Reinforced concrete beams with web openings A state of the art review.pdf

Materials and Design 40 (2012) 90–102

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Materials and Design j o u r n a l h o m e p a g e :  :  w w w . e l s e v i e r . c o m / l o c a t e / m a t d e s

Reinforced concrete beams with web openings: A state of the art review A. Ahmed a, M.M. Fayyadh b, S. Naganathan a, , K. Nasharuddin a ⇑

a b

Department of Civil Engineering, Universiti Tenaga Nasional, 43000 Selangor, Malaysia Department of Civil Engineering, University Malaya, 50603 KL, Malaysia

a r t i c l e

i n f o

 Article history: Received 8 November 2011 Accepted 2 March 2012 Available online 16 March 2012 Keywords: A. Concrete and composite G. Destructive testing H. Failure analysis

a b s t r a c t

The construction of modern buildings requires many pipes and ducts in order to accommodate essential services services such as air condition conditioning, ing, electrici electricity, ty, telephone telephone,, and compute computerr network. network. Web openings openings in concrete concrete beam beamss enabl enable e the instal installa latio tion n of these these servic services. es. A numbe numberr of studie studiess have have been been conduc conducted ted with with regard regardss to reinforced concrete beams which contain web openings. The present paper aims to compile this state of  the art work on the behaviour, analysis and design of Reinforced Concrete (RC) beams with transverse web openings. A variety of aspects will be highlighted and discussed including the classification of openings, guidelines guidelines for opening opening location, location, and the structura structurall behaviou behaviourr of RC beams beams with web openings openings.. Various Various design design approac approaches hes will also be detailed detailed,, for example example the America American n Concrete Concrete Institute (ACI) approach, the Architectural Institute of Japan (AIJ) approach and the strut and tie method. Moreover, the strengthening of RC beams with openings using Fibre Reinforced Polymer (FRP) material and steel plates is presented. Finally, directions for future research based on the gaps which exist in the present work are presented.   2012 Elsevier Ltd. All rights reserved.

1. Introduction In modern building construction, transverse openings in reinforced concrete beams are often provided for the passage of utility ducts and pipes. These ducts are necessary in order to accommodate essential services such as water supply, electricity, telephone, and computer network. These ducts and pipes are usually placed underneath the soffit of the beam and for aesthetic reasons, are covere covered d by a suspe suspend nded ed ceilin ceiling, g, thus creatin creating g a dead dead space. space. In each each floor, the height of this dead space adds to the overall height of the building depending on the number and depth of ducts. Therefore the web openings enable the designer to reduce the height of the structure, especially with regard to tall building construction, thus leading to a highly economical design. The presenc presence e of transver transverse se ope opening ningss will transfor transform m simple simple beam behaviour into a more complex behaviour, as they induce a sudden change in the dimension of the beam’s cross section. However, as the opening represents a source of weakness, the failure plane always passes through the opening. The ultimate strength, shear shear strength, strength, crack width width and stiffness stiffness may may also be seriousl seriously y affected. Furtherm Furthermore, ore, the provisio provision n of ope opening ningss produce producess disconti discontinuiti nuities es or distu disturb rbanc ances es in the norm normal al flow flow of stress stresses, es, thus thus leadi leading ng to stress stress concentration and early cracking around the opening region. Similar to any discontinuity, special reinforcement or enclosing of the

⇑

Corresponding author. Tel.: +60 126848463.  [email protected] (S.  (S. Naganathan). E-mail address:  [email protected]

0261-3069/$ - see front matter    2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.matdes.2012.03.001

opening close opening close to its perip periphery hery,, should should therefore therefore be provi provided ded in sufficient quantity to control crack widths and prevent possible premature failure of the beam [1] beam  [1]..

2. Opening classification This section presents the classification of Reinforced Concrete (RC) beams with web openings based on the opening’s size and position. Openings are classified as small or big openings and the best best posit position ion of the open opening ing is decid decided ed based based on its size. size. We Web b open open-ings have been found to take many shapes such as circular, rectanrectangular, diamond, triangular, trapezoidal and even irregular shapes. However, circular and rectangular openings are the most common ones in practice [2] [2].. With regards to the size of openings, many research researchers ers use the terms terms ‘‘small’’ ‘‘small’’ and ‘‘large’’ ‘‘large’’ without without drawin drawing g any clear clear-cu -cutt dema demarca rcatio tion n line. line. Small Small openi openings ngs are are defin defined ed as those those which are circular, square or nearly square in shape  [1,3]  [1,3].. In contrast, and according to Somes and Corley [4] [4],, a circular opening may be considered as large when its diameter exceeds 0.25 times the depth of the web. The author however feels that the essence of classifying an opening as either small or large lies in the structural response of the beam. When the opening is small enough to maintain the beam-type behaviour, or in other words, if the usual beam theory applies, then the opening may be termed as small. When beam-type behaviour ceases to exist due to the provision of openings, then the opening may be classified as a large opening. By assumin assuming g the prevale prevalence nce of Viernde Vierndeel el action action and considering considering the fact that failure occurs occurs after the format formation ion of a four-hin four-hinge ge

 A. Ahmed et al. / Materials and Design 40 (2012) 90–102

mechanism, Mansur [5], recommended certain criteria with which to classify the size of an opening as either large or small. It can be assumed that hinges form in the chord members at a distance of  h/2  from the vertical faces of the opening. This is shown in  Fig. 1, where h is the overall depth of a chord member, and the subscripts t  and  b  refer to the top and bottom chords, respectively.

  Small opening,  l o 6 hmax   Large opening,  l o  > hmax where  h max  is the larger of  h t  and  h b. That is, when the length of opening  I 0  is less than or equal to hmax, it may be defined as a small opening. For large openings, I 0 > hmax. In this definition, it is assumed that the members above and below the opening have adequate depth to accommodate the reinforcement scheme. In the case of circular openings, the circle should be replaced by an equivalent square for the determination of the value of  h max. Mansur and Tan [6] provided guidelines to facilitate the selection of the size and location of web openings as illustrated in Fig. 2. (i) For T-beams, openings should preferably be positioned flush with the flange for ease in construction. In the case of rectangular beams, openings are commonly placed at mid-depth of  the section, but must also be placed eccentrically with respect to depth. Care must be exercisedto provide sufficient concrete cover to the reinforcement for the chord members above and below the opening. The compression chord should also have a sufficient concrete area to develop the ultimate compression block in flexure and should also have adequate depth to provide effective shear reinforcement. (ii) Openings should not be located closer than one-half of the beam’s depth D   to the supports. This is in order to avoid the critical region for shear failure and reinforcement congestion. Similarly, the positioning of an opening closer than 0.5D  to any concentrated load should be avoided. (iii) Depth of openings should be limited to 50% of the overall beam depth. (iv) The factors which limit the length of an opening are the stability of the chord members, in particular the compression chord, and the serviceability requirement of deflection. When the opening becomes bigger, it is preferable to use multiple openings providing the same passageway instead of using a single opening. (v) When multiple openings are used, the post separating two adjacent openings should not be less than 0.5D   to ensure that each opening behaves independently. Basedon theaforementionedreview, it is clear that openings can take manyshapes andsizes. The actualtypeand locationof anopening must be clearly decided before the design specification stage.

91

3. Small openings design approaches This section presents the design approaches which have been used for RC beams with small openings. The traditional method, the Architectural Institute of Japan (AIJ) method, the plasticity method and the strut and tie method are the main design approaches which are presented in this paper. Mansur and Tan  [6] recommended that a circular, square, or nearly square opening may be considered a small opening provided that the depth (or diameter) of the opening is in realistic proportion to the beam size, that is, less than approximately 40% of the overall beam depth. In such a case, beam action may be assumed to prevail. Therefore, the analysis and design of a beam with small openings may follow a similar course of action to that of a solid beam.

 3.1. Traditional design approach It was recommended that in the case of pure bending, because the concrete there would have cracked anyway in flexure at ultimate, the placement of an opening completely within the tension zone does not change the load-carrying mechanism of the beam [7,8]. This was illustrated through worked examples supported by test evidence. Thus, the ultimate moment capacity of a beam is not affected by the presence of an opening as long as the minimum depth of the compression chord,  h c , is greater than or equal to the depth of the ultimate compressive stress block, that is, when:

hc  6

 As f  y 0:85 f c 0 b

ð1Þ

where As is the area of flexural reinforcement, f  y is the yieldstrength of flexural reinforcement,  f c 0  is the compressive strength of concrete and b  is the beam width. Tests have been conducted with a small opening enclosed by reinforcement and introduced into a region subjected to predominant shear  [4,9–12]. As shown in Fig. 3, the beam may fail in two distinctly different modes. The first type is labelled  beam-type  failure which is typical of the failure commonly observed in solid beams except that the failure plane passes through the centre of  the opening (as shown in Fig. 3a). Conversely, in the second type labelled  frame-type  failure, the formation of two particular diagonal cracks, one in each member bridging the two solid beam segments, leads to the failure (as shown in  Fig. 3b). It was suggested that these types of failures require separate treatment for complete design [6]. Similar to the traditional shear design approach, in both the cases it may be assumed that the nominal shear resistance,  V n, is the sum of two components V c  and V s   (attributable to concrete and shear reinforcement across the failure plane).

V n  ¼  V c  þ V s

ð2Þ

 3.1.1. Beam-type failure Similar to a solid beam, a 45   inclined failure plane may be assumed when designing for beam-type failure, with the plane being traversed through the centre of the opening, as shown in Fig. 4 [6]. A simplified approach namely the American Concrete Institute (ACI) Code  [13]   can be followed to estimate the shear resistance V c  provided by the concrete:

V c  ¼

Fig. 1.  Forming of hinge in RC beam with opening [5].

 1  f 0 bw ðd  do Þ: 6 c 

q  ffiffi

ð3Þ

where bw  is the web width, d  is the effective depth and do   is the diameter of opening. For shear reinforcement contribution, V s   should be calculated by using Eq. (3). It may be seen that the stirrups available to resist

A. Ahmed et al./ Materials and Design 40 (2012) 90–102

92

Fig. 2.   Guidelines for the location of openings [6].

Fig. 3.  The two modes of shear failure at small openings [6].

resisted by the usual bending mechanism, that is, by the couple formed by the compressive and tensile stress resultants, N u, in the chord members above and below the opening. These stress resultants may be obtained by:

ðN u Þt  ¼

Fig. 4.   Shear resistance ‘Vs’ provided by shear reinforcement at an opening [6].

shear across the failure plane are those by the sides of the opening within a distance of (dv  do):

V s  ¼

 A  f  y ðd   do Þ s v 

v 

v 

ð4Þ

where  d v  is the distance between the top and bottom longitudinal rebars and do  is the diameter (or depth) of opening; Av  = area of  vertical legs of stirrups per spacing s; f  yv  = yield strength of stirrups. The total amount of web reinforced and thus calculated should be contained within a distance (dv   do)/2, or preferably be lumped together on either side of the opening.

 3.1.2. Frame-type failure This type occurs due to the formation of two independent diagonal cracks, one on each of the chord members above and below the opening, as shown in Fig. 3b. It appears that each member behaves independently similar to the members in a framed structure. Design reinforcement was recommended for this mode of failure and it was also suggested that the chord member requires independent treatment [6]. Let us consider the free-body diagram at the beam opening, as shown in Fig. 5. Clearly, the applied factored moment, M u, at the centre of the opening from the global action is

M u  ¼ ðN u Þb d   a2

 

ð5Þ

subject to the restrictions imposed by Eq.  (5). In this equation,  d  is the effective depth of the beam,  a  is the depth of equivalent rectangular stress block and the subscripts t  and b denote the top and bottom cross members of the opening, respectively. It was recommended that the applied shear,  V u, may be distributed between the top and the bottom chords in proportion to their cross-sectional areas [14]. Thus:

 At  ðV u Þt  ¼  V u  At  þ Ab





ðV u Þb  ¼  V u   ðV u Þt 

ð6Þ

ð7Þ

Knowing the factored shear and axial forces, each member can be independently designed for combined shear and axial force by the usual procedure for solid beams.

Fig. 5.   Free-body diagram at beam opening [5].

 A. Ahmed et al. / Materials and Design 40 (2012) 90–102

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 3.2. AIJ approach A formula has been suggested by the Architectural Institute of   Japan (AIJ),  [15], and is the standard for structural calculation of  reinforced concrete structures to evaluate the shear capacity V n of beams which contain a small opening. This empirical formula is considered similar to the traditional approach where the total shear resistance is provided by both concretes and the steel crosses a 45  failure plane passing through the centre of the opening as shown in Fig. 6. The formula given is as follows:

"

q  ffiffi ffi ffi ffi ffi ffi#

0:092K u K  p ð f c 0 þ 17:7Þ 1:61do V n  ¼ 1 þ 0:846 q0w f  y M  h 0:12  þ Vd





v 

bd

v 

Fig. 7.   Determination of  K u.

ð8Þ

where k p   = 0.82 (100 As/bd)0.23, do   is the diameter of the circular opening or diameter of the circumscribed circle in the case of a square opening, which should be taken as less than or equal to h/3; h is the overall depth of the beam, and  M /(V d) is taken as less than or equal to 3. The term K  is a function of the effective depth d to account for the size effects in shear and has a value of between 0.72 and 1.0 as shown in Fig. 7.  f  yv  is the yield strength of web reinforcement. Where dv is the distance between the top and bottom longitudinal bars, the term  p w  in Eq. (8) refers to the ratio of web reinforcement placed within a longitudinal distance  dv/2 from the centre of  the opening as shown in Fig. 6, and defined as:

q0w  ¼

 A ðsin a þ cos aÞ bd v 

ð9Þ

v 

where A v  is the area of web reinforcement.

 3.3. Plasticity method In a beam with openings, however, it is difficult to develop an arch mechanism, and consequently, the applied shear is transferred by means of a truss mechanism. A beam has a circular opening only when the beam is reinforced transversely by vertical stirrups  [16], as shown in Fig. 8. Note that  u s  is the angle of concrete compression strut in the upper and lower chord members. The horizontal arrows show bond stress and the vertical arrows represent forces acting on the concrete due to the forces in the stirrups. The unshaded portion shows the zone where the diagonal compressive stress field is not formed. The diagonal compressive stress in concrete around the opening becomes larger as the unshaded portionwidensor as the openingbecomelarge. The effective depth dtw  for the truss mechanism is defined as:

 Atw  ¼  d   v 

do  S   tanØs cosØs v 

ð10Þ Fig. 8.  Truss action in beam with opening [16].

where do is the diameter of the circular opening (or that of a circumscribed circle in the case of a square or rectangular opening), and  sv is the spacing between the two stirrups, one on each side adjacent to the opening. Assuming yielding of stirrups, the concrete compressive stress in the shaded portion is given by:

 f cw  ¼  q  f  y ð1 þ cot 2 Øs Þ v 

Fig. 6.   Effective web reinforcement for opening [15].

v 

ð11Þ

where p v  is the ratio of shear reinforcement placed adjacent to the opening and  f  yv  is the yield strength of the stirrups. The value of  f cw  in Eq. (11) equating to the effective compressive strength of concrete vf c  gives the value of  u s  as:

A. Ahmed et al./ Materials and Design 40 (2012) 90–102

94

s  ffiffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 0

cotØs  ¼

Vf c  q  f  y   1

ð12Þ

v 

v 

The shear strength of the beam with an opening is given by:

V n  ¼  bd tw q  f  y  cot Øs v 

ð13Þ

v 

This applies when diagonal steel reinforcement bars are provided and the development length is anchored outside of the stirrups adjacent to the opening as shown in  Fig. 9. Where the compressive stress in concrete is relatively low, the contribution of the diagonal bars to the shear capacity is given by:

V nd  ¼  A d f  yd sin hd

ð14Þ

where hd is the angle of inclination to the axis of the beam, and Ad is the cross-sectional area of the diagonal bars. Thetotal shear strength ofthe beamcan begivenby addingthe value of vnd   to the value of vn  .

 3.4. Strut and Tie model The strut and tie method was suggested in order to design a beam with a small opening. This model takes into account the fact that the applied loads are transmitted through the member to the support by means of a system of tension and compression struts provided by the steel reinforcement and concrete, as shown in Fig. 10   [17]. They are interconnected at the nodes. Such a truss model assumes or requires that: (i) Forces in the truss member are in equilibrium. (ii) Concrete resists only compression and has effective compressive strength  f ce  equal to  f c , where the effectiveness factor v  is usually less than 1.0. (iii) Steel reinforcement is required to resist all tensile force. (iv) The centred axis of each truss member and the lines of  action of all externally applied loads at a joint must meet at the nodes. (v) Failure occurs when a concrete compressive strut crushes, or when a sufficient number of steel tension ties yield to produce a mechanism. The traditional method is an American method based on the ACI Code equations and can be considered a simple method as it is based on the same considerations for RC beams without openings.

(a) Arrangement of diagonal reinforcement around opening

(b) truss action Fig. 9.   Beam with small opening reinforced by diagonal bars [16].

Fig. 10.  Strut and Tie force [17].

The AIJ method is a Japanese method which is based on an empirical equation where a constant K  value must be found from a chart. The plasticity method is based on the load trend between the applied load on the beam top surface and the supports. The strut and tie method considers the transmission of the applied load through a member to the supports by means of a struts system. Each approach has been applied by a different researcher but as yet no work has been conducted on a comparison between these methods. Such a study would be useful in order to establish the most convenient approach for RC beams with small openings.

4. Large opening design approaches The presence of large openings in reinforced concrete beams requires special attention in the analysis and design phase because of the reduction in both strength and stiffness of the beam and excessive cracking at the opening due to high stress concentration [18]. This section presents existing approaches which have been used for the design of RC beams with large openings. The plastic hinge method, with its three revisions, is presented as well as the plasticity method.

4.1. Plastic hinge method This approach was proposed by Mansur and Tan  [6] in 1996 for the design of RC beams with large openings. In the ten years following this milestone, authors have revised these approaches and in 2006 produced the third revision of the original method. This section presents the mathematical equations of these approaches.

4.1.1. Plastic hinge method I  Similar to a beam with small openings, the incorporation of a large opening in the pure bending zone of a beam will not affect its moment capacity provided that the depth of ultimate compressive stress block is smaller than or equal to the depth of the compression chord, and that instability failure of the compression chord is prevented by limiting the length of the opening [2]. In practice, openings are located near the supports where shear is predominant. Experiments have shown that a beam with insufficient reinforcement and improper detailing around the opening region fails prematurely in a brittle manner  [19]. When a suitable scheme consisting of additional longitudinal bars near the top and bottom faces of the bottom and top chords, and short stirrups in both the chords are furnished, then the chord members behave in a manner similar to a Vierendeel panel and failure occurs in a ductile manner. The failure of such a beam is shown in   Fig. 11. Clearly, the failure mechanism consists of four hinges, one at each end of the top and bottom chords. The experimental observations of the finalmode of failure developed a method of analysis for predicting the ultimate strength of a beam with a large rectangular opening  [20]. It is based on the

 A. Ahmed et al. / Materials and Design 40 (2012) 90–102

collapse load analysis in which the basic requirements of equilibrium, yield and mechanism are satisfied simultaneously. The main ingredients of this method, which yields a closed-form solution for the collapse load, are briefly described below for a simply-supported beam subject to a point load,  P , at a solid section distance,  X , from its right support, as shown in  Fig. 12. For the beam in   Fig. 12a, the free-body diagram through the opening centre and those of the chord members above and below it maybe represented by Fig. 12b andc, respectively. It may benoted that the unknown actions at the centre of the opening are the axial forces (N t  and N b), the bending moments (M t  and M b), and the shear forces (V t  and  V b) in the chord members. Three equilibrium equations relate these six unknowns in which M m and V m are the applied moment and shear force, respectively, at the centre of the opening. Thus, the beam is statically indeterminate to the third degree. In a general situation, the problem is statically indeterminate to the third degree. Equilibrium provides only three equations. Therefore, three additional equations must be formulated in order to solve the three unknown actions.

M t  þ M b  þ N z  ¼ M m

ð15Þ

N t  þ N b  ¼  0

ð16Þ

V t  þ V b  ¼  V m

ð17Þ

This may be accomplished as outlined below. When the chord members are symmetrically reinforced then the moments at the two ends of each chord member (potential hinge location) are numerically identical at plastic collapse. That is, M 1 = M 2 and M 3 = M 4. From the free-body diagram of the chord members (as shown in Fig. 12c), it may be readily shown that the contra flexure points occur at the midpoint of the chord members. This means that M t  = 0 and  M b = 0. Eq. (15) then reduces to:

N z  ¼ M m

ð18Þ

if the total shear,  V m, through the centre of the opening due to global action is suitably apportioned between the chord members, that is, if:

V t  ¼  k V m

ð19Þ

v 

where kv  is a known value, then the problem reduces to a statically determinate one and the critical sections at the ends of the chord members which are subject to combined bending, shear and axial force can be designed in the usual way following the provisions of  any current building codes. There are, however, three schools of thought regarding the distribution of applied shear between the chord members at an opening. The first, as proposed by Lorensten   [21], assumes that the compression chord carries the total shear and the tension chord merely acts as a link carrying no shear. This is probably true when the opening is near the bottom. The second proposal by Nasser et al. [14] and Ragan and Warwaruk  [22], distributes the total shear between the chord members in proportion to their cross-sectional areas. The third, suggested by Barney et al.  [12], distributes the shear force in proportion to the flexural stiffness of the chord members. Accordingly:

Fig. 11.   Failure mode [20].

95

Lorenstenð½21Þ  :  k ¼ 0

ð20Þ

Nasser etal:ð½14Þ :  k v  ¼  At =ð At  þ AbÞ

ð21Þ

Barney etal:ð½12Þ :  k v  ¼  It =ðIt  þ IbÞ

ð22Þ

Clearly, the three proposals would lead to widely varying amounts of shear being assigned to each chord. However, such an assumption is not necessary if the chord members are symmetrically reinforced. The salient points in such a design process are described in the flowing steps [23]. Due to the introduction of an opening, additional reinforcement must be provided in the chord member so as to retain the original strength of the beam. This additional reinforcement may be conveniently arranged in a symmetrical manner for the top (compression) chord member as shown in Fig. 13. For the bottom (tension) chord member, which has already been provided with a relatively large amount of reinforcement near the bottom face, it is difficult to reinforce in a symmetrical manner because of the danger of steel congestion and over-reinforcement. Hence it is most likely that the bottom chord will be asymmetrically reinforced. With the assumption that the top chord is symmetrically reinforced, the interaction diagrams for positive and negative bending will be numerically identical. The nondimensional interaction chart for the top (compression) chord may be obtained by using the method of equilibrium and strain compatibility. A typical linearised chart, corresponding to the case where the distance between the centroid of the two layers of reinforcement is presented in Fig. 14. The chart has been developed using the stress–strain relationship for steel and the compressive stress block for concrete as recommended in the ACI code   [24], and is valid for f c 0   < 30MPa and f  y  = 400 MPa. The curves in this chart are labelled with values of (l q g) where l =  f  y/0.85  f c 0  and q g  = 2 As/bh; b  and h  are the overall depths of the chord members, and they are expressed in terms of  nominal strength with a capacity reduction factor of 0.9. The interaction chart for the bottom (tension) chord may be obtained in a similar manner.  Fig. 15 shows a chart for a typical value of  c b   and for equal concrete cover for top and bottom reinforcement. For unequal concrete covers, a similar chart may be plotted by varying the position of the top reinforcement as defined by c 0b , keeping  c b  at a fixed value. In  Fig. 15 each linearised curve with a particular (l q  g) is subdivided into three curves labelled 0 0 0 with different values of  a  where  a  ¼ A s =ð As  þ As Þ,  q  g  ¼ ð As  þ As Þ= 0 bh and  A s  = area of reinforcement for negative bending, and they are expressed in terms of ultimate axial load and moment also using a capacity reduction factor U of 0.9. The design steps involved in this simplified method are as follows:

 Z  ¼

 ðht  þ hb Þ þ do þ ð0:5  aÞcb hb 2

ð24Þ

where N o = axial load capacity in direct tension and M o  = pure bending moment capacity in positive and negative bending. The value of bending moments at the two ends (M u)b,3 and (M u)b,4  at collapse are calculated from:

ðM u Þb  ¼ ½ðN o Þb   ðN u Þb ðM o Þb =ðN o Þb  ¼ ðM u Þb;3or 4

ð25Þ

ðM 0u Þb  ¼ ½ðN o Þb   ðN u Þb ðM 0o Þb =ðN o Þb  ¼ ðM u Þb;3or 4

ð26Þ

ðV u Þb  ¼ ½ðM Þb;4   ðM Þb;3 =‘o

ð27Þ

ðV u Þt  ¼  V m   ðV u Þb

ð28Þ

ðN u Þt  ¼ ðN u Þb

ð29Þ

ðM u Þt ;2  ¼

 ðV u Þt ‘o ¼ ðM u Þt ;1 2

ð30Þ

96

A. Ahmed et al./ Materials and Design 40 (2012) 90–102

Fig. 12.  Beam with a large opening under bending and shear [20].

Fig. 13.  Beam before and after introduction of the opening [6].

4.1.2. Plastic hinge method II  If the bottom chord is also assumed to be symmetrically reinforced, the number of design charts may be minimised, leading to

a considerable simplification of the overall design process [6]. Inthis case, the moment-tension interaction diagram would be numerically identical for positive and negative bending, as represented by

 A. Ahmed et al. / Materials and Design 40 (2012) 90–102

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depend on the amount of longitudinal reinforcement and whether the depth is sufficient to provide effective shear reinforcement. Thus, if the opening is provided in a T-beam just below the flange, and the flange thickness is inadequate for the placement of stirrups, the entire shear should be assigned to the bottom chord. Similarly, a situation may arise where the opening is near the bottom of the beam and the bottom chord member is very shallow compared to the top chord. In such a case, the top chord member should be designed to carry the total shear force. For equally sized chord members, however, assignment of less than half of the external shear to the bottom (tension) chord member leads to a more economical design. Once a suitable shear force is assigned to the bottom chord, (M u)b,4  can be calculated from Eq.  (31) and the required reinforcement can be obtained from the appropriate chart in  Fig. 15. The design of the top chord member is identical to that used in simplified method 1.

4.1.3. Plastic hinge method III  Mansur [25] has stated that his old method of  [20] is relatively complex and requires the development of new sets of design charts. The complexity arises mainly due to the consideration of  the generalised arrangement of reinforcement in the chords. However, the design solution may be considerably simplified if the chord members are reinforced symmetrically. This is a feasible option because opening length represents only a small fraction of  the total span of the beam. Fig. 14.   The interaction diagram for compression chord [6].

Step 1:  Determine longitudinal reinforcement for the compression chord. First, assuming that the beam contains no openings, design the longitudinal reinforcement. If the beam is subject to a sagging moment, the main reinforcement,  A s, will be at the bottom. The top reinforcement will be lighter than the bottom reinforcement and the same amount is usually continued throughout the length of  the beam, including the opening region. Thus, the top reinforcement in the compression chord is known. Use the same amount and the arrangement at its bottom as additional reinforcement is required to restore the strength and avoid brittle failure of the beam due to the provision of openings.

Step 2:   Determine the shear force carried by the compression chord.

Fig. 15.   The interaction diagram for tension chord [6].

the solid line in Fig. 15, and the contra flexure point would occur at mid span. With this in mind, Eqs. (27) and (23) reduce to:

ðV u Þb  ¼  2ðM u Þb;4 =‘o

ð31Þ

 M m  Z 

ð32Þ

ðN u Þb  ¼

Eq. (32) gives the magnitude of axial force in the chord member directly irrespective of the amount of reinforcement. However, to proceed with the design, it is necessary either to assume a certain quantity of reinforcement or to assign a fraction of the total shear to be carried by the bottom chord. The latter approach is suggested because it offers great flexibility. When assigning the shear force, it should be kept in mind that the shear carrying capacity of a chord member depends on the moment capacities of the critical end sections. These in turn

Since the amount and arrangement of longitudinal reinforcement in the compression chord are known, and as the axial force acting on it is given by Eq. (18), the moment capacity of the section may be estimated in the usual manner. Because of symmetry, the capacity in positive and negative bending will be numerically identical. Therefore, fromthe free-body diagram of Fig. 12c, the amount of shear force which can be transmitted through the compression chord at ultimate may be obtained as:

ðV u Þt  ¼  2ðM u Þt ;2 =‘o

ð33Þ

where l o  is the length of opening.

Step 3:   Determine the moments and forces at critical sections and design the tension chord. The shear force carried by the tension chord will be the difference between the applied shear and that carried by the compression chord in accordance with Eq.   (17). Due to reinforcement symmetry, the contra flexure point will be at mid-span. The moment at the critical end section is then given by:

A. Ahmed et al./ Materials and Design 40 (2012) 90–102

98

shown in this figure by the direction of arrows with blank circles as the targets.

4.2. Plasticity method This approach was proposed by the Architectural Institute of   Japan (AIJ) in 1988 for the design of RC beams with large openings. In small openings in plasticity truss models, shear is resisted by a beam through a combination of the arch and truss mechanism. In this case, where the top and bottom chord members are of equal depth and where the distance between the top and bottom longitudinal of reinforcement is dvs. It has been recommended by the Architectural Institute of Japan  [26]  that vertical stirrups be provided uniformly through the chord members and for a distance equal to dvs cot Us   on each side of the opening, where  u s  is the angle of inclination of the compression concrete struts in each chord member, as shown in   Fig. 17. The longitudinal reinforcement adjacent to the opening should be extended beyond the above vertical stirrups and provided with anchorage hooks bent inside. The shear force carried by each truss mechanism in the chord members is:

V t  ¼  bd  p  f  y  cotØs v 

where  b  = width of section, qv  = shear reinforcement ratio of stirrups, and f  yv  = yield strength of stirrups. Thus, the shear capacity of the beam can be obtained as:

Fig. 16.   Typical design chart [25].

ðM u Þb  ¼ ðV u Þb

‘o 2

ð35Þ

v 

v 

ð34Þ

V u  ¼  2bd s q  f  y  cotØs v 

ð36Þ

v 

v 

where With the axial tension given by Eq.  (18), the required amount of  longitudinal reinforcement can be obtained by following the standard design procedure. Reinforcement which has already been determined from the global action can now be taken into account to obtain the desired symmetrical arrangement of reinforcement in the tension chord. Design for shear is identical to a solid beam. Use of design charts, similar to a column, may expedite the design process. A typical design chart (using the capacity reduction factor u = 0.9) for symmetrical arrangement of reinforcement, approximated by straight lines, is shown in   Fig. 16, where  l  =  f  y/0.85  f c 0 and q g  = 2 As/bh. The simple design steps, as outlined above, are

s  ffiffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 0

cotØs  ¼

Vf c  162 q  f  y v 

ð37Þ

v 

and

q  f  y v 

v 

6

0

Vf c 

where  f  yv  = yield compressive strength and for the compressive strength of concrete:

V  ¼  0:7 f c 0 =200

Fig. 17.   Truss action in beam with reinforced rectangular opening [26].

ð38Þ v 

= effectiveness factor

ð39Þ

 A. Ahmed et al. / Materials and Design 40 (2012) 90–102

Fig. 18.  Truss action in beam with rectangular opening reinforced with diagonal bars [26].

The lower limit for cot Us  as indicated in Eq.   (37)  is to ensure aggregate interlocking in crack and to prevent excessive crack widths. The required force in the longitudinal reinforcement near the opening in each chord member is given by:

T sn  ¼  A sn f  y  ¼

 V u l0 2d s

ð40Þ

 V u ‘o 2d s

ð41Þ

v 

T sn  ¼  A sn f  y  ¼

v 

and that in the longitudinal reinforcement away from the opening is:

T sf  ¼ A st  f  y  ¼

 V u ð‘o  þ d s cotØs Þ 2d s v 

ð42Þ

v 

where Asn  and Asf  are the area of longitudinal reinforcement in each chord member near and away, respectively, from the opening. Where diagonal reinforcement is provided as shown in  Fig. 18, the shear resistance provided is given by:

 Ad  ¼  A d f  yd sin hd

ð43Þ

where A d  is the area,  f  yd  is yield strength, and  hd  angle of inclination to the beam axis of the diagonal bars. The plastic hinge approaches are based on the ACI Code equations using a columns chart while the plasticity approach is based on the force trend between the applied load on the beam top surface and the supports. The difference between the three approaches to theplastic hinge method is the portion of shear carried by the top and bottom chords. No research has been conducted to compare these three plastic hinge methods nor has the method been compared with the plasticity approach.

5. Performance of beams with openings Among the earliest works on beams with openings, Nasser et al. [12] in 1967 studied the behaviour of rectangular RC beams with large web openings. They found that the top and the bottom chord of large opening beams behaved like a Vierendeel panel and the contra flexure points occurred at their mid point. They also assumed that the apportionment of applied shear was distributed between the top and bottom chord in proportion to their cross-sec-

99

tional area. Test data exists regarding what happens when a small opening is introduced in a region subjected to predominant shear and the opening is enclosed by reinforcement to investigate the beam’s failure mode [4,9,10]. A method has been developed to predict the ultimate strength based on the collapse load analysis of a reinforced concrete beam with large rectangular opening subjected to a point load [2]. They also assumed that the solid sections of the beam were rigid and that collapse results from the formation of a mechanism with four plastic hinges, one at each end of the top and bottom chords. They found that the ultimate strength increased with a decrease in the moment to shear ratio at the centre of the opening and the amount of external shear carried by the top and bottom chord depends not just on the cross-sectional properties, but also on the opening size (length and depth) and location of  the opening. Later in 1985, a rational design method was used for reinforced concrete beams with large rectangular openings   [27]. Twelve beams were tested under one point load and subjected to bending and shear force. They observed that an increase in the opening size (length and depth) or moment–shear ratio at the centre of the opening increased both crack width and maximum deflection. They also stated that the diagonal bars for corner reinforcement were more effective in both controlling crack width and reducing beam deflection, as shown in  Table 1. In 1991, the behaviour of reinforced concrete continuous beams with large web openings was studied   [28]. Eight beams were tested and it was found that the location of the opening has very little influence on the cracking load, but that openings located in a relatively high moment region yield smaller collapse loadand large deflection. The deflection brought about by Vierendeel action and the mode of  collapse remain virtually unaffected by the location of opening. An analysis of the service load of reinforced concrete with large web openings in the analytical modelling was proposed by Mansur et al. [17]. They assumed that the beam was treated as a non prismatic member with two different cross–sectional properties (the soild section and the opening section by replacing the chord members through equivalent continuous medium). A total of 15 reinforced concrete T-beams containing large web openings were tested  [29], each simulating either negative or positive moment. They found that the presence of a web opening led to a decrease in both cracking and ultimate strength. They also found that the external shear may be distributed between chords in accordance with their flexure stiffness based on either gross or cracked transformed section. The critical lateral buckling load of deep slender rectangular beams containing openings along the centre-lines of  the beams has been studied by Tbevendran and Shandugam [30]. The numerical method which was proposed to predict the critical load was outlined in detail; cantilever beams and simply supported beams were considered. The critical loads evaluated numerically using the energy approach were compared with those values obtained experimentally and a good agreement was achieved. The nonlinear analysis and design of statically loaded simply supported post tensioned and pre-stressed concrete beams and girders with rectangular openings was investigated by Kennedy and Abdalla  [31]. Several design parameters were varied such as: opening length and depth, vertical and horizontal locations of the opening, location of the applied load, type of cross section, and opening reinforcement. The results from the analytical study were substantiated by test results from 12 post tensioned pre-stressed concrete beams, eight of which had rectangular sections, whilst the remaining four were T-section beams. A rational method of distributing the shear between the top and bottom chords of the opening was also proposed, together with a design procedure against the cracking of such chords. The influence of openings on the response of  hybrid reinforced concrete T-beams, with the beams subjected to acyclic load applied at mid–span was investigated by Tanijaya and Hardjito   [32]. The experimental work involved testing and

A. Ahmed et al./ Materials and Design 40 (2012) 90–102

100  Table 1

Beams details  [27]. Specimen

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12

Design ultimate load P UD   (KN)

204 162 132 107 89 164 92 138 144 137 131 127

l  (mm)

400 600 800 1000 1200 800 800 800 800 800 800 800

Dt  (mm)

110 110 110 110 110 130 90 120 130 110 110 110

do   (mm)

180 180 180 180 180 140 220 180 180 180 180 180

investigating the structural responses of three hybrid reinforced concrete T-beams, partially constructed using light weight concrete. An opening was provided on the web of each beam, with the exception of the third one, that is, the beam was provided with an opening in the low flexural moment-high shear region and in the high flexural moment-shear region. The behaviour of the specimens was discussed based on the observed degradation of  strength and stiffness as well as the energy dissipation capability. Test results indicated that the presence of web opening caused a decrease in both cracking and ultimate strengths. The deflection ductility of a beam with a web opening in the low flexural moment-high shear region was seen as approximately equal to that of a beam with an opening in the high flexure-shear region. A three-dimensional nonlinear finite element model suitable for the analysis of reinforced concrete beams with large openings under flexure was developed by Al-Shaarbaf et al.  [33]. Numerical studies including some material parameters such as concrete compressive strength, amount of longitudinal tensile reinforcement and opening size on the load–deflection response were conducted. The finite element solution revealed that the ultimate load and post-cracking stiffness increased with an increase in the concrete compressive strength and the extent of the bottom steel reinforcement decreased with an increase in the length or depth of the opening. The bond anchorage and the shear carrying behaviour of pre-stressed concrete beams made of Ultra-High Performance Concrete (UHPC) with and without web openings was investigated by Hegger and Bertram [34]. Results of the shear tests with multiple openings in the web showed that the remaining shear resistance was approximately 60–65% compared to solid beams. The effect of small circular openings on the shear, flexural, and ultimate strength of beams made by normal and high strength concrete has also been studied   [35]. The main factors of this test were the changes of diameter, the position of opening and the type and location of reinforcement around the opening as well as changes in the strength of the concrete. In this investigation nine of the beams tested were made up of normal concrete and five of the beams were made up of high strength concrete; all of which were tested. The testing beams were loaded as simple beams with two concentrated and symmetrical loads. The effect of concrete strength depends on parameters such as diameter and the position of the opening. The load under which the first flexural crack is induced does not depend on the presence or the lack of opening or its situation, but shear cracks around the opening will induce sooner than shear cracks around a similar area in a solid beam. The increase of  the diameter of the opening in beams with an opening made of  normal concrete will cause a change in the pattern of cracks and the type of failure from flexural failure to frame type or beam type shear failure. The increase of strength in concrete does not have much influence on the ultimate strength, but it will increase the

Db   (mm)

110 110 110 110 110 130 90 100 90 110 110 110

eo   (mm)

0 0 0 0 0 0 0 10 20 0 0 0

a  + l/2 (mm)

1000 1000 1000 1000 1000 1000 1000 1000 1000 800 1000 1200

Stirrups spacing

S b   (mm)

S t   (mm)

40 40 40 40 40 50 30 35 30 40 40 40

40 40 40 40 40 50 30 45 50 40 40 40

stiffness and improve the serviceability of the beams. The most critical position of an opening in order to reach the ultimate strength in beams made of normal concrete is near the support. In addition to this, the best place for the location of the opening in these beams is in the middle of a distance between the place of applied load and support (in the middle of the shear span). The presence of longitudinal bars on the top and bottom of the opening is necessary in order to control the cracks and flexural strains around the opening. The installation of diagonal bars and small stirrups at the top and bottom of the openings will increase the ultimate strength of the beams with opening.

6. Strengthening with Fibre Reinforced Polymer (FRP) materials This section presents the use of externally bonded material for strengthening and updating the performance of RC beams with openings. A total of 10 beams were tested under static loading, simulating the negative moment regions of reinforced concrete T-beams [36]. Of these beams nine were fabricated with large openings through the web whilst the other beams had a solid web. They studied the effect of tension reinforcement area, reinforcement around the opening, strength of concrete and the shear span to depth ratio on the strength of such beams. In general, the presence of web openings decreased both the cracking and ultimate strength, as well as the stiffness of the beams. Both the shear span to depth ratio and concrete compressive strength of T-beams with openings had a pronounced effect on the load bearing capacity of the tested beams. Both the top chord and bottom chord member independently resisted the shear force of tee beams with openings made from high strength concrete. The predicted ultimate loads based on this rule agreed with the experimental ultimate loads for beams with a shear span to depthratioequal to 2. In general, thebehaviour of high strength concrete beams with openings was shown to be quite different from the behaviour of normal strength concrete beams with openings.   Table 2  presents the results compiled by Zainab [36]. The possibilityof reducing theeffect of existingcircular web openings in solid beams on both strength and stiffness by means of strengthening the openings with Glass Fibre Reinforced Plastics was investigated in  [37]. The experimental program consisted of testing 11 simply supported reinforced concrete rectangular beams. The first beam was solid with no opening and served as the reference beam. Each of the remaining beams was provided with one or two circular web openings with different diameters. Table 3 presents the results found in [37]. Several design parameters including the opening width and depth, as well as amount and configuration of the Fibre Reinforced Polymer (FRP) sheets in the vicinity of the opening were investigated by Abdalla  [38]. The experimental program included the testing of 10 reinforced

 A. Ahmed et al. / Materials and Design 40 (2012) 90–102

101

 Table 2

Beams results  [36]. Beam no.

P cr 

P u

P uth

P u/P uth

Mode of failure

R

10

25.4

24.4

1.036

Shear failure at opening

A

A1 A2 A3

4 4 4.5

19 19.6 20

20 20 20

0.95 0.98 1

Shear failure at opening Shear failure at opening Shear failure at opening

B

B1 B2

4 4

18.3 19.2

20 20

0.915 0.96

Shear failure at opening Shear failure at opening

C

C1 C2

6 3

17.6 14

20 20

0.88 0.7

Flexural shear failure Flexural tension failure

D

D1 D2

2 4

14.8 19

14 20

1.057 0.95

Shear failure at opening Shear failure at opening

Group no.

 Table 3

Beams test result  [37]. Specimen

Dimensions of openings

W   (mm) SB1 RO2 RO3 RO4 RO5 RO6 UO7 UO8 UO9 UO10 a

a

NA 100 200 300 300 300 100 200 300 300

H   (mm) NAa 100 100 100 150 150 100 100 100 150

Concrete strength  f cu   (Mpa)

Configuration of CFRP strengthing

Cracking load P cr  (kN)

Ultimate load  P u (kN)

Mode of failure

49 52 49 51 49 49 43 49 52 42

NAa Type Type Type Type Type NAa NAa NAa NAa

30 25 20 20 15 20 25 13 20 5

83 86 73 62 35 34 41 43 41 22

Flexure at mid-span Flexure at mid-span Flexure at mid-span Shear at opening Shear at opening Shear at opening Shear at opening Shear at opening Shear at opening Shear at opening

1 1 1 1 1

NA denotes not applicable.

concrete beams, five of which were strengthened with FRP sheets around the opening, whilst four were tested without strengthening, and the remaining beam was solid without an opening and was consideredas the control beam. The efficiency of using Carbon Fibre Reinforced Polymer (CFRP) sheets to control local cracks around openings and to resist excessive shear stresses in the opening chords was examined. Based on the results of this investigation, they found that the presence of an un-strengthened opening in the shear zone of a reinforced concrete beam significantly decreased its ultimate capacity. With an un-strengthened opening with a height of 0.6, the beamdepth may reduce the beamcapacity by 75%. The application of CFRP sheets according to the arrangement presented in this research greatly decreases beam deflection, controls cracks around the opening, and increases the ultimate capacity of the beam. The use of FRP sheets to strengthen the area around openings may retrieve the full capacity of the beam for relatively small openings. The shear failure at the opening chords of strengthened openings occurs due to a combination of shear cracking of concrete and bond failure of the FRP sheets glued to the concrete. The method was shown to be capable of estimating the shear capacity of reinforced concrete beams with CFRP strengthened openings. There are two patterns of strengthening by FRP rod. The first involvesFRP rods enclosing the openingand the other involvesplacing FRP rods diagonally throughout the entire depth of the beam. Both were investigated by Pimanmas  [39]. The author found that simply placing FRP rods around the opening was not fully effective because a diagonal crack can propagate through the beam with the crack path diverted to avoid intersecting with the FRP rod. When FRP rods were placed throughout the entire beam’s depth, a significant improvement in loading capacity and ductility was achieved, similar to strengthening by pre-fabricated internal steel bars. The flexural failure mode was restored. An experimental study to

investigate the efficiency of external strengthening of such beams when provided with large openings within their shear zones was conducted by Allam [40]. It was found that both types of material used for strengthening namely steel plates and CFRP sheets as well as its configuration scheme, significantly affected the efficiency of  strengthening in terms of beam deflection, steel strain, cracking, ultimate capacity and failure mode of the beam. Test results revealedthat the efficiency of external strengthening of beams with openings increased significantly when such strengthening was applied to both the inside and outside edges of the beam opening. The increase was also found to be more significant than in the case of only strengthening the outside edges. Furthermore, it was discovered that using steel plates for strengthening beams with openings was much more efficient than in the case of CFRP sheets.

Fig. 19.   Photos of specimens strengthened with CFRP at failure [41].

102

A. Ahmed et al./ Materials and Design 40 (2012) 90–102

Theinterfacialshear stresses found to be influenced by the geometryparameters suchas thicknessof theFRP plate andadhesive layer in range of the different degrees  [41], where the interfacial shear stress concentrations and levels increase obviously with the increase of the thickness of the FRP plate. A typical failure of a specimen strengthenedwith CFRPis shown inFig.19. Thecross-sectional shape found to hasa significant influence on the effectiveness of the CFRP-confinement under concentric loading   [42], where member with the circular cross-section benefited the most, followed by the member with the square cross-section andthe gain in load capacity of RC members with rectangular cross-sections due to CFRPconfinement depends on the aspect ratio of the cross-section.

7. Conclusion and future directions This paper has reviewed the existing work related to RC beams with openings. Based on the aforementioned review, it is evident that there are gaps in the previous research which must be investigated. Below are the main points which can be considered as conclusions and directions for future work in order to fill the gaps which exist in the work carried out thus far. 1. Since no research has been conducted regarding the comparison of existing design approaches of RC beams with small openings, future work must take into consideration such comparisons based on a number of experiential samples. 2. Future work must also cover the comparison between the proposed three plastic hinge approaches as well as the plasticity approach based on a number of experimental samples. 3. Strengthening of RC beams with externally bonded steel plates must be investigated corresponding to different parameters such as the steel plates’ mechanical and geometrical properties as well as their configurations. 4. Strengthening of RC beams with externally bonded FRP materials, namely CFRP sheets and fabric, GFRP sheets and fabric and so on, must be investigated with different parameters such as the FRP materials’ mechanical and geometrical properties and configurations. 5. Since no repair work has been conducted on RC beams with openings, it would be extremely beneficial were an experimental study to be conducted for the repairing of damaged RC beams with openings using externally bonded FRP material or steel plates with different parameters such as materials’ mechanical and geometrical properties and configurations.

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