Three-Dimensional Nonlinear Finite Element Analysis of Reinforced Concrete Horizontally Curved Deep Beams

Three-Dimensional Nonlinear Finite Element Analysis of Reinforced Concrete Horizontally Curved Deep Beams Ammar Y Ali University of Babylon Ali N Attiyah University of Kufa Haider A A Al-Tameemi University of Kufa Abstract This research deals with the analysis of reinforced concrete horizontally curved deep beams, loaded transversely to its plane, using a three-dimensional nonlinear finite element model in the pre and post cracking levels and up to the ultimate load The 20-node isoparametric brick element with sixty degrees of freedom is employed to model the concrete, while the reinforcing bars are modeled as axial members embedded within the concrete brick element Perfect bond between the concrete and the reinforcing bars is assumed The behavior of concrete in compression is simulated by an elasto-plastic work hardening model followed by a perfect plastic response, which is terminated at the onset of crushing In tension, a fixed smeared crack model has been used with a tension-stiffening model to represent the retained post-cracking tensile stress Also, a shear retention model that modifies the shear modulus after cracking is used Numerical study is carried out to investigate some of effects on the behavior of reinforced concrete horizontally curved beams such as the shear length to effective depth ratio (a/d) on the ultimate load resisted by curved beams and the effect of the central subtended angle, boundary conditions, amount of transverse reinforcement, and using additional longitudinal bars (horizontal shear reinforcement) on the behavior of reinforced concrete horizontally curved beams with different shear length to effective depth ratios (a/d) Keywords :Finite Element; Nonlinear Analysis; Reinforced Concrete; Curved Beam; Deep Beam ‫الخلصاة‬ ‫تتناول هذه الدراسة تحليل العتبات الخرسانية المسلحة العميقة المقوسة أفقيا تحت تأثير أحمممال‬ ‫مسلطة بصورة عمودية على مستوى تقوسها باستخدام نموذجا ل خطيا ا ثلثي البعاد للعناصر المحددة فممي‬ ‫ تم استخدام العنصر الطابوقي ذي العشممرين‬ ‫مراحل التحميل قبل وبعد التشقق و إلى حد الحمل القصى‬ ‫عقدة مع ستون درجة حرية لتمثيل الخرسانة أما حديد التسليح فقد مثل بعناصممر محوريممة مطمممورة داخممل‬ ‫ اعتبر تصرف الخرسممانة فممي‬.‫العناصر الطابوقية مع افتراض وجود ترابط تام بين الخرسانة وحديد التسليح‬ ‫ أما لتمثيل سمملوك الخرسممانة‬.‫ًلدنا ا يتبعه تصرفا ا لدنا ا تاما ا ينتهي عند تهشم الخرسانة‬-‫النضغاط تصرفا ا مرناا‬ (Fixed Smeared Crack Model)‫تحت تأثيرات اجهادات الشد فقد تم تبني نموذج التشقق المنتشر الثابت‬ ‫( لحساب اجهادات الشد المتبقي بعممد حممدوث‬Tension Stiffening Model) ‫واستعمل النموذج تصلب الشد‬ ‫(مم والذي يقوم بتخفيممض قيمممة معامممل‬Shear Retention Model) ‫التشقق وتم تبني أنموذج احتباس القص‬ ‫القص المتبقي مع استمرار التحميل في مرحلة ما بعد التشقق‬ ‫تم إجراء دراسة تحليلية على العتبات الخرسانية المسلحة المقوسة أفقيا لدراسة تأثير تغير نسبة‬ ‫ و كذلك تم دراسة تممأثير‬.‫( على التصرف والحمل القصى لتلك العتبات‬a/d) ‫فضاء القص إلى العمق الفعال‬ ‫ بالضإممافة إلممى اسممتعمال قضممبان حديممد‬,‫ كمية حديممد التسممليح المسممتعرض‬,‫ طريقة السناد‬,‫زاوية التقوس‬ ‫تسليح طولية إضإافية لتتصرف كتسليح قص أفقي على العتبات الخرسانية المسلحة المقوسة أفقيا لنسممب‬ ‫( مختلفة‬a/d) Introduction Reinforced concrete horizontally curved beams are extensively used in many fields, such as in the construction of modern highway intersections, elevated freeways, the rounded corners of buildings, circular balconies,….etc In some of these cases, large depths are needed for curved beams in order to resist high loads or to fulfill some aesthetic purposes The analytical analysis of such members is very complex due to the fact that those members are subjected to combined action of bending, shear and torsion Furthermore, non-homogeneous nature of the materials involved contributes to the complexity of the problem Therefore, it becomes necessary to employ numerical analysis procedures, such as the finite element method, to satisfy the safety and the economy requirements A horizontally curved beam, loaded transversely to its plane, is subjected to torsion in addition to bending and shear Furthermore, in deep beam the plane section does not remain plane after bending because of high stresses and warping occurs Therefore, special features of analysis and design for horizontally curved deep beams is necessary to include the effect of above mentioned factors Several methods of collapse analysis (Khalifa 1972, Jordaan et al 1974, Badawy et al 1977, Hsu et al 1978, and Abul Mansur and Rangan 1981 ) were proposed for analysis of specific cases of reinforced concrete curved beams However, till yet studies concerning reinforced concrete horizontally curved deep beams are rare At present, with the application of digital computers beside the development of numerical methods, the mathematical difficulties associated with curved deep beam have been largely overcome One of the most effective numerical methods utilized for analyzing reinforced concrete members is the finite element method Using this method, many aspects of the phenomenological behavior of reinforced concrete structures can be modeled rationally These aspects include the tension-stiffening, non-linear multiaxial material properties, modeling of cracking and crushing, and many other properties related to the behavior of reinforced concrete members under stresses An important utilization of the finite element method is the modeling of the degradation of concrete compressive strength in the presence of transverse tensile straining as happens in members subjected dominantly to torsion or shear stresses Therefore, the present study adopted a three dimensional non-linear finite element model to investigate the behavior and the load carrying capacity of reinforced concrete horizontally curved deep beams 2.Finite Element Model The 20-node isoparametric brick element shown in Fig.1 is used in the current study to model the concrete Each node of this element has three degrees of freedom (u, v, and w) in the (x, y, and z) directions, respectively The isoparametric definition of the brick element is(Al- Shaarbaf, 1990): 20 20 20 i 1 i 1 i 1 u  , ,    N i  , ,  ui , v , ,    N i  , ,   vi , w , ,    N i  , ,   wi (1) where Ni (ξ, η, ζ) is the shape function at the i-th node and ui, vi, wi are the corresponding nodal displacements The shape functions for the 20 node brick element which are adopted to map the element are given in Table The Gauss-Legender quadrature numerical integration scheme has been found to be accurate and a convenient technique to carry out the finite element analysis The integration rule, which has been used in this study , is the 15-point rule The weights and abscissa of the sampling points are listed in Table The relative distribution of the Gaussion points over the element is given in Fig Table(1)Shape functions of the quadratic 20-node brick element (Cook,1974, Carlos, 2004) Location Corner nodes Mid-side nodes Mid-side nodes Mid-side nodes ξ η ζ Ni (ξ, η, ζ) ±1 ±1 ±1 (1+ ξ ξi)(1+ η ηi)(1+ ζ ζ i) (ξ ξi+ η ηi + ζ ζi-2 )/8 ±1 ±1 (1- ξ2)(1+ η ηi )(1+ ζ ζi ) /4 ±1 ±1 (1- η2 )(1+ ξ ξi)(1+ ζ ζi )/4 ±1 ±1 (1- ζ2 )(1+ ξ ξi)(1+ η ηi )/4 Figure (1) 20-node brick element Table (2) Weights and abscissa of sampling points (Al- Shaarbaf, 1990) Integration rule 15a-point rule Sampling point number 2,3 4,5 6,7 8-15 Natural coordinates    Weight 0.0 1.0 0.0 0.0 0.6714 0.0 0.0 0.0 1.0 0.6714 0.0 0.0 1.0 0.0 0.6714 1.56444 0.35556 0.35556 0.35556  Figure(2)Distribution of sampling points (Al- Shaarbaf, 1990) Modeling Of Material Properties The material model used in the present work is suitable for the three-dimensional nonlinear analysis of reinforced concrete structures under monotonically increasing load The behavior of concrete in compression is presented by an elastic-plastic work hardening model followed by a perfectly plastic response, which is terminated at the initiation of crushing The growth of subsequent loading surfaces is described by an isotropic hardening rule A parabolic equivalent uniaxial stress-strain curve shown in Fig.3 has been used to represent work hardening stage of behavior and the plastic straining is controlled by an associated flow rule A yield criterion suitable for analyzing reinforced concrete members has been used This criterion was used successfully can be expressed as(Al- Shaarbaf, 1990): f   cI    cI1   3 J2  12  (2) Where c and β are material parameters to be determined by fitting biaxial test results.Using the uniaxial compression test and the biaxial test under equal compressive stresses I1 and J2 are the first stress and second deviatoric stress invariants and σ0 is the equivalent effective stress taken from uniaxial tests In tension, linear elastic behavior is assumed to occur prior to cracking Crack initiation is controlled by a maximum tensile stress criterion A smeared crack model with fixed orthogonal cracks has been adopted to represent the behavior of cracked sampling points The retained post-cracking tensile stress and the reduced shear modulus are calculated according to Fig.4 and Fig.5 respectively Details of the plasticity based model in compression and the smeared crack model in tension can be found elsewhere(Al-Tameemi, 2005) Figure(3) Uniaxial stress-strain curve for concrete(Al- Shaarbaf, 1990)   1.0 2  3     1  n   3 1    cr  2 3 n Figure (4) Post-cracking model for concrete (Al- Shaarbaf, 1990)  cr   cr Fig.(5) Shear retention model for concrete (Al- Shaarbaf, 1990( Analysis Of Reinforced Concrete Horizontally Curved Beams In this section, reinforced concrete horizontally curved beams subjected to single load have been analyzed using the finite element technique and the models discussed in the pervious sections The computer program 3DNFEA (3-Dimensional Nonlinear Finite Element Analysis) has been used in the present study This program has been originally developed by Al-Shaarbaf (Al- Shaarbaf,1990) The analytical results are compared with the available experimental results on load-deflection curves In the following sections a description of the concrete horizontally curved beams and the validity of the finite element analysis are presented 4.1 Jordaan et al (1974) Reinforced Concrete Horizontally Curved Beam In this study, a reinforced concrete horizontally curved beam subjected to single point load tested by (Jordaan et al., 1974) was selected for the analysis using the present computer program The geometry and loading conditions for this beam are shown in Fig The beam was fully fixed at the two supports Fig 10 shows the cross section details The total length of the beam was considered in the finite element analysis The beam was modeled using 20-quadratic brick elements mesh The finite element meshes used, boundary conditions and loading arrangement are shown in Fig The external force was modeled as line loads distributed across the width of the beam The material properties adopted in the analysis are given in Table Table (3) Material properties used in the analysis of Jordaan et al (1974) curved beam Concrete Young's modulus, Ec (MPa) Compressive strength ( * f c' ) (MPa) Tensile strength ( f t )(MPa) Poisson's ratio (  ) Steel 29725 Young's modulus, Es (MPa) 40 0.2 200000 Longitudinal Bars Yield stress (MPa) 384 Diameter of longitudinal bars (mm) 22 Stirrups * E c 4700  f c' Yield stress (MPa) 240 Diameter of stirrups bars (mm) 6.35 all dimensions in mm 22Figure(6) Ø Jordaan et al.(1974) reinforced concrete horizontally curved beam, dimensions and loading 1@ Ø mm 22 Ø2 mm251 mm503 229 Figure(7) Cross section details of Jordaan et al.(1974) reinforced concrete horizontally curved beam all dimensions in mm Figure(8) Finite element idealization of Jordaan et al.(1974) curved beam 4.1.1 Results of The Analysis The experimental and numerical load-deflection curves obtained for curved beam tested by (Jordaan et al.,1974) is shown in Fig This figure generally reveal that the finite element solution is in good agreement with the experimental results throughout the entire range of behavior It can be noted that, the behavior was relatively more brittle compared with the experimental results On the other hand, it was observed that the numerical ultimate load was lower than the experimental ultimate load by (1%) However, the computed failure load is very close to the corresponding experimental ultimate load 01@ 1.7 Ø Figure( 9) Experimental and numerical load-deflection curves of Jordaan et al (1974) curved beam 4.2 Badawy et al.(1977) Reinforced Concrete Horizontally Curved Beam In this study, a reinforced concrete horizontally curved beam subjected to single point load tested by (Badawy et al., 1977)was selected for the analysis using the present computer program The geometry and loading conditions for this beam are shown in Fig 10 The beam was completely fixed at one end, while the flexural and torsional fixity were removed at the other end (simple support) Figure 11 shows the cross section details The total length of the beam was considered in the finite element analysis The beam was modeled using 20quadratic brick elements mesh The finite element meshes used, boundary conditions and loading arrangement are shown in Fig 12 The external force was modeled as line loads distributed across the width of the beam The material properties adopted in the analysis are given in Table 91 Ø all dimensions in mm Figure(10) Badawy et al.(1977) reinforced concrete horizontally curved beam, dimensions, boundary conditions and loading 229m503 m mm 91 Ø2 mm251 Figure(11) Cross section details of Badawy et al.(1977) reinforced concrete horizontally curved beam all dimensions in mm Figure(12) Finite element idealization of Badawy et al (1977)curved beam Table (4) Material properties used in the analysis of Badawy et al (1977)curved beam Concrete Young's modulus, Ec (MPa)* Compressive strength ( f c' ) (MPa) Tensile strength ( f t )(MPa) Poisson's ratio (  ) * E c 4700  f c' Steel 25743 30.04 Young's modulus, Es (MPa) Longitudinal Bars 2.7 Yield stress (MPa) 475 0.2 Diameter of longitudinal bars (mm) 19 Stirrups Yield stress (MPa) Diameter of stirrups bars (mm) 4.2.1 Results of The Analysis 200000 300.3 7.1 The experimental and numerical load-deflection curves obtained for curved beam tested by Badawy et al.(1977) is shown in Fig 13 This figure generally reveal that the finite element solution gives good agreement with the experimental results throughout the entire range of behavior The computed failure load is close to the corresponding experimental ultimate load Furthermore, experimental results showed more ductile behavior of the tested beam than the numerical behavior A relatively stiffener response has been obtained in the initial cracking stage of behavior for curved beam On the other hand, it was observed that the numerical ultimate load was lower than the experimental ultimate load by (4%) Figure(13) Experimental and numerical load-deflection curves of Badawy et al.(1977)curved beam Numerical Study Of Reinforced Concrete Horizontally Curved Deep Beams This section illustrates a numerical study that was carried out on reinforced concrete horizontally curved beams with different depths to investigate the effect of some important parameters on the load-deflection response of curved beams and the ultimate load resisted by those beams The parameters included in this study were the total depth of the beam, subtended angle, boundary conditions, amount of transverse steel reinforcement ,use additional longitudinal bars, besides change the location of load The reinforced concrete horizontally curved beam tested by (Jordaan et al., 1974), subjected to single point load was adopted in this numerical study 5.1 The Influence of the Depth of the Beam The effect of increasing the total depth (h) on the load-deflection response and the ultimate load was investigated In this section the total depth (h) was increased from (305 mm) to (400 mm), (500 mm), (600 mm), (700 mm), and (750 mm) The result of this study leads to the conclusion that increasing the total depth has a significant rule on load-deflection and ultimate load of curved beams This effect of increasing the total depth becomes more significant when the total depth exceeds 600 mm Table shows the results of the ultimate load for different total depths with their ratios of the shear length (length of curved segment of beam)to the effective depth (a/d) Calculation of shear length for curved beams is shown in Fig 14 Table (5) Effect of increasing depth (h) on the ultimate load Total depth (h) (mm) a/d 305 4.3 400 500 600 700 750 3.21 2.51 1.75 1.628 Ultimate load (kN) 159 252 335 425 570 610 % of increase in ultimate load 58 112 167 258 284 Figure(14) Shows the calculation of shear length (a) Fig 15 shows the influence of total length (h) increasing on the loaddeflection response for the curved beams This figure reveals that both initial and post cracking stiffness and the ultimate load are significantly increased as the total depth increased This can be attributed to the fact that when the total depth is increased, the internal lever arm between the compression force in the concrete and the tensile force in tension reinforcement is significantly increased Also , the capacity of the curved beam cross section in shear and torsion is increased as the area enclosed by the centerline of stirrups legs increases Fig 16 shows the influence of the ratio of (a/d) on the ultimate load of curved beam It can be concluded according to this figure that the ultimate load resisted by curved beams increases with decreases (a/d) For values of (a/d) lower than two (h >600 mm), the ultimate load increasing at a sharp slope with decreasing (a/d) This can be attributed to the effect of arch action on the behavior of the reinforced concrete curved beams Consequently, curved beam with (a/d) ratio less than two can be considered as a deep beam Figure(15) Effect of depth (h) on load-deflection Figure(16) Influence of (a/d) ratio on the ultimate load behavior of curved beams 5.2 Effect of the Central Subtended Angle This section studies the effect of increasing or decreasing the central subtended angle of curved beams on the load-deflection response and the ultimate load of curved beams for different (a/d) ratios Numerical tests were carried out on three curved beams with (a/d) ratio of (4.36), (2.51) and (1.75) In these tests the central subtended angle was increased from (86 degree) to (120 degree), and decreased to (60 degree) Different central subtended angles were achieved by varying the radius of the curved beams, since length of those beams were kept constant Table shows the percentage of increase or decrease in the ultimate load sustained by curved beams due to changing of central subtended angle (curvature) for different (a/d) ratios Fig 17 shows the influence of increasing and decreasing the subtended angle on the load-deflection response of curved beams for different (a/d) ratios These results reveal that as the central subtended angle is decreased, the post cracking stiffness for curved beams is significantly increased This can be attributed to the fact that when the central subtended angle (or the curvature) is decreased, the internal torsion acting on the beam cross section is decreased Also, it can be noted that the effect of (a/d) ratio is clear in the case of the central angle is 120 degree, whereas the internal torsion is dominate It can be concluded that the effect of internal torsion is degrading with the decrease of the (a/d) ratio These effects on the ultimate load, may be due to the increase of torsional capacity of a cross section with increasing the area enclosed by the centerline of stirrups legs Table (6) Effect of changing central subtended angle on the ultimate load with different (a/d) ratios a/d =4.36 Subtended angle (degree) Curvature 1/R* a/d =1.75 Ultimate load (kN) % of change in ultimate load Ultimate load (kN) % of change in ultimate load Ultimate load (kN) % of change in ultimate load +9.4 347 +3.58 638 11.93 335 570 290 -13.4 543.4 -4.67 (m-1) 60 0.3157 174 86 0.4524 159 120 0.6314 120 200 a/d =2.51 -24.5 * R : radius of the curved beam 400 180 350 160 300 120 Load (kN) Load (kN) 140 100 80 60 40 20 10 150 Angle=60 degree Angle=86 degree Angle=120 degree 100 Angle=86 degree 50 Angle=120 degree 15 20 Deflection under load (m m ) A (a/d=4.36) 200 Angle=60 degree 0 250 25 0 10 15 Deflection under load(m m ) B (a/d=2.51) 20 700 600 Load (kN) 500 400 300 Angle=60 degree 200 Angle=86 degree 100 Angle=120 degree 0 10 15 20 Deflection under load (mm) C (a/d=1.75) Figure(17) Effect of changing central subtended angle on behavior of curved beams with different (a/d) ratios Influence of the Transverse Reinforcement 5.3 In order to study the effect of the amount of the transverse reinforcement on the loaddeflection response and the ultimate load of curved beams, numerical tests were carried out on the curved beams with different (a/d) ratios The (a/d) ratios were (4.36), (2.51) and (1.75) In these tests the stirrup's diameter is increased from (6.35 mm) to (10 mm) and (12 mm) Fig 18 shows the effect of the variation of the amount of transverse reinforcement on the load-deflection response and ultimate load of curved beams with different (a/d) ratios This result reveals that the ultimate load is increased with increase in the transverse steel amount Table shows the percentage of increase in the ultimate load for different (a/d) ratios The table indicated that the effect of increase in the amount of transverse reinforcement on the ultimate load is lower in the case of ratio (a/d=1.75) The decrease in the effect of transverse reinforcement on the ultimate load can be attributed to the fact that the diagonal cracks in beams are more steeper in slope for this case Therefore, the share of vertical steel (stirrups) is decreased Table(7) Effect of amount of transverse reinforcement on the ultimate load with different (a/d) ratios a/d =4.36 Diameter of the transverse reinforcement (mm) Ultimate load (kN) 6.35 159 10 164 12 % of increasing of ultimate load 3.14 166 4.4 % of increasing of ultimate load Ultimate load (kN) % of increasing of ultimate load 335 570 355 590 3.5 9.1 600 5.2 365 160 350 140 300 120 250 100 Exp Diam eter=6.35 m m 80 60 Diam eter=6.35 m m 40 Diam eter=10 m m 20 Diam eter=12 m m a/d =1.75 Ultimate load (kN) 400 Load (kN) Load (kN) 180 a/d =2.51 200 Diam eter=6.35 m m 150 Diam eter=10 m m 100 Diam eter=12 m m 50 0 10 15 20 Deflection under load (m m ) 25 30 10 15 Deflection under load (m m ) 20 A (a/d=4.36) B (a/d=2.51) C (a/d=1.75) Figure(18) Effect of amount transverse reinforcement on load-deflection behavior of curved beams with different (a/d) ratios 5.4 Influence of the longitudinal Reinforcement In order to investigate the effect of the amount and the distribution of longitudinal reinforcement on the load-deflection response and the ultimate load numerical tests carried out on the curved beams with different (a/d) ratios The (a/d) ratios were (4.36), (2.51) and (1.75) In these tests, additional longitudinal bars were used along the curved beam Two bars diameter (12 mm) at the center of vertical legs of stirrups are used in one case, and four bar diameter (12 mm) at the third points of the vertical legs of stirrups were used in the other case, as shown in Fig 19 Fig 20 shows the influence of using additional longitudinal bars along the curved beams on the load-deflection response and ultimate load for different (a/d) ratios Table shows the percentage of increase in the ultimate load for different (a/d) ratios These results reveal that using additional longitudinal bars leads to increasing the ultimate load for all (a/d) ratios It can be noticed that the effect of using additional longitudinal bars is augment with decreasing of (a/d) ratios This can be attributed to the fact that the additional bars operate as a horizontal shear reinforcement to resist the shear forces produced due to the increase in sharpness of diagonal cracks slope which increase with decreasing of (a/d) ratio Figure(19) shows cases of the additional bars (diameter 12 mm) Table (8) Effect of using additional longitudinal bars on the ultimate load of curved beams with different (a/d) ratios a/d =4.36 State of using additional longitudinal bars Ultimate load (kN) Without 159 Using two bars diameter=12 mm 162 Using four bars diameter=12 mm 165 a/d =2.51 % of increasing in ultimate load a/d =1.75 Ultimate load (kN) % of change in ultimate load Ultimate load (kN) % of increasing in ultimate load 335 570 1.88 345 606 6.3 3.77 355 630 10.5 180 160 140 Load (kN) 120 100 80 Exp Without 60 With out 40 2-diam eter=12 m m 20 4-diam eter=12 m m 0 10 15 20 25 Deflection under load (m m ) 30 A (a/d=4.36) 400 700 350 600 500 250 Load (kN) Load (kN) 300 200 150 50 300 Without 200 Without 2-diam eter=12 m m 4-diam eter=12 m m 100 400 2-diam eter=12 m m 100 4-diam eter=12 m m 0 10 15 20 Deflection under load (m m ) 25 10 15 Deflection under load (m m ) 20 B (a/d=2.51) (a/d=1.75) C Figure(20) Effect of using additional longitudinal bars on the load-deflection behavior for curved beams with different (a/d) ratios 5.5 The Influence of the Location of Load The effect of changing the location of load (change the central angle limited between the support and the position of load) on the load-deflection response and the ultimate load was investigated In this section, this angle was decreased from (43 degree at the center of curved beam) to (30 degree), (15 degree) , and (7.5 degree) The result of this study leads to the conclusion that decreasing the angle of load has a significant rule on load-deflection and ultimate load of curved beams Table shows the results of the ultimate load for different angle of load with their ratios of the shear length (length of curved segment of beam)to the effective depth (a/d) Calculation of shear length for curved beams is shown in Fig 14 Fig 21 shows the influence of angle of load increasing on the load-deflection response for the curved beams This figure reveals that both initial and post cracking stiffness and the ultimate load are significantly increased as the angle of load decreased This can be attributed to the fact that when the angle of load is decreased, the arm of bending and torsion moments decreased Fig 22 shows the influence of the ratio of (a/d) on the ultimate load of curved beam It can be concluded according to this figure that the ultimate load resisted by curved beams increases with decreases (a/d) For values of (a/d) lower than two the ultimate load increasing at a sharp slope with decreasing (a/d) This can be attributed to the effect of arch action on the behavior of the reinforced concrete curved beams Consequently, curved beam with (a/d) ratio less than two can be considered as a deep beam Table (9) Effect of decreasing angle of load on the ultimate load Angle of load At 7.5 degree (degree) 300 43 At 15 degree a/d 250 At 30 degree Ultimate load (kN) Load (kN) 200 300 (At center) 15 7.5 6.25 250 4.36 2.18 1.1 155 200 159 197 280 150 2.6 27 80.6 At 43 degree (Center) % of increase in ultimate load 150 30 100 100 50 50 0 10 15 20 Deflection under load (m m ) 25 (a/d) ratio Fig.(22) Influence of (a/d) ratio on the Fig.(21) Effect of angle of load on load- ultimate load due to change the angle of load deflection behavior of curved beams Conclusions Based on the finite element analysis curried out in the present research on the behavior of reinforced concrete horizontally curved deep beams, the following conclusions can be drawn: The three-dimensional nonlinear finite element model, adopted in the present work, is suitable to predict the behavior of the reinforced concrete curved deep beams Application of this model is beneficial in spite of the difficulties arising due to the difference in representation of data between the polar coordinate system and Cartesian coordinate system The numerical results are in good agreement with available experimental load-deflection results throughout the entire range of behavior The ultimate load resisted by curved beams increased as the shear length to effective depth ratio (a/d) decreased This increase of ultimate load becomes more effective when the shear length to effective depth ratio (a/d) was lower than two Consequently, curved beam with (a/d) ratio less than two can be considered as a deep beams Varying the central subtended angle (curvature) of curved beams, while the length of these beams is kept constant, affects the ultimate load significantly Decreasing the central subtended angle is found to cause an increase the ultimate load resisted by curved beams There is no clear effect for changing the (a/d) ratios on the increase of ultimate load due to decrease of curvature The effect of internal torsion, acting in the cross section of curved beams, on the ultimate load decreased as the (a/d) ratio decreased The ultimate load resisted by curved beams decreases due to releasing the torsional restraint at one curved beam ends by 22% for (a/d= 4.36 ), while the decrease is 12% for (a/d= 1.75 ) The increase in the amount of transverse reinforcement causes a corresponding increase in the ultimate load resisted by curved beams The effect of the amount of transverse reinforcement on the ultimate load have been reduced when the (a/d) ratio is less than two The increase in the ultimate load resisted by curved beams, due to varying the stirrup's diameter from 6.35 mm to 12 mm, decreases from 9.1% for (a/d= 2.51 ) to 5.2% for (a/d= 1.75 ) Using additional longitudinal bars as a horizontal shear reinforcement leads to increase in the ultimate load of curved beams This effect of using additional longitudinal bars is increase with decreasing of (a/d) ratio The increase in the ultimate load resisted by curved beams, due to using four additional longitudinal bars(diameter 12 mm), increases from 3.77% for (a/d= 4.36 ) to 10.5% for (a/d= 1.75 ) References Al- Shaarbaf, I.A.S., 1990, "Three-Dimensional Non-Linear Finite Element Analysis of Reinforced Concrete Beams in Torsion", PH D Thesis University of Bradford, U.K Al- Tameemi, H.A.A., 2005, "Three-Dimensional Non-Linear Finite Element Analysis of Reinforced Concrete Horizotally Curved Deep Beams", M.Sc Thesis University of Kufa Al-Mahiadi R.S.H., 1979, "Non-linear Finite Element Analysis of Reinforced Concrete Deep Beams", Report No 79, Dept of Structure Engineering, Cornell University Badawy, H.E.I, Jordaan, I.J., and McMullen, A.E., 1977 "Effect of Shear on Collapse of Curved Beams", Journal of the Structural Division, ASCE, Vol 103, No ST9, Proc Paper 13185, September, pp 1849-1866 Carlos A.F., 2004, "Introduction To Finite Element Methods", Department of Aerospace Engineering Sciencesand Center for Aerospace StructuresUniversity of Colorado, U.S.A Cervenka, V., 1985, "Constitutive Model for Cracked Reinforced Concrete", ACI Journal, Vol 82, No 6, November-December, pp 877-882 Chen, 1982, W.F., and Saleeb, A.F., "Constitutive Equations for Engineering Materials: Elasticity and Modeling", Vol.1, John Wiley and Sons, New York, U.S.A Cook, R.D.,1974, "Concept and Application of Finite Element Analysis", John Wiley and Sons, Inc., New York, U.S.A Hsu, T.T.C., Inan, M., and Fonticiella, L., 1978, "Behavior of Reinforced Concrete Horizontally Curved Beams", ACI Journal, Vol 75, No 4, April, pp 112-123 Jordaan, I.J., Khalifa, M.M.A., 1974, and McMullen, A E., "Collapse of Curved Reinforced Concrete Beams", Proceedings, ASCE, Vol 100, ST11, November, pp 2255-2269 Khalifa, M.M.A., "Collapse of Reinforced Concrete Beams Curved in Plan", thesis presented to the University of Calgary, at Calgary, Alberta, Canada, in 1972, in partial fulfillment of requirements for the degree of Master of Science Mansur, M A., and Rangan, B V., 1981, "Study of Design Methods for Reinforced Concrete Curved Beams", ACI Journal, Vol 78 No.3, May-July, pp 226-231 Naville, A M and Brooks, J J., 1987, "Concrete Technology", Longman scientific and Technical, U.K Notations The following symbols are used in this paper Scalar a Shear length of curved beam b Width of section Cp Plasticity coefficient d Distance from extreme compression fiber of concrete to centroid of tensile reinforcement d Plastic multiplier E Modulus of elasticity Ec Modulus of elasticity of concrete Es Modulus of elasticity of steel f Function fc ’ Uniaxial compressive strength of concrete (cylinder test) G Shear modulus h Height of section I1 First stress invariant I'1 First strain invariant J Jacobian J2 Second deviator stress invariant J'2 Second deviator strain invariant V Volume W Weight of a sampling point I, m, n Direction cosine of principal stresses u , v, w Displacement components, in x , y and z Cartesian coordinates x,y,z Cartesian coordinates o Ratio of plastic torsional capacity to plastic bending capacity for cross section 1 , 2 Tension-stiffening parameters  Shear retention factor  Shear strain  Shear retention parameters  Strain cu Ultimate strain p Effective plastic strain o Strain corresponding to peak uniaxial compressive stress 'o Total strain corresponding to the parabolic part of uniaxial compressive stress-strain curve  Stress o Effective stress at onset of plastic deformation ' Effective stress  Shear stress  Curvilinear coordinate set ... Non-Linear Finite Element Analysis of Reinforced Concrete Horizotally Curved Deep Beams" , M.Sc Thesis University of Kufa Al-Mahiadi R.S.H., 1979, "Non-linear Finite Element Analysis of Reinforced Concrete. .. a description of the concrete horizontally curved beams and the validity of the finite element analysis are presented 4.1 Jordaan et al (1974) Reinforced Concrete Horizontally Curved Beam In... angle of load deflection behavior of curved beams Conclusions Based on the finite element analysis curried out in the present research on the behavior of reinforced concrete horizontally curved deep