Analytical model for steel fiber concrete composite short-coupling beam

Composites: Part B 56 (2014) 318–329

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Analytical model for steel fiber concrete composite short-coupling beam E.S. Khalifa ⇑ Department of Civil Engineering, Higher Technological Institute (HTI), Egypt

a r t i c l e

i n f o

Article history: Received 6 June 2013 Received in revised form 11 July 2013 Accepted 12 August 2013 Available online 22 August 2013 Keywords: A. Fibers B. Strength C. Analytical modeling D. Mechanical testing E. Casting

a b s t r a c t Lateral stiffness of a high-rise building is significantly influenced by the design of coupling beams to spread plasticity over the system height. Design and reinforcement detailing should be performed to retain strength and a significant percentage of stiffness during large deformations into a plastic range. The cracking and low toughness problems of high-strength concrete can be overcome by the addition of short randomly distributed steel fibers. These steel fibers provide a crack bridging the interference plan between shear walls and coupling beam. An alternative design is proposed in this paper, using an analytical model for high-strength fiber reinforced concrete (HFC). This is to reduce the reinforcement congestion and construction difficulties. In this study, the fiber composite enables the use of straight bars as partial or total replacement of diagonal bars. An analytical relationship is proposed, herein, to generate the complete stress–strain curve of HFC subjected to uniaxial compression. The fiber generates a passive confinement inside the composite that prevents the concrete from spilling-out during cycles of seismic load. Based on nonlinear fracture mechanics, a continuum approach is developed, as a linear elastic-strain softening material, for modeling the tensile behavior of HFC. The model accounts for composite inelasticity and ductility. It also slows down crack growth, fiber debonding and pullout mechanisms, and also attenuates fracture energy and element size effect. There is a wide variation in the code limit for predicting maximum shear stress. For this reason, based on experimental results, a proposed strut-and-tie model is developed to determine the contribution of fiber composite in the shear resistance of short-coupling beams. Comparing the analytical results with experimental results, the adopted analytical model shows a good agreement. A non-linear finite element model is proposed to examine the effect of using HFC on forty stories high-rise building. Ó 2013 Elsevier Ltd. All rights reserved.

1. Research significance The aim of this paper is to use fiber composite as total or partial replacement of diagonal bars. For nonlinear analysis of HFC shear wall-short coupling beam system, analytical relationship is proposed. This is to generate a complete stress–strain curve of fibrous concrete under uniaxial compression and tension. Based on experimental results, a strut-and-tie model is developed to account for the contribution of fiber composite in shear resistance of shortcoupling beams. To test hypothesis, the analytically predicted model was compared with corresponding experimental results, where a good coincidence was obtained. A non-linear finite element program has been adopted in this paper to study apply the use of HFC on forty stories high-rise building. 2. Introduction Shear walls-short coupling beams are the most common method in structures resisting seismic loads. Windows, doors and ser⇑ Tel.: +20 1001479986. E-mail address: [email protected] 1359-8368/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2013.08.050

vice ducts require that shear walls be provided with openings. A number of empirical expressions [1–5] were performed to examine seismic performance of coupled shear walls made from highstrength and fiber concrete. Numerical models were published [6–10] to analyze the behavior coupled shear wall system subjected to seismic load. The test results of high strength fiber reinforced concrete coupling beams examined [4,5]. The traditional reinforcement of concrete walls subjected to little bending is not less than 0.25% of the wall cross section area. Such an arrangement does not efficiently utilize the steel at ultimate moment because many bars operate on a relatively small internal lever arm. Moreover, the ultimate curvature ductility becomes considerably reduced when a large amount of flexural steel is used in this form [1]. The typical reinforcement details of shear walls with coupling beam are shown in Fig. 1. The most common reinforcement type is represented as conventional reinforcement with longitudinal bars and transverse reinforcement or diagonally reinforcement [3] as shown in Fig. 2. Shear stress-story drift relationship for a fibrous and non-fibrous coupling beam having diagonally reinforcement is shown in Fig. 3. The non-fibrous specimen SP1 is compared with 1.5% (volume ratio) of steel fibers specimen SP4. For fibrous coupling beam (SP4), hysteretic loops have characteristics of a ductile

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Fig. 1. Typical reinforcement details of a coupling beam [3].

member with excellent distribution of cracks [4]. Strength degradation occurs only when the buckling of compression bars commences. When load reversal occurs, these bars can take up tension and straighten themselves. The process leads to the whole breaking up of the concrete around the compression bars, to further loss of restraint against buckling and consequent loss of strength. Lateral stiffness and strength of a system is significantly influenced by design of coupling beams to spreading plasticity over the system height. During application of load reversal, these bars become subjected to large compression stresses. Since equal amounts of steel are provided in both diagonal bands, the loss of contribution of concrete provides unstable diagonal compression bars do not become unstable. The cracking and low toughness problems of high-strength concrete can be overcome by the addition of short randomly distributed steel fibers. These fibers provide a crack, bridging the interference plan, between shear walls and coupling beam. In the last two decades, high rise buildings showed clear trends towards performance-based seismic designs. Such designs can be looked at as explicit designs for multiple limit states [11]. Analyzing a high-rise structure for various levels of earthquake intensity and checking some local and/or global criteria for each level has been a popular academic exercise. It is realized that while structures built in countries aware of seismic risks are in general adequately safe, the cost resulting from business interruption can be difficult to tolerate. This is one of the issues that should be considered in the performance-based design [12]. Inelastic analysis is recently gaining popularity as a reason for appropriate anal-

Fig. 3. Shear stress-drift relationship for fibrous and non-fibrous diagonally reinforced coupling beam [4].

Fig. 2. Typical reinforcement configuration of diagonally reinforced concrete coupling beam [3].

ysis tools required for performing static pushover or dynamic time-history. The CEN-2003 code retains the limit state of design for ultimate and damage-limitation, so that inelastic analysis can be used in the design procedure [13]. A significant step towards

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the use of inelastic analysis in the practical context was the ASCE– FEMA pre-standard for seismic rehabilitation of buildings [14]. The seismic performance design and requirements for a high-rise concrete building has been proposed [15]. Design and reinforcement details are performed in order to retain strength and a significant percentage of stiffness during large deformations into a plastic range of behavior. Current design provisions in the ACI building code [16] are shown for reinforced concrete coupling beams in seismic resistance structures. But, this still requires substation reinforcement details to ensure stable seismic behavior, without leading to reinforcement congestion and construction difficulties. In this paper, an alternative design analytical model for HFC is proposed to reduce the reinforcement congestion and construction difficulties. In this study, the fiber composite enables the use of straight bars as partial or total replacement of diagonal bars. For nonlinear analysis of HFC shear wall-short coupling beam system, analytical relationship was proposed to generate the entire stress– strain curve of HFC composite under uniaxial compression. The fiber generates passive confinement inside the composite that prevents the concrete from spilling-out during cycles of seismic loads due to high increase in composite ductility. The proposed maximum shear stress values in the code had wide variations. For this reason, based on experimental results, a proposed strutand-tie model is developed to account for the contribution of fiber composite in shear resistance of short-coupling beams. Comparing the numerical results with experimental ones, the adopted analytical model showed a good agreement. Based on nonlinear fracture mechanics, a continuum approach has been developed for modeling the tensile behavior of steel fiber reinforced concrete HFC as a linear elastic-strain softening material. The model accounts for composite inelasticity and ductility, slow crack growth, fiber debonding pullout mechanisms, HFC fracture energy, and element size effect. In the cracked range, the strain softening law is obtained from post-cracking tensile strength, critical fracture energy, characteristic element length and ultimate strain. A non-linear finite element analysis is adopted to study the effect of HFC on the behavior of forty stories high-rise building.

3. Behavior of shear wall-coupling beam In the coupled shear walls, there are three critical zones, as shown in Fig. 4. These can be listed as follows: 1. The previous theoretical studies were performed to ensure the displacement ductility factor of 5 [2]. The ductility demand on the coupling beams may have to be very high. It is necessary to examine available experimental evidence to determine wither such ductility demands can be met. 2. One of the walls is subjected to considerable tension, flexure and shear. The applied vertical load may adversely affect the diagonal tension capacity of a shear wall. In this case, the design code should be followed. 3. The total horizontal shear also must be transferred across horizontal construction joints. Large axial tension may exist across such joints. The shear wall has a large cross section area and thus the effect of axial compressive loads is considerably smaller than causing balanced failure condition. As a result, the moment capacity is usually increased by gravity forces in shear walls. However, it should be remembered that axial compression would then reduce ductility. When it is desirable to increase ductility of a cantilever shear wall, the concrete in the compression zone must be confined. This is particularly important over the region of a possible plastic hinge, which may extend over a full story height or more. The mechanism

Fig. 4. Shear wall with coupling beam deformed shape due to seismic load, with critical areas.

of aggregate interlock or shear friction should be considered. The use of HFC makes pronounced enhancement of the three critical zones. Average shear stresses that can be safely transferred across wall-prepared rough horizontal joint, as a function of axial force Pu and reinforcement yield stress fy [2], are at least:

V uj ¼

P u þ As v f y Ag

ð1Þ

Asv is the total vertical steel to be utilized for required clamping force and vuj is the nominal shear strength transmitted across construction joint. 4. Traditional model for analysis of coupled shear wall In the past, for design information when only small computers were available, a laminar analysis or continuum approach was used. The coupling system was consisting of a number of short coupling beams. These beams would transmit shearing forces from one wall to another as shown in Fig. 5a, thus subjecting the coupling beam for flexure and shear. Because of the small span to depth ratios of these beams, shear deformations may become very significant. The relative stiffness of the two shear walls connecting with a coupling beam is the most powerful issue for straining action distribution as shown in Fig. 5b. To evaluate various aspects of behavior of coupled shear walls, the concept of laminar analysis have used [2]. The most important activity of an external lateral load, provide the overturning moment, must be resisted at any horizontal section across the shear wall, as shown in Fig. 5c. The corresponding equilibrium statement at level x can be written as follows:

Mo ¼ M1 þ M2 þ LT

ð2Þ

where Mo is the total external moment, M1, M2 are the internal moments generated in wall 1 or wall 2, T is the axial force induced in the walls i.e. tension in wall 1 and compression in wall 2, and where L is the distance between centroids of the two walls. The laminates are subjected to shearing forces q(x) and axial forces p(x) at the midspan points. The shear force, of the total external load resisted by wall 1 is kW 1 . The conditions of compatible deformations need establish of second order differential equation. This equation yields to the laminar shear forces q(x) over the full height of the coupling shear wall structure.

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moved. When the reversal of load is applied during an earthquake, these bars become subjected to large compression stresses before the previously formed cracks remove. As shown in Fig. 6, the developed forces can be derived as follows:

T u ¼ C u ¼ As f y

ð3Þ

V u ¼ 2T u sin a

ð4Þ

Vu 2f y sin a

ð5Þ

h  2d Ls

ð6Þ

Asd ¼

tan a ¼

The resisting moment at the supports of the beam (Fig. 6b), may be established from the shear force as:

Mu ¼

Fig. 5. Coupled shear wall and their mathematical model: (a) prototype structure, (b) mathematical model and (c) internal and external actions.

5. Design code-limits of shear wall-coupling beam The enhancement in ductility and strength of coupling beams can be considerably improved if the principal reinforcement is placed diagonally. The design of such a beam can be based on the premise that the shearing force resolves itself into diagonal compression and tension forces. So, forces are intersecting each other at mid-span, where no moments are resisted, as shown in Fig. 6. Initially, the diagonal compression transmitted by the concrete and the compression steel make an insignificant contribution. After the first excursion of the diagonal tension bars into the yield range, a significant crack form and remain open when the load re-

V u Ls ¼ Ls T u sin a 2

ð7Þ

Since equal amounts of steel are being provided in both diagonal bands, the loss of contribution of concrete is without consequence, provided that the diagonal compression bars do not become unstable. For seismic type loading, it is important to have ample ties around the diagonal compression bars to retain the concrete around the bars. The main purpose of retaining the concrete is to furnish some lateral flexural rigidity to the diagonal strut, thus enable yielding of the diagonal compression bars to take place. For any structure member designed to resist earthquake E, the strength reduction factor for shear / shall be equal 0.60 if the nominal shear strength of the member is less than the shear strength corresponding to the development of the nominal flexural strength of the member. The nominal flexure strength is determined considering the most critical factored axial loads and including earthquake load E. For diagonally reinforced coupling beams (Fig. 6), the strength reduction factor shall be 0.85 [16]. The distributed wall reinforcement rations ql and qt shall not less than 0.0025 of the concrete area expect if ultimate shear stress Vu does not exceed the following:

V u 6 0:083Acv

qffiffiffiffi fc0

ð8Þ

where Acv is the gross area of concrete section. At least two curtains of reinforcement shall be used if Vu exceeds the following:

V u P 0:17Acv

qffiffiffiffi fc0

ð9Þ

The shear wall effective length should be permitted to be 0.80 of the wall length lw. The design force Vu shall be obtained from lateral load analysis according to the factored load combination. The nominal shear strength Vn of the structure wall shall not exceed the following:

V n ¼ Acv ðac

Fig. 6. Modeling of diagonally reinforced coupling beam (a) geometry of diagonal reinforcement, (b) external actions and (c) internal forces.

qffiffiffiffi fc0 þ qt fy Þ

ð10Þ

where the coefficient ac is 0.25 for hw lw 6 1:5 is 0.17 for hw =lw P 2:0 and varies linearly between 0.25 and 0.17 for hw/lw between 1.50 and 2.0, respectively. If the factored shear force, at a given level, in the structure is resisted by several walls of perforated wall, the average units shear strength isp then ffiffiffiffi assumed for the total available cross-sectional limited to 0:66 fc0 with the requirement padditional ffiffiffiffi for unit shear strength not exceeding 0:83 fc0 . The limit of strength to be assigned to any one member is imposed to limit the degree of redistribution of shear force [16]. Two design approaches are proposed for evaluating detailing requirements at the wall boundaries. The first one considers the reinforcement uniform through the wall, while the second one is considering boundary confinement zones [2]. From a displace-

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ment-based approach [17] the compression zones shall be reinforced with special boundary elements C, as follows:

CP

lw 600ðdu =hw Þ

ð11Þ

where C is corresponding to the largest neutral axis depth calculated for factored axial force and nominal moment strength consistent with the design displacement du. Ratio of du/hw shall not be taken less than 0.007. Coupling beams with aspect ratio, Ls =h P 4:0, shall satisfy the intermediate moment resisting frame previsions, as shown in Fig. 7. The coupling beams with an aspect ratio Ls/h less than 4.0, shall be permitted to be reinforced with two intersection groups of diagonally placed bars symmetrical about the mid-span. The couplingp beams with aspect ratio less than 2.0 and with Vu exceedffiffiffiffi ing 0:33 fc0 Acw shall be reinforced with two intersection groups of diagonally placed bars symmetrical about mid-span. This can apply unless the loss of stiffness and strength of the coupling beam do not affect the vertical load-carrying capacity of the structure. Each group of diagonally placed bars shall consist of minimum four bars assembled in a core. The core has sides measured to the outside of transverse reinforcement not smaller than bw/2 and perpendicular to the plan of the beam. In this case, bw/5 in the plan of the beam and is perpendicular to the diagonal bars, as shown in Fig. 7. The nominal shear Vn, shall be determined as follows:

qffiffiffiffi V n ¼ 2Av d fy sin a 6 0:83 fc0 Acw

ð12Þ

composite inelasticity and ductility, compressive strain softening, effect of randomly distributed and oriented steel fiber parameters. The proposed expression for the normalized complete stress–strain relationship under uniaxial compression is illustrated schematically in Fig. 8. The ascending branch simulates the inelastic prepeak response of HFC, while the descending portion activates the compressive strain softening regime. The curve shows that HFC exhibits large deformations until the ultimate failure indicates a significant increase in ductility. The compressive strength and the corresponding strain are used to normalize, respectively, the stress and strain at any point on the curve. The subscripts o, p, i and t of the normalized stress strain in Fig. 8 refer respectively to the peak stress point (xp, yp), inflection point (xi, yi) and a far point on the descending tail of the curve (xt, yt). The expression for unconfined HFRC can be represented by the following equations:

y¼

bx b  1 þ xb

for 0 6 x 6 xi

y ¼ yi EXP½aðx  xi Þ for xi 6 x 6 xt b¼ 1

a

Efo efp Efo efp  ffp

ð14Þ ð15Þ

  yt yi

ð16Þ

and y ¼ f =ffp

ð17Þ

¼ ðxt  xi Þ ln

x ¼ e=efp

for b P 1:0

ð13Þ

where a is the angle between the diagonally placed bars and the longitudinal axis of the coupling beam and Acw, is the cross sectional area that resist shear of coupling beam. The diagonally placed bars should be developed for tension in the wall and should be considered to contribute in resisting the applied moment of coupling beam. The area of shear reinforcement perpendicular to the flexural tension reinforcement, Av shall not be less than 0.0025 bws, and s shall not exceed the smaller of d/5 and 300 mm. Meanwhile, the area of shear reinforcement parallel to the flexural tension reinforcement, Avh shall not be less than 0.0015 bws2, and s2 shall not exceed the smaller of d/5 and 300 mm.

where b and a are constants depending on the shape of the stress strain curve and on concrete strength. On the other hand, y and x are the normalized stress and strain, respectively. The stress and strain in general are represented by f and e. Meanwhile, ffp and efp are the peak stresses of HFC and corresponding strain. The other five characteristic concrete parameters are the initial tangent modulus Efo and the normalized stress and strain at an inflection point (xi, yi) and at the tail point (xt, yt). The model can also, account for stress degradation at any drift strain level, as shown in Fig. 8. During the load cycles, the walls are making restraining action for the coupling beam.

6. Behavior of coupling beam using high performance fiberreinforced concrete

6.2. Tensile behavior

In the following subsections, the adopted analytical model for high-strength fiber reinforced concrete is clearly described. The adopted nonlinear models are accounted for compression and tension. A proposed model has been developed, in this paper, to account shear behavior of short coupling beam.

Based on nonlinear fracture mechanics, a continuum approach is developed for modeling the tensile behavior of HFC as a linear

6.1. Uniaxial compression behavior For nonlinear analysis of concrete structures, an analytical relationship is proposed to predict the complete stress–strain curve of unconfined HFC under uniaxial compression [18]. It accounts for

Fig. 7. Applied shear force on short-coupling beams with Ls =h 6 2:0.

Fig. 8. Proposed constitutive model for HFRC under uniaxial compression.

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elastic-strain softening material. The model accounts for composite inelasticity and ductility, slow crack growth, fiber debonding and pullout mechanisms, as well as HFC fracture energy and gauge size effect [19]. In the elastic range, empirical expressions for composite elasticity modulus, tensile strength and cracking strain are derived in relation to tensile strength of the concrete matrix and fiber reinforcing dependency functions. In the cracked range, the strain softening law is obtained due to post-cracking tensile strength, critical fracture energy, characteristic gauge length and ultimate strain. Using semi-empirical relations, the model predicts these post-cracking parameters as functions of the fiber reinforcing index and interfacial bond strength. The model generates the full curves of the proposed stress–crack displacement relation and the stress–strain expression. For the equivalent continuum tension model of HFC, the proposed expression for total normalized stress– strain relation under uniaxial monotonic tension is illustrated in Fig. 9. The model can also, account for strain softening at any drift strain level during seismic cycles of load. The tensile strength fp and the crack initiation strain ep are used to normalize the stress f and strain e, respectively, at any point on the curve (x, y). The subscripts p and pp stand respectively for the peak stress point (xp, yp) and post-cracking point(xpp = xp, ypp). The expression is represented by the following equations:

y ¼ x for 0 6 x 6 xp

ð18Þ

ypp ¼ fpp =fp

for x ¼ xp

ð19Þ

h e i p y ¼ ypp EXP  ðx  xp Þ for x > xp B

ð20Þ

x ¼ e=ep

ð21Þ

and

y ¼ f =fp

The softening constant B is given [19] as functions of fracture energy and specimen size, respectively Gc, Lch and fpp:

B¼

Gc Lch fpp

ð22Þ

A shear crack is developed during cycles of loads and the separation occurs between the coupling beams and shear wall and can be resisted by shear friction of HFC. The crack displacements and thus, strain can be developed. Concrete softening strength for tension can be conducted using constitutive model. The fibers are bridging the cracks up to tensile tension displacement capacity equal lf/2.

323

6.3. Shear strength of high performance fiber concrete An analytical model to predict the shear strength of short-coupling beam is proposed in this subsection. The short-coupling beam is assigned for beams with span/depth ratio less than 2. The ACI-318-08 permit using strut and tie model (Appendix A), for shear friction members of span to depth ratio less than 2 [16]. The use of the deep beam design method in the ACI provisions leads to very conservative shear designs for coupling beams. The ACI shear stress limitation for deep beams, which is a function of the span-depth ratio and imposed to guard against a diagonal compression failure, does not necessarily limit the load-carrying capacity of stocky link beams supported over their heights. The strutand-tie model used in ACI code, permits design of reinforced concrete link beams for substantially higher loads than use of ACI sectional design methods. A wide variation in the maximum shear stress limit is found in the codes of practice. The difference is more thanp a ffiffiffifactor of two between sectional design models in ACI [16] is ffi 0:83 fc0 , the Canadian [20] and AASHTO LRFD [21] codes are 0:25fc0 . The higher the shear stress limits (Canadian and AASHTO LRFD codes) were found to be appropriate for short link beams supported over their heights at their ends by continuous wall piers. Due to shear-sliding force, these beams, at the interface plan of shear wall, resisted by shear friction, as shown in Fig. 3. This model presents a stable performance of strut and ties model and is sustainable for large deformation due to high ductility of HFC. When sliding cracks develop, the fibers are crossing the interface failure plan between wall face and coupling beam. The fibers are bridging the cracks and enable stable crack growth. It founds formation of small diagonal distributed cracks with small width instead of a single wide crack. The model is also taking in consideration the high compression ductility due to passive confinement of fibers. The applied loads acting on the interface of short-coupling beams with the wall face are shown in Fig. 10. The half span of short-coupling beam has a downward force Vu that forms the diagonal compression strut, while the flexure force at mid-span of the short-coupling beam is approximately equal zero. The internal horizontal shear force Vh is the resultant of horizontal force in concrete Cc and tension force on the main reinforcing steel Ts as follows:

V h ¼ T s ¼ Cc

ð23Þ

The distance H between the compression force in concrete and tension force in the reinforcing bars is considered as:

H ¼ d  z=3

ð24Þ

The proposed macro-mechanical strut-and tie model proposed in this paper is a modified model developed by Khalifa [20]. The downward vertical force Vu is forming a compression strut with the angle of inclination h equal to tan1 (H/ac). After the development of cracks, the steel bars that are subjected to tension force and concrete act as compression strut. As shown from Fig. 10, statically

Fig. 9. Proposed constitutive model for HFRC under uniaxial tension.

Fig. 10. Proposed strut and tie model for prediction of shear strength for HFC shortcoupling beams.

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Table 1 Comparison between experimental and numerical results. qffiffiffiffiffi ffp

b mm

d mm

vf (%)

lf/u

As mm2

fy MPa

ffp MPa

Vu Exp. kN

Vu Num. kN

qsf MPa

qsf =

200 200 152.5 155 154 153 156 153 153.5 154 153.5 154.1 152 152 153 153

550 550 123 124 125 123.5 122 122.5 123.5 120.2 124 122.1 122 121 121 118

1.50 2.00 1.66 1.66 0.74 1.75 1.50 2.00 2.50 2.00 2.25 2.50 0.70 2.10 0.70 0.70

80 80 60 60 60 60 60 60 60 60 60 60 92 92 92 92

603 603 226 226 100 226 220 220 220 270 260 220 157 220 101 226

420 420 340 340 340 340 340 340 340 340 340 340 400 400 400 400

57.0 63.0 39.4 41.4 36.2 36.8 35.7 32.2 40.2 38.6 36.9 35.2 45.6 45.9 45.1 54.3

580 705 153 160 92 126 118 126.5 171.5 132.5 144.5 164.5 133 171.2 125.8 179

572 695 160.8 163.1 85.6 124.1 120.3 132.1 180.1 138.4 150.2 162.2 122.3 205.5 129.4 163.6

2.97 4.11 4.06 4.32 3.01 2.60 2.27 2.76 5.10 2.20 2.95 4.77 3.78 4.52 4.61 4.91

0.393 0.517 0.647 0.672 0.501 0.429 0.379 0.486 0.804 0.354 0.485 0.803 0.561 0.668 0.687 0.665

If

Ref.

1.20 1.60 0.99 0.99 0.44 1.05 0.90 1.20 1.50 1.20 1.35 1.50 0.64 1.93 0.64 0.64

[5] [5] [23] [23] [23] [24] [24] [24] [24] [24] [24] [24] [25] [25] [26] [26]

derived from fibrous high-strength concrete in tension and can be analyzed in this paper as being equal to the tension force. This can be carried by fibers and can be calculated as a function of compression stress block and a short-coupling beam width b as follows:

  T h ¼ ð2=3Þðfp Þbðd  a=0:8Þ J t

ð25Þ

The effective force of the diagonal compression strut Dc can be calculated as follows:

Dc ¼ ½ffp abJ c

ð26Þ

where Jt and Jc are the adjustment factors for tension tie and compression strut, respectively. The value of Jt and Jc are considered as 0.80 and 0.70, respectively through this study while, the value of a can be derived elsewhere [22]. The equilibrium of the diagonal compression strut can be rewritten [20] in the form of a vertical shear force Vu as follows: Fig. 11. The proposed design equation with experimental results.

ð27Þ

The ratios of the contribution of diagonal compression strut and horizontal tension tie to carry the vertical shear force Vu can be calculated [22] as follows:

b n As Kfb

V u ¼ Dc sin h þ T h tan h

Dc sin h : T h tan h ¼ Rc : Rt

(d-z) N.A. z

Fig. 12. Proposed analytical model to calculate the effective stiffness of HFC coupling-short beam.

indeterminate load paths are proposed [20] for the macro-mechanical strut and tie model. The development of a compression strut is formed between loading point and the supported. This strut is failed by crushing of concrete while the fibers work as a tension tie. There are three paths for the load, as shown in Fig. 10, vice; the vertical, horizontal and diagonal. In this case, the short-coupling beam transmits the vertical load through diagonal and horizontal directions. The diagonal mechanism compression strut is inclined by angle h while the tension tie is horizontal. The tension force is resisted by fibrous high-strength concrete, in place of diagonal bars. The horizontal mechanism is composed of two flat-struts and a horizontal tie, as shown in Fig. 10. The horizontal tie is resisted by a force Th

ð28Þ

where Rc and Rt are the ratios of the coupling beam shear resisted by diagonal compression and tension tie, respectively [22]. The predicted shear stress in this case can be calculated based on experimental results [5,23–26], as shown in Table 1. Comparing the proposed strut-and tie model numerical results with those of experimental results, showed a good agreement. The fitting curve and equation can be conducted as based on experimental results, as shown in Fig. 11. The adopted empirical design shear strength qsf equation is obtained as a function of fiber reinforcement index If as follows:

 0:047 qffiffiffiffiffi qsf ¼ 0:547 If ffp

If ¼ v f

lf

u

ð29Þ

ð30Þ

The R-squared value of the proposed fitting curve equation is equal to 0.005, which shows a good agreement with experimental results (Fig. 11).

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Fig. 13. Typical flooring plan structure arrangement and concrete dimensions.

Fig. 15. Proposed design response spectrum curve according to IBC code [27].

Fig. 14. Proposed numerical model with adopted finite element meshing.

7. Adopted structure modifiers Owning to the shear wall-coupling beam stiffness, the coupling beams are sensitive to relative movements of their built-in supports. For this reason, the axial deformations of the coupled shear walls, which are responsible for such movements, may have a considerable impact on the overall behavior. Because of the very large differences between the stiffness of the components, and the drastic loss of stiffness in coupling system after diagonal cracking, a

75–100% increase in both the deflections and the wall moments is obtained in case of studies in which allowance is made for cracking [2]. Many assumptions are supposed throughout the analysis, with respect to loss of stiffness caused by cracking in the walls and coupling beam system. The existence of fibers mainly improves the post-cracking tensile performance of concrete and has a little effect on its compressive strength [22]. Every single fiber in the tension zone of the coupling beam may be considered as a little longitudinal rebar of area af. The distribution of fiber in the tension zone of the link beam is random and depends on fiber volume fraction, fiber diameter/length ratio, boundary conditions, mixing and placement method. For these reasons, fibers expected to orient randomly in the three dimensions but, the vibration of high strength concrete

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are particularly clear. The depth of neutral axes z can be determined as follows: 2

bz =2 ¼ nAs ðd  zÞ þ K f bðd  zÞ

ð32Þ

The value of n is the modular ration of reinforcing steel and equal to the modulus of elasticity of reinforcing steel to the modulus of elasticity of concrete Es/Ec. Where Kf is the fiber parameter and can be derived in conjunction with Eq. (31) as:

K f ¼ mbV f

ð33Þ

The value of m is the modular ration of fibers and equal to the modulus of elasticity of fibers to the modulus of elasticity of concrete Ef/ Ec. 8. Application to high-rise buildings The proposed design methodology is applied, in the present work, to a high-rise reinforced concrete residential structure project composed of forty floors (total height of 120 m). The building is designed according to standard code procedures of IBC-2009 [27] and ACI-318-08 [16]. The geometry of typical flooring plan is shown in Fig. 13. The structure is symmetric in X and Y directions, but the difference is located in the core-wall zone due to requirements of architectural openings. The seismic load, applied on the building, is equivalent to 0.20 of gravitational acceleration. The cylinder concrete strength was adopted to be 40 MPa for horizontal element and 56 MPa for vertical element. The building was incorporated into finite element program [28], considering nonlinearity of load displacement analysis. It has been considered as a residential building with applied live load of 2 kN/m2 and superimposed dead load of 1.5 kN/m2, as per ASCE-load provision [29]. The flooring slab system is a post-tensioned flat slab without boundary beams, in order to accelerate the flooring slab construction. The adopted finite element meshing and elements are presented in Fig. 14. The columns represented frame elements while the RC walls and slabs represented shell elements. The building seismic analysis is carried out according to IBC [27] code of design to follow the response spectrum analytical curve as shown in Fig. 15. The seismic load E, where the effect of gravity and seismic ground motion are additive, is as follows:

Fig. 16. Deformed shape due to earthquake in Y direction.

during mixing and placement process make fiber re-orient in the horizontal plan. The final form is intermediate between two and three dimensions. The orientation factor b has different values [22] according to geometric conditions. Thus, the number of effective fibers per unit cross section of the link beam S could be evaluated as:

S ¼ bðV f =af Þ

ð31Þ

Therefore, the area of the fibers per unit area of the section (S.af) is equal to bVf. The proposed analytical model to calculate the effective fibrous section properties of HFC is shown in Fig. 12. The representation of flexural reinforcement and the fibers in tension zone

E ¼ qQ E þ 0:2SDS D

ð34Þ

E ¼ qQ E  0:2SDS D

ð35Þ

where D is the effect of dead load, E is the combined effect of horizontal and vertical earthquake-induced forces, q is redundancy coefficient, QE is the effect of horizontal seismic forces and SDS is the design spectral response acceleration at short periods. The seismic base shear V is determined in accordance with the following equation:

V¼

1:2SDS W R

ð36Þ

Table 2 Results of different finite element models. Model identification

Model description of high-rise building

Time period (S)

Maximum elastic displacement (mm)

Story drift ratio (d=h)

HR–NC–C HR–NC–F HR–NC–S HR–NC–M

Reinforced concrete, without coupling beam Reinforced concrete, with coupling beam-frame element Reinforced concrete, with coupling beam-spandrel element Reinforced concrete, with coupling beam-continuous wall meshing FRC, without coupling beam FRC, with coupling beam-shell element

8.14 6.74 6.34 6.29

144.21 87.44 78.12 75.20

1.55  103 9.20  104 8.54  104 8.30  104

7.85 6.02

137.97 70.42

1.50  103 7.68  104

HR–FC–C HR–FC–S

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The empirical allowable limit for structure maximum displacement dmax can be applied as a function of the total structure height, using the following equation:

dmax 6 H=500

Fig. 17. Maximum elastic story displacement relationship at each story level for different numerical models.

Fig. 18. Maximum elastic story displacement plots for different numerical models.

Fig. 19. Maximum elastic story drift ratio (d/h) level for different models.

where R the response modification is factor and W is the effective seismic weight of the structure, including total dead load and code percentage of live load [27]. The allowable story drift is 2% of floor height. The elastic building displacement due to seismic loading de can then be obtained as a function of inelastic displacement di, using the following equation:

de ¼

0:70di R

ð37Þ

ð38Þ

Six numerical finite element models were performed, using finite element program [28], to study the lateral stability with different finite element configurations. All load combinations were considered according to requirements of IBC code [27]. The coupling beams were numerically represented using three types of finite element representation. The first type is using frame element, the second model is using spandrel element, while the third model represents using shell element. The deformed shape due to seismic load in Y direction is shown in Fig. 16. The results of the six numerical model to compare the behavior of normal and HFC are presented in Table 2. The coupling beam of concrete dimension 1000 mm thickness and 300 widths was designed using diagonal reinforcement 4T25. The span to depth ratio of the coupling beam was less than 2. Other types of coupling beam were designed, using straight reinforcement detailing of 4T25 with HFC with fiber volume fraction (vf) 1% and fiber aspect ratio of (lf/u) 60. The use of HFC increases the structure stiffness Eof, and mechanical properties in tension and mechanical properties in tension and compression. This also simplifies the construction methodology by making total replacement of diagonal main reinforcement bars with straight ones. The strut-and-tie model was used to predict the maximum shear stress for the short-coupling beam case. The structure modifiers were calculated, using the equations proposed in this paper. It has been found that the structure modifiers for HFC short-coupling beam increased by about 27% than in case of non-fibrous shortcoupling beams. Comparing the numerical models results with and without the use of coupling beams, showed that the use of coupling beams applies more uniform structure plasticity over its height. The maximum elastic story displacement relationship extracted from numerical models is shown in Fig. 17. Finite element representation of coupling beam by the frame element produced a much softening model than spandrel and shell element representations. Spandrel and shell element versions are almost producing the same results but, the shell element gains stiffer numericalmodel results due to clamping of walls with shear walls at different points. The reduction in the structure period was about 8% using the shell element, compared to frame element while reduced by 2% than the model with spandrel simulation. The maximum elastic story displacement was also decreased by 14% than the model with shell element representation than frame element, as shown in Fig. 18. The effect of the pair walls would reduce the vertical transverse expansion. The pair walls also provided confinement that enabled the short-coupling beams to support large compressive stresses at their ends. Furthermore, they enabled a more uniform field of diagonal compression and vertical distribution of shear over depth of the member, throughout the entire length of the coupling beam. The effect of using HFC on the behavior of high-rise was also studied by incorporating the fiber properties into the finite element numerical model, HR–FC–S, using a shell element representation. Comparing the two numerical models with and without fibers, it was evident that the structure period was reduced by 5%, while the maximum story displacement was reduced by 6%. The story drift ratio (d/h) plot for HFC and normal reinforced concrete short-coupling beams are shown in Fig. 19. On addition of 1% fiber volume, the story drift ratio was reduced by 8% than high-rise without fibers. It can be concluded from Fig. 19 that the drift curve is pronouncedly more stable and represents the curve of elastic– plastic ductile material. This means that the fiber produces ductile plasticity in the high-rise structure. Strength degradation occurs

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only when a buckling of compression bars commences. Lateral stiffness and strength of the wall is therefore significantly influenced by the design of coupling beams to spread plasticity over the system height. The design is thus performed to retain strength together with a significant percentage of stiffness during large deformations into the plastic range of behavior. These fibers provide a crack bridging the interference plan, between shear walls and coupling beams, and enhance the concrete to be sustainable during seismic load cycles-degradation of the concrete fiber composite. This would provide a more stable and ductile story drift behavior.

9. Conclusion In this paper, for nonlinear analysis of HFC structures, analytical models are developed to generate full analysis of HFC under uniaxial compression, and tension, as well as prediction of shear strength, using the strut-and-tie model. The developed analytical results of HFC were incorporated into a finite element program of forty floor high-rise building. The finite element analysis took into consideration the load–displacement nonlinearities, and the following concluding points could be revealed: 1. An analytical relationship is performed for predicting the complete stress–strain curve of HFC under uniaxial compression. It accounts for composite inelasticity and ductility, compressive strain softening, and effect of randomly distributed and oriented steel fibers parameters. The model can also determine stress degradation, at any drift deformation level, during making restraining action for the coupling beam by RC walls during the load cycles. 2. A strain softening elasticity model is developed on the basis of composite fracture concepts of smearing cracks discontinuities into an energy equivalent continuum. This model can be applied through the finite element numerical model. The material properties are modified to account for the localized damage due to HFC cracking. The post-cracking HFC response was found to be highly sensitive to variations in fiber length lf and characteristic element length Lch. Thus, it can be concluded that the ductility of HFC increases with the increase of fiber length. This could be attributed to fibers pullout resistance and critical fracture energy increase with the increase of embedment length of fibers. The combined effect of lf and Lch can be better considered on using the proposed maximum cracking strain value as lf/2Lch. 3. The use of the deep beam design method in the ACI provisions leads to very conservative shear designs for coupling beams. The ACI shear stress limitation for deep beams, which is a function of the span-depth ratio and imposed to guard against a diagonal compression failure, unnecessarily limits the load-carrying capacity of stocky link beams supported over their heights. The strut-and-tie model used in the ACI code permits the design of reinforced concrete link beams with substantially higher loads than ordinary design method. A wide variation was found in the maximum shear stress limit between the ACI, Canadian and AASHTO–LRFD codes. 4. Due to shear-sliding force, at the interface plan of a shear wall and a coupling beam, is resisted by shear friction. This model presents stable performance of strut-and-tie model and sustainable for large deformation due to high ductility of HFC. When sliding cracks developed the fibers are crossing the interface failure plan between the wall face and the coupling beam. The fibers bridging the cracks enable stable crack growth. The model is also considering high compression ductility due to passive confinement of fibers. Wide variation of the maximum shear stress found in design codes. For this reason, based on

experimental results, a proposed strut-and-tie model is developed to estimate the contribution of fiber composite in shear resistance of short-coupling beams. A comparison between the experimental results with the present adopted analytical model showed a good agreement. 5. The structure modifiers can be also calculated, using the equations proposed, in this paper. It is revealed that the structure modifiers for HFC short-coupling beam are increased by about 27% than with non-fibrous short coupling beams. 6. In the finite element numerical models proposed herein, spandrel and shell element versions are almost producing the same results but, the shell element gains stiffer numerical model results due to clamping of walls at different points with shear walls. The period of the structure is reduced by about 8% by using shell element than frame element while reduced by 2% than the model with spandrel simulation. Also, the maximum elastic story displacement decreased by 14% of the model with shell element representation than frame element 7. The effect of using HFC on the behavior of high-rise buildings can be also studied by incorporating the fiber properties into the finite element numerical model, HR–FC–S, using shell element representation. Two numerical models with and without fiber incorporated. It conducted that the period of the structure reduced by 5% while the maximum story displacement reduced by 6%. The use of fiber volume 1% a reduction in the story drift ratio by about 8% is adopted.

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