DEFLECTIONS OF COMPOSITE BEAMS WITH WEB OPENINGS

By Manuel A. Benitez; Member, ASCE, David Darwin,:Z Fellow, ASCE,and Rex C. Donahey,3 Member, ASCE

ABSTRACT: Procedures for calculating the deflections of composite beams with web openings are described.Initially, a matrix formulation is used. Modeling assumptions are verified by comparison with experimental data,and recommendations for practical application of the matrix analysis procedures are made. The results of thecomparison are used to develop a design aid for estimating the maximum deflection of beams with web openingsand an expression for calculating the deflection across a web opening. The work demonstrates that, in mostcases, a single web opening often has little effect on the total deflection of a composite beam. There are, however,important cases where the effect can be significant. The effects of an opening and of shear deflections are ofthe same order. Ignoring both the web opening and shear deformation can lead to significant error. The matrixstiffness method, the design aid, and closed form equations provide reasonable estimates of both total deflectionand deflection across an opening.

INTRODUCTION

Most multistory steel buildings use composite membersconsisting of a concrete deck integrally connected to a steelbeam. The depth of the floor system can be reduced if webopenings are used to pass utilities through the steel sections.The result is reduced building height and overall cost savings.The introduction of web openings, however, can significantlyaffect beam behavior by increasing both the total deflectionsand the differential deflections across the openings. In mostcases, the influence of a single web opening is small, but forthose cases where the increase in deflections is unacceptable,a procedure capable of accurately predicting the deflections isneeded.

Since the early 1970s, a number of procedures have beendeveloped to calculate deflections for flexural members withweb openings. Three methods address steel beams (ASCE1971; McCormick 1972; Dougherty 1980), and one methodcovers both steel and composite beams (Donahey and Darwin1986; Donahey 1987). The first three procedures require cal­culation of the deflection due to the web opening; the totaldeflection is then obtained by adding this deflection to thedeflection of the beam without an opening. The fourth methoddirectly calculates the total deflection.

Web openings reduce the stiffness of a member at an open­ing by (1) lowering the gross moment of inertia at the opening;(2) eliminating strain compatibility between the top and bot­tom tees (regions above and below the opening); and (3)reducing the cross-sectional area available for carrying shear.The lower gross moment of inertia results in increased cur­vature at openings subjected to bending. The elimination ofstrain compatibility and the loss of material for carrying shearresult in differential vertical (or Vierendeel) deflections be­tween the ends of openings subjected to shear. The use ofreinforcement at an opening can restore the gross moment ofinertia and, thus, limit the increase in curvature. Reinforcementhas much less effect on the Vierendeel deflections.

This paper describes procedures for finding both total anddifferential deflections of composite beams with web openings

'Vice Pres., Metric Engrg., Inc., Miami, FL 33186.2Deane E. Ackers Prof. of Civ. Engrg. and Dir., Struct. Engrg. and

Mat. Lab., Univ. of Kansas, Lawrence, KS 66045. E-mail:daved@ukans.edu

'Dir. of Engrg. and Res., Composite Techno!. Corp., Ames, IA 50010.Note. Associate Editor: Samuel Easterling. Discussion open until

March I, 1999. To extend the closing date one month, a written requestmust be filed with the ASCE Manager of Journals. The manuscript forthis paper was submitted for review and possible publication on Novem­ber 10, 1997. This paper is part of the Journal of Structural Engineer­ing, Vo!. 124, No. 10, October, 1998. ©ASCE,ISSN 0733-9445/98-0010­1139-1147/$8.00 + $.50 per page. Paper No. 16963.

(e.g., Fig. 1). The procedures can be used for steel beams aswell. The methods represent an extension of the work by Don­ahey and Darwin (1986) and Donahey (1987). Initially, a ma­trix formulation is used. Modeling assumptions are verifiedusing comparisons with experimental data, and recommenda­tions for the practical application of the matrix analyses aremade. The results of the comparison are used to develop adesign aid for conservatively calculating the maximum deflec­tion of beams with web openings and an expression for cal­culating the deflection across a web opening. Full details ofthe study are presented by Benitez et al. (1990).

MATRIX ANALYSIS

Deflections are calculated using the stiffness method of ma­trix analysis. The top and bottom tees at an opening, as wellas the nonperforated sections adjacent to the opening, are mod­eled using standard 6 degree of freedom (DOP) beam ele­ments. The local element stiffness matrix, [K.l, is (Severn1970)

A0 0

-A- - 0 0i3 i3

L L- 0 -1 -2 2

L2 -L L2

-+'YJ 0 - 6"-'YJ£[3 3 2[K,J =-A

(1)L - 0 0

(3-L2

L 2

- + TJ3

in which E = modulus of elasticity; j3 = //(L2/12 + 11); 11 =EI/(AyG); Ay =effective shear area; A =gross transformed areafor axial deformations; L =element length; and I =momentof inertia of the transformed section.

The element is capable of incorporating shear deformations.as well as axial and bending deformations. Shear deformationscan be neglected by setting TJ = O.

[K.] is derived considering bending of a beam about its ownneutral axis. For the nonperforated sections of the beam, thelocal and global DOF are coincident. Thus, the local andglobal element stiffness matrices, [K.J and [Kg], are identical.

At an opening, the neutral axes of the tees and the adjacentnonperforated section are not coincident. Therefore, the localDOF for the top and bottom tees do not correspond to theglobal DOF for the structure. By assuming that the webs ad-

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Xg

Bottom Tee Elemen

00

rTop Te. Element

f

~1tt

2i-tb

I1 t

RIgid LInks(Typ. loch I

FIG. 2. Beam Element with Two Rigid Links (Donahey andDarwin 1986; Donahey 1987): (a) Global and Local Axes; (b)Global and Local Degrees of Freedom

FIG. 3. Web Opening Element Geometry and Axes Orienta­tions (Donahey and Darwin 1986; Donahey 1987)

(2)

(3)

~=~,...., ,...., ,...., ,...., ,...., ,...., ,....,

=~Yg Yt

Beam Element

IXttLO

Ls(a)

2Xg

*.Lr-- be -----j Ixl exl

Ts I I I I (a)

1 T }-'if'T.JLt~d Vtl Vt2

J r-JL tfBeam Element

Sb --LUt2L

./bf-l P- Utl 0t2(b)

FIG. 1. Composite Beam with Web Opening (Donahey and Vgl Vg2

Darwin 1986; Donahey 1987): (a) Beam Schematic; (b) Opening Ug2Detail -Xg

Ugl

and the global DOF are

{ug } T :::: Lug, VgI 6g1Ug2Vg26g2l

An eccentricity transformation, [8], is used to relate the lo­cal DOF and the global DOE

jacent to the opening are infinitely rigid, however, the nodesof the individual tees can be connected to the nodes of thenonperforated section by rigid links in the matrix model. Therigid links relate the local DOF for the top and bottom tees tothe global DOF at the ends of the opening.

Fig. 2 shows a beam element, in this case representing thetop tee, with 2 rigid links that connect the top tee to the cen­troid of the nonperforated section in the matrix model of thebeam. The local (related to the beam element) and global (re­lated to nodes on the nonperforated portion of the beam) axesare parallel, and eccentricities exist in both the x and y direc­tions. The local DOF are

{Uf} :::: [S]{Ug } (4a)P/ 2

l (12

[81 {

0 0 0

-t]<D 0-eyl

1 @ @ 11 exl 0 00 01 0

(4b)FIG. 4. Deflection Analysis Model for Beam with Web Opening

1 ex2 (Donahey and Darwin 1986; Donahey 1987)1

in which eyl and e y2 are the local y eccentricities; and exl andex2 are the local x eccentricities at nodes 1 and 2, respectively.

The global stiffness matrix, [Kg], for an eccentric beam el­ement is

(5)

The coordinate transformation given above is derived for abeam element with eccentricities at both ends. For some mod­els, it is necessary to use two beam elements to represent thetop tee. For these cases, the interior node at the centerline ofthe opening must be condensed out of the local stiffness matrixfor the top tee prior to the coordinate transformation.

The global stiffness matrices for the individual tees can becombined to form a web opening element. The web openingelement consists of top and bottom tee elements connected byfour rigid links (Fig. 3). Nodes 1 and 2 of the web openingelement are located so that the positive global coordinate axis,

xg , passes through the nodes. The nodes of the web openingelement are connected to the ends of the top and bottom teesby rigid links of length I, and lb' respectively. Local eccentric­ities eXl :::: e x2 :::: 0 for both the top and bottom tees, whereaseyl :::: e y2 :::: /, for the top tee and eyl :::: e y2 :::: -h for the bottomtee. The global stiffness matrix for the web opening element,[Kg]wo, is the sum of the global stiffness matrices (Kg] for theindividual tees. The global stiffness matrix for the web open­ing element can then be added directly to the global structurestiffness matrix, which consists of uniform beam elements oneither side of the web opening. An example structure incor­porating a web opening element is shown in Fig. 4.

Modeling Assumptions

The element stiffness matrices discussed above are a func­tion of the material properties and certain section parameters.The modeling assumptions used to calculate the element ma-

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(7)

(OO,b)

trices for both the perforated and nonperforated sections of thebeam are now discussed.

Moment of Inertia for Nonperjorated Sections of Beam

The moments of inertia for the nonperforated sections of abeam are computed using either the effective moment of in­ertia, Ieff (Specification 1989; Commentary 1993), or the lowerbound moment of inertia, lib (Load 1993). For both I eff and lib'the concrete slab is transformed into an equivalent steel areaand is assumed to act compositely with the steel section.

For models using I eff, the concrete deck is replaced by anequivalent steel area based on the modular ratio E.~/Ecorv;rc~'

I eff is calculated using the provisions of the AISC Specificationfor structural steel buildings (1989)

rv:.Ieff = Is + vi (Itr - Is); V~ 5; Vh

in which Is = moment of inertia for the steel section; ItT =moment of inertia of the transformed composite section; V~ =sum of the shear stud capacities between the point of maxi­mum moment and the nearest point of zero moment; and Vh

=smaller of the tensile yield capacity of the net steel sectionor the crushing strength of the concrete slab. Except for smalldifferences in study capacity for concrete compressivestrengths greater than 4 ksi, the expression for I eff in (6) isidentical to that used in the Commentary to the AISC LRFDSpecification (1993).

For models using lib, the concrete deck is replaced by anequivalent steel area based on the ratio of the crushing strengthof the concrete to the yield strength of the steel, 0.85 f;/Fy • libis calculated using the provisions of Part 4 of the AISC Loadand resistance factor design manual of steel construction(1993) and is

(d)2 2: Qn

lib = Is + As YENA - '2 +~ (d + Y2 - YENAf

in which As = cross-sectional area of the steel section; YENA =distance from the bottom of the beam to the elastic neutralaxis; d =depth of the beam; L Qn =the smaller of the tensileyield capacity of the gross steel section, the crushing capacityof the concrete slab, or the sum of the shear stud capacitiesbetween the point of maximum moment and the nearest pointof zero moment; Fy =yield stress of the steel; and Y2 =distancefrom the concrete flange force to the beam top flange.

Moments of Inertia at Web Opening

The bending stiffness of the bottom tee is represented usingthe elastic moment of inertia, whereas the bending stiffness ofthe top tee is represented using three different modeling as­sumptions. In each of the three cases, both I eff and lib are con­sidered.

For Modell, the representation used by Donahey and Dar­win (1986) and Donahey (1987), it is assumed that the con­crete above an opening does not significantly contribute to thebending stiffness of the top tee. For this model, the momentof inertia of the top tee is calculated based on the steel teeonly.

For Model 2, the concrete above the opening is included inthe bending stiffness of the top tee. For this model, the mo­ment of inertia for the top tee is computed based on both thesteel tee and the transformed area of concrete above the open­ing using the procedures discussed above.

For Model 3, two beam elements (one steel and one com­posite) of equal length are used to represent the top tee (seetop tee in Fig. 5). At the low moment end of the opening, theconcrete is neglected and only the steel tee is considered in

Top Tee Element

Bottom Tee Element

y

4-.FIG. 5. Model of Region at Web Opening (Simplified Model 3)

calculating the moment of inertia. At the high moment end,the bending stiffness of the top tee is calculated consideringboth the steel tee and the transformed area of the concretedeck. Of the three models, Model 3 should provide the mostrealistic results, because the concrete slab tends to crack intension over the low moment end of the opening in caseswhere the opening region is subjected to a shear (Clawson andDarwin 1980; Donahey and Darwin 1986).

Effective Area for Carrying Axial Force

For all three models, the effective area for carrying axialforce within the top tee, At, is taken as the area of the steeltee (including reinforcement, if used) plus the transformed areaof the concrete deck. The centroid of the top tee is taken asthe centroid of the transformed section. Therefore, the lengthof the rigid link, 1" is taken as the distance from the centroidof the nonperforated section to the centroid of the transformedsection for the top tee.

Effective Area for Carrying Shear Force

Only the webs of the steel sections are considered effectivein carrying the shear. Away from the opening, Ay = dtw, inwhich d =depth of the steel section and tw = thickness of theweb. At an opening, the effective shear areas for the top andbottom tees are Ayt = sttw and AYb = Sbtw, in which St and Sb =depths of the top and bottom steel tees, respectively.

Material Properties

Steel sections are modeled using E = 200 GPa (29,000 ksi)and G = 76.9 GPa (11,150 ksi). The elastic modulus for con­crete is taken as E = 4.7 Y1e GPa, withf; in MPa (57 v'1cksi, with f; in psi).

Comparison with Test Results

To evaluate the three models, calculated deflections for themodels are compared with deflections for twenty-five testspecimens. Models of entire beams, as well as models of onlythe region at the web opening, are used. Models of test beamsinclude rigid links at the supports as well as at the openingsand incorporate beam elements to represent nonperforated sec­tions of the beams (Fig. 4). The rigid links at the supports inthe model connect the bottom flange of the steel section to the

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centroid of the transformed section to accurately represent thetest beams, which were all supported at the bottom flange.Models of only the region at the web opening include a fixedsupport at the low moment end (Fig. 5). As will be explainedlater, the fixed support has an initial rotation equal to the ro­tation at the position of the low moment end of the openingin the nonperforated beam under full loading.

The deflection data used in the parametric study include 13tests by Donahey and Darwin (1986), six tests by Clawsonand Darwin (1980), two tests by Granade (1968), and fourtests by Redwood and Wong (1982). The beams are desig­nated, respectively, with numbers only, and with the letters c,g, and r. Data on the two g beams include maximum deflec­tion, but not deflection across the opening.

The test beams were designed primarily to obtain informa­tion on the strength at web openings. The beams were rela­tively short, and opening locations were predominately in highshear regions. For this reason, the relative importance of sheardeflections and the deflections through the opening are greaterthan. for beams in which flexural deformations play a greaterrole. To study the importance of shear deformation in totalbeam deflection, analyses that account for shear deformationsthroughout the span, V, are compared with analyses that ignoreshear deformations throughout the span, NV. Comparisons aremade at loads equal to 30 and 60% of the ultimate appliedload. As will be shown in the following discussion, Model 2(concrete included in bending stiffness for full top tee) givesthe best results for the deflection at the point of maximummoment, whereas Model 3 (concrete included in bending stiff­ness at the high moment end of the top tee only) performsbest in predicting the deflection across the opening.

Deflections at Point of Maximum Moment

The mean ratios of calculated deflection to measured testdeflection, along with the standard deviations and coefficientsof variation for the models, are shown in Table I. Review ofTable I shows that ignoring shear deformations, NV, causesthe models to be too stiff and thus unconservative (mean ratiosof calculated to test deflections less than 1.0). From a practicalpoint of view, it is better to have the calculated deflectionssomewhat above the actual to provide for adequate stiffnessin the structure. By including shear deformation, V, the modelsare IS to 24% more flexible than models that ignore sheardeformation. Likewise, models using lib are 12 to 19% moreflexible than models using lerr.

At an applied load of 30% of ultimate, models includingshear, V, have mean ratios of calculated to measured deflec­tions of 1.13, 0.94, and 0.97 using leff and 1.26, 1.08, and 1.11using lib for Models I, 2, and 3, respectively. Ignoring shear

deformations, NV, the respective models have mean ratios of0.97, 0.76, and 0.80 using leff and 1.10, 0.90, and 0.94 usinglIb' At an applied load of 60% of ultimate, V models have meanratios of 1.01, 0.84, and 0.87 using leff and 1.13, 0.97, and1.00 using lib' whereas NV models have mean ratios of 0.86,0.68, and 0.72 using leff and 0.98, 0.81, and 0.84 using lib'

The models that use lIb and include shear deformations, V,provide the best overall agreement with the test data for totaldeflection. Of these models, Model 2 provides the best per­formance and least scatter in the data. Figs. 6 and 7 comparedeflections calculated using Model 2 (lIb' V) to the test deflec­tions at the point of maximum moment at 30 and 60% ofultimate, respectively. At 30% of ultimate, the model has amean ratio of calculated to measured deflection of 1.08, witha standard deviation of 0.18 and a coefficient of variation(COV) of 16.5%. At 60% of ultimate, the model has a meanratio of 0.97, with a standard deviation of 0.15 and a COY of15.1 %. Overall, deflections at the point of maximum momentcalculated using Model 2 (lIb' V) are in close agreement withthe measured deflections.

The decrease in the mean ratio of calculated to test deflec­tion as the load is increased from 30 to 60% of ultimate is areflection of the relatively early onset of yielding around theopening.

Deflections across Opening Using Full-Beam Model

For deflection across the opening (i.e., the difference in thedeflections measured at the high and low moment ends of the

12.00

• Test 1 tbru 9b

~• Test cl tbru c6• Test ..0 tbru r3

9.00 + Test gl tbru g4Ii=:eell

IClell 6.00Q

1ell

-=u 3.00'iJU

0.00

0.00 3.00 6.00 9.00 12.00

Measured Deflection, mm

FIG. 6. Calculated versus Measured Deflection at Point ofMaximum Moment, Model 2 (lib' V), at 30% of Ultimate Load

TABLE 1 Mean Ratios of Calculated to Measured Deflection at Point of Maximum Moment

MODEL 1 MODEL 2 MODEL 3

Ie. lib Ie. I'b Ie. I'bParameter V

INV V

I NV VI

NV VI

NV VI

NV VI

NV(1 ) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11 ) (12) (13)

MeanStandard deviationCoefficient of variation

0.900.18

19.84

0.970.16

16.90

0.800.16

19.49

1.110.19

17.22

0.940.18

19.01

Mean 1.01 0.86 1.13 0.98 0.84 0.68 0.97 0.81 0.87 I 0.72 11.00 0.84Standard deviation 0.22 0.20 0.23 0.22 0.12 0.13 0.15 0.16 0.12 0.13 0.15 0.15Coefficient of variation 21.53 23.53 20.27 21.98 13.90 19.36 15.13 19.35 14.02 17.46 15.16 17.62

Note: Modell: Bending stiffness of the top tee neglects any contnbutlOn of the concrete slab. Model 2: Bendmg stIffness of the top tee Includescontribution of the concrete slab. Model 3: Bending stiffness of top tee includes contribution of the concrete slab at the hIgh moment end only. V =shear included in the model; NV = shear not included in the model.

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Measured Deflection, mm

FIG. 7. Calculated versus Measured Deflection at Point ofMaximum Moment, Model 2 (lIb' V), at 60% of Ultimate Load

opening), agreement between calculated and experimental val­ues is not as good as that obtained for deflection at the pointof maximum moment. The comparisons are summarized inTable 2 and include the mean ratios of calculated to measureddeflection, along with the standard deviations and coefficientsof variation. All of the models exhibit a large amount of scatterwhen compared to the measured deflections, as reflected bythe large coefficients of variation. As for the total deflections,the models that use lib and include shear deformation, V, pro­vide the best agreement with the test data.

The calculated deflections across the opening are sensitiveto the assumptions made concerning composite behavior overthe opening. Model 1, in which composite action over theopening is neglected, is too flexible and overly conservative.Model 2, however, which assumes composite action over theentire length of the opening, is too stiff and unconservative.At 30% of ultimate load and including shear deformations, themean ratios of calculated to measured deflection for Model Iare 1.59 using leff and 1.64 using lib' For model 2, the meanratios are 0.74 and 0.83 using leff and lib' respectively. At 60%of ultimate, the mean ratios are 1.15 and 1.24 using Model 1and 0.58 and 0.65 using Model 2.

Model 3 gives the best results for deflection across the open­ing. This model considers composite action over the openingonly at the high moment end; whereas the moment of inertiafor the top tee at the low moment end is calculated using thesteel tee only. Deflections calculated using this model are still

2.50

•0.00

0.00

2.50

Deflections across Opening Using Model of Web OpeningOnly

To evaluate models of the opening only, the assumptionsused to represent the top and bottom tees at the opening forModel 3, the best of the three models for deflection throughthe opening, are used. An initial rotation, e, equal to the ro­tation at the position of the low moment end of the openingin the nonperforated beam under full loading, is assumed (Fig.5).

A comparison with the same test results as used for the fullmodel shows that considering only the region around the open­ing gives a model that is slightly more flexible than the fullbeam model, with the simplified and full models giving meanratios (and Cays) of 1.03 (43.5%) and 1.00 (43.7%) at 30%of ultimate and 0.79 (44.3%) and 0.77 (45.1%) at 60% ofultimate, respectively. The data scatter is virtually identical forthe two models. The calculated deflections for the simplifiedmodel are compared to the test deflections at 30 and 60% ofultimate load in Figs. 8 and 9, respectively.

• Test 1 thna 9b• Test cl thru c6

0.50 1.00 1.50 2.00

Measured Deflection, mm

FIG. 8. Calculated versus Measured Deflection across Open­ing, Simplified Version of Model 3 (~b' V), at 30% of UltimateLoad

generally lower than the test deflections, but the scatter in thedata is less than that for Model 1. At 30% of ultimate load,the model has a mean ratio of 1.0, with a standard deviationof 0.44 and a cav of 43.7%. At 60% of ultimate, the meanratio is 0.77, with a standard deviation of 0.35 and a cav of45.1%.

10.00 15.00 20.00 15.005.00

• Test 1 thru 9b• Test cl thru c6• Test rO thna r3+ Test gl thna g2

0.000.00

15.00

TABLE 2. Mean Ratios of Calculated to Measured Deflection across Opening

MODEL 1 MODEL 2 MODEL 3

Ion lib I•• lib I•• Ib

Parameter V I NV VI

NV V I NV VI

NV V I NV VI

NV(1 ) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11 ) (12) (13)

(a) 30% of ultimate load

Mean 1.59 1.42 1.64 1.49 0.74 0.46 0.83 0.56 0.91 0.68 1.00 0.76Standard deviation 0.88 0.82 0.91 0.83 0.31 0.21 0.34 0.25 0.41 0.31 0.44 0.35Coefficient of variation 54.93 57.72 55.27 55.99 41.77 44.99 41.02 44.16 44.30 45.81 43.71 46.20

(b) 60% of ultimate load

Mean 1.15 1.08 1.24 1.13 0.58 0.37 0.65 0.45 0.71 0.52 0.77 0.59Standard deviation 0.55 0.51 0.58 0.53 0.27 0.18 0.30 0.22 0.32 0.24 0.35 0.28Coefficient of deviation 47.74 47.68 46.55 46.76 45.81 50.30 45.46 49.12 44.89 46.59 45.06 47.11

Note: Modell: Bending stiffness of the top tee neglects any contribution of the concrete slab. Model 2: Bending stiffness of the top tee includescontribution of the concrete slab. Model 3: Bending stiffness of top tee includes contribution of the concrete slab at the high moment end only. V =shear included in the model; NV =shear not included in the model.

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6.00

• Test 1 thru 9b• Test cl thru c6• Test rO thru r3

•

•0.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Measured Deflection, mm

FIG. 9. Calculated versus Measured Deflection across Open­ing, Simplified Version of Model 3 (I,., V), at 60% of UltimateLoad

DESIGN APPLICATIONS

The comparisons discussed above show that matrix analysisprocedures provide close agreement with test results for totaldeflection and are reasonably accurate for predicting the de­flection across the opening. However, most of the test beamshad relatively short spans, and all of the beams were loadedwith point loads. Since this is not typical of actual construc­tion, two studies were undertaken to investigate the effect ofweb openings on uniformly loaded beams of longer span, withthe goal of developing procedures for determining deflectionsthat will not require a matrix model (Benitez et aI. 1990).

The studies treated the total deflection and the deflection

1.16

across the opening separately. Model 2 was used to predict thetotal deflection, and the simplified version for Model 3 (onlythe region around the opening was investigated) was used topredict the deflection across the opening. lib was used for themoment of inertia.

The beams in the studies had a total slab thickness of 114mm (4.5 in.) placed on a 51 mm (2 in.) ribbed deck and werespaced 2.7 m (9 ft) on center. Three steel sections were con­sidered: (1) W61Ox82 (W24x55) with an 11.9 m (39 ft) spanand uniform load, W = 25.7 kN/m (1.76 kJft); (2) W460x52(WI8x35) with a 9.1 m (30 ft) span and w = 25.2 kN/m (1.73kJft); and (3) W360x32.9 (W14x22) with a 6.4 m (21 ft) spanand w = 25.1 kN/m (1.72 kJft). Three opening height to steelbeam depth ratios (ho/d), 0.3, 0.5, and 0.7, three openinglength to opening height ratios (ao/ho), 1, 2, and 3, and fivelocations for the opening centerline with respect to the support,Lo =I1t6, lfs, lf4, 3fs, and Ih of the span length L.. were considered,for a total of 45 combinations for each beam. The beams weremodeled with simple supports located at the centroid of thetransformed section.

Deflection at Point of Maximum Moment

For calculations involving deflection at the point of maxi­mum moment, the studies revealed that deflection is alwaysmaximum when an opening is located at the span centerline.These results are illustrated for the W360x32.9 (WI4x22)beam in Fig. 10. In the figure, the total deflection at the pointof maximum moment calculated using Model 2 and includingshear deformations throughout the span .lm, is normalized withrespect to the maximum deflection of a uniformly loaded beamwithout an opening calculated using classical beam theory .lb+ .l.. in which .lb = 5wL:/(384El) and .ls = wL;/(8AyG). Asshown in Fig. 10, opening size can have a greater effect ontotal deflection than opening placement.

For the three beams studied, the ratio .In/.lb ranges between

1.12

,.-..

'"<I+

S 1.08-J3

1.04

1.00

1/16 1/8

d Q ...2hh ...0.5, 0 0o

1/4 3/8

h .0.5d. Qo·hoo

h ..0.3d. Qo·2hoo

1/2

Location of Opening Centerline (LolLs)

FIG. 10. Location of Opening Centerline versusl1",1(l1b + 11.) for W360x32.9 (W14x22) Beam

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... 'LaFIG. 11. Design Aid for Estimating Maximum Deflection InBeam with Web Opening-Deflection Calculated for OpeningLocated at Beam Centerline

(12)

Deflection across Opening

For the typical case, where the web opening centerline doesnot coincide with the span centerline, the differential deflectionfrom one side of the opening to the other should also be con­sidered when checking the serviceability of a beam. For thispurpose, the deflection can be divided into three parts: (1) thedeflection due to a rotation a; (2) the deflection due to a shearload P; and (3) the deflection due to a moment M. The sim­plified version of Model 3 can be used to develop expressionsthat do not require the use of matrix analysis to calculate thedeflection across the opening, ay •

In the simplified version of Model 3, the rotation at thelower moment end of the opening is taken as the rotation, a,in an equivalent nonperforated member. In this case, the con­tribution to deflection due to a is

a~8) =aoa (10)

The deflection across the opening due to shear force, P,(Fig. 5) consists of two components: (1) the deflection due tosecondary bending, a~7, and (2) the deflection due to sheardeformation aC;:l. In general, for a large opening, the deflectionwill primarily be due to secondary bending. For smaller open­ings, shear deformations dominate. Separate equations are de­rived for the two cases. Using Model 3 as a basis, the deflec­tion due to secondary bending can be conservativelyapproximated as (Benitez et al. 1990)

A<P) a~ [ 4(14/eq + I b ) 1]ayb = - 2 2 + P (11)

. 12£ (8/eq + h)(l,A, + IbAb) (8/eq + I b)

in which leq = l,l,i(14/'l + Ia); 1'1 = moment of inertia for thetop tee at the low moment end of the opening; la =momentof inertia for the top tee at the high moment end of the open­ing; and I b = moment of inertia for the bottom tee.

Eq. (11) can be simplified by considering the upper andlower bounds on the ratio between the moments of inertia ofthe top and bottom tees. For openings with very stiff slabs,the value of I b will be small in comparison to 8/eq. In this case,the ratio (14/eq + Ib)/(8/eq + Ib ) will approach an upper boundof 1.75. Conversely, as the stiffness of the slab becomes neg­ligible, taking 8/eq = Ib, the ratio (14/eq + Ib)/(8/eq + Ib) willhave an effective lower bound of 1.375. Using the medianvalue, 1.5625, in (11) gives

A<P) a~ [6.25 1]ay,b =- 2 2 + P

12£ (I,A, + IbAb) (8/eq + Ib)

Deflections calculated using (12) are most conservative(high relative to the matrix solution) for small openings andapproach the matrix solution as the ratio hold increases (Ben­itez et al. 1990). For openings with hid = 0.3, (12) overesti­mates the deflection by 19.3%, 14.7%, and 11% for theW61Ox82 (W24x55), W460x52 (W18x35), and W360x32.9(W14x22) sections, respectively. However, for hold = 0.7 (12)overestimates the matrix solution by only 4.3%, 2.3%, and1.35% for these sections.

Although (12) is conservative for small openings, the totaldeflection across the opening is not severely affected. This isbecause the deflections due to secondary bending, a<;J, becomesignificant only as the opening size increases. For small open­ings, the effects of secondary bending are negligible. Takingthe W61Ox82 (W24x55) section with hold =0.3 and ao/ho = 1as an example, the deflection a~7, is only 0.56% of ayo How­ever, for the same beam with hold = 0.7 and aolho = 3, a<;J is60.4% of avo

As for a~:2, a simplified expression a~:l can be derived.

a<1') _...!:...- Ai' (13)y,s - Ay,G

(8a)

(8b)

0.150.100.051.00

0.00

1.25

1.50

1.75

1.38 and 1.22 (Benitez et al. 1990), while the ratio aml(ab +a.) ranges between 1.001 and 1.144. For web openings withhold s 0.3 or aolho S 1, the web opening causes an increasein maximum deflection less than 5%, independent of openinglocation.

The results of this study, as illustrated in Fig. 10, show thata conservative estimate of the maximum deflection can be ob­tained by assuming that the opening is located at the spancenterline. For this case, taking the moment of inertia for thebeam in the region of the opening Iwo as the moment of inertiaof the top and bottom tees about the centroid of the trans­formed section without an opening, the ratio of the maximumbending deflection of a uniformly loaded beam with an open­ing to the maximum deflection of a beam without an opening,am.Ja b, is (Benitez et al. 1990)

where ao = opening length; L. = span length; Ig = moment ofinertia for the nonperforated section; and I wo = moment ofinertia in the region of the web opening.

Eq. (8) can be used to develop a design aid based on theratios Igllwo and aolL.. as shown in Fig. 11. Each curve in Fig.11 represents a constant value of am,blabo Given am,bla b andthe deflection for the nonperforated beam, the total bendingdeflection for the perforated beam can be readily calculated.

Since Fig. 11 considers deflection due to bending only, theestimate for deflection can be improved by including sheardeformation in the analysis. This can be done with sufficientaccuracy by adding a.. the shear displacement for the non­perforated beam, giving

(9) in which

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(14)

A = ~eq~eq + ~b

8/eq 8/eq~eq = [(a~112) + TJeql' TJeq =Ay,G

Ib £lb~b = [(a~112) + TJbl' TJb = AybG

Deflections due to shear defonnation have the greatest effecton small openings. For the three beams used in the case stud­ies, (13) overestimates the value of df:; by less than 2%.

The deflections across the opening due to moment M canbe approximated by (Benitez et al. 1990)

A(M) _ a~ [ (12/eq + I b) ]J.J.yb - 22 M

. 2£ (8/eq + Ib)(l,A, + IbAb)

Eq. (14) can be simplified by considering that the ratio of(12/eq + Ib )/(8/eq + Ib) will have an upper bound of 1.5 andan effective lower bound of 1.25. Taking the median value,1.375, d~~ can be expressed as

(M) a~ [ 1.375 ]d Y•b =2£ 17A, + I~Ab M (15)

For openings with an hold ratio of 0.3, (15) overestimatesthe matrix solution by 16.6%, 11.6%, and 7.7% for theW61Ox82 (W24x55), W460x52 (WI8x35), and W360x32.9(WI4x22) sections, respectively (Benitez et al. 1990). Foropenings with hold = 0.7, (15) underestimates the matrix so­lution by 3.3%, 4.9%, and 5.9%, respectively.

In most cases, the deflection across the opening due to Madds little to d y • However, this component of the deflectiondoes increase as the size of the opening increases and as theopening is located progressively closer to the span centerline.The importance of including d~~ in the analysis therefore in­creases as the relative size of the opening increases. For theW24x55 section with hold = 0.3 and aolho = 1, d~~ only ac­counts for 0.8% through 8.2% of dY•b as the location of theopening ranges from 1116 Ls to % Ls• For the same beam withhid =0.7 and aiho=3, d~~ accounts for 5.2% through 57.4%of d y•b ' Therefore, for small openings, overestimating the de­flection d~~ will not significantly affect the total deflectionacross the opening, dy•b '

For design purposes, the total deflection across the openingcan be calculated by combining (10), (12), (13), and (15)

(16)

Because (10) represents the deflection across the openingdue to rotation of the nonperforated beam, only the last threetenns in (16) represent the real effects of the opening on localdeflection.

Values for the total deflection calculated using (16) werecompared to the measured deflections across the openings forthe test beams (Benitez et al. 1990). At 30% of ultimate, theratios of calculated to measured ratios have a mean of 1.075,with a standard deviation of 0.483 and a COV of 45.0%. At60% of ultimate load the mean ratio decreases to 0.832, witha standard deviation of 0.382 and a COY of 45.9%. Theseresults show that (16) provides accuracy that is equivalent tothe matrix solutions.

SUMMARY AND CONCLUSIONS

A web opening often has little effect on the deflections ofa composite beam. However, there are important cases wherethe effect can be significant. Application of the stiffnessmethod of matrix analysis provides a reasonable estimate oftotal deflection and deflection across the opening when com-

1146/ JOURNAL OF STRUCTURAL ENGINEERING / OCTOBER 1998

pared with test data. The best match with experimental resultsis obtained by using the lower bound moment of inertia andincluding shear defonnations in the analysis. For total deflec­tion, the top tee is best modeled by including composite actionover the full length of the tee, while deflection across the open­ing is best modeled by including composite action only overthe high moment end of the opening.

Generally, the effects of the web opening and shear deflec­tions over the span will be of the same order. Ignoring boththe web opening and shear defonnation can lead to significanterror. For small openings, it is more important to considershear deflections than the effects of a web opening. A conser­vative estimate of the effects of an opening can be obtainedby analyzing a beam with the opening located at the spancenterline. Fig. 11 can be used to obtain the ratio of the totalbending deflection of the perforated beam to the maximumbending deflection of the nonperforated beam, dm,bldb, for agiven opening size. Shear defonnations throughout the spanmust be added to get the best estimate.

A simplified matrix model for the deflections across theopening has been developed that considers only the regionaround the opening. Modified equations for the total deflectionacross the opening are derived from the matrix model andevaluated. The results show that, for large openings, the effectsof shear defonnation are negligible. Conversely, for smallopenings, bending defonnations are negligible.

APPENDIX I. REFERENCES

ASCE Subcommittee on Beams with Web Openings. (1971). "Suggesteddesign guides for beams with web holes." J. Struct. Div., ASCE,97(11), 2702-2728.

Benitez, M. A., Darwin, D., and Donahey, R. C. (1990). "Deflections ofcomposite beams with web openings." SL Rep. No. 90-3, Univ. ofKansas Center for Research, Lawrence, Kan.

Clawson, W. C., and Darwin, D. (1980). "Composite beams with webopenings." SM Rep. No.4, Univ. of Kansas Center for Research,Lawrence, Kan.

Commentary to the load and resistance factor specification for structuralsteel buildings. (1993). American Institute of Steel Construction, Chi­cago, III.

Donahey, R. C. (1987). "Deflections of composite beams with web open­ings." Build. Struct.: Proc., ASCE Struct. Congress, D. R. Shennan,ed., ASCE, Reston, Va., 404-417.

Donahey, R. C., and Darwin, D. (1986). "Performance and design ofcomposite beams with web openings." SM Rep. No. 18, Univ. of Kan­sas Center for Research, Lawrence, Kan.

Dougherty, B. K. (1980). "Elastic defonnation of beams with web open­ings." J. Struct. Div., ASCE, 106(1), 301-312.

Granade, C. J. (1968). "An investigation of composite beams havinglarge rectangular openings in their webs," MS thesis, University ofAlabama, Tuscaloosa, Ala.

Load and resistance factor design manual of steel construction. (1993).2nd Ed., American Institute of Steel Construction, Chicago, III.

Lopez, L. A., Dodds Jr., R. H., Rehak:, D. R., and Schmidt, R. J. (1994)."Polo Finite: a structural mechanics system for linear and nonlinearanalysis." Tech. Rep., University of Illinois, Urbana-Champaign, III.

McConnick, M. M. (1972). "Open web beams-behavior, analysis, anddesign." BHP Melbourne Res. Lab. Rep. MRL 17/18, Melbourne Res.Lab., The Broken Hill Proprietary Co. Ltd., Clayton, Australia.

Redwood, R. G., and Wong, P. K. (1982). "Web holes in compositebeams with steel deck." Proc., 8th Canadian Struct. Engrg. Con/.,Canadian Steel Construction Council, Willowdale, Ontario, Canada.

Severn, R. T. (1970). "Inclusion of shear deflection in the stiffness matrixfor a beam element." J. Strain Analysis, London, U.K., 5(4), 239­241.

Specification for structural steel buildings. (1989). American Institute ofSteel Construction, Chicago, III.

APPENDIX II. NOTATION

The following symbols are used in this paper:

A area;Ay effective shear area;

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ao = opening length;d = depth of steel beam;E = modulus of elasticity;e = local eccentricity;

Fy = yield strength of steel;.fc = compressive strength of concrete;G = shear modulus of steel;ho = opening depth;I = moment of inertia;

[K] = stiffness matrix;L = length of element or member;t = length of top or bottom tee;

M = bending moment;P = shear force;

{u} = displacement vector;w =uniform load;A = deflection;11 = ElI(AG); ande = rotation at low moment end of web opening.

Subscripts

b, m, s, t =bottom tee or bending, model, shear, top tee;e or t, g, wo = local element, global or gross, web opening;

eff, eg, lb =effective, equivalent for top tee, lower bound;s, 0 = span, opening; andl, 2 = low moment end or node, high moment end or

node.

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