Steel beam with large web openning

Journal of Constructional Steel Research 59 (2003) 1177–1200www.elsevier.com/locate/jcsr

Steel beams with large web openings of variousshapes and sizes: an empirical design methodusing a generalised moment-shear interaction

curve

K.F. Chunga,∗, C.H. Liu b, A.C.H. Ko a

a Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, HongKong, China

b Manchester School of Engineering, University of Manchester, Simon Building, Oxford Road,Manchester M13 9PL, UK

Received 9 April 2002; accepted 12 February 2003

Abstract

Vierendeel mechanism is always critical in steel beams with single large web openings.While the depth of web openings controls both the shear and the flexural failures of the perfor-ated sections, it is the length of the web openings that governs the ‘Vierendeel’ mechanismwhich in turn depends on the local shear and moment capacities of the tee sections above andbelow the web opening. A comprehensive finite element investigation on steel beams withweb openings of various shapes and sizes was reported in a complementary paper, and theprimary structural characteristics of those steel beams were presented in detail.

Comparison on the global moment-shear interaction curves of those steel beams shows thatthey are similar to each other in shape, and thus, it is possible to develop a generalised moment-shear interaction curve to assess the load capacities of all steel beams with web openings ofvarious shapes and sizes. As the global shear forces cause both shear failure and ‘Vierendeel’mechanism in perforated sections, the effect of local ‘Vierendeel’ moments acting onto thetee-sections above and below the web openings may be incorporated through a reduction tothe global shear capacities of the perforated sections. A global coupled shear capacity is thusestablished and its values for web openings of various shapes and sizes are obtained directlyfrom the finite element investigation. Details of the design method are fully presented inthis paper.

Moreover, an indicative parameter, the ‘Vierendeel’ parameter, is established to assess the

∗ Corresponding author. Tel:+1-852-2766-6063.E-mail address: [email protected] (K.F. Chung).

0143-974X/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0143-974X(03)00029-4

1178 K.F. Chung et al. / Journal of Constructional Steel Research 59 (2003) 1177–1200

performance of ‘Vierendeel’ mechanism in perforated sections. Through comparison amongthe moment and the shear utilisation ratios, m and v, and the ‘Vierendeel’ parameter, vi , thecritical modes of failure in perforated sections under different moment-shear ratios may bereadily assessed. 2003 Elsevier Science Ltd. All rights reserved.

Keywords: ‘Vierendeel’ mechanism; Perforated sections; Openings of various shapes; Moment-shear inter-action; Design development

1. Introduction

Modern multistorey buildings always have a stringent requirement on headroom.In order to accommodate building services within the constructional depth of a floor,it is common practice to provide web openings in structural floor beams for passageof services. A large amount of research efforts on the structural behaviour of steelbeams with web openings have been reported in the literature, primarily for steelbeams with multiple hexagonal web openings, and also for composite beams withsingle rectangular web openings, both with or without reinforcement. Rectangularweb openings were often formed with aspect ratios ranging from 1.0 to 3.0 whilethe opening depth, do, was commonly restricted to about 50% of the overall sectionheight, h. Circular web openings were also popular in commercial buildings withhigh specifications in building services due to easy installation of water pipes.

In a perforated section under a global moment Mo,Sd and a global shear forceVo,Sd , three local actions are induced in the tee-sections above and below the webopening as shown in Fig. 1:

� Axial force in the tee-section, NT, due to the global moment Mo,Sd.� Shear force in the tee-section, VT, due to the global shear force, Vo,Sd.� Local moment in the tee-section, MT, due to the transfer of shear force Vo,Sd

across the opening length.

For beams with given loading and support conditions, the magnitudes of theselocal actions depend on the shapes, the sizes, and also the locations of the openings.As reported by Lawson [1], Darwin [2], Redwood [3], and Oehlers and Bradford[4], the presence of web openings may have a severe penalty on the load carryingcapacities of structural members, depending on the configurations of the web open-ings. An overall review on the design recommendations [1–4] shows that in general,there are two design approaches in assessing the structural behaviour of steel beamswith web openings:

� Tee section approach: In this approach, the perforated section is considered to bebuilt up of two tee sections which are separated by a distance according to theheight of the web opening, and all the global actions are re-presented as localforces and moments. The structural adequacy of the steel beams depends on the

1179K.F. Chung et al. / Journal of Constructional Steel Research 59 (2003) 1177–1200

Fig. 1. Force distribution in a perforated section.

section capacities of the tee sections under co-existing axial and shear forces, andlocal moments. In general, the design methods with this approach are complicatedand the calculation effort is considerable. The accuracy of the methods dependson the accuracy of a number of design rules against respective failure modes.However, due to the complexity of the problems, approximate design expressionsare often presented to reduce the calculation effort, leading to conservative results.

� Perforated section approach: In this approach, the perforated section is the criticalsection to be considered in design, and the structural adequacy of the steel beamsdepends on the section capacities of the perforated sections under co-existingglobal shear forces and moments. Simple and empirical moment-shear interactioncurves are often used, and thus, the design methods are generally considered tobe simple, straight forwards, and suitable for engineers in their practical design.However, the design methods are somehow restrictive with limited applications,and often, they are very conservative [5].

In order to provide design guidance for engineers to design steel beams with webopenings of various shapes and sizes for full integration with building services, it is


highly desirable to develop an empirical design method which is structurally efficientand applicable to web openings of various shapes and sizes [6]. This is bestaccomplished through extensive parametric studies of non-linear finite element analy-ses on steel beams with web openings followed by development of an empiricaldesign method using a generalised moment-shear interaction curve.

2. Scope of investigation

The project may be divided into the following parts of activities:

1. Part I. Finite element investigation. Based on the finite element models withmaterial and geometrical non-linearity established for steel beams with circularweb openings, a comprehensive parametric study was carried out to investigateand compare the load carrying capacities of steel beams with web openings ofvarious shapes and sizes. A total of eight different opening shapes with threedifferent opening sizes in steel beams of four different section sizes were covered.The structural behaviour of the perforated sections in terms of deformation charac-teristics, moment shear interaction curves, and yield patterns were fully reportedand discussed in a complimentary paper [7]. However, for ease of reference, themoment-shear interaction curves of perforated sections are presented in this paper.

2. Part II. Development of empirical design method. Based on the results of thefinite element investigation, an empirical design method for steel beams with webopenings of various shapes and sizes was developed through the use of a general-ised moment-shear interaction curve. Basic section capacities of the perforatedsections were first established, and then the effect of local ‘Vierendeel’ momentsacting on the tee-sections above and below the web openings was incorporatedthrough a reduction to the global shear and moment capacities of the perforatedsections. The design method is fully presented in this paper supplemented withworked examples in the Appendices.

In the present investigation, all steel beams are hot rolled steel I sections of class1 or 2 (plastic or compact). All web openings are concentric to the mid-height ofthe sections with diameters between 0.5 and 0.75 h where h is the section depth; noreinforcement is considered. The formulation is presented in accordance with Euroc-ode 3 for easy reference. It should be noted that both the bending moment, Mo,Sd,and the shear force, Vo,Sd, due to global actions are evaluated at the centre of theweb openings, as shown in Fig. 1. The geometric configurations of all the web open-ings covered in the finite element investigation are presented in Fig. 2.

3. Moment–shear interaction curves from finite element investigation

The moment-shear interaction curves obtained from the finite element investi-gation are presented in Fig. 3. The moment-shear interaction curves for eight opening


Fig. 2. Geometric configuration of web openings.

shapes with three opening sizes subjected to various moment-to-shear force ratiosin steel beams of four section sizes are arranged in a rational manner for easy com-parison and reference. The global shear force, Vo,Sd, and the global moment, Mo,Sd,at the centre-line of the perforated sections at failure are non-dimensionalised withrespect to the global section capacities of the perforated sections, namely, Vo,Rd andMo,Rd. All the interaction curves are shown to be similar in pattern. This confirmsthe suitability of a generalised moment-shear interaction curve for the design ofperforated sections with various shapes and sizes.

It should be noted that despite the variation in sizes and shapes of web openings,


Fig. 3. Moment–shear interaction curves obtained from finite element investigation.

all the curves converge to the same x-intercept, i.e. having the same momentcapacities under zero global shear force, provided that the web openings have thesame depths. However, all the curves have different y-intercepts, i.e. different shearcapacities at perforated sections under zero global moment, probably due to different


Fig. 3 (continued)

local ‘Vierendeel’ moments acting onto the tee-sections above and below the webopenings.

In order to understand the effects of both the shapes and the sizes of web openingson the structural performance of perforated sections, it is important to relate boththe opening depth, do, and the critical opening length, c, to the following:


Fig. 3 (continued)

� global shear force and moment acting on the perforated sections, and� local co-existing axial and shear forces, and moment acting onto the tee-sections

above and below the web openings.

In general, an increase in the opening depth, do, always reduces both the shear


Fig. 3 (continued)

and the moment capacities of the perforated sections, and thus, both shear and flex-ural failures of perforated sections are primarily controlled by the magnitude of do.However, while an increase in the opening length, c, does not affect the shear andthe moment capacities of the perforated sections, it increases directly the local ‘Vier-endeel’ moments acting onto the tee-sections, and thus promotes the ‘Vierendeel’


mechanism in the perforated sections. Consequently, for web openings with samevalues of do but with different values of c, the load capacities of the perforatedsections are inversely proportional to the values of c.

It may be convenient to consider that both the shear failure and the ‘Vierendeel’mechanism in perforated sections are primarily caused by global shear forces. Theglobal shear capacities of the perforated sections attain their maximum values in theabsence of axial forces due to global moment. Any increase in the global momentat the perforated sections will induce local axial forces in the tee-sections, promotinglocal yielding of the tee-sections and hence collapse of the perforated sections, asshown in Fig. 4. For perforated sections under large global moment, the global shearcapacities will be greatly reduced.

4. Proposed design method

The whole range of behaviour of perforated sections is characterised by threeactions: global bending action, global shear action, and local ‘Vierendeel’ action.Design rules for basic moment and shear capacities of perforated sections arepresented first. The effect of ‘Vierendeel’ mechanism is then incorporated into theshear capacities of perforated sections, giving the ‘coupled shear capacities’ of per-forated sections. Finally, a generalised moment-shear interaction curve is rec-ommended to allow for the presence of co-existing global shear force and momenton perforated sections with web openings of various shapes and sizes.

4.1. Basic moment capacity

Consider a perforated section, and an opening of depth, do, is formed at the mid-height of the web. The applied global moment and shear force at the centre of the

Fig. 4. Reduction in moment-shear interaction curve of perforated sections due to coupled ‘Vierendeel’mechanism.


web opening are Mo,Sd and Vo,Sd respectively. The moment resistance of the perfor-ated section, Mo,Rd , is given by:

Mo,Rd � fy Wo,pl � Mo,Sd Wo,pl � Wpl �d2

otw

4(1)

where Wpl is the plastic modulus of the un-perforated section; tw is the web thickness;do is the opening depth; fyis the design yield strength of the steel beam.

4.2. Basic shear capacity

In general, the shear area of an I-section is defined as h × tw based on simpleplastic section analysis and h is the overall section depth. This approximation iswidely adopted in modern steel codes because of its simplicity. Moreover, as theflanges are assigned to resist bending moment while the web is assigned to resistshear force, the contribution of the flanges to the shear capacity of the entire sectionmay often be neglected without causing any significant error. However, in an I-section with a large web opening, the shear area of the web is substantially reduced,and thus the shear areas of the flanges become significant in assessing the shearcapacity of the perforated section.

During the finite element investigation, the shear capacities of perforated sectionswere consistently found to be larger than those predicted from current design rules,suggesting that parts of the flange areas should be incorporated in order to assessthe shear capacities of perforated sections accurately. As shown in Fig. 5, the equival-ent shear area of a flange, Avf, after calibration against finite element results, isgiven by:

Avf � tf � (0.375 tf � tw � 0.375 tf)

where tw is the web thickness; tf is the flange thickness.Consequently, the plastic shear resistance of the perforated section, Vo,Rd, is

given by:

Fig. 5. Equivalent shear area in tee section based on finite element investigation.


Vo,Rd � fvAvo�Vo,Sd (2a)

Avo � Av�do � tw (2b)

where fv is the shear strength of the steel beam taken as 0.577 fy/gMo; gMo is a materialfactor taken as 1.0; Av is the shear area of the un-perforated section,

� h � tw � 2 � (0.75 tf � tf) (2c)

For a typical beam section such as UB 457 × 152 × 52 S275, the shear area ofthe flanges contributes an increase of 10% in the shear capacity of a perforatedsection with a web opening of do/h equal to 0.50. For a perforated section with do/hequal to 0.75, the increase in the shear capacity is 21%. It should be noted that forbeam sections with thick flanges, the increase in shear capacities is likely to exceed30%, and Table 1 summarises the increases of the shear capacities in four steelbeams with web openings of three different do/h ratios.

4.3. Coupled shear capacity allowing for Vierendeel mechanism

After determining the basic shear capacity of a perforated section, Vo,Rd, it isimportant to consider the effect of local ‘Vierendeel’ moments acting at the tee-sections above and below the web opening on the global shear capacity of the perfor-ated section. It should be noted that both the global shear force and the local ‘Vieren-deel’ moment are ‘coupled’ as they act on the perforated section simultaneously.The extents of coupling in perforated sections are obviously very complicated,depending not only on the shapes and sizes of web openings, but also on the appliedglobal shear forces and moments at the perforated sections. As reported in the comp-lementary paper, the ‘Vierendeel’ mechanism has been studied extensively in thefinite element investigation. The global shear capacities allowing for ‘Vierendeel’mechanism, or the global coupled shear capacities, Vo,Rd,Vi, of perforated sectionscovered in the present study may be obtained directly from the moment-shear interac-tion curves presented in Fig. 3. For design purposes, a coupled shear capacity ratiofor perforated sections, v̄, is defined as follows:

v̄ �Vo,Rd,Vi

Vo,Rd

(3)

The coupled shear capacity ratio governs the global shear capacity of a perforated

Table 1Summary of increase in pure shear capacities of perforated sections due to shear areas of flanges

Perforated sections do /h = 0.50 do /h = 0.67 do /h = 0.75

UB 457 × 152 × 52 10% 16% 21%UB 457 × 152 × 82 22% 33% 43%UB 610 × 229 × 101 10% 16% 21%UB 610 × 229 × 140 18% 27% 36%


section under coupled shear failure and ‘Vierendeel’ mechanism in the absence ofglobal moment. Table 2 summarises the values of v̄ for perforated sections coveredin the present study, which are obtained directly from the finite element investigation.It is shown that for web openings with small opening lengths such as c-hexagonalweb openings, the values of v̄ are close to unity for various do/h ratios, showinglittle coupling effect of the ‘Vierendeel’ mechanism on the perforated sections. How-ever, for web openings with large opening lengths such as rectangular and elongatedweb openings, the ‘Vierendeel’ mechanism is apparent and the values of v̄ are oftenless than 0.5.

4.4. Generalised moment-shear interaction curve

In general, an interaction curve with an elliptical expression may be used to allowfor interaction between moment and shear force in solid rectangular plates:

m2 � v2 � 1 (4a)

or

v � �1 � m2 (4b)

where

v is the shear utilisation ratio �Vo,Sd

Vo,Rd(5a)

m is the moment utilisation ratio �Mo,Sd

Mo,Rd

. (5b)

However, for perforated sections, this interaction curve should be modified toallow for the presence of ‘Vierendeel’ mechanism as shown in Fig. 4. In general,for perforated sections under zero global moment, the shear utilisation ratios are

Table 2Summary of coupled shear capacity ratios, v̄, for perforated sections with web openings of various shapesand sizes

Opening shapes Opening sizes

do /h = 0.50 do /h = 0.67 do /h = 0.75

C-hexagon 1.00 0.96 0.91Circle 0.95 0.87 0.80Regular octagon 0.90 0.78 0.68Regular hexagon 0.82 0.66 0.55Square 0.66 0.42 0.32Elongated circle 2do 0.56 0.35 0.26Rectangle 2:1 0.38 0.21 0.16Elongated circle 3do 0.35 0.19 0.13


reduced from unity to the coupled shear capacity ratios, v̄. In the presence of globalmoments, the shear utilisation ratios of the perforated sections will diminish gradu-ally, depending on the magnitudes of the global moments. After careful calibrationwith the moment-shear interaction curves obtained from the finite element investi-gation, a generalised moment-shear interaction curve is recommended as follows:

For v̄ � 2 /3 �vv̄�2

� m2 � 1 (6a)

For v̄ � 2 /3 �v � ( v̄ � 2 /3 )2 /3 �2

� m2 � 1 (6b)

where

m is the coupled moment capacity ratio, �Mo,Rd,Vi

Mo,Rd

. (5c)

For analysis, the shear utilisation ratio, v, and the moment coupled momentcapacity ratio, m, are given by:

For v̄ � 2 /3 v � v̄ �1 � m2 � v or m � �1 � (v /v)2 (6c)

For v̄ �2/3 v � v̄�23

�23�1 � m2 � v or (6d)

m � �1��v �( v � 2/3 )2 /3 �2

.

In all cases, the shear utilisation ratio, v, should not exceed the coupled shearcapacity ratio, v̄. Fig. 6 plots the proposed moment-shear interaction curves for webopenings with various shapes and sizes, and they are considered to be applicable tosteel beams with practical section sizes. The design curves are also plotted in Fig.3 for direct comparison with the finite element results. It is shown that for perforatedsections with large web openings, the proposed design interaction curves followclosely the finite element results for the whole ranges of moment and shear ratios.However, for perforated sections with small web openings, the proposed designcurves are considered to be relatively conservative when interactions betweenmoment and shear forces are significant.

It is important to note that, based on the three ratios, namely, the shear utilisationratio, v, the moment utilisation ratio, m, and the coupled shear capacity ratio, v̄, theload carrying capacities of steel beams with web openings of various shapes andsizes may be obtained readily through the proposed moment-shear interaction curve.The design procedures are fully illustrated in the two worked examples given inAppendices A and B. Web openings with different shapes and sizes are selected toillustrate the design procedure of the proposed design method in meeting the require-ments on the load carrying capacities of steel beams.

It should be noted that the proposed design method is very effective as it has been


Fig. 6. Proposed moment-shear interaction curves.

carefully calibrated against non-linear finite element results. The design expressionis simple and easy to use. Consequently, the proposed design method is suitable forengineers in their practical design.

5. ‘Vierendeel’ parameter

After assessing both the global shear force and moment capacities of perforatedsections, it may be useful in some cases to know the relative importance of ‘Vieren-deel’ mechanism in perforated sections with web openings of various shapes andsizes. Comparison on the results of the finite element investigation reveals that:

� For web openings with small opening lengths under high shear force, shear failureis apparent in the perforated section, especially for deep web openings.

� For web openings with large opening lengths under high shear force, ‘Vierendeel’mechanism is dominant in the perforated section.

� For web openings with large opening depths under low shear force, flexural failurein the perforated section may be critical.


In order to assess the performance of ‘Vierendeel’ mechanism in perforated sec-tions, an indicative parameter, namely, the ‘Vierendeel’ parameter, vi , is establishedand defined as follows:

vi �Vo,Rd,Vi

4 MT,Rd / c(7)

where MT,Rd is the basic moment capacity of tee-sections under zero axial and shearforces; Vo,Rd,Vi is the global coupled shear capacity of perforated sections obtainedfrom the finite element investigation; and c is the critical opening length.

For perforated sections under zero global moment, the ‘Vierendeel’ parameter, vi,is equal to unity provided that plastic hinges are fully developed at the ends of thetee-sections above and below the web openings.

6. Relative importance of ‘Vierendeel’ mechanism

In order to illustrate the importance of ‘Vierendeel’ mechanism in perforated sec-tions with web openings of various shapes and sizes, a plot of the ‘Vierendeel’parameter, vi , against the critical opening length ratio, c/do, for perforated sectionsunder zero global moment is presented in Fig. 7. It is shown that for web openings

Fig. 7. Typical values of ‘Vierendeel’ parameter in perforated sections with various opening shapes andsizes. UB457 × 152 × 52 S275 and Mo,Sd = 0.


with large c, such as square, rectangular and elongated openings, vi is always closeto unity, showing the importance of the ‘Vierendeel’ mechanism. For web openingswith small c, such as c-hexagonal, circular, octagonal and hexagonal web openings,vi is generally small, in particular for those web openings with do/h equal to 0.50.

In Fig. 8, vi is plotted for three different shapes of web openings located at variouslocations along a 12 m long simply supported beam of UB 457 × 152 × 52 S275.Both the shear and the moment utilisation ratios of the perforated sections, v andm, are also plotted on the same graph for direct comparison. It is shown that:

1. For perforated sections with rectangular web openings with do = 0.5 h, vi is verylarge where the sections are under high to medium shear force. In perforatedsections beyond 4 m from the support, m increases quickly at the expense of vi,demonstrating a sudden change of critical failure mode from ‘Vierendeel’ mech-anism to flexural failure.

2. For perforated sections with regular hexagonal web openings with do = 0.67 h,vi is very large only in sections under high shear force. In perforated sectionsbeyond 2 m from the support, vi diminishes quickly, but m increases sharplytowards unity showing the significant reduction in moment capacities of the per-forated sections due to the presence of web openings.

3. For perforated sections with c-hexagonal web openings with do = 0.75 h, the valueof v is always larger than vi along the entire length of the beam, showing thatshear failure is always more critical than ‘Vieredneel’ mechanism as the criticalopening length is small. However, the switch of critical failure mode for shearfailure to flexural failure takes place at about 1.5 m from the support, showingthe severe reduction in moment capacities of the perforated sections due to thepresence of large web openings.

7. Conclusions

Based on the results of a comprehensive finite element investigation on steel beamswith web openings of various shapes and sizes, a design method using a generalisedmoment-shear interaction curve is proposed for determining the load capacities ofsteel beams with web openings of various shapes and sizes. The design method isconsidered to be simple, straight forwards, and highly efficient in structural economyfor engineers in their practical design.

It should be noted that

1. Based on the finite element investigation, it is found that the basic shear capacitiesof perforated I-sections are consistently larger than those predicted from currentdesign rules, and thus, the shear areas of the flanges should be included. Conse-quently, a revised design rule for basic shear capacities of perforated sectionsincluding both the shear areas of the web and the flanges is proposed after carefulcalibration against finite element results.

2. As the global shear forces in perforated sections may cause both shear failure and


Fig. 8. vi, v and m of perforated sections with various opening shapes and sizes along beam spanUB457 × 152 × 52 S275.


‘Vierendeel’ mechanism in perforated sections simultaneously, the effect of local‘Vierendeel’ moments acting onto the tee-sections above and below the web open-ings is incorporated through a reduction to the global shear capacities of the per-forated sections, giving rise to the global coupled shear capacity, Vo,Rd,Vi, of per-forated sections. The values of global coupled shear capacities for perforatedsections covered in the present study may be obtained directly from the moment-shear interaction curves in Fig. 3. The normalised values of the global coupledshear capacities for perforated sections covered in the present study are summar-ised in Table 2.

3. It is important to note that, based on the three ratios, namely, the shear utilisationratio, v, the moment utilisation ratio, m , and the coupled shear capacity ratio,v̄, the load carrying capacities of steel beams with web openings of various shapesand sizes may be obtained readily through the proposed moment-shear interactioncurve. Moreover, the proposed design method is very effective as it has beencarefully calibrated against non-linear finite element results, and the designexpression is simple and easy to use. Consequently, the proposed design methodis suitable for engineers in their practical design.

4. For perforated sections with large web openings, the proposed moment-shearinteraction curve is considered to follow closely the finite element results for thewhole ranges of moment and shear ratios. However, for perforated sections withsmall web openings, the proposed curve is slightly conservative when interactionsbetween moment and shear forces are significant.

5. An indicative parameter, the ‘Vierendeel’ parameter, is established to assess theperformance of ‘Vierendeel’ mechanism in perforated sections. Through compari-son among the moment and the shear utilisation ratios, m and v, and the ‘Vieren-deel’ parameter, vi, the critical modes of failure in perforated sections under differ-ent moment-shear ratios may be readily assessed.

Acknowledgements

The research project leading to the publication of the paper is supported by theResearch Grant Council of the Government of Hong Kong Special AdministrationRegion (Project No. PolyU5085/97E).

Appendix A. Worked example 1

Load carrying capacity of a steel beam with multiple web openings based onempirical design rule

A 7.5-m span simply supported beam of UB 457 × 152 × 52 S275 with six circularweb openings (do = 0.75 h) is subject to a uniformly distributed load of 35 kN/m.The openings are placed symmetrically about the center of the beam with an intervalof 1.0 m starting from the supports. Check the structural adequacy of the steel beam.


Data

L, 7.5 m; xo, 1.0, 2.0, 3.0 m; h, 449.8 mm; tw, 7.6 mm; bf, 152.4 mm; tf, 10.9 mm;do, 337.35 mm; Wpl, 1096 × 103 mm3; v̄ = 0.8 for circular opening.

The load carrying capacity of the beam without web openings, wo, is 42.9 kN/mfor L = 7.5 m.

Steel beams with circular web openingsStep 1: Determine the shear and the moment capacities of the perforated section

Eq. (1) Mo,Rd � fyWo,pl � fy(Wpl�twd2

o

4)

� 275 � (1096�7.6 � 337.352

4 � 1000) � 10�3 � 241.9 kNm

Eq. (2) Avo � tw(h�do) � 2 (0.75t2f )

� 7.6 � (449.8�337.35) � 2 � 0.75 � 10.92 � 1032.8 mm2

Vo,Rd � 0.577fyAvo � 0.577 � 275 � 1032.8 � 10�3 � 163.9k N

Step 2: Determine the applied shear force and moment at the center of the webopenings

xo (m) vVSd = w(L2

�xo) (kN) MSd =wxo(L�xo)

2(kNm)

1.0 96.3 0.587 113.82.0 61.3 0.374 192.53.0 26.3 0.160 236.3

Step 3: Determine the moment capacities at the perforated sections under highshear


For xo � 1.0 m, v � 0.587:

Eq. (4) M1,Rd � Mo,Rd�1�v2 � 241.9�1�0.5872 � 195.8 kNm

Noting that v̄ � 0.8 �23,

Eq. (6) M2,Rd � Mo,Rd�1�[v � (v � 2/3)

2/3]2 � Mo,Rd�1�[�0.587 � (0.8 � 2/3)

2 /3 �2

� 0.733 Mo,Rd � 177.3 kNm

So Mo,Rd,Vi � 177.3 kNm � MSd � 113.8 kNm

m̄ � �1��v��v̄�23�

23

�Similarly,

For xo � 2.0 m, Mo,Rd,Vi � 0.928 Mo,Rd � 224.4 kNm � MSd � 192.5 kNm� OK

For xo � 3.0 m, Mo,Rd,Vi � 0.987 Mo,Rd � 238.8 kNm � MSd � 236.3 kNm� OK

If the opening shape is changed, the load carrying capacity of the beam will bedifferent. Suppose the opening shape is changed to octagon and c-hexagon, repeatingthe above steps will give the following results.

Steel beams with octagonal web openings

xo do/h w v̄ VSd v MSd Mo,Rd,Vi CheckingMSd

Mo,Rd,Vi(m) (kN/m) (kN) (kNm) (kNm)

1 0.75 35 0.68 96.3 0.587 113.8 123.3 0.923 OK2 0.75 35 0.68 61.3 0.374 192.5 203.6 0.946 OK3 0.67 35 0.78 26.3 0.160 236.3 251.9 0.938 OK


Steel beams with C-hexagonal web openings



1 0.75 35 0.91 96.3 0.587 113.8 195.9 0.581 OK2 0.75 35 0.91 61.3 0.374 192.5 224.4 0.858 OK3 0.75 35 0.91 26.3 0.160 236.3 238.8 0.989 OK

Note: All web openings are spaced apart without any interaction.

Appendix B. Worked example 2

A 12-m span simply supported beam of UB 610 × 229 × 140 S355 with six squareweb openings (do = 0.5 h) is subject to a uniformly distributed load of 80 kN/m.The openings are placed symmetrically about the center of the beam with an intervalof 1.2 m starting from the supports. Increase the opening sizes by changing theopening shapes, if appropriate, without reducing the load carrying capacity.

Data

L, 12 m; xo, 1.5, 3.0, 4.5 m; h, 617.2 mm; tw, 13.1 mm; bf, 230.2 mm; tf, 22.1 mm;do, 308.6 mm; Wpl, 4142 × 103 mm3.

The load carrying capacity of the beam without web openings, wo, is 81.6 kN/mfor L = 12 m.

The following tables list the results of which the square web openings are changedto other shapes at each location.


Steel beams with octagonal web openings



1.2 0.67 80 0.78 384.0 0.551 518.4 959.3 0.540 OK2.4 0.67 80 0.78 288.0 0.413 921.6 1135.7 0.811 OK3.6 0.67 80 0.78 192.0 0.275 1209.6 1222.4 0.990 OK

Steel beams with circular web openings


Mvo,Rd,Vi(m) (kN/m) (kN) (kNm) (kNm)

1.2 0.75 80 0.80 384.0 0.551 518.4 698.2 0.742 OK2.4 0.75 80 0.80 288.0 0.413 921.6 1007.4 0.915 OK3.6 0.67 80 0.87 192.0 0.275 1209.6 1222.4 0.990 OK


Steel beams with c-hexagonal web openings


Mvo,Rd,Vi(m) (kN/m) (kN) (kNm) (kNm)

1.2 0.75 80 0.91 384.0 0.551 518.4 895.1 0.579 OK2.4 0.75 80 0.91 288.0 0.413 921.6 1050.4 0.877 OK3.6 0.67 80 0.96 192.0 0.275 1209.6 1222.4 0.990 OK

Thus, if the opening shape is to be changed at various specific locations alongthe beam without reducing the original load carrying capacity, then the followingopening configuration is one possibility:

References

[1] Lawson RM. Design for openings in the webs of composite beams. CIRIA/Steel Construction Institute,1987 CIRIA Special Publication and SCI Publication 068.

[2] Darwin D. Steel and composite beams with web openings. In: Steel design guide series No. 2.Chicago, IL, USA: American Institute of Steel Construction; 1990.

[3] Redwood RG, Cho SH. Design of steel and composite beams with web openings. J Construct SteelRes 1993;25:23–41.

[4] Oehlers DJ, Bradford MA. Composite steel and concrete structural members: Fundamental behaviour.Pergamon, 1995.

[5] Ko CH, Chung KF. A comparative study on existing design rules for steel beams with circular webopenings. In: Yang YB, Leu LL, Hsieh SH, editors. Proceedings of the First International Conferenceon Structural Stability and Dynamics, Taipei. 2000. p. 733–8.

[6] Chung KF, Liu TCH, Ko ACH. Investigation on Vierendeel mechanism in steel beams with circularweb openings. J Construct Steel Res 2001;57(5):467–90.

[7] Chung KF, Liu TCH, Ko ACH. Steel beams with large web openings of various shapes and sizes:Finite element investigation. doi:10.1016/S0143-974X(03)00030-0.

Documents

Steel beam with large web openning