 1198750than when 119875 gt 119875
0because of
heavier loading So heavier load can be considered forcamber design to guarantee less energy consumptionswhen overloading
The research results of this paper can also provide a greatreference value to cutting girder web along a certain curve forprefabrication
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was financially supported by National Founda-tion of General Administration of Quality Supervision andInspection (2012QK178) Program of Science Foundation ofGeneral Administration of Quality Supervision and Inspec-tion of Jiangsu Province (KJ103708) and ldquoexcellence plans-zijin starrdquo Foundation of Nanjing University of Science Alsothe work is preresearch of theNational Natural Science Foun-dation of China (research on energy consumption modellingand methodology of energy-saving design for cranes) Thesupports are gratefully acknowledged
Mathematical Problems in Engineering 9
References
[1] P R Patel and V K Patel ldquoA review on structural analysisof overhead crane girder using FEA techniquerdquo InternationalJournal of Engineering Science and Innovative Technology vol 2no 4 pp 41ndash44 2013
[2] T Yifei Y Wei Y Zhen L Dongbo and L XiangdongldquoResearch on multidisciplinary optimization design of bridgecranerdquoMathematical Problems in Engineering vol 2013 ArticleID 763545 10 pages 2013
[3] S G Lee and N Q Hoang ldquoEnergy-based approach forcontroller design of overhead cranes a comparative studyrdquoApplied Mechanics and Materials vol 365-366 pp 784ndash7872010
[4] C Li-Feng ldquoMethods and analysis of bridge camber anddeformation of main girder in gantry cranerdquo Equipment Manu-facturing Technology no 5 pp 124ndash125 2010
[5] C R Bradlee ldquoMethod of measuring camberrdquo Us Patent4794773 1989
[6] F Rongbo ldquoCamber amp pre arch curve of cranesrdquo Hoisting andConveying Machinery vol 2 pp 16ndash18 1990
[7] G Shen X Li D Li and C Zhou ldquoReaseach on energy-savingdesign of arch curve of bridge crane girderrdquo in Proceedings ofthe International Conference on Remote Sensing Environmentand Transportation Engineering (RSETE rsquo11) pp 1448ndash1450Nanjing China June 2011
[8] T Jiantao Load Statistical Analisys and Reliable IndicatorResearch of Bridge Crane Taiyuan University of Science andTechnology Taiyuan China 2011
[9] S Zivanovic R P Johnson H V Dang and J Dobric ldquoDesignand construction of a very lively bridgerdquo in Topics in Dynamicsof Civil Structures vol 4 pp 371ndash380 2013
[10] X Gelin GBT3811-2008 Design Rules for Cranes ChineseStandard Press Beijing China 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
5000
10000
15000
20000
200 400 600 800 1000P0
39657 340102
E(P
0)
Figure 10 Energy consumptions distribution with probability load1198750under 119876
uncertain load status level119876 can be obtained as well as energyconsumptions distribution shown in Figure 10
119876
1198641198750 = 000049 sdot (int
1198750
0
119890minus(119875minus275)
216200
(119875 + 157) (1198750minus 119875) 119889119875
+ int
500
1198750
119890minus(119875minus275)
216200
times (119875 + 157) (119875 minus 1198750) 119889119875)
= minus775587 + 274307119890(0034minus00000621198750)1198750 + 232119875
0
+ (122226 minus 41621198750)Erf (2166 minus 0006119875
0)
(18)
FromFigure 10 energy consumption distribution law canbe found when load spectrum coefficient is uncertain When1198750= 0 that is without camber energy consumptions are
greatly much With 1198750acting on the girder and increasing
1198641198750 decreasesWhen119875
0approaches 39657 KN theminimum
of 1198641198750 occurs as 340102 J If 1198750continues to increase 1198641198750 will
increaseDifferent 119875
0determines the corresponding reasonable
camber to obtain the least energy consumption for climbingAccording to formula (9) and 120572camber + 120572 = 0 then
120572camber = minus120572 =1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878) (19)
Because 120572camber is very small
119889119910camber119889119909
= 119905119892120572camber asymp 120572camber (20)
119889119910camber = [1198751198782
3119864119868
119883
119878(1 minus
119883
119878)(1 minus
2119883
119878)]119889119909
(21)
After integral solution of formula (21) the camber curvecan be expressed as
119910camber =1198751198783
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(22)
Combined with the weight of the trolley 119875trolley thecamber curve can be obtained and expressed as
119910camber =(1198750+ 119875trolley) 119878
3
6119864119868(119883
119878)
2
(1 minus119883
119878)
2
(23)
5 Conclusions
Just as expressed in formula (23) the camber curve can bedesigned However in previous researches on camber design1198750is designed as fixed or rated loads In fact the hoisting
weight of overhead crane is not fixed and energy consump-tions under certain camber with different hoistingload willvary Camber design based on the fixed or rated loadmay notget the optimal energy-savingThis research seeks to get ideal1198750for trolley moving with probabilistic loads to obtain the
least energy consumption Then according to formula (23)energy-saving camber design of overhead travelling cranecan be obtained Besides some conclusions can be drawn asbelow
(1) With the load status level improved the hoistingloadmean gradually increases and the load distributionbecomes more concentrated The loads of 1198764 aredistributed around the rated load with a certainprobability of overloading
(2) The minimum energy consumptions occur when 119875approaches 119875
0and the energy consumptions are less
when 119875 lt 1198750than when 119875 gt 119875
0because of
heavier loading So heavier load can be considered forcamber design to guarantee less energy consumptionswhen overloading
The research results of this paper can also provide a greatreference value to cutting girder web along a certain curve forprefabrication
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was financially supported by National Founda-tion of General Administration of Quality Supervision andInspection (2012QK178) Program of Science Foundation ofGeneral Administration of Quality Supervision and Inspec-tion of Jiangsu Province (KJ103708) and ldquoexcellence plans-zijin starrdquo Foundation of Nanjing University of Science Alsothe work is preresearch of theNational Natural Science Foun-dation of China (research on energy consumption modellingand methodology of energy-saving design for cranes) Thesupports are gratefully acknowledged
Mathematical Problems in Engineering 9
References
[1] P R Patel and V K Patel ldquoA review on structural analysisof overhead crane girder using FEA techniquerdquo InternationalJournal of Engineering Science and Innovative Technology vol 2no 4 pp 41ndash44 2013
[2] T Yifei Y Wei Y Zhen L Dongbo and L XiangdongldquoResearch on multidisciplinary optimization design of bridgecranerdquoMathematical Problems in Engineering vol 2013 ArticleID 763545 10 pages 2013
[3] S G Lee and N Q Hoang ldquoEnergy-based approach forcontroller design of overhead cranes a comparative studyrdquoApplied Mechanics and Materials vol 365-366 pp 784ndash7872010
[4] C Li-Feng ldquoMethods and analysis of bridge camber anddeformation of main girder in gantry cranerdquo Equipment Manu-facturing Technology no 5 pp 124ndash125 2010
[5] C R Bradlee ldquoMethod of measuring camberrdquo Us Patent4794773 1989
[6] F Rongbo ldquoCamber amp pre arch curve of cranesrdquo Hoisting andConveying Machinery vol 2 pp 16ndash18 1990
[7] G Shen X Li D Li and C Zhou ldquoReaseach on energy-savingdesign of arch curve of bridge crane girderrdquo in Proceedings ofthe International Conference on Remote Sensing Environmentand Transportation Engineering (RSETE rsquo11) pp 1448ndash1450Nanjing China June 2011
[8] T Jiantao Load Statistical Analisys and Reliable IndicatorResearch of Bridge Crane Taiyuan University of Science andTechnology Taiyuan China 2011
[9] S Zivanovic R P Johnson H V Dang and J Dobric ldquoDesignand construction of a very lively bridgerdquo in Topics in Dynamicsof Civil Structures vol 4 pp 371ndash380 2013
[10] X Gelin GBT3811-2008 Design Rules for Cranes ChineseStandard Press Beijing China 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
References
[1] P R Patel and V K Patel ldquoA review on structural analysisof overhead crane girder using FEA techniquerdquo InternationalJournal of Engineering Science and Innovative Technology vol 2no 4 pp 41ndash44 2013
[2] T Yifei Y Wei Y Zhen L Dongbo and L XiangdongldquoResearch on multidisciplinary optimization design of bridgecranerdquoMathematical Problems in Engineering vol 2013 ArticleID 763545 10 pages 2013
[3] S G Lee and N Q Hoang ldquoEnergy-based approach forcontroller design of overhead cranes a comparative studyrdquoApplied Mechanics and Materials vol 365-366 pp 784ndash7872010
[4] C Li-Feng ldquoMethods and analysis of bridge camber anddeformation of main girder in gantry cranerdquo Equipment Manu-facturing Technology no 5 pp 124ndash125 2010
[5] C R Bradlee ldquoMethod of measuring camberrdquo Us Patent4794773 1989
[6] F Rongbo ldquoCamber amp pre arch curve of cranesrdquo Hoisting andConveying Machinery vol 2 pp 16ndash18 1990
[7] G Shen X Li D Li and C Zhou ldquoReaseach on energy-savingdesign of arch curve of bridge crane girderrdquo in Proceedings ofthe International Conference on Remote Sensing Environmentand Transportation Engineering (RSETE rsquo11) pp 1448ndash1450Nanjing China June 2011
[8] T Jiantao Load Statistical Analisys and Reliable IndicatorResearch of Bridge Crane Taiyuan University of Science andTechnology Taiyuan China 2011
[9] S Zivanovic R P Johnson H V Dang and J Dobric ldquoDesignand construction of a very lively bridgerdquo in Topics in Dynamicsof Civil Structures vol 4 pp 371ndash380 2013
[10] X Gelin GBT3811-2008 Design Rules for Cranes ChineseStandard Press Beijing China 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
