Research ArticleProbability Analysis of Crane Load andLoad Combination Actions

Jitao Yao, Hui Gu , and Liuzhuo Chen

School of Civil Engineering, Xi’an University of Architecture and Technology, Xi’an 710055, China

Correspondence should be addressed to Hui Gu; guhuiyatou@163.com

Received 11 April 2018; Revised 13 June 2018; Accepted 4 July 2018; Published 17 July 2018

Academic Editor: Luis Martınez

Copyright © 2018 Jitao Yao et al.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This study presents probability models of crane load and load combination actions for reliability analysis of the industry buildings.Crane load values are simplified here only varying in time, and load combination actions only vary in space. With the feasiblesurvey program and K-S test, the Gumbel distribution is chosen as the probability distribution of the arbitrary point-in-time craneload values with the statistical investigation data. With the simple stationary binomial random process model hypothesis and theblock maxima method, it determined that the maximum value of crane load during the design reference period also follows theGumbel distribution, with the peaks-over-thresholdmethod evaluation. Formultiple crane load combination actions, themodifiedTurkstra rule is improved to determine the probability distribution of the actions as the Gumbel distribution by considering theoccurrence probability, numbers, location, and values of each crane load acting on the influence line, evaluated by the Monte Carlosimulations. Design standard values for crane loads and load combination actions, as specified in building codes, are evaluatedbased on these distributions in probability significance. The calculation results illustrate that the design standard values for craneloads are relatively safe and conservative.

1. Introduction

Crane load is a free dynamic action, which has arbitraryspatial distributions over crane supporting structures withingiven limits and causes significant structural acceleration.Crane load is a random variable varying in time and spacedimensionswithin the design reference period and is denotedas random field model [1, 2]. General specifications stipulatethe design value of the crane load and load capabilities,such as EN1991-3: Eurocode 1-actions on structures-Part 3:actions induced by cranes and machinery [1], ACSE/SEI 7-05: minimum design loads for buildings and other struc-tures [3], and GB50009-2012: load code for the design ofbuilding structures [4]. However, these codes do not givean acceptable load probability models except GB50009-2012[4] which have done some traditional statistical surveys andload experiments while lacking theoretical analysis to giveinstructions on the validity of these data.That is to say, for thedynamic action in these codes, there are no acceptable loadprobability models for crane load and load actions have beenapproved for reliability analysis. It means the actual reliability

of the industry buildings designed with these above codes ofconsidering the crane loads is unknown.

In the Probabilistic model code of the Joint Committeeon Structural Safety (JCSS) [5], it proved the basis methodon building the probability models for load and load actions.That is, determining the probability distribution of the arbi-trary point-in-time values with the statistical investigationdata in independent statistical interval, and the probabilitydistribution of the maximum values during the design ref-erence period by being simplified to the stationary binomialrandom process. When more than one variable loads actin combination, simplified rules such as JCSS or Turkstracombination rule applied to the reliability calculation of thefirst-order second-moment method (FOSM) are applied tothe load action probability model. But crane load is muchdifferent with other live loads when using the model code. Inindustry building, crane load and their combination actionsare functions of both time and space [6]. The time variableshows the different crane load values in different pointsin time. The space variable includes the position of thecrane and the crab (Figure 1). To determine the probability

HindawiMathematical Problems in EngineeringVolume 2018, Article ID 6527307, 13 pageshttps://doi.org/10.1155/2018/6527307

2 Mathematical Problems in Engineering

l

KB

KB

？ＧＣＨ

１Ｌ,Ｇ；Ｒ １Ｌ,Ｇ；Ｒ ∑１Ｌ,Ｇ；Ｒ

１Ｂ,ＨＩＧ

１Ｌ,(Ｇ；Ｒ) １Ｌ,(Ｇ；Ｒ)∑１Ｌ,(Ｇ；Ｒ)

②③

①②

③

①

(a) Load arrangement of the loaded crane use to obtain the maximum vertical load on the runway beam

l

K BK

B

g+Q

G（４,1

（４,1 （４,2

（４,2

（，,1 （，,2

②

①

②

①

(b) Load arrangement of longitudinal and transverse horizontal wheel forces caused by acceleration anddeceleration

Figure 1: Schematic diagram of load arrangements of the vertical load and horizontal force, whereA is the crab;B is the crane bridge;C isthe crane beam. Qr,max is the maximum load per wheel of the loaded crane. Qr,(max) is the accompanying load per wheel of the loaded crane.∑Qr,max is the sum of the maximum loads Qr,max per runway of the loaded crane.∑Qr,(max) is the sum of the accompanying maximum loadsQr,(max) per runway of the loaded crane. Qh,nom is the nominal hoist load. emin is the minimum distance between the hoist point and the cranewheel. 𝑙 is the span of the crane bridge. K is the wheel distance of the crane. B is the width of the crane. g is the self-weight of the crab. Gis the self-weight of the crane. Q is the hoist load. HT,𝑖 (𝑖 = 1, 2) is the transverse horizontal wheel forces. 𝐻L,𝑖 (𝑖 = 1, 2) is the longitudinalhorizontal wheel forces.

distribution of the point-in-time crane load values, the worst-case situation should be considered in the simplified analysis,considering the actual measured maximum load for thecrab worst-case position (in Figure 1), but the independentstatistical interval is too long to do the investigation andobtain enough samples (with the operating period as 20years). And to determine the probability distribution of thecrane load actions with multiple cranes acting on a beam, thetraditional JCSS or Turkstra combination rule consideringthe different load actions is not effective, for multiple craneswith respective crane loads acting different with multipleloads. It is much more dependent on the actual productionprocess, and with little uncertainty with survey results. Withactual investigation, if multiple cranes are acting on beam,the synchronous occurrence probability of two cranes on thespan beam is 1 (it means in the operating period, two craneswill simultaneously act on the same span beam more thanonce) and the synchronous occurrence probability of threecranes is almost 0. That is to say, two cranes are certainlysynchronous at least once. If two cranes are synchronous, thepossible computational maximum load (with ‘full load’ foreach crane) is almost zero.When determining the probabilitymodel and statistical parameters for crane load combinationactions, these above two points need to be considered.

In this paper, the available method was improved toobtain the enough samples and certificate the availability

of the investigation data of [4], to obtain the probabilitydistribution of the arbitrary point-in-time crane load valueswith the time-space exchange method theory and the time-space-dimensional variable separation method. And it mod-ified the Turkstra rule to determine the probability modelof the multiple crane load combination actions based onthe actual operating case. The probability model of craneload and load combination actions can be improved asfollows. Firstly, simplifying a crane load action process asthe stationary binomial random process, the design referenceperiod is divided into equal independent intervals, usingthe time-space exchange method theory and the time-space-dimensional variable separation method to determine theexecutable observation interval. With the actual survey data,the probability distribution of the arbitrary point-in-timecrane load values is determined, considering only time andcrab position variability. Secondly, the probability distribu-tion of the randommaximum process load during the designreference period is derived with the stationary binomial ran-domprocess theory and the blockmaximamethod [7]. Lastly,the crane position variable is considered with the occurrenceprobability along the crane beam influence line during theoperation period [4]; the modified Turkstra rule is improvedto calculate the multiple crane load combination actions con-sidering the occurrence probability, occurrence load values,and occurrence position.The influence line is ensured by the

Mathematical Problems in Engineering 3

worst-case effect of the load actions (Figure 1(a)). What ismore, the above simple methods obtained the distributionexpression formulas of the crane loads and multiple craneload combination actions can be directly used for FOSMreliability calculation.With the peaks-over-thresholdmethod(more precise but could not prove expression formula) toverify the distribution of the crane loads and theMonte Carlosimulations (more precise but could not prove expressionformula) to verify the distribution of the multiple craneload combination actions; these above simple methods anddistribution functions are acceptable.

The rest of this paper is arranged as follows. Section 2states the arbitrary point-in-time distribution and the max-imum value distribution during the crane load design refer-ence period and improves the distribution of multiple craneload combination actions using the influence line method.Section 3 constructs the evaluated results of the above dis-tribution with the peaks-over-threshold method for craneloads and Monte Carlo simulations for multiple crane loadcombination actions. The probability evaluated significanceof the design standard crane load and load combinationmethods specified in Chinese load code are introduced inSection 4. Finally, Section 5 presents the conclusions.

2. Probability Models of Crane Load and LoadCombination Actions

2.1. Methods of Building Crane Load Model. Multiple esti-mation methods for building load models during the designreference period have been developed, including the blockmaxima method [8, 9] and peaks-over-threshold (POT)method [8, 10]. The block maxima method is based on thestationary binomial random process model hypothesis andthe extreme value distribution theory of random variables. Itdivides the complete time history sample into equally sizedblocks and extracts the maximum load values from eachblock. Assuming that the block maximum is independentlyand identically distributed, the distribution of extremes isthen determined. In order to ensure model accuracy, theload, as a broad stationary stochastic process, should beindependent during the block maximum [9, 11, 12]. Thisblock maxima method can prove an expression formulaof the maximum value distribution during the crane loaddesign reference period for FOSM reliability calculated. ThePOT method deals with multiple independent peaks abovea preselected high threshold, and the extreme value theorywarrants use of a generalized Pareto distribution (GPD) as anadequate model for peaks over the threshold [10, 13, 14]. ThePOT method contains a more precise theory than the blockmaxima method, but with much more complex calculationsand without expression formula [15, 16]. In this paper, theblockmaximamethodwas used to build the crane loadmodelwith an expression formula, and the POT method was usedto evaluate the model result.

Crane load is only determined using the hoist weightand the crab position from the survey results. To simplify,the crane load value only considers time randomness and isseldom of immediate relevance to the crane position. Andcrane load actions are affected by crane position, with the

location of each crane having only space randomness. Thatis to say, the macroscopic statistical law for the crane action(the number and location on the influence line) has nothingto do with time, but the position of adjacent cranes withrespect to each other is restricted. Considering the abovefeatures of crane load through time, it can be simplified toa stationary binomial random process to improve probabil-ity distributions of the arbitrary point-in-time values andmaximum values during the design reference period. Andconsidering the space randomness features of crane loadaction, crane load combination actions can be calculatedusing the combination method on the influence line [4].

This study only discusses the overhead traveling craneand crane beams to illustrate this method. With differentcharacteristics and load conditions for different crane clas-sifications, eight working levels of overhead traveling cranesare used in China. Within the same working level, craneshave different weight classes, hoisting heights, and workingspeeds. To ensure a proper probability analysis for crane loadand actions, different working levels should be distinguished.Crane load includes a vertical load and horizontal force withdifferent load directions. The vertical load is composed ofthe weight of the hoisted block and the hoist load, generallyshown as vertical wheel loads on a runway beam [4]. Themaximumvertical load is determined by considering the loadarrangements shown in Figure 1(a). Horizontal forces includethe longitudinal horizontal force and transverse horizontalforce. The longitudinal horizontal force acts horizontally atthe traction surface of the runway beam in either directionparallel to the beam. The transverse horizontal force actshorizontally at the traction surface of the runway beam ineither direction, perpendicular to the beam, and is equallydistributed on each side of the crane bridge (Figure 1(b)).Here we only consider the transverse horizontal force.

2.2. Crane Load Arbitrary Point-In-Time Distribution. Whenconsidering a single crane load on the influence line, thefollowing assumptions of a stationary binomial randomprocess for the crane load (vertical loads and horizontalforces) are necessary:

(1) The design reference period 𝑇 is divided into 𝑟 equalstatistical intervals 𝜏 (𝑟 = 𝑇/𝜏), and the maximum crane loadin each interval is independent.

(2) In each statistical interval, the occurrence probabilityfor the maximum crane load is 1, while the nonoccurrenceprobability is 0.

(3) The probability distribution for the maximum craneload 𝑄max(𝜏) in interval 𝜏 can be denoted with

𝐹𝑄𝜏 (𝑥) = 𝑃 [𝑄max (𝜏) ≤ 𝑥] (1)

where the function 𝐹𝑄𝜏(𝑥) can be called the arbitrary point-in-time value probability distribution, which is determinedby the magnitudes of the maximum crane load in differentintervals.

For a general crane beam in China, the design referenceperiod is commonly 50 years [4]. The longer the design life,the higher the probability that the maximum load occurs.This is related to the design reference period and is affected

4 Mathematical Problems in Engineering

by the frequency of crane replacement, the operating cycle,vacancy time, nominal hoist load, and crane working level.Generally, crane vacancy is not considered during the designlife. The replacement frequency and operating cycle aredetermined according to survey results. To ensure indepen-dence of the maximum crane load in each interval 𝜏, theoperating period should be regarded as the statistical intervalof crane replacement. The operating period, including thenonoperating and operating period of time, depended on thecrane replacement time. According to the design code forcranes: theGB/T 3811-2008 [17], theD\bf 27584-88 [18], andtheD\bf 22827-85 [19], the operating period for an overheadtraveling crane is 20 years. Thus, the equal statistical interval𝜏 should be taken as 20 years.

Normally, during each operating period, the nominalhoist load and work level of the crane and other maintechnical parameters do not change after replacement [4].Crane load values and maximum crane load values arerandom and need to be considered for structural reliabilityanalysis. To simplify, the maximum crane load value in eachstatistical interval is assumed to follow the same probabilitydistribution [12]. Meaning the maximum crane load 𝑄max(𝜏)in each interval is a nonnegative random variable, and theprobability distribution during different intervals is identical.Load through, in theory, should be obtained with the maxi-mumcrane load values in each statistical interval by statisticalinvestigation, but the statistical interval 𝜏 is too long to obtainenough samples. A feasible method for effective investigationshould be put forward.

As a tool of moving and transporting material, cranes areused in various fields to handle heavy objects [6]. Generally,when the crane structure is engaged in regular heavy work,there is a small observation interval Δ𝜏 in which all possiblecrane operating conditions within a complete operatingperiod can be observed. The observation interval Δ𝜏 shouldbe determined according to the nominal hoist load, worklevel, and production cycle to ensure that crane loads inthis small observation interval provide the sufficient recordwith a complete load history. Additionally, the chosen Δ𝜏should ensure that crane loads in every observation intervalare identical and completely relevant. The values of Δ𝜏 areempirically estimated from observations and consultationswith operators and factory employees. The maximum craneload in the observation interval is determined by statisticalsurveys and load experiments.

According to the time-space exchange method theory,the maximum crane load samples during the statisticalintervals 𝜏 (actually measured in observation interval) formany similar crane types (but not the same crane) in theirrespective working plants with spatial-dimension statistics,can be approximated using samples from several statisticalintervals of the same crane through time. The distributioncan be determined from these samples and validated usingthe K-S test. Reference [4] did the actual survey on 57cranes with the light, medium, and heavy working levelin different factories with different production process inBeijing, Shanghai, Shenyang, Anshan, Dalian, and so on. Itchoose the maximum crane load in about 5 years (means theobservation interval Δ𝜏 here is taken as 5) as the statistical

data. Using the above-mentioned method, the statisticalresults from [4] can be used to improve the probabilitydistribution of the arbitrary point-in-time crane load values.

For the vertical load and transverse horizontal force ofoverhead traveling cranes, the probability distribution of thearbitrary point-in-time crane load values are expressed as𝐹VS(𝑥) and 𝐹HS(𝑥). These two loads, respectively, refer tothe maximum wheel pressure and the maximum transversehorizontal force caused by acceleration or deceleration ofthe crab in relation to its movement along the crane bridge.Actually, the transverse horizontal force here includes theforce caused by the skewing of the crane in relation to itsmovement along the runway beams. At different workinglevels, nominal hoist loads result in large divergences, andthe ‘unified standard for reliability of building structures’uses dimensionless quantities to normalize the original datawith actual measured values (𝑊Cmeasured

or 𝐷HKmeasured) for the

corresponding standard design value (WC or DHK) [20]. WCis the nominal maximum wheel pressure; DHK is the sum ofthe nominal hoist load QHK; and g is the self-weight of thecrab [17]. Statistical results for different working level cranesare discussed below. According to survey results in [4] and K-S test, the Cumulative Distribution Functions (CDFs) of the𝐹VS(𝑥) and 𝐹HS(𝑥) are expressed as

𝐹VS (𝑥) = exp{− exp[−𝛼VS (𝑥 − 𝛽VS𝑊C)𝑊C]} (2)

𝐹HS (𝑥) = exp{− exp[−𝛼HS (𝑥 − 𝛽HS𝐷HK)𝐷HK]} (3)

where the subscripts VS and HS refer to the arbitrary point-in-time values of the vertical load and the transverse hori-zontal force; 𝛼VS, 𝛼HS and 𝛽VS, 𝛽HS are the scale and locationparameters, respectively, calculated as 𝛼VS = 1.2825/𝜎VS,𝛽VS = 𝜇VS − 0.5772/𝛼VS, 𝛼HS = 1.2825/𝜎HS, and 𝛽HS =𝜇HS − 0.5772/𝛼HS, where 𝜇VS, 𝜇HS, and 𝜎VS, 𝜎HS are the totalmeans and standard deviations approximated by the samplemeans and standard deviations.The values of these statisticalresults andmodel parameters for different working levels andnominal hoist loads are listed in Table 1 [4]. Considering thefact that transverse horizontal forces are primarily affected bythe nominal hoist load, the statistics of the working level arenot distinguished.

2.3.MaximumCrane LoadDistribution during theDesign Ref-erence Period. Using the stationary binomial randomprocessmodel described in Sections 2.1 and 2.2, the crane load samplefunction can be represented as a rectangle wave functionwithequal intervals [5]. The maximum value in random process𝑄max(𝑇) during the design reference period 𝑇 should be themaximum statistical interval 𝑄max(𝜏). 𝑄max(𝑇) is a randomvariable of 𝑄max(𝑇) = max0≤𝑡≤𝑇[𝑄max(𝜏)], whose probabilitydistribution is [5, 20]

𝐹𝑄𝑇 (𝑥) = 𝑃 {𝑄max (𝑇) ≤ 𝑥}= 𝑃{max0≤𝜏≤𝑇

[𝑄max (𝜏)] ≤ 𝑥}

= 𝑟∏𝑗=1

𝑃 {𝑄max (𝜏) ≤ 𝑥, 𝜏 ∈ 𝜏𝑗} ≈ [𝐹𝑄𝜏 (𝑥)]𝑇/𝜏(4)

Mathematical Problems in Engineering 5

Table1:Cr

aneloadprob

abilitymod

elandsta

tistic

alparameters.

Working

level

QHK/t

Thea

rbitrarypo

int-in-tim

evalue

distrib

ution

Them

axim

umvalued

istrib

ution(𝑇

=50𝑎)

statistic

alresults

[4]

mod

elparameters

Mom

ents

mod

elparameters

𝜇 VS𝜎 VS

𝛼 VS𝛽 VS

𝜇 VM𝜎 VM

𝛼 VM𝛽 VM

THEVER

TICA

LLO

AD

LightA

1-A3

20.520

0.055

23.318

0.495

0.559

0.055

23.318

0.535

30.46

00.071

18.063

0.428

0.511

0.071

18.063

0.479

50.370

0.069

18.587

0.339

0.419

0.069

18.587

0.388

Medium

A4-A5

50.610

0.037

34.662

0.593

0.636

0.037

34.662

0.620

100.520

0.045

28.500

0.500

0.552

0.045

28.500

0.532

15/3

0.490

0.050

25.650

0.46

70.526

0.050

25.650

0.503

20/5

0.420

0.04

131.280

0.402

0.44

90.04

131.280

0.431

Heavy

A6-A7

50.64

00.071

18.063

0.608

0.691

0.071

18.063

0.659

100.490

0.069

18.587

0.459

0.539

0.069

18.587

0.508

15/3

0.500

0.056

22.902

0.475

0.540

0.056

22.902

0.515

30/5

0.46

00.065

19.731

0.431

0.506

0.065

19.731

0.477

50/10

0.380

0.050

25.650

0.357

0.416

0.050

25.650

0.393

THEHORIZO

NTA

LFO

RCES

QHK/t

𝜇 HS𝜎 HS

𝛼 HS𝛽 HS

𝜇 HM𝜎 HM

𝛼 HM𝛽 HM

50.079

0.028

45.804

0.06

60.099

0.028

45.804

0.086

100.053

0.016

80.15

60.04

60.06

40.016

80.15

60.057

200.04

60.014

91.607

0.04

00.056

0.014

91.607

0.050

300.043

0.011

116.591

0.038

0.051

0.011

116.591

0.04

675

0.038

0.013

98.654

0.032

0.047

0.013

98.654

0.041

Note.Th

eactualstatistic

alresults

forv

ertic

alload

andtransverse

horiz

ontalforce

shou

ldbe

multip

liedby

WCandDHK,respectively

.Thec

rane

nominalho

istload

isrepresentedby

QHK.Th

ehorizon

talforcesh

ere

refertothetotalforces

onthetwosid

esof

thec

rane

bridge.

6 Mathematical Problems in Engineering

CA B

The first crane

L

The second crane

＂1 ＂2

＋1 ＋2

Q1 Q1 Q2 Q2＞ＧＣn

(a) The position of the two cranes in a combination action

CA BThe first crane The second crane

QＧ；Ｒ,４ QＧ；Ｒ,４

Q0.5,Ｇ；Ｒ, (or Q0,Ｇ；Ｒ,)

(b) The values of two cranes' combination with modified Turkstra method

Figure 2: Schematic diagram of two cranes acting in combination on a runway beam. Note: L is the adjacent column spacing (the span of thecrane beam); point 𝐴 and point 𝐵 are the crane beam bearing connections; point 𝐶 is the mid-span position of the beam; Bi (i = 1, 2) is thewidth of the crane; Ki (i = 1, 2) is the wheel distance of the crane; and 𝑄𝑖(𝑥) (𝑖 = 1, 2) expresses the load (vertical load or horizontal force)for each crane.

The maximum value probability distribution of verticalload and horizontal force during the design reference period,obtained using (4) and the corresponding arbitrary point-in-time value distribution given by (2) and (3), is expressed as aGumbel (type Ι) distribution of 𝐹VM(𝑥) and 𝐹HM(𝑥)

𝐹VM (𝑥) = [𝐹VS (𝑥)]𝑇/𝜏

= exp{− exp[−𝛼VM (𝑥 − 𝛽VM𝑊C)𝑊C]} (5)

𝐹HM (𝑥) = [𝐹HS (𝑥)]𝑇/𝜏

= exp{− exp[−𝛼HM (𝑥 − 𝛽HM𝐷HK)𝐷HK]} (6)

where the subscripts VM and HM represent the mean maxi-mum value of vertical load and horizontal force, respectively,during the design reference period. 𝛼VM, 𝛼HM and 𝛽VM, 𝛽HMare the scale and location parameters, calculated as 𝛼VM =𝛼VS = 1.2825/𝜎VS, 𝛼HM = 𝛼HS = 1.2825/𝜎HS, 𝛽VM = 𝛽VS +ln(𝑇/𝜏)/𝛼VS, and 𝛽HM = 𝛽HS+ ln(𝑇/𝜏)/𝛼HS. Mean values andstandard deviations 𝜇VM, 𝜇HM and 𝜎VM, 𝜎HM are calculatedusing 𝜇VM = 𝛽VM + 0.5772/𝛼VM, 𝜇HM = 𝛽HM + 0.5772/𝛼HM,𝜎VM = 1.2825/𝛼VM, and 𝜎HM = 1.2825/𝛼HM. Parametersfor different working levels and nominal hoist loads are listedin Table 1. The design reference period 𝑇 and the statisticalintervals 𝜏 are taken as 50 years and 20 years, respectively.

2.4. Probability Distribution of Multiple Crane Load Com-bination Actions. To determine the probability distributionof multiple crane load combination actions, the occurrenceprobability as well as the number and emergence location

on the influence line must be confirmed. Then a combi-nation probability model using influence line method canbe determined. This paper uses the influence line of thecrane beammid-span bending moment (point 𝐶 in Figure 2)as an example to illustrate the theoretical method. Othercombination crane actions on the crane beam or columnshould be the same.

2.4.1. Number and Location of Cranes. Although we assumethat the random number of cranes on the influence linehas nothing to do with the time interval, the most possiblecase with the number of cranes on a crane beam shouldbe obtained by observation over each observation interval.The number of cranes on a span crane beam is an uncertainvariable; however, it is simplified as the certain number aboutthe most frequent maximum number in the observationinterval for reliability analysis [20]. Cranes that operatetogether should be treated as a single crane to determine thecombination actions. If several cranes operate independently,meaning the combination action varies in time, the numberof cranes to be considered in the worst-case position maybe specified by an engineering survey. Considering thedesign load such as EN1991-3 [1], ASCE/SEI 7-05 [3], andGB50009-2012 [4], and the survey of [21] in China, two-crane combination on the same crane beam is considered.Additionally, if the maximum number in each observationinterval is recommended as the sample for investigation, thetwo-crane load occurrence probability is approximately 1.This would be the worst-case statistical interval for analysis,having nothing to do with time.Therefore, two cranes shouldbe consideredwhen determining the distribution of the craneload combination actions on the bending moment influenceline [21].

Mathematical Problems in Engineering 7

One crane is taken as always located at the worst-caseposition on the crane beam, and the second crane movesbased on a uniform distribution along the adjacent operatingrange (Figure 2(a)).

2.4.2. The Modified Turkstra Method for Crane Load Combi-nation Actions. When multiple cranes act on the influenceline, it is almost impossible that each crane operated attheir respective maximum load values. If these maximumvalues are used together, the combination result will betoo conservative. Therefore, when determining the designcombination coefficient of multiple crane loads, short-termcrane loads should be taken into account [21]. The heavyworking level (A6-A7) cranes are considered to be “onefull and one-half”; medium and light working level (A1–A5)cranes are considered as “one full and one empty.” Here the“full,” “half,” and “empty” terms refer to the crane hoist loadbeing full, half, and zero compared to the nominal hoist loadQHK [2, 4].This suggestion comes from engineering research[21] and takes the impact of the crane working level intoaccount.

To determine the distribution for multiple crane loadcombination actions, the Turkstra method is modified asfollows. With two cranes acting on the beam, the main crane(commonly with the larger crane load) is taken as the firstcrane located at the worst-case position, and the adjacentcrane is taken as the second crane, whose position is regardedas a uniform distribution along the operating range. Theoccurrence probability of each crane on the influence line is1.The first crane takes the “full load,” meaning the maximumload during the design reference period 𝑇. The second cranetakes the “half load” or “empty load,” respectively, for A6-A7 or A1–A5 working levels, meaning reduced arbitrarypoint-in-time values within the statistical interval 𝜏. ThemodifiedTurkstra combinationmethod shows theworst-casecombination of the two cranes for the bendingmoment of themid-span (point 𝐶), where 𝑄max,𝑇 is the maximum value of“full load” during the design reference period (the verticalload 𝑉max,𝑇 and the horizontal force 𝐻max,𝑇) (Figure 2(b)).𝑄0.5,max,𝜏 and 𝑄0,max,𝜏 are the “half” and ‘empty’ loads of thereduction arbitrary point-in-time values (the vertical load𝑉0.5,max,𝜏,𝑉0,max,𝜏 and the horizontal force𝐻0.5,max,𝜏,𝐻0,max,𝜏).

2.4.3. Probability Models for Multiple Crane Combinations.When the first and second cranes are located at the worst-case position along the influence line, the sumof the influenceline ordinates is expressed as 𝑦1,max = (L−K1)/2 and 𝑦2,max =(L − K2)/2 − dmin. Considering the probability combination,the actual values are expressed as 𝑦1 and 𝑦2. With themodified Turkstra method, 𝑦1 = 𝑦1,max, and 𝑦2 is takenas a random value in the random interval [0, 𝑦2,max]. Whenthe second crane position follows a uniform distributionwithin its operating range, 𝑦2 follows a piecewise uniformdistribution, expressed as a function of distance 𝑑 betweenthe adjacent wheels of the two different cranes. Based onthe analysis in Section 2.4.2, the first crane is fixed, andthe second crane follows a uniform distribution of distanceU[dmin, L/2], where dmin = (B1 + B2 − K1 − K2)/2 is the

minimum distance between the adjacent wheels of the twodifferent cranes, depending on the size of the cranes (Biand Ki in Figure 2). The means and standard deviations ofthe distance 𝑑 are calculated as 𝜇𝑑 = (L + 2dmin)/4 and𝜎𝑑 = (L − 2dmin)/4√3, respectively. The outer wheel of thefirst crane near the second crane is always located at themaximum bearing moment value position on the bearingmoment influence line (point C) (Figure 3). The influenceline ordinate of the first crane is expressed as 𝑦1 = 𝑦1,max =(L−K1)/2. The influence line ordinate of the second crane 𝑦2follows a piecewise uniform distribution:

𝑦2 ={{{{{

12 (L − 2𝑑 − K2) 𝑑 ∈ [dmin, L2 − K2)14 (L − 2𝑑) 𝑑 ∈ [L2 − K2, L2 ]

(7)

These two stages are shown in Figures 3(a) and 3(b),respectively. If the case probability shown in Figure 3(a) iscalculated as𝑃1 = 1−2K2/(L−2dmin), and the case probabilityshown in Figure 3(b) is calculated as𝑃2 = 2K2/(L−2dmin), for𝑃1+𝑃2 = 1, the influence line ordinates𝑦2 can be transformedas

𝑦2 = 𝑃12 (L − 2𝑑 − K2) + 𝑃24 (L − 2𝑑)= (1 + 𝑃1) L4 − 𝑃1K22 − (1 + 𝑃1) 𝑑2

(8)

The mean and standard deviation of 𝑦2 can be calculated as

𝜇𝑦2 = (1 + 𝑃1) L4 − 𝑃1K22 − (1 + 𝑃1) 𝜇𝑑2= L − 2dmin − K24 − K2 (L − 2dmin − 2K2)2 (L − 2dmin)= (L − 2dmin)4 − 3K24 + K2

2

(L − 2dmin)

(9)

𝜎𝑦2 = (1 + 𝑃1) 𝜎𝑑2 = L − 2dmin − K24√3 (10)

The crane load combination action 𝑆 applied by multiplecranes on the crane beam is a function of the crane loadsand their corresponding influence line ordinates. Probabilitydistributions and moment parameters of these independentvariables are known. To calculate reliability index, thesevariables can be considered as independent randomvariables,or simplified into a comprehensive variable 𝑆 for convenience:𝑆= {(𝑦1𝑄max,𝑇 + 𝑦2𝑄0.5,max,𝜏) 𝑜𝑟 (𝑦1𝑄max,𝑇 + 𝑦2𝑄0,max,𝜏)} (11)

All random variables in (11) are assumed as mutu-ally independent and uncorrelated. The 𝑆 distribution typedepends on the distribution types of these independentvariables. Considering the fact that crane load plays a decisiverole in the distribution type of the combination action, the

8 Mathematical Problems in Engineering

CA B

L/4

The first crane The second crane

L/4-d/2

Q1 Q1 Q2 Q2

Q1 Q2 Q2

＞ＧＣn

，/2-＋2

，/4-＋1/2 ，/4-d/2-＋2/2

(a) 𝑑 ∈ [dmin, L/2 − K2)

CA B

L/4

The first crane The second crane

L/4-d/2

L/2

Q1Q1 Q2 Q2

Q1 Q2Q2

，/2-＋2

，/4-＋1/2 ，/4-d/2-＋2/2

(b) 𝑑 ∈ [L/2 − K2, L/2]

Figure 3: Schematic diagram showing the two stages function of the influence line ordinate 𝑦2.

Gumbel (type Ι) distribution is assumed as the distributiontype, which simplifies the type of crane load combinationaction 𝑆 for FORM analysis [5]. Distribution parameters

are determined according to the error transfer formula.The assumed distribution and calculated parameters areevaluated in Section 3.2.

𝜇𝑆 = {(𝑦1𝜇𝑄max,𝑇+ 𝜇𝑦2𝜇𝑄0.5,max,𝜏

) 𝑜𝑟 (𝑦1𝜇𝑄max,𝑇+ 𝜇𝑦2𝜇𝑄0,max,𝜏

)} (12)

𝜎𝑆 = {(𝑦12𝜎𝑄max,𝑇2 + 𝜇𝑦22𝜎𝑄0.5,max,𝜏

2 + 𝜇𝑄0.5,max,𝜏2𝜎𝑦22)0.5 𝑜𝑟 (𝑦12𝜎𝑄max,𝑇

2 + 𝜇𝑦22𝜎𝑄0,max,𝜏2 + 𝜇𝑄0,max,𝜏

2𝜎𝑦22)0.5} (13)

3. Assessment of the Assumed Distributionand Calculated Parameters of 𝑆

In Section 2, the maximum value distribution of the craneloads during the design reference period is established. Usingthe modified Turkstra method, the probability model ofmultiple crane combination actions on the crane beam isimproved. In this section, these expression formulas distribu-tions will be estimated by the POT andMonte Carlomethods(much more precise theory but without expression formulamethod), respectively.

3.1. Checking the Maximum Crane Load Value DistributionUsing the Peaks-Over-Threshold Method. As a discontinu-ous stochastic process through time, crane load should beregarded as Poisson rectangular wave if the load amplitudechanges to a rectangular form during the design periodand the amplitude is constant for each time interval. Thisprocess assumes that both the amplitude change (𝑘) duringthe design period and action time interval (𝑡𝑖 = 𝜏𝑖) arerandom variables. The average number of action changes perunit-time is represented as 𝜆 = 1/𝜏.

Mathematical Problems in Engineering 9

Table 2: Statistics of combination actions.

Crane loadcategory

Experimental results Calculated resultsMean relative

errorSTD relative

errorMeanSTD

(StandardDeviation)

Skewness Median MeanSTD

(StandardDeviation)

vertical loads 202.87 18.91 0.28 202.01 200.11 22.66 0.014 0.198horizontal forces 26.86 7.64 0.98 25.73 28.04 7.90 0.044 0.034

Themagnitude of the maximum crane load is a nonnega-tive value in each operating time, whose probability distri-butions in different time intervals are identical. Maximumcrane load values during each action time interval (𝜏𝑖) arechangeless, and the probability distributions in each 𝜏𝑖 arethe same. The number of action changes 𝑘 during the designreference period follows a Poisson process [22]

𝑃 [𝑁 (𝑇) = 𝑘] = (𝜆𝑇)𝑘𝑘! exp (−𝜆𝑇) (14)

If the arbitrary point-in-time value probability distribu-tion is 𝐹𝑄𝜏(𝑥), the mean upcrossing rate at any threshold 𝑥per unit-time is

𝜐+ (𝑥) = 𝜆 (1 − 𝐹𝑄𝜏 (𝑥)) (15)

Invoking the Poisson assumption of upcrossings, theextreme value distribution during the design reference period𝑇 is calculated in terms of a generalized extreme value (GEV)distribution [23]. The maximum crane load is a pulse-typeload not a continuous-type load during each action timeinterval (𝜏𝑖); the cumulative distribution function (CDF) ofthe maximum crane load during the design reference period𝐹𝑄𝑇(𝑥) is

𝐹𝑄𝑇 (𝑥) = Prob {𝑄max (𝑇) ≤ 𝑥} = exp [−𝜐+ (𝑥) 𝑇]= exp [−𝜆𝑇 (1 − 𝐹𝑄𝜏 (𝑥))]

(16)

With (16), the maximum value CDFs of the verticalload and horizontal force calculated by (5) and (6) can beevaluated. Here a crane load with a medium working level(A5) and 𝑄HK=5t is chosen as an example for analysis. Withthis crane designed by the Dalian Lifting Machinery Group,the corresponding performance parameters are 𝑙 = 19.5 m, B= 4.77m, K = 4m, G = 13.9 t, g = 1.698t, andWC = 69 kN [24].The CDFs and PDFs (Probability Distribution Functions) ofthe arbitrary point-in-time value andmaximum value duringthe design reference period are determined by the maximumvalue method in Section 2.3. The POT method for verticalload and horizontal force is shown in Figure 4. Other cranecases demonstrate similar results.

In Figure 4, the distribution of the arbitrary point-in-timevalue is plotted in (2) and (3) with the statistical parameterslisted in Table 1 (the black line). The distribution of themaximum value during the design reference period 𝑇 = 50years is plotted using (5) and (6) with the maximum valuemethod (the red line, using the stationary binomial randomprocess) and (16) with the POT method (the blue line,

using the Poisson rectangular wave process). The maximumvalue is larger than the arbitrary point-in-time value for thesame guarantee rate. The maximum value method and thePOT method for the tail of the distribution functions are inapproximate agreement.Themaximumvaluemethod has thehighest quotient representing the most reliable result.

3.2. Checking Crane Load Combination Distribution ActionsDistribution Using Monte Carlo Simulation. To estimate theassumed distribution and the calculated parameters of thecrane load combination action 𝑆, some adequate sampledata are directly collected from the random data using theMonte Carlo simulation (MCS) method. Considering theprobability distributions of these random variables in (11)and the actual cases in Figure 3, as well as using the examplecrane mentioned in Section 3.1, let two identical cranes acton a beam with a span of 12 m. Through calculations, thereduction factors for the arbitrary point-in-time values ofthe vertical loads and the horizontal forces with the “emptyload” considering about A5 working levels here are 0.61 and0.25, respectively. We wrote a MATLAB program to obtainthe combination action date. To ensure simulation error isless than 0.005, the simulation must be larger than 1537times [2]. For 10000 times simulation and 10000 empiricalsamples, the empirical histograms and CDFs are shown inFigure 5. The assumed distributions (PDFs and CDFs) withthe calculated parameters are shown in Figure 5. Using theChi-square goodness-of-fit test, the data does not reject thenull hypothesis of the Gumbel (type Ι) distribution at thedefault 5% significance level. Distribution parameters of therandomexperimental combination actions and the calculatedresults are summarized in Table 2. For the comparison resultsof the vertical load in Figure 5(a), when the crane loadcombination action 𝑆 is less than 225, the probability of 𝑆appearing is slightly higher than in the Gumbel distributionmode; and when it is more than 225, the probability is slightlylower. That means with the improved load model here, thecalculated value of reliable indicator is slightly lower thanactual value with smaller load values (conservative for theproject), and slightly higher with larger load values (notconservative for the project).With the conservative reliabilitycalculation method of FOSM [23], and the less deviation ofthe distribution parameters (in Table 2), the final calculationshould be acceptable. For the comparison results of thehorizontal force in Figure 5(b), the improved load model andthe actual value are almost consistent. So, the Gumbel (typeΙ) distribution and the distribution parameters determinedin Section 2.4.3 for the crane load combination actions aresatisfactory.

10 Mathematical Problems in Engineering

PDF

Line 1Line 2Line 3

CDF

Line 1Line 2Line 3

40 50 60 7030value of the vertical loads

0

0.2

0.4

0.6

0.8

1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

40 50 60 7030value of the vertical loads

(a) Validation of the vertical load model

PDF

Line 1Line 2Line 3

CDF

Line 1Line 2Line 3

10 15 205value of the horizontal forces

0

0.05

0.1

0.15

0.2

0.25

10 15 205value of the horizontal forces

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Validation of the horizontal force model

Figure 4: CDF and PDF of the arbitrary point-in-time distribution, the maximum value distribution during the design reference perioddetermined by (a) the maximum value method and (b) the POT method for the vertical load and the horizontal force (for interpretation ofthe references to color in this figure legend, the reader is referred to the web version of this article). Note. Line 1: distribution of the arbitrarypoint-in-time value. Line 2: distribution of the maximum value determined by POT method. Line 3: distribution of the maximum valuedetermined by the maximum value method.

Mathematical Problems in Engineering 11

PDF

Empirical HistogramGumbel PDF

0

0.012

150 200 250 300 350 400100the vertical loads combination action the vertical loads combination action

0

300

X: 225Y: 0.8718

CDF

Empirical CDFGumbel CDF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F(x)

350150 200 250 300 400100

(a) The comparison results of the vertical load

PDF

Empirical HistogramGumbel PDF

20 40 60 80 1000the horizontal force combination action

0

0.035

0.07

0

500

1000 CDF

Empirical CDFGumbel CDF

0

0.2

0.4

0.6

0.8

1

F(x)

20 40 60 80 1000the horizontal force combination action

(b) The comparison results of the horizontal force

Figure 5: The distributions of multiple crane load combination actions simulated using the MCS method (empirical histogram and CDF(the blue line)) and the approximate method described in Section 2 (Gumbel PDF and CDF (the red line)) for the (a) vertical load and (b)horizontal force (for interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

4. Code Design Standard Values andQuota Levels

China “load code for the design of building structures” [4]provides the design load and their actions for structuredesign and engineering projects. The load code specifiesthe value principle for crane load and multiple crane load

combination actions for runway beam design. For overheadtraveling cranes, the nominal maximum wheel pressure WCprovided by the crane supplier is taken as the design standardvertical load value. WC is calculated as the sum of the bridgeweight, the rated capacity, the trolley weight, and the actualmaximum load effect (Figure 1). Considering the inducedvertical impact or vibration force, WC is increased by a

12 Mathematical Problems in Engineering

dynamic factor, about 1.05 for A1-A5 working level cranesand 1.1 for A6-A8 working level cranes. The design standardvalue of the longitudinal horizontal force is calculated as 10%of the maximum wheel loads of the crane. The transversehorizontal force for electrically powered trolleys is calculatedby the following percentages for the sum of the rated capacityand the weight of the hoist and trolley [4].

For soft hook cranes: QHK ≤ 10t 12%10t< 𝑄𝐻𝐾 <75t 10%QHK ≥ 75t 8%

For hard hook cranes: 20%When calculating multiple crane load combination

actions for beam design, two cranes simultaneously locatedin the worst-case positions are considered. Moreover, consid-ering the occurrence probability in this case, the combinationresults are reduced by a coefficient, approximately 0.9 for A1-A5 working level cranes and 0.95 for A6-A8 working levelcranes.

With the specified crane load and combination principlesfor runway beam design, characteristic design values can becalculated. With the determined distribution of crane loadand multiple crane load combination actions, the designvalues can be evaluated with a quote level for exceedingprobability towards an unfavourable value during the refer-ence period [25]. Considering the example in Section 3, thestandard design values of the vertical load and the transversehorizontal force reach 99.98% and 80.69%of quota levels, andthe design crane load combination actions of the vertical loadand the transverse horizontal force reach 99.9% and 97.43%of quota levels. The characteristic value of vertical load isexcessively conservative.The standard value of the horizontalforce is somewhat excessive and hazardous because forcescaused by skewing of the crane in relation to itsmovement arenot considered. However, the crane load combination actionis much safer and conservative.

5. Summary and Conclusion

Crane load, as a two-dimensional stochastic process, isanalysed using probability methods. The arbitrary point-in-time value distribution of the crane load values varyingin time follow the Gumbel (type Ι) distribution improvedwith engineering surveys and K-S test results. The maximumvalue distribution of the crane load values during the designreference period is determined as the Gumbel (type ) dis-tribution using the block maxima method of the stationarybinomial random process hypothesis and evaluated with thePOTmethod for the filtrationPoissonprocess.Themaximumvalue is larger than the arbitrary point-in-time value with thesame guarantee rate level.

Theprobability distribution ofmultiple crane load combi-nation actions is taken as Gumbel (type ) distribution usingthe modified Turkstra rule and the influence line methods.The probability distribution is accepted by evaluation withthe Monte Carlo simulations method and the Chi-squaregoodness-of-fit test.

The model parameters of the above distribution withthe used assumptions and the chosen value parameters aredetermined by statistical surveys and load experiments in

China. If some of the used assumptions are modified or someof the different chosen value parameters are changed withreliable survey results, it can be calculated with the samemethod.

After crane load probability models improved, the designstandard values specified in China load code for runwaybeam design are assessed with probability significance. Withhigh quota levels, the specified crane load and combinationactions are conservative and redundant for design. For thestandard codes used in China, the combination coefficientsto calculate the design value of the multiple crane loadcombination actions could be reduced appropriately afteranalysis of the design reliability calibration results of the cranebeams and columns with different plant spans, componentsizes, various load combinations, and crane working levels.It will be discussed in further research and another paper.

Data Availability

The data used to support the findings of this study areincluded within the article and [4].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The support for this work was provided in part by theNational Key R & D Program of China Grant no. 2016YFC0701301-01. This support is greatly appreciated.

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