Flange and web limit states in beams subjected to patch loading

Journal of Constructional Steel Research 63 (2007) 45–54www.elsevier.com/locate/jcsr

Flange and web limit states in beams subjected to patch loading

Lyle P. Cardena, Gokhan Pekcanb,∗, Ahmad M. Itanib

a Martin & Chock, Inc., 1132 Bishop St., Suite 1550, Honolulu, HI 96813, USAb University of Nevada, Reno, Department of Civil and Environmental Engineering, MS258, Reno, NV 89557, USA

Received 22 October 2005; accepted 1 March 2006

Abstract

Recent field observations during bridge construction revealed excessive localized deformations in sill and cap beams that support falseworkposts. Hence, possible limit states associated with flange and web deformations in W and HP sections subjected to patch loading are investigatedin this paper. Critical limit states were found to be related to flange bending, post compression, and the interactions associated with patch loadingbetween a beam flange and timber or steel post. Other limit states are associated with web deformations. Two alternative methods based on yieldline analyses that establish upper-bound limiting loads for these limit states are proposed. The first method accounts for an interaction betweenflange bending and post compression strength, whereas the second method uses an effective bearing area of the post. The former method is mosteffective in characterizing flange–timber posts interactions, while the effective bearing area method was found to be more accurate for steel posts.For beams with relatively thick webs, such as those typically used in bridge falsework, the web was found to have a greater capacity than theflange and post, with the critical web limit state due to web yielding. It was shown that current recommended equations were appropriate withmodifications to account for the patch loading condition.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: Bridge falsework; Steel beam; Post; Flange bending; Web yielding; Web crippling; Web buckling

1. Introduction

Falsework used in the construction of cast-in-place boxgirder bridges typically consists of: timber or concretefoundation pads; timber corbels; sand jacks and wedges; steelsill beams; timber or round hollow steel posts; steel cap beams,stringer beams and timber joists. Lateral stability is providedusing timber elements or cable braces. Two examples showingeach of these components are given in Fig. 1.

Recent field observations during bridge constructionrevealed localized bending in some of the beam flanges, atthe interface between the cap and sill beams and posts, asillustrated in Fig. 2. While local web yielding of the beam maybe considered during design, the flange local bending capacityrequired to resist the effects of patch loading from a timberor steel post is not generally checked. Stiffeners are generallynot used in bridge falsework due to it being a temporarystructure and, with section sizes generally governed by what the

∗ Corresponding author.E-mail address: [email protected] (G. Pekcan).

0143-974X/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2006.03.005

contractor has readily available; contractors generally prefer touse larger beams to prevent the use of stiffeners.

The AISC LRFD specifications for steel buildings [1]recommends procedures for the evaluation of various limitstates associated with beam flange and web deformations.Accordingly, the flange local bending capacity is determinedfrom yield line analysis based on a tensile line load and uniformstress distribution of a flange [2]. However, the recommendedequation may be very conservative if applied to flange bendingdue to compressive patch loading, as the region of flangeavailable to resist the load would be greater for a patch loadthan a line load, and a compression load allows a concentrationof stresses around the web.

Past studies on the effect of patch loading [3–7] havegenerally assumed “rigid patch loads” which have not resultedin flange bending other than that required for deformationof the web. A rigid patch load is considered one where theload is applied through an axially rigid post, thus deformationoccurs mostly in the loaded member rather than the elementapplying the load and the stress distribution from the patchload changes as the loaded member deforms. This is in contrastto a flexible patch load, in which the post applying the patch

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46 L.P. Carden et al. / Journal of Constructional Steel Research 63 (2007) 45–54

Fig. 1. Bridge falsework (a) with steel posts and (b) with timber posts for typical bridges in California.

Fig. 2. Flange bending in sill beam (J. Lammers, Caltrans, Personalcommunication).

load deforms axially as the loaded member deforms, resultingin a more uniform distribution of applied stress. Two possiblelimit states associated with web deformation due to patchloading are: (1) local web yielding that governs for stockywebs, summarized by Galambos [8], and (2) web crippling thatgoverns for more slender webs, as summarized by Elgaaly [5].Chen and Newlin [9] developed an expression for the localizedbuckling of beam webs for an equal and opposite concentratedload applied to both edges of the web. As with the equationdeveloped for flange bending, the equation proposed by Chenand Newlin [9] is expected to be conservative for an appliedpatch loading, where the area of web involved in resisting theload is considerably greater than for concentrated loads. Thebuckling equation developed by these researchers assumes thatboth flanges are restrained from lateral deformation. If thereare no stiffeners along a beam, buckling of the web with lateraldeformation of one of the flanges may be possible.

The case of a flexible patch load provided by a timberpost, as found in bridge falsework, has not been consideredand, therefore, there is no method for predicting the resultingcapacity of the beam flange. The objectives of this paper areto investigate the critical flange and web limit states due topatch loading, assess the effectiveness of existing equationsand, where necessary develop new equations that can be usedto predict the ultimate load for the various limit states, withvalidation from finite element analyses.

2. Flange bending limit state

From a yield line solution by Graham et al. [2],beam–column connection capacity, Rn , associated with thecolumn flange subjected to tensile line loads from the adjoiningbeam flanges is given by [1]

Rn = 6.25t2f Fy f (1)

where t f = thickness of the flange and Fy f = minimumspecified yield stress of the flange. A similar mechanism to thatused in developing Eq. (1) can be used to calculate the ultimateload for flange local bending due to a patch load on a falseworksill or cap beam. The resulting ultimate load for beam flangebending, R f , is given by

R f = βt2f Fy f (2)

in which β is determined from a yield line analysis. Assuming apatch load with dimensions, bp and dp, as well as the yield linepattern and a uniform pressure, as shown in Fig. 3, equating theinternal and external virtual work for the patch load gives

β =bpbfeff

2b2peff

(dp

bfeff+

4sin 2θ

+2

tan θ

)(3)

where θ = angle of the yield line shown in Fig. 3, bfeff =

effective width of the flange (Fig. 3) given by

bfeff =b f

2− k1 −

t f

2(4)

where k1 = distance from the center of the web to the edge ofthe fillet and b f = width of the flange. The effective width ofthe post, bpeff, is given by

bpeff =bp

2− k1 −

t f

2. (5)

The expression for β can be minimized with respect to α

giving a minimum value at angle of 55◦. However, assumingan angle of 45◦ gives a result for the last two terms of Eq. (3)

L.P. Carden et al. / Journal of Constructional Steel Research 63 (2007) 45–54 47

Fig. 3. Yield line pattern on beams for a patch load assuming a uniform stress distribution in the post.

that is within 6% of the minimum value, therefore α = 45◦ isassumed, allowing Eq. (3) to simplify to

β =bpbfeff

2b2peff

(dp

bfeff+ 6

). (6)

Note that if dp tends to zero, as for a line load, bp is equalto b f and therefore bpeff is equal to bfeff, and bfeff is assumed tobe 0.96b f /2, then β simplifies to 6.25 as in Eq. (1). For typicalfalsework applications, where nominal 300×300 mm (bp ×dp)

posts are used, the value of β for typical beam sizes are givenin Table 1. The values of β range from 10.9 to 18.1 for thedifferent beam sizes considered, which are all greater than 6.25as expected due to the finite width of the post instead of the lineload assumed in Eq. (1).

It is later shown that a uniform stress distribution in thepost may be unrealistic for many beams, particular those withsmaller flanges, as stresses are concentrated at the web of thebeam. While, in reality a complex stress distribution exists, thisis approximated by a triangular stress distribution as shown inFig. 4 to reflect the concentration of stresses around the weband an assumed linear reduction towards the edge of the beamand post. This gives a new expression for β where

β =3b2

pbfeff

8b3peff

(dp

bfeff+ 6

). (7)

The value of β for the different beam sizes assuminga triangular stress distribution is given in Table 1. Thevalues range from 20.1 to 41.9, indicating an ultimate loadthat is approximately twice that for the assumed rectangulardistribution. It is noted that the β value for the heaviest sectionconsidered based on the uniform stress distribution (β = 18.1)

is close to the β value for the lightest section consideredbased on the triangular stress distribution (β = 20.1). It isexpected that for the heavier sections a rectangular distributionis most appropriate as the flanges will deflect a relatively smallamount allowing the stresses to be uniformly distributed in thepost, while for the lighter sections the flanges will deflect a

Fig. 4. Cross section of a beam for a patch load assuming a triangular stressdistribution in the post.

greater amount leading to triangular stress distribution. Bothfor heavier sections with a uniform stress distribution and thelighter sections with a triangular stress distribution, β convergesto a value of between 18 and 20.

3. Post compression strength in the beam–post joint region

The strength of a timber post in compression is definedbased on NDS specifications. To be consistent with the otherlimit states, the NDS LRFD specifications [10] are used. In thebeam–post joint region, the nominal compressive strength, Rp,is considered over a short length of the post and is given by

Rp = F ′c Ap (8)

where Ap = nominal cross sectional area of the post and F ′c =

nominal stress in the post after modification factors are appliedfor moisture content, temperature, size, preservative treatment,fire-retardant treatment and column stability [10]. Because thecompression capacity is considered over a short length, thecolumn stability factor in this calculation is equal to 1.0. It isnoted that stability of the full length of the column should beconsidered as a separate limit state in the design of the column.The axial capacity of a steel post in the bearing region canbe calculated as defined in the AISC specifications for a shortlength column [1].


Table 1β values for beams assuming uniform and triangular stress distributions and 300 mm square patch load

Section HP12 × 53 HP14 × 73 HP14 × 89 W14 × 90 HP14 × 117 W14 × 120

Uniform stress 10.9 13.1 13.6 16.2 14.9 18.1Triangular stress 20.1 24.4 26.1 35.3 29.9 41.9

4. Interaction between flange bending and post compres-sion limit states

The force at which bending occurs in the beam flange isrelated to that for crushing of the post where in contact withthe beam. The flange will not bend unless the post deformsto accommodate the bending and, conversely, the compressionstrength of the post will be reduced, if flange bending occurs.Therefore, it is helpful to consider an interaction betweenthe flange bending and post crushing limit states. Whileinteraction is typically applied to different actions on the samemember, this interaction considers a region that involves twomembers, therefore the concept applies to the joint region.A number of interaction equations have been proposed fordifferent applications, such as for interaction between axialloads and bending moments in the design of columns, or forthe interaction between bending and torsional moments inmembers [1], for the interaction between post crushing andflange bending an elliptical relationship is assumed, such that(

Ru

R f

)2

+

(Ru

Rp

)2

≤ 1 (9)

where Ru = applied axial load in the post. The applicationof this equation and comparisons to results obtained fromfinite element analyses is demonstrated later in the paper. Anextensive experimental study also conducted by the authors ispresented elsewhere [11].

5. Effective post bearing area for calculating flange–postcapacity

This section, instead of focusing on the beam flange indetermining the flange–post response, proposes an alternativemodel which focuses on the post capacity for calculating theultimate load in the flange–post joint region. For this model,an effective cross sectional area of the post is considered tocarry the entire axial load, as shown in Fig. 5, calculated usingsimilar assumptions to those used for the web yielding equationin the AISC specifications [1]. The effective area of the post isdetermined by the width of the post that is effective in carryingthe load, multiplied by the thickness of the post within thiswidth (Fig. 6). An equation for the capacity of the region ofbearing between the post and beam, Rn , for a timber post issubsequently given by

Rn = (αt f + 2k1)dp F∗c (10)

where α = constant which depends on the slope of the stressgradient assumed through the flange, which is determinedbased on the experimental observations [11] and finite elementanalysis, F∗

c = nominal compressive stress perpendicular to

Fig. 5. Effective area of (a) timber post and (b) steel post for calculation ofultimate post load.

Fig. 6. Yield line pattern for the bottom flange of sill beams with a patch loadfrom the supports assuming a uniform stress distribution.

the grain and the other variables are as previously defined. Asimilar equation for the capacity of the bearing region for a steelpost is given by

Rn = (α(t f + tbp) + 2k1)2tp Fy (11)

where tbp = thickness of the baseplate of the post, and tp =

wall thickness of the round hollow steel section. The 2tp allowsfor the transfer of axial load on two sides of the round hollowsteel section.

6. Bottom flange bending of a sill beam

A similar methodology to that used in developing the flangelocal bending equation for the top flange of a sill beam orbottom flange of a cap beam can also be used for the bottomflange of the sill beam which would result in a yield linepattern shown in Fig. 6. However, formation of this yield patternrequires deformation in the corbels and elements such as thewedges and sandjacks between the sill beam and the corbels.Analysis shows that the capacity of the bottom flange–corbelregion is best characterized by the strength of the corbels [11],although the corbel crushing limit state is not considered in thispaper. Thus, the bottom flange bending limit state in a sill beam


Fig. 7. Effective web yielding region for (a) timber post and (b) steel post.

is not considered a critical limit state as corbel failure will occurprior to failure of the flange.

7. Web yielding and crippling

The critical limit states for the web of beams in bearingjoint regions are web yielding and web crippling. The ultimatecapacity of the web associated with the web yielding limit state,Pw, due to loading from a timber post can be derived as

Pw = (αk + dp)Fywtw (12)

where k = distance from the outer face of the flange to the toe ofthe web (Fig. 7(a)) and tw = thickness of the web. Based on theCaltrans Falsework Manual [12], α is equal to 2 for a 1:1 stressgradient. However, based on the AISC specifications, α is equalto 5 for an assumed 2.5:1 stress gradient through the flange andtoe of the web. This value is investigated in Section 9. A similarequation when the load is due to a steel post bearing is given by

Pw = 2(α(k + tbp) + tp)tw Fyw (13)

where in this case, tp = wall thickness of the round hollow steelsection (Fig. 7(b)). This allows for yielding of the web in twolocations, directly under each side of the post.

The AISC specifications [1] also specify equations for webcrippling. However application of these equations to the typicalbeams used in bridge falsework with relatively thick webs,results consistently in web crippling loads that are larger thanthe web yielding loads, therefore the web crippling equationsare not discussed in this paper.

8. Lateral web buckling

In a typical bridge falsework, particularly sill beams aregenerally not braced, therefore there is a possibility that theflanges loaded with the timber or steel posts could displacesideways through buckling of the web. Unlike the other limitstates where deformation is localized at the beam–post bearingregion, sideways web buckling occurs over a long length of thebeam resulting in a failure as shown in Fig. 8. The web of abeam can be treated like a column assuming that the effectivelength of the beam is equal to the tributary length associatedwith each post, approximately equal to the spacing of the posts.Based on the column buckling equations given in the AISC

Fig. 8. Unbraced sill beam resulting in sideways lateral buckling (J. Lammers,Caltrans, Personal communication).

specifications [1], the capacity of the web of a beam, Pw, canbe determined from

Pw = twleff Fcr (14a)

where leff = effective tributary length of beam for a particularpost and Fcr = 0.877Fe, in which Fe is the Euler buckling load

Fe =π2 E( K h

r

)2 (14b)

where E = elastic modulus of the steel, K = effective lengthfactor, which will range between 1.2 and 2.1 for a member thatthe is free to deflect laterally at one end, h = clear height of theweb between the flanges less the fillet radius and r = radius ofgyration for the web. This assumes that the web is slender forcompression buckling such that Fe < 0.44Fy , which will bethe case for the web of almost all beam sections. For typicalbeams used in bridge falsework with a relatively thick web,web buckling is unlikely to govern design. However, for doublestacked beams that are commonly used, h in Eq. (14b) will beeffectively 2h, resulting in a reduction in Fe by a factor of 4. Inthis case, even beams with relatively heavy webs will becomesusceptible to lateral web buckling.

9. Comparison of calculated limit states with finite elementanalysis

In order to investigate response characteristics and the limitstates experimentally, a total of 32 subassemblies of variousbeam sizes, timber and steel posts were tested and reportedelsewhere [11]. Subsequently, a series of finite element modelswere developed using ABAQUS [13] and calibrated to theexperimental data. These finite element models allowed a largerrange of beams and configurations to be considered, withoutmaterial variability affecting the results. There were four groupsof finite element models. The first group used a 1220 mmlong post bearing against a 1220 mm long section of beamto determine the localized flange bending capacity for a postpatch load (Fig. 9(a)). The second group replaced the timberpost with a section of 457 mm diameter, 10 mm thick steelpipe and a 12 mm base plate, which was assumed for the largerbeams (Fig. 9(b)). In the third group, a rigid 300 × 300 mmpatch load was modeled on top of the beam to determine criticallocalized web yielding and crippling limit states (Fig. 9(c)).


Fig. 9. Finite element models for (a) a timber post and beam, (b) a steel postand beam, (c) a beam with a rigid patch load and (d) long double stacked beamand post.

In the final group a 3.05 m long single or double stackedbeam was modeled with a 3.05 m long post, to study lateralweb buckling limits states (Fig. 9(d)). This length beam isconsidered for typical post spacing in bridge falsework. Thesefigures show the post bearing onto a sill beam, although thesystem could be inverted for the bearing connection regionbetween a post and cap beam. The top of the post was assumedto be fixed, with the longer post used for to model the lateralweb buckling limit state. While contact elements were used ininitial models, no separation between the beam and post wasobserved, thus the beam was connected to the post throughnodal constraints in the models described. Similarly the flangeswere connected between the double stacked beams. The steelmaterial was assumed to be elasto-plastic with a yield stressequal to the expected yield stress of 375 MPa. This waswithin 2% of the average yield stress observed during couponexperiments of 367 MPa. The wood material was modeled,based on measured properties of the posts during experiments,using a tri-linear model with an initial stiffness of 3800 MPaand yield strength of 18.6 MPa, post-yield stiffness of 760 MPawith an ultimate strength of 22.3 MPa, followed by a stiffnessof −220 MPa to reflect crushing of the post. A comparison ofthe force–displacement curve for the finite element model of apost compared to two experiments is shown in Fig. 10(a).

For each configuration considered, the axial force in thepost or force due to the patch load was plotted againstthe axial displacement at the free end of the post or patchload. For each experiment, the ultimate load was definedfrom the force–displacement curve at a point where the

Fig. 10. (a) Force–displacement curves for finite element model compared toexperimental data for (a) a 1220 mm long axial loaded timber post, and (b) anHP12 × 53 beam with flange bending and timber post crushing.

tangential stiffness reduced to 50% of the initial stiffness. Thiswas considered an appropriate ultimate limit state to give aconsistent definition of failure for the different configurations,some which reached their maximum load soon after firstyield was observed, and others which yielded gradually. Thecalculated ultimate load is illustrated for the finite elementmodel and experiments on the timber posts in Fig. 10(a). Asimilar comparison for an HP12 × 53 beam assembly whichfailed through beam flange bending and post crushing is shownin Fig. 10(b).

As loads were applied to the beams with 1220 mm longtimber posts, the concentrations in stresses were observed in theposts around the beam webs with a stress distribution at the endof the post similar to the triangular distribution assumed in thedevelopment of Eq. (7). Once the middle of the post yieldedthe stresses propagated towards of the outside of the post.Meanwhile, first yield in the beam flange was observed at thetop flange under the center of the post soon after first yielding ofthe post was observed. This propagated longitudinally along thebeam resulting in the deformation pattern shown in Fig. 9(a).With a steel post, localized yielding was first observed inthe post above the beam web. This progressed to inelasticlocalized buckling or crippling of the web after which flangedeformations and flange yielding was observed (Fig. 9(b)).


Fig. 11. (a) Comparison of (a) flange bending capacity, (b) post compression capacity, (c) interaction between flange bending and post compression capacity and(d) post capacity using effective area versus ultimate load from finite element models for different beams.

Unlike with the timber post, with the steel post flange yieldingwas not concurrent with inelastic actions in the post, but insteadwas a consequence of post deformation.

The calculated flange bending capacity (Eq. (2)), as a ratioof the capacity calculated from the finite element analysis,is plotted in Fig. 11(a) for different beam sizes. The flangebending capacity was calculated with three different methodsfor calculating β; (1) using Eq. (6) assuming a uniform stressdistribution from the patch load, (2) using Eq. (7) assuminga triangular stress distribution, and (3) assuming a constantvalue of β = 18, based on convergence of β from the uniformstress distribution for the largest beam and the triangular stressdistribution for the smallest beam. In Fig. 11(a), HP12 × 53,HP14 × 73, HP12 × 89 and W14 × 90 beams are modeledwith timber posts and W14 × 90, HP14 × 117 and W14 × 120beams are modeled with steel posts. In calculating the flangebending capacity with the steel post, the post was assumed tohave an effective thickness tp (Fig. 7) equal to the diameterof the post (457 mm) and effective width, bp, equal to thewidth of the flange. The capacities of the beams with timberposts are shown with the shaded markers, while the beamswith steel posts are shown with unshaded markers. Fig. 11(a)shows that the calculated capacity of the flanges for the smallestbeams is conservative compared to the capacity of the systemcalculated from the finite element analysis, while for the largerbeams the calculated capacity is unconservative, regardless of

the method for calculating β. Therefore, the flange bendingcapacity calculations alone did not give a good estimate of thecapacity of the joint region.

The nominal axial capacity for a short length of post (Eq.(8)) is shown in Fig. 11(b), as a ratio of the capacity of thebeam–post bearing joints for the different beams from finiteelement analysis. The figure shows that the calculated capacityof the post in all cases is greater than the capacity of thebeam–post joint. Regardless of the method used to calculateβ, the calculated flange bending capacity and post capacityare both greater than the capacity of the system for all but thesmallest two beams. Thus, the use of these limit states resultsin a generally unconservative estimate of the ultimate load.

When Eq. (9) is used to model the interaction betweenthe post capacity and flange capacity, Fig. 11(c) shows thatestimated capacity of the beams is in good agreement with thatcalculated from the finite element analysis for the timber post.Using the interaction equation and assuming a constant β = 18,resulted in the best estimate of the beam–post joint capacitywith timber posts. The estimate was between 80% and 103%of the ultimate capacity from the finite element analysis. Thisis relatively accurate yet generally conservative, thus use of theconstant β value is justified. However, the interaction equationwas not effective and is shown to overestimate the capacity ofthe joints with the steel post. This is attributed to the high axial


Fig. 12. (a) Web yielding, and crippling capacity and (b) lateral web bucklingcapacity versus ultimate load from finite element models for different beams.

stiffness of the steel post which does not allow the end of thepost to deform as is the case for the timber post.

As an alternative to the interaction equation, the ultimatecapacity of the beam–post joint region is calculated usingthe effective bearing area formulation (Eqs. (10) and (11)).A value of α = 5 is assumed, based on the same 2.5:1stress gradient assumed for web yielding in beam–columnconnections recommended by the AISC specifications [1].Resulting capacities are compared to the finite element analysisin Fig. 11(d) for the different beam–post regions. This figureshows that with the timber post the method is excessivelyconservative with an estimated capacity of between 54% and70% of the capacity from the finite element analysis. While itmay be possible to adjust the α value to increase the accuracyin calculating the capacity of the beam–post joint region witha timber post using this method, it is uncertain how differentratios of beam and post stiffness and strength will affect α.Therefore, the interaction equation model is considered moreappropriate than the bearing area model when using a timberpost. For the steel post, the bearing area method is moreaccurate but still conservative, with capacities between 84%and 95% of those from the finite element analysis. The effectivebearing area method is therefore considered more effective thanthe interaction equation method with the steel post.

With the rigid post web yielding was first observed underthe edges of the post, propagating to yielding under the widthof the post. After a large area under the web had yieldedinelastic buckling was observed as shown in Fig. 9(c). Thecalculated web yielding (Eq. (12)), and crippling [1] capacitiesare compared in Fig. 12(a) to the capacity from the finiteelement models with a rigid patch load. In these models, thebeam flanges were laterally restrained, therefore web bucklingwith lateral displacement of the flange was not considered.Furthermore, it is noted that the web crippling limit state is notlikely to govern the design of bridge falsework because of therelatively large web thicknesses of typical cap and sill beams.Therefore, finite element models were expanded to include alarger range of sections with thinner webs to investigate theweb crippling limit state. The additional sections included:W12 × 26, W12 × 40, W12 × 53, W14 × 22, W14 × 30,W14 × 43, W14 × 61 and W14 × 132. It is shown, however,that even for the lightest 300 and 350 mm deep sections, webyielding using an effective bearing width of 2k+dp resulted in acapacity, approximately equal to or less than the web cripplingcapacity. Fig. 12(a) shows that web yielding, assuming α = 2,conservatively and reasonably accurately predicts the ultimateweb capacity of the beams, with calculated capacities between85% and 91% of that from the finite element analyses. For therigid patch load, α = 5 typically over-predicts the web yieldingcapacity by up to 15% compared to the finite element model.Although, lateral deformation of the web is observed, indicativeof web crippling as shown in Fig. 9(c), Fig. 12(a) shows thatweb crippling generally occurs at forces greater than the webyielding capacity for these beams.

With the long timber posts and beams with more slenderwebs, or double stacked beams, elastic lateral buckling of thebeams was observed. The calculated capacity (Eq. (14)) of thesingle or double stacked long beams without lateral restraintof the loaded flange (Fig. 9(d)) is compared to the capacityof those exhibiting lateral buckling from the finite elementanalyses in Fig. 12(b). In the single stacked configurationW12 × 26, W12 × 40, W14 × 22, W14 × 30, W14 × 43 werefound to buckle laterally. In the double stacked configuration,where only the larger beams were considered, the HP12 × 53,HP14×73 and W14×90 beams were found to laterally buckle.In calculating the capacity of the beams, K = 1.7 is assumedto ensure the calculated capacities are less than those from thefinite element analyses. Fig. 12(b) shows that the calculatedcapacity is between 51% and 98% of the ultimate load fromthe finite element model. The variation is attributed to thesensitivity of the effective length in the resulting buckling load,for example the most conservative factor could be accuratelycalculated assuming K = 1.2. In this case a larger amountof rotational restraint is provided to the beam from the post.However it is difficult to predict the level of rotational restraintthat will be present in each beam as it is related to the rotationalstiffness of the post, length of the post and torsional stiffness ofthe laterally deflecting flange. Assuming K = 1.7 is shown tobe conservative for these practical cases. With a steel post, therotational restraint provided to the flange by the post is greater


Table 2Ultimate loads of typical falsework beams for different limit states (kN) assuming an expected steel strength of 380 MPa and timber post strength as calculated fromexperiments

Section HP12 × 53 HP14 × 73 HP14 × 89 W14 × 90 W14 × 90 HP14 × 117 W14 × 120Post type Timber Timber Timber Timber Steel Steel Steel

Flange bendinga 853 1145 1700 2220 2220 2890 3890Nominal post compression 1570 1570 1570 1570 3950 3950 3950Flange–post interactionb 749 924 1150 1280 1370 1330 1580Web yieldingc 1480 1730 2140 1530 2100 4110 3180Web crippling 3530 4260 6210 2490 2490 10400 4420Lateral web buckling 3160 3620 6480 2320 2320 d 5610Lateral web buckling for a double stacked beam 791 905 1620 579 579 3630 1400

a Flange bending capacity is based on β = 18.b Based on interaction equation for timber post and effective bearing area with α = 5 for steel post.c α = 2 for timber post and α = 5 for steel post.d Pcr > 0.44Fy and therefore web would buckle inelastically. Web buckling will not govern.

than for the timber post, therefore using the same assumptionsas for the timber post will be conservative.

A bending moment in the web, provided by an eccentricitybetween the centroid of the post and centroid of the beam, mayalso result in deformation of the web and affect the bucklingcapacity. However, analyses show that if the eccentricity is lessthan a critical eccentricity, calculated at 2tw for a typical case,it has minimal effect on the capacity of the web [11].

10. Comparison of limit states for typical cap and sill beams

The ultimate loads for the different critical limit statescalculated using the different equations are compared fora range of typical falsework beams given in Table 2. Thedifferent limit states considered are: (1) flange bending andpost compression strength, calculated using the interactionequation and assuming a constant value for β = 18 for thetimber post and using the effective bearing area for the steelpost; (2) web yielding; (3) web crippling, and; (4) lateral webbuckling, assuming 3.05 m long beams. Table 2 shows that theflange–post capacity is smaller than the web capacity for all butthe double stacked beams in which lateral buckling may governthe capacity. This shows that for beams with thick webs, similarto those typically used in bridge falsework, web limit statesare unlikely to control the capacity of the beam–post regions.However, double stacked beams should be laterally braced toprevent lateral web buckling if it is calculated to govern thecapacity.

11. Summary and conclusions

Limit states that potentially control the design of beam–postjoint regions, similar to those typically found in falseworkfor cast-in-place concrete bridges, were investigated in thispaper. The limit states considered are associated with: (1)beam flange bending; (2) post compression, for timber orsteel posts; (3) an interaction between beam flange bendingand post compression; (4) post bearing; (5) web yielding;(6) web crippling and (7) lateral web buckling. A companionstudy was conducted by the authors [11] to establish responsecharacteristics and limit states experimentally. Subsequently,

a large number of finite element models were validatedand allowed this comprehensive analytical investigation.Accordingly, the critical limit states were found to be due toan interaction between flange bending and post compression,except for cases where beams were stacked on top of each other.Careful observations of the deformation and formation of yieldlines suggested two alternative methods to calculate limitingloads. The first method focuses on flange bending capacityassuming both uniform and triangular stress distributions inthe post. Based on the finite element analyses, this methodwith a constant β value of 18 is recommended for use incalculating the beam flange capacity with a timber post. Thesecond method utilizes an effective bearing area to establishthe capacity of the joint region and was more appropriate whenapplied patch load was due to a steel post. This method assumesa similar mechanism to that for web yielding with a 2.5:1 stressgradient. The critical web limit state was due to web yielding.In consideration of the different interactions between post andbeam, 1:1 stress gradient was found to be most appropriate andconservative for a timber post, whereas 2.5:1 stress gradientcharacterized more accurately the joint capacity with a steelpost in which yielding took place under each side of the steelpost. An equation for web buckling was also developed, whichmay govern in certain circumstances. It was demonstrated thatweb crippling was unlikely to govern the design of a beam witha relatively thick web. Lateral web buckling, however, will notgovern the design of single beams but may govern the designfor double stacked beams.

Acknowledgements

The authors would like to extend their gratitude to theCalifornia Department for Transportation for funding of thisstudy under contract number 59A0445, with special thanks toJohn Lammers and Peter Lee for their assistance and direction.

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Flange and web limit states in beams subjected to patch loading

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