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Nonlinear frame analysis

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2011 Elsevier Ltd. All rights reserved.

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Many studies have shown that failures of reinforced concretestructures that occur during their construction phase are oftentraceable to the collapse of formwork shoring systems [1,2]. Theeconomic and legal consequences of such structural failures canbe disturbing. The Guangxi (China) Medical University library acci-dent in 2007, in which seven construction workers were killed [3],

abilities). Herein, structural reliability is dened as the probabilitythat the structure will meet its performance objectives for strengthor serviceability over a given period of time.

A limited number of structural reliability analyses have beenconducted for temporary structures during construction. Gromala[13] reviewed the reliability requirements for wood-based accessscaffold planks in codes and standards, and related those require-ments to allowable stresses published by lumber grading agencies.The statistical information of load and resistance used in the reli-ability calculation are based on assumptions rather than experi-mental data. Charuvisit et al. [14] studied risk of access scaffolds

Corresponding author. Tel.: +61 2 93513923; fax: +61 2 93513343.E-mail addresses: hao.zhang@sydney.edu.au (H. Zhang), kim.rasmussen@syd

Engineering Structures 36 (2012) 8189

Contents lists available at

Engineering

lseney.edu.au (K.J.R. Rasmussen), bruce.ellingwood@ce.gatech.edu (B.R. Ellingwood).gers (horizontal members), braces and adjustable jacks. Theuprights are connected to the ledgers by means of various types ofconnections such as cuplok joints or wedge-type joints. Note thatanother type of scaffolds commonly used in construction is the ac-cess scaffolds, which are mainly used to support light to moderateloads fromworkers, small constructionmaterial and equipments forsafe working space. The access scaffold is not considered in thisstudy.

details [12]. Unfortunately, even with accurate structural modelsthat have been calibrated by experimental tests, the actual perfor-mance of steel scaffolds cannot be predicted with certaintybecause large uncertainties in structural properties and appliedloads will always be present. Recent advances in structural reliabil-ity theory permit these uncertainties to be quantied using proba-bilistic methods, and the safety of scaffold structures to berationally assessed in terms of their limit-state probabilities (reli-Probability-based designScaffoldShoringSteelStructural engineeringStructural reliability

1. Introduction

Steel scaffolds are commonly usestruction as shoring systems to supwith other shoring systems such as wfold-type systems have the advantalarger panel sizes. A steel scaffold syrights (tubular columns, referred to a0141-0296/$ - see front matter 2011 Elsevier Ltd. Adoi:10.1016/j.engstruct.2011.11.027inforced concrete con-rmwork. As comparedmetal posts, steel scaf-increased strength andormally consists of up-dards in industry), led-

is a recent example of a catastrophic failure of steel scaffoldshoring system. Such failures have prompted numerous efforts toinvestigate the structural performance of steel scaffolds bothexperimentally and numerically [411]. More recently, second-order inelastic structural analysis (advanced analysis) has beenused to predict the behavior and limit state load carrying capacityof steel scaffold structures, including material and geometric non-linearities, initial geometric imperfections, and semi-rigid jointConstruction formworkLoads (forces)

the analyzed scaffold structures. The reliability framework can be used to improve the current workingload limit basis for the design of steel scaffold structures and make scaffold design risk-consistent.Reliability assessment of steel scaffold sh

Hao Zhang a,, Kim J.R. Rasmussen a, Bruce R. Ellingwa School of Civil Engineering, University of Sydney, NSW 2006, Australiab School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlant

a r t i c l e i n f o

Article history:Received 9 March 2011Revised 21 November 2011Accepted 21 November 2011Available online 26 December 2011

Keywords:

a b s t r a c t

Many failures of concrete ssystems. A reliability analrecent survey data on geominelastic structural analysiticmodels for loads acting oure, the effects of different

journal homepage: www.ell rights reserved.ring structures for concrete formwork

d b

30332, USA

tures during construction are traceable to the collapse of formwork shoringfor typical steel scaffold shoring structures is presented herein utilizingic and mechanical properties of steel scaffold members, and a second-orderdel. Published shore load surveys were analyzed to develop the probabilis-caffolds during concrete placement. The paper investigates themode of fail-dom variables on the variability of structural strength, and the reliability of

SciVerse ScienceDirect

Structures

vier .com/ locate /engstruct

judgmental in nature. The reliability associated with current steel

ity efciently using FORM. If the structural resistance and loads aremutually statistically independent normal variables, the limit-stateprobability Pf is given by

Pf U R D Lr2R r2D r2L

q0B@

1CA Ub 3

where U() is the standard normal distribution function. In thispaper, Xn, X, rx and VX denote the nominal, mean, standard devia-tion and coefcient of variation (COV) for a random variable X,respectively. The reliability index, b, has been customarily used as

crete. Fig. 1 shows a typical history of a shore load (a combination

g Structures 36 (2012) 8189scaffold design practice is unknown, making it difcult to extendscaffold design beyond current practices to meet the emergingneeds of the construction industry.

This paper presents a reliability assessment of a typical steelscaffold structure designed in accordance with the WLL design for-mat. The reliability implications for the scaffolds designed byadvanced nonlinear analysis and the customary elastic analysisare compared. The probabilistic studies utilize the available litera-ture and data on shore loads, the recent survey data on geometricand mechanical properties of steel scaffold members, and a sec-ond-order inelastic structural model. The effect on structural reli-ability of human error is also discussed.

2. Structural reliability analysis

We consider the ultimate strength of a scaffold structure, inwhich the system limit state is dened as failure by overload. Steelscaffolds are designed towithstand vertical loads as well as the hor-izontal loads due to lateral concrete pressure orwind, etc. The form-work is usually restrained externally by the completed part of thepermanent structure, to which the horizontal loads are transmitted.Thus the design of steel scaffolds is generally governed by the ver-tical loads. This study only considers the combination of dead andlive load due to construction. The limit-state probability, Pf, is de-ned as:

Pf PR D L 6 0 2in which P() = probability of the event in the bracket, R = ultimatestructural strength (resistance), D = applied dead load effect, andL = live load effect. Since the loads vary with time, D + L in Eq. (2)is interpreted as the maximum total load effect during the construc-tion process.

Eq. (2) can be evaluated using standard reliability analysismethods such as Monte Carlo simulation or rst-order reliability-method (FORM) if the probabilistic information of structural resis-tance and loads is available. This study assumes that the loads act-ing on each upright of a scaffold at the point in the process whereunder strong wind by wind tunnel test and eld measurement ofwind loads acting on actual scaffolds. The study focused on thesafety of the ties connecting the access scaffolds to external build-ing walls. Epaarachchi et al. [15,16] calculated the probability ofstructural failure during the construction of typical multistoryreinforced concrete buildings. It was assumed that system failureoccurs when the slab strength (in exure or shear) limit state isviolated. Failure of shores/reshores was not considered in theirstudy.

At present, industry practice for the design of steel scaffoldstructures are based on the working load limit (WLL) philosophy.The basic WLL design equation has the format

Rn P F:S:Dn Ln 1in which Rn = nominal system strength computed using the analysisand design methods accepted by the design code under consider-ation for the nominal values of the material and sectional proper-ties, F.S. = factor of safety, Dn and Ln are nominal dead and liveloads due to construction. Rn/F.S. represents the allowable workingload. The construction dead load includes the weight of concreteand formwork. The construction live loads acting on scaffoldsinclude the weight of construction personnel, equipment, stackedmaterial, and an allowance for localized mounding during concreteplacing. The factor of safety, typically ranging from 2.0 to 3.0, is

82 H. Zhang et al. / Engineerinthey reach their maximum values are perfectly correlated. Thus,only two random variables are required to describe the dead andlive loads. This assumptionmakes it possible to analyze the reliabil-of the dead and live load effects) [19,20, c.f.]. The maximum shoreload typically occurs during the period of concrete placement. Aswill be described in Sections 3.1 and 3.2, the dead load is the dom-inant component of the shore load during concrete placement.Therefore, the maximum shore load effect occurs when the deadload is fully developed while the live load assumes its instanta-neous value, which may be less than the maximum live load[19,20]. At the end of the placement, the live load effects diminishbecause construction activities lessen. The scaffold then supportsonly the weight of the slab. The difference between the maximumshore load and the load just after the casting provides an estima-tion of the live load effect. The construction live load may increasesignicantly after the pouring due to increased constructionan alternative to Pf in probability-based limit-states design [17].For correlated non-normal random variables, a Rosenblatt transfor-mation may be applied to produce equivalent independent normalvariables for use with FORM. See Ref. [18] for details.

In this study, the probabilistic load models for scaffolds wereobtained from the load survey data published in the literature.The statistical characteristics of the structural resistance werederived using simulations and an advanced nonlinear nite ele-ment analysis (FEA). The simulation studies also provide usefulinformation on the relative signicance of different sources ofuncertainty to the overall variability in structural strength. Theload and resistance data were then synthesized using FORM toestimate the structural reliability.

3. Probabilistic load models for scaffolds

The vertical loads acting on a scaffold can be categorized asdead and live loads. The dead load is due to the weight of concreteand formwork. It increases as the concrete placement progresses,and reaches its maximum at the end of the placement. The liveload consists of the weight of construction personnel, equipment,stacked material, and the effects of any impact during concreteplacement. The magnitudes of the live loads depend on the stagesof construction, i.e., before, during, and after placement of con-Fig. 1. Typical shore load history.

conducted for construction loads. Several analytical models [2225] have been proposed to study shoreslab interactions in multi-

g Ststory constructions. Those studies were focused on determiningthe loads on partially cured slabs. In [26] and [27], constructionlive loads on slab formworks before and after concrete placementwere surveyed. Other load surveys measured directly the loadeffects (axial loads) on formwork supporting elements[19,21,28,20,29]. A brief summary of pertinent shore load surveysis given below.

3.1. Live loads

The live load of interest in this study is the companion live loadwhen the dead load reaches its maximum at the end of the pour.The companion live load can be estimated by subtracting the max-imum shore load by the load just after the pour. This companionlive load was examined by Fattal [19] and Ikheimonen [20]. Fromthe data reported in [20], it is found that: (1) the load due to theweight of workmen and equipment is very small in comparisonwith the dead load; (2) when concrete is placed by pumping, theimpact load is insignicant; (3) when the crane-and-bucket meth-od is used for concreting, the COV VL equals 0.7, and the mean liveload L equals approximately 0.3 Ln if the nominal live load Ln is cal-culated using a design formwork live load of 2.4 kPa (50 psf) asspecied in ACI 347 [30]. The investigation conducted by Fattal[19], in which the crane-and-bucket method was used for concret-ing, suggested the similar result: L=Ln 0:31 and VL = 0.71 for thetributary area method, and L=Ln 0:27 and VL = 0.54 if the moreaccurate distributed method was used to calculate the shore loads.Based on this limited data, it is assumed in the present study thatthe live load on scaffolds has a Type I extreme distribution, withVL = 0.7 and L=Ln 0:3 for a design formwork live load of 2.4 kPa(50 psf). It is worth emphasizing that L=Ln depends on the designformwork live load, and that the value may vary considerably fromstandard to standard. For example, the American Concrete Institutespecies a design live load of 2.4 kPa (50 psf) for all constructionactivity and storage of material. However, the shore loads tend todecrease as the concrete slab gains partial strength and starts car-rying part of the applied loads [20,21]. Therefore, the most criticaltime for scaffold shoring system is typically during the process ofconcrete placement. A survey of falsework collapses [1] has shownthat 74% of falsework failures occurred during concrete placementoperations. This stage of construction represents the critical condi-tion for scaffold design, and it is this stage of construction that thepresent paper addresses.

During the stage of concrete placement, the newly poured slabhas no rigidity, and all loads are supported by the scaffolds. The ef-fect on structural reliability of human error is also discussed. In de-sign practice, the vertical loads acting on the uprights of a scaffoldare most often calculated using the tributary area method, assum-ing that the uprights support the weight of the newly poured slaband a code-specied basic design live load (on the formwork).Alternatively, the shore load can be obtained by calculating thesupport reactions for the bearer beams loaded by concentratedloads from the joists (referred to as distributed method in [19]and beammethod in [20]). It has been shown that the distributedmethod or beam method is only slightly more accurate than thetributary area method [19,20]. In the following discussion, unlessnoted otherwise, the shore load was calculated using the tributaryarea method.

As compared to the substantial work that has been done foroccupancy loads in buildings, relatively limited studies have been

H. Zhang et al. / Engineerinstages [30], while in the Australian Standard for formwork for con-crete AS3610 [31], the design live load varies for different con-struction stages; it is 1.0 kPa for the phase of concrete placement.3.2. Dead load

Theoretically, the dead load transmitted to the uprights of a scaf-fold can be calculated with reasonable accuracy as the variability inconcrete weight is relatively low (a COV on the order of 0.060.09[17]). However, many studies have shown that the measured-to-calculated value for the dead load applied to scaffolds has arelatively large variation [1921,29].

Fattal [19] studied a six-storey at plate ofce building with aslab thickness of 203 mm (8 in.). One-bay steel scaffold portalframes consisting of two uprights and three horizontal cross-barswere used as shores. Eleven instrumented uprights were placedwithin an interior bay under the formwork for the fourth oor slab.Itwas observed that the load in uprightNo. 4was signicantly lowerthan its theoretical share of the superimposed load, and the adjacentupright No. 10 picked up the additional load. The author suggestedthis might be due to premature yielding or misalignment of uprightNo. 4. If uprights Nos. 4 and 10 are excluded from the survey data,D/Dn has amean of 0.92 and a COV of 0.35 for the tributary area meth-od, and amean of 0.93 and a COV of 0.32 for the distributedmethod.

Rosowsky et al. [21] andKothekar et al. [32] reported a shore loadsurvey at a Federal prison site in Beckley, WV. Two identical smallpour areas (8.10 m2 each) and one large pour area (51.28 m2) weresurveyed during both the casting and curing phases. Each of thesmall pour areas was supported by a steel scaffold tower with 4 up-rights. The formwork shoring system for the large pour area con-sisted of 16 steel posts, eight of which were instrumented. COVs of33.3% and 28.5% were observed among the dead loads for the smallpour areas and the large pour area, respectively. It was also foundthat the tributary area load was close to the average of the actualshore loads.

Ikheimonen [20] surveyed shore loads at four bridge and veresidential building sites. The slab thickness varied from 950 mmto 1310 mm for the bridges, and 150 mm to 350 mm for the resi-dential buildings. A total of 66 shores were instrumented. Mea-sured shore loads were subdivided into load due to formwork,load due to concrete, and the maximum load during the placement.The measured-to-calculated value for the dead load was found tohave a mean of 0.9 and a COV of 0.29 for the tributary area method,and a mean of 0.99 and a COV of 0.30 for the beam method.

Puente et al. [29] investigated the construction of a concretebuilding with seven stories and four levels of underground parking.The thickness of the slab was 250 mm. A total of 34 instrumentedsteel post shores were placed on a typical 7.2 5.65 m interiorbay of each parking level. The study suggested that the dead loadD has a mean value approximately equal to the nominal Dn and aCOV around 0.25.

On the basis of the above survey results, it appears that D Dn,and VD ranges from 0.25 to 0.3. Note that the dead load and occu-pancy live load for ordinary buildings have typical COVs of 0.1 and0.25, respectively [17]. It is evident that the variability of dead loadon scaffolds ismuchgreater than thevariability associatedwith con-crete weight. The reason for this additional uncertainty is not fullyunderstood but it might be related to differential settlement of theuprights, or imperfections in the scaffold installations such as lackof bearing between the uprights and the formwork/bearer beams[19,20,33,34]. Also, the tributary area concept might be a source ofthe discrepancy since strictly speaking, the tributary area conceptis valid only when the soft formwork is very rigid. It appears thatunder current normal construction practice, a COV of 0.30 is repre-sentative for the dead load effect on the scaffolds. Thus, it is assumedin this study that the dead load is normally distributed, withD=Dn 1:0 and VD = 0.30. It should be emphasized that VD includes

ructures 36 (2012) 8189 83the effect incurred by minor imperfections in construction withinthe limits of practical tolerances; it does not account for the effectof (gross) human error.

4. Strength of scaffold structure

4.1. Example structure

A typical steel scaffold tower as shown in Fig. 2 was considered.The system features 3 stories in a 1 bay 1 bay arrangement, withequal storey height of 1.5 m and a bay width of 1.829 m in bothdirections. The uprights and ledgers are connected via cuplokjoints. A cuplok joint consists of a xed bottom cup that is weldedto the uprights and a sliding upper cup. The blade ends of the hor-izontal members (ledgers) are placed into the bottom cup. Toengage the joint, the top cup is slid down over the blades and ro-tated by striking its lugs with a hammer. This type of connectionrequires no bolting or welding and allows for simple and speedyerection. Experimental tests have shown that cuplok joints aresemi-rigid and generally show looseness with small stiffness atthe beginning of loading due to a lack of t [35]. The jacks at thebottom and top of the frame are adjustable and are assumed tobe of the same length. Two extreme values for the jack extensionsare considered, i.e., 100 mm and 600 mm. The correspondingframes are referred to as Scaffolds 1 and 2, respectively. Theuprights, ledgers and braces are steel circular hollow sections(CHS). The jacks are solid steel rods. Table 1 summarizes the nom-inal section dimensions and yield stresses for the members.

The mean and COV in the member out-of-straightness and loadeccentricity are taken from the results of the eld measurements

[35]. The out-of-straightness for uprights with spigot joints (i.e.,splices) is modelled by a lognormal distribution with a mean of

Table 1Nominal member dimension and yield stress.

Dimension (mm) Fy (MPa)

Upright 48.3 4 450Ledger 48.3 3.2 350Brace 48.3 3.2 400Jack 36 430

Table 2Description of basic random variables.

Random variable Nominal value meannominal COV Distribution Refs.

Fy (upright) 450 MPa 1.05 0.1 normal [36]Fy (jack) 430 MPa 1.05 0.1 normal [36]Thickness (upright) 4 mm 1.0 0.08 lognormal [17]Diameter (jack) 36 mm 1.0 0.08 lognormal [17]Joint stiffness (k2) 77.6 kNm/rad 1.0 0.2 normal [35]Load eccentricity b/4 a 1.0 0.55 lognormal [35]Out-of-straightness L/500 0.65 0.6 lognormal [35]

a b = width of the bearer beam.

84 H. Zhang et al. / Engineering Structures 36 (2012) 81894.2. Random properties of steel scaffold members

Seven basic random parameters were identied in the presentstudy: (1) yield stress of the uprights, (2) yield stress of the jacks,(3) initial out-of-straightness of the uprights, (4) load eccentricity,(5) thickness of the CHS of the uprights, (6) diameter of the jacks,and (7) joint stiffness of the semi-rigid cuplok joints. All randomvariables are assumed to be mutually statistically independent.The following paragraphs and Table 2 summarize the statisticalinformation for the basic random variables. Detailed informationmay be obtained from the references cited.Fig. 2. A steel scaffold towL/770 and a COV of 0.6. The nominal value for out-of-straightnessis assumed to be L/500, a typical value adopted in industry. Themagnitude of load eccentricity is modelled by a lognormal distri-bution with a mean of b/4 and a COV of 0.55, in which b is thewidth of the bearer beam; this mean value is equal to the nominaldesign value specied in the Australian Standard for formwork forconcrete AS3610 [31]. It is assumed that the measured data forout-of-straightness and load eccentricity are representative of nor-mal quality construction.

The stability of a steel frame is affected not only by the magni-tudes of load eccentricity and column out-of-straightness but alsotheir patterns. As specied in AISC 360-10 [37], the most unfavour-able pattern should be assumed for the initial geometric imperfec-tions such that they provide the greatest destabilizing effect. Theeld survey [35] suggested that the pattern of load eccentricity ap-er with cuplok joints.

pears to be random and that the load eccentricities of uprights thusmay be assumed to be uncorrelated. However, this conclusion isdifcult to generalize without additional eld data. In the presentstudy, then, only the magnitudes of the out-of-straightness andload eccentricity are taken to be random variables. All load eccen-tricities are oriented in the same direction to maximize the desta-bilizing effect. The pattern of member out-of-straightness isdetermined using an elastic buckling analysis. The rst elasticbuckling mode is scaled and added to the perfect geometry todetermine the form of member out-of-straightness [38].

For ordinary steel structures fabricated with average qualitycontrol, it is reasonable to assume that the mean sectional proper-ties are equal to the Handbook values, with a typical COV of 0.05[17]. However, since steel scaffold members are reused from onejob to another and new members are mixed with old ones, manybelieve that this gives rise to additional uncertainty in sectionalproperties. Accordingly, it is assumed that themean-to-nominal va-

relation for cuplok joints. The joint stiffness appears to be depen-

H. Zhang et al. / Engineering Stdent on the number of ledgers connected at the joint. Three casescan be identied, i.e., 4-way (interior joint), 3-way (edge joint)and 2-way joint (corner joint). Only the 2-way joint is applicableto the scaffolds considered in this study since they are one bay byone bay. Among the three stiffness parameters (k1, k2 and k3), thevariations of k1 and k3 are relatively small. Thus they were treatedas deterministic and set equal to 40.9 and 4.6 kN m/rad, respec-tively. k2 was taken to be a normal variable with a mean of 77.6kN m/rad and a COV of 0.2.

4.3. Statistical characteristics of system strengths

To derive the statistical information for the strength of the steelscaffolds, simulation approach was employed. This approach re-quires an accurate means to predict the actual system strength,and knowledge of the probability distributions for each basic ran-dom parameter. In this study, a three-dimensional second-orderinelastic nite elementmodel [12] was used in simulation to obtainlue is 1.0 and the COV is 0.08 for the thickness of the uprights andthe diameter of the jacks. Researchers at the School of Civil Engi-neering at the University of Sydney measured the sectional proper-ties for a sample of 54 uprights taken from stocks of material in use.That investigation showed that the statistics for sectional proper-ties assumed above appear to be on the conservative side.

The relation between the moment and rotation (Mh) of the cu-plok joints is semi-rigid. Based on joint tests [35], a simple tri-linearmodel as shown in Fig. 3 is adopted in this study to idealize theMhFig. 3. Idealized Mh curve for two-way cuplok joint.accurate theoretical predictions of system strength. The cuplokjoints were modelled as semi-rigid using the tri-linear Mh modelas shown in Fig. 3. The stressstrain relation for the steel in allCHS is assumed elastic perfectly-plastic. For simplicity, a pinnedconnection is assumed for the base connection, i.e., exural rota-tions are free to occur while twist rotations are prevented. Alterna-tively, the base connection may be modelled as semi-rigid using anelastic rotational spring krb [38]. The top of each upright is re-strained from translation in the horizontal directions. This assump-tion is justied primarily because in construction practice, thehorizontal formwork is usually restrained externally by the (par-tially completed) permanent structure. The bearer beam (seeFig. 2) generally places a constraint on the rotation of the U-head,which can be modelled by an elastic rotational spring krt appliedat the top of each upright [12]. Values of krb and krt depend onmanyfactors such as the applied load and properties of the bearers, andare difcult to quantify and generalize. In the present study, the ef-fects of krb and krt are not included. It is believed that neglectingtheir effects is conservative.

Concentrated vertical loads of equal magnitude are applied atthe top of each upright at an eccentricity with respect to the cen-troid axis, as described in Section 4.2. The loads were increasedincrementally until system failed by instability. Details of thestructural modelling can be found elsewhere [12].

Statistics of system strengths were simulated using the LatinHypercube (LHC) sampling method and the basic random variablessummarized in Table 2. In each simulation, the ultimate systemstrength was obtained using the second-order inelastic analysis.The LHCmethod is a stratied sampling scheme to derive the statis-tical properties of a response variable, and is highly efcient for acomplex system. Comparedwith the direct Monte Carlo simulation,fewer samples are required in LHC method to cover the probabilityspace and to achieve a desired mean-square error. Several investi-gations were performed for each scaffold. The overall uncertaintyin scaffold capacity is obtained by treating all parameters as ran-dom. Treating one parameter as a random variable and the remain-ing parameters as deterministic and equal to their respectivenominal values provides an indication of the relative signicanceof different random properties to the variability in system strength.For simplicity, perfect correlation is assumed for the material andgeometric properties between upright-to-upright, jack-to-jack,and for the stiffness between joint-to-joint. No correlation existsbetween upright-to-jack, upright-to-joint, or jack-to-joint. Typi-cally, 300 LHC simulations were performed for each uncertaintyanalysis of each scaffold.

The nominal strengths of the two scaffold structures were rstevaluated as reference points using the second-order inelastic FEA,and the nominal values for the material, geometric and stiffnessproperties, as summarized in Table 2. The nominal load eccentric-ity is taken as b/4 = 20 mm, assuming that the width of the bearerbeam is 80 mm.

4.4. Scaffold 1100 mm jack extension

With the nominal properties, the ultimate load that can be car-ried by Scaffold 1 is found to be 107.2 kN per upright. The uprightsbuckled in an S curvature with negligible overall sway as the nalfailure shape, indicating the failure of the system is due to inelasticexural buckling of the upright members. The maximum yield ra-tio, dened as the percentage of cross-section that has yielded, isabout 40% for the uprights at the rst storey. This justies the useof an inelastic analysis. The jacks, ledgers and bracings are all withintheir elastic limits at failure.

ructures 36 (2012) 8189 85Table 3 presents the simulated strength statistics for Scaffold 1considering each random variable separately as described above.The results show that R=Rn is around unity for all cases. The

variability in system strength mainly arises from the uncertaintiesassociated with load eccentricity, material (Fy) and geometric prop-erties (thickness) of the uprights. The strength histogram associatedwith random load eccentricity is shown in Fig. 4, with a COV of12.9%. This is not surprising considering the large variability as-sumed in the load eccentricity (a COV of 55%). Note that the histo-gram is negatively skewed. Since Scaffold 1 with nominalproperties failed due to inelastic exural buckling of the uprights,it is expected that the system strength will be sensitive to the vari-ations of those parameters inuencing the inelastic buckling

diameter is given in Fig. 6 for Scaffolds 1 and 2. The slope of thecurve is a measure of the sensitivity of strength to the jack diame-ter. From Fig. 6 it is evident that the strength of Scaffold 2 is verysensitive to the jack diameter. In the case of Scaffold 1, the curveis essentially at when the jack diameter is greater than approxi-mately 33 mm since the failure mode, inelastic buckling of the up-rights, is not affected by the jack diameter. However, a suddenchange in the slope of the curve is observed when the jackdiameter becomes less than 33 mm. This is because the failure

(b)

Fig. 4. Histogram for system strength under random load eccentricity. (a) Scaffold 1and (b) Scaffold 2.

2.5

3

3.5

1

1.5

2

2.5

3

3.5

3

0

0.5

1

1.5

2

2.5

3

3.5

0.37 0.55 0.74 0.92 1.11 1.29

Prob

abilit

y de

nsity

Normalized system strength (R/Rn)

Fig. 5. Histogram for Scaffold 1 strength with all random variables (tted with athree-parameter Weibull distribution).

86 H. Zhang et al. / Engineering St4.5. Scaffold 2600 mm jack extension

The nominal model of Scaffold 2 failed at 44.64 kN, indicatingan almost 60% reduction to the system nominal capacity whenthe top and bottom jack extensions increased from 100 mm to600 mm. A different failure mode, in which the lateral deforma-tions were essentially conned to the jacks and the scaffolding inbetween shows little deformation, was observed in this case. Theuprights and ledgers experienced only small curvature andremained elastic. The system strength is governed by the bucklingof the jacks and the strength of other members and connectionstiffness of the cuplok joints are not being utilized.

Table 3 presents the results for Scaffold 2 obtained from simu-lation studies. The diameter of the jack and the load eccentricityare the most inuential factors for the variability in structuralstrength. It is worth noting that the variability in jack diameter(COV = 8%) led to the greatest variability in system strength(VR = 12.7%). For all other basic random variables, however, thestrength variabilities are smaller than those associated with therandom properties. The variation of system strength with jack

Table 3Effects of random variables on system strength R.

Random variable Scaffold 1 Scaffold 2

R=Rn VR R=Rn VR

Fy (upright) 1.02 4.85% 1.0 0Fy (jack) 1.0 0 1.0 1.28%Thickness (upright) 1.0 5.08% 1.0 2.70%Diameter (jack) 0.99 2.48% 0.99 12.67%Out-of-straightness 1.03 4.0% 1.0 0Load eccentricity 0.99 12.91% 1.0 8.90%strength of the uprights. For example, VR takes the values of 4.9%and 5.1% for the random yield stress and random thickness of theuprights, respectively. On the other hand, the joint stiffness andout-of-straightness appear to have modest impacts on the variabil-ity in system strength. Large changes in these two random proper-ties are associatedwith relatively small changes in system strength.The yield strength of the jacks has an inconsequential effect on VRsince at failure the jacks were in elastic range and buckling of thejacks did not occur.

Fig. 5 shows the strength histogram for Scaffold 1 when all therandom variables are included, assuming that they are statisticallyindependent. The mean strength R equals 1.02Rn (108.8 kN) with aCOV of 17%. The histogram is noticeably skewed to the left, a similartrend observed in the strength histogram associated with randomload eccentricity (see Fig. 4). Four probability distribution functionswere examined to represent the strength distribution of Scaffold 1:the normal, lognormal, Beta and Weibull distributions. Two statis-tical hypothesis tests, namely KolmogorovSmirnov test andAndersonDarling test, were performed using the simulated data.It was found that the strength of Scaffold 1 can be best tted by athree-parameter Weibull distribution.Joint stiffness 1.0 2.10% 1.0 1.48%All random variables 1.02 17% 1.0 16%(a)

ructures 36 (2012) 8189mode for Scaffold 1 changes to inelastic buckling of the jacks; thus,the jack diameter becomes an inuential factor for systemstrength.

reliability than Scaffold 1, its nominal strength is only about 40%of the latter. Considering that Scaffolds 1 and 2 are identical except

Fig. 6. System strength vs. diameter of jack.

H. Zhang et al. / Engineering Structures 36 (2012) 8189 87Fig. 4 shows the strength histogram for Scaffold 2 under randomload eccentricity. The system strength vs. load eccentricity relationfor Scaffolds 1 and 2 are presented in Fig. 7 for comparison. Fig. 7suggests that the strength of Scaffold 1 is more sensitive to the var-iation of load eccentricity than the strength of Scaffold 2.

The yield stress and initial out-of-straightness of the uprightshave no effects on VR for Scaffold 2. This is due to the fact thatthe uprights remain elastic and essentially straight within eachstorey when the jacks buckle. The joint stiffness does not inuencethe system strength signicantly as the rotation of the cuplokjoints was relatively small. The effect on VR of the variation in jackyield stress is minor because only a small portion (less than 20%) ofthe jack cross-section reached the yield stress.

The strength of Scaffold 2 has a mean-to-nominal ratio of 1.0and a COV of 16% when all random variables are considered simul-taneously. The strength histogram is shown in Fig. 8. The strengthis described by a three-parameter Weibull distribution in the sub-sequent reliability analysis.

5. Scaffold reliability

5.1. Design with advanced analysis

Table 4 summarizes the statistical data of the loads (D and L)and the system resistance (R). Note that the nominal system resis-tance, Rn, in Table 4 was computed by the advanced nonlinear anal-ysis. With this information at hand, the structural reliability ofScaffolds 1 and 2 can readily be evaluated using FORM. Using the

WLL design equation (Eq. 1) and a factor of safety F.S. = 2.0, the

Fig. 7. System strength vs. load eccentricity.allowable working load, Rn/F.S., are 53.6 kN and 22.32 kN for Scaf-folds 1 and 2, respectively. These values are half of the nominalsystem strengths obtained using advanced analysis, as describedin Sections 4.4 and 4.5. Fig. 9 plots the reliability index, b, withLn/Dn for Scaffolds 1 and 2 when all the basic random variablesare taken into account. For a design live load on formwork of2.4 kPa (50 psf), Ln/Dn for the supporting scaffolds has a practicalrange of 0.30.7, since the slab thickness ranges from 150 to300 mm (612 in.) in typical concrete residential and commercialbuildings. Representative values for b are 2.6 and 3.1 for Scaffolds1 and 2, respectively, for the loading condition Ln/Dn = 0.5. Thesetwo reliability indices correspond to probabilities of failure of4.7 103 and 9.7 104, respectively.

From Fig. 9, it can be seen that b for Scaffold 2 is somewhathigher than that of Scaffold 1, although the two structures havesimilar strength statistics. This result can be explained by the factthat the strength distribution of Scaffold 1 is negatively skewedwith a long lower tail which governs the limit-state probability,while for Scaffold 2, its strengths are relatively evenly distributedon both sides of the mean. This highlights the importance of incor-porating information about probability distributions in the reliabil-ity analysis.

It should be recognized that although Scaffold 2 has a higher

Fig. 8. Histogram for Scaffold 2 strength with all random variables (tted with athree-parameter Weibull distribution).for the length of jack extensions (100 mm vs. 600 mm), Scaffold 1is a much more efcient design from the point of view of materialutilization.

Fig. 9 indicates that b increases as Ln/Dn increases. This observa-tion is different from the trend observed for typical steel structuralmembers whose reliability generally decreases as Ln/Dn increases[17]. The discrepancy is a consequence of two characteristics ofthe probabilistic loadmodels for scaffolds: (1) dead load is the dom-inant component in the total loads; and (2) although the live loadhas a larger COV, itsmean-to-nominal ratio ismuch smaller in com-

Table 4Statistical information for the loads and resistance.

Variable Mean COV Distribution

D 1.0Dn 0.3 NormalL 0.3Ln 0.7 Extreme type IR (Scaffold 1) 1.02Rn 0.17 3-P WeibullR (Scaffold 2) 1.0Rn 0.16 3-P Weibull

Based on a design formwork live load of 2.4 kPa (50 psf).

occurs (and is undetected), and FjEi = failure of the structure given

88 H. Zhang et al. / Engineering Stparison with the dead load, i.e., L=Ln 0:3 vs. D=Dn 1:0. It appearsthat for scaffold structures, dead load has a more dominant inu-ence on the structural limit-state probability than the live load.

5.2. Design with elastic analysis

The above reliability analysis is based on the nominal systemstrengths Rn that are computed using the advanced nonlinear FEA.Although design by second-order inelastic analysis is permitted inthe Australian steel design standard AS 4100-1998 [39] and theAmerican specication for structural steel buildings AISC 360-10[37], the current routine design procedure is to use a two-dimen-sional elastic analysis and evaluate the limit states in accordancewith the Specication equations. Table 5 summarizes the nominalstrengths for Scaffold 1 computed by elastic analysis conformingto the Australian steel specication [39]. The design was controlledby the inelastic exural buckling of the uprights. Three elastic de-signmethods, denoted EA1, EA2 and EA3, are presented. The cuplokconnections between uprights and ledgers are assumed to be rigidin EA 1 and EA 2, and semi-rigid in EA 3. The effective length factorfor the uprights was conservatively taken as 1.0 in EA 1, but wasdetermined more rationally by a frame elastic buckling analysisin EA 2 and EA 3. While method EA1 is the simplest and most likely

Fig. 9. Reliability indices for Scaffolds 1 and 2 designed by the advanced analysiswith F.S. = 2.0.to be used by the design engineers, it is the least accurate andunderestimates the strength by 25% as compared with the ad-vanced analysis. The last column of Table 5 compares the mean-to-nominal ratio R=Rn for different analysis and design methods.

Similar elastic designs were performed for Scaffold 2, and theresults are presented in Table 6. The strength of Scaffold 2 is gov-erned by the buckling of the jacks. It can be seen that the resultsfrom elastic analysis are very accurate in this case. This is to be ex-pected since Scaffold 2 is largely in its elastic range when the veryslender jacks buckle.

Table 7 compares the reliability indices for Scaffolds 1 and 2when their nominal strengths are computed by the advanced anal-

Table 5Nominal strength (Rn) for Scaffold 1 computed using different methods.

Methods Upright-ledger joint ke Rn R=Rn

Advanced FEA semi-rigid 107.2 kN 1.02EA 1 rigid 1.0 80 kN 1.36EA 2 rigid 0.75 a 108 kN 1.0EA 3 semi-rigid 0.83a 91.2 kN 1.19

ke: effective length factor for uprights.a Determined by a frame elastic buckling analysis.ysis and elastic analysis, assuming a typical loading condition Ln/Dn = 0.5. For Scaffold 1, method EA 1 gives the most conservativenominal strength, thus it has the highest reliability (b = 2.93). ForScaffold 2, the reliability indices for the designs by advanced FEAand by elastic analysis are all around 3.0.

5.3. Reliability in the presence of human error

The reliability assessment described above addresses structuralfailures that are caused by stochastic variability in loads and struc-tural strength. In addition to these uncertainties, the safety of scaf-folds may be compromised by various types of human errors[2,4043].

Considering the possibility of n human error scenarios, theprobability of structural failure due to human error can be derivedfrom the theorem of total probability [40]:

PFE Xni1

PFjEiPEi 4

in which FE = structural failure due to human error, Ei = ith error

Table 6Nominal strength (Rn) for Scaffold 2 computed using different methods.

Methods Upright-ledger joint ke Rn R=Rn

Advanced FEA semi-rigid 44.6 kN 1.0EA 1 rigid 2.35a 47 kN 0.95EA 2 rigid 2.15b 48.0 kN 0.93EA 3 semi-rigid 2.3 b 44.0 kN 1.01

ke: effective length factor for jacks.a Determined from the alignment charts for ke for sway members.b Determined by a frame elastic buckling analysis.

Table 7Reliability indices, b, for Scaffolds 1 and2designedwith differentmethods (Ln/Dn = 0.5).

Methods Scaffold 1 Scaffold 2

Advanced FEA 2.6 3.12EA 1 2.93 2.95EA 2 2.6 2.88EA 3 2.78 3.17

ructures 36 (2012) 8189that ith error occurs.It appears from Eq. (4) that the failure probability P(FE) can be

managed by controlling the consequence of an error if it doesoccur, i.e., reducing the term P(Fj Ei). In light of this, one mightattempt to use classic reliability theory to evaluate Eq. (4), and de-rive a more conservative factor of safety (or load and resistancefactors) to compensate for the effect of human errors. This practice,however, has proved to be ineffective and problematic [40]. It isvery difcult to evaluate Eq. (4) given the fact that there arenumerous sources of human errors, and available statistics ofhuman errors is very scarce and often biased. Even in the caseswhere the variety of errors is limited or only one error is consid-ered at a time, as suggested in [43], determining the magnitudeand frequency of human error is problematic since they may varygreatly among individuals and organizations. Using a larger factorof safety to account for human error (say, defective workmanship)will unfairly penalize those who implement good constructionquality control. Instead, the risk of structural failure shall be miti-gated by reducing the incidence of human error(s), i.e., reducingthe magnitude of P(Ei). This may be achieved by quality assur-ance/quality control strategies, vocational education and training,and organization and management measures [2,4042]. Theirimportance cannot be overemphasized.

H. Zhang et al. / Engineering St6. Conclusions

A reliability analysis of steel scaffold shoring systems was pre-sented. Scaffold towers with steel cuplok connections and differentjack extensions were considered. The limit state was dened asinstability failure by overload. The study focused on the construc-tion stage of concrete placement as this is the most critical time forthe safety of supporting scaffolds. Analyses of available literatureon shoring loads measured during concrete placement found that:(1) Shoring load is mainly due to dead load effect. Live load, in gen-eral, is small in this phase of construction. (2) The live load has aCOV of approximately 0.60.7. Its mean-to-nominal ratio equals0.3 if the design live load (50 psf) given in ACI 347 [30] is used.(3) The dead load has a mean-to-nominal ratio of 1.0 and a COVaround 0.3.

Simulation studies showed that for Scaffold 1, the variability instructural strength mainly arises from the uncertainties in loadeccentricity, the yield strength and thickness of the uprights. ForScaffold 2, the jack diameter and load eccentricity become domi-nant inuences on the structural strength. These results indicatethe importance of using accurate structural models to capturethe mode of failure.

For a typical loading ratio Ln/Dn = 0.5, the reliability index forScaffold 1 is found to be 2.6 with a factor of safety of 2.0 if its nom-inal ultimate strength is determined by an advanced nonlinearstructural analysis. If the nominal system strength is computedusing the customary elastic methods, the reliability index rangesfrom 2.6 to 2.93, depending on the modelling of the cuplok jointsand the method for determining the effective length factor. ForScaffold 2, the elastic analysis appears to be sufciently accuratefor evaluating the system strength, as the design is controlled bythe elastic buckling of the jacks. In this case, the reliability indicesfor the designs by advanced FEA and by elastic analysis are similar,all being approximately 3.0.

It appears that with a factor of safety of 2.0, the calculated reli-ability indices for the scaffolds are lower than the reliability indicesfor typical steel structural members under gravity loads, which areabout 4.04.2 on an annual basis [17]. While it is often argued thatthe safety requirement for temporary structures need not be ashigh as for permanent structures, risk acceptance and requiredsafety levels for temporary structures warrant further examina-tion, as does the role of human error in control of scaffold safety.

Acknowledgements

This research is supported by Australian Research Council underLinkage Grant LP0884156. This support is gratefully acknowledged.The writers would also like to thank Dr. igo Puente (Universidadde Navarra) for making the shore load survey data available.

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ructures 36 (2012) 8189 89

Reliability assessment of steel scaffold shoring structures for concrete formwork1 Introduction2 Structural reliability analysis3 Probabilistic load models for scaffolds3.1 Live loads3.2 Dead load

4 Strength of scaffold structure4.1 Example structure4.2 Random properties of steel scaffold members4.3 Statistical characteristics of system strengths4.4 Scaffold 1100mm jack extension4.5 Scaffold 2600mm jack extension

5 Scaffold reliability5.1 Design with advanced analysis5.2 Design with elastic analysis5.3 Reliability in the presence of human error

6 ConclusionsAcknowledgementsReferences