Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 108 ADVANCED MODEL FOR THE EFFECTIVE MOMENT OF INERTIA TAKING INTO ACCOUNT SHEAR DEFORMATIONS EFFECT Dr.Haider K. Ammash in Civil Engineering, University of Al-Qadisiya Mr. Muthana H. Muhaisin in Civil Engineering, University of Al-Qadisiya Abstract The study aims to present anew form of the effective moment of inertia by enhancement Branson's model to take the effect of several factors such load type(concentrated, uniformly distribution, and two points) loads, shear deformations affect are also considered. These deformations depend on the span to depth ratio. The results of the presented model were compared with (experimental results, Branson's model results, and results of other models). The results of the present model give best agreement with experimental results than Branson's and the other models; the results showed that the effective moment of inertia reduced by about 27% for span to depth ratio of (20 to 5) due to shear deformation effects. Keywords: beams, reinforced concrete, deflection, effective moment of inertia, shear deformation. ��ر ����� ا� � ه�ت ا������� ذج ��م ���م ا� �� ر ا�"ا�' ا�&��ل $�#" ! �� ا�� 8� �� 8 �� 9 :� 2� ا��2; =�> ا�3 � ا���د������ 2ر آ�?> ��1ش.د� 2� ا��2; =�> ا�3 ����� ا���د ا�*()� �ر ��ة ��ا ' =,�ف ا��را�Y إ�� =�E �I�J d$�P$�ة � Aم ا��4Pر ا�&ا=� ا�) �ل R��5�/ ���ذج 5�ا��Rن < ��Cا �025 &- G' ��ع ا���' )Aآ� �زع �0���5م ، ، /��QP� � Aآ� ( � Rfو� Z�Rfا� �fء إ��gf(ا� � Rf� ��� د����C�5 hPو=��ه�ت ا� U��Rا�� �$��) U��Rا�� �$�� iIgfا� �fPQ2 �f U��Rا�� �$�� � R� ا��� ا�� �PQ2 � .( Lf [fر��j م�fPذج ا���f�2ا� kM�f�� ) ����� kM��� ، ى�f-ذج أ�f�� kM�f�� ن و�Rا��ذج 5��� kM��� .( ذج�f�� /f �f��� ا� kM�f�2�� ب�fjأ kM�f�� �fQ $ م�fPذج ا���f�2ا� � ، 5�ا��Rن وا���2ذج أ-f�ى Rf25 'fP$ ل�f (ا� �ا=&f�4ر ا�fPم ا�Af� إن kM�f�2ا� [f2�5 Z�&fء % ٢٧آ�gf � Rf2�\ /f Z�Rfا� .����� =��ه�ت ا�hP) ٥ا��20( Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 109 Notations b1=top flange width. b2=bottom flange width. bw=web width. d = Effective depth of tension region. d' = Effective depth of compression region fr=modulus of rupture H=total depth of beam. Ie : Effective moment of inertia. Icr : Cracked moment of inertia. Ig : Gross moment of inertia. Ma: Applied external moment. Mcr: Cracking moment n=modular ratio. =ρ Ratio of steel area at tension region. =ρ′ Ratio of steel area at compression region Introduction Deflections of reinforced concrete flexural members were the focus of several research activities for many years. It is prime importance in the determination of the deflection of beams is calculation of the moment of inertia (I) of the beam, since its value changes along the span length from (Ig) for uncracked sections to (Icr) for cracked sections. Branson developed a well known expression for the effective moment of inertia (Ie) over the entire length of the simply supported beam in the following form: cr 3 a cr g 3 a cr e I M M 1I M M I ×               −+×      = (1) The ACI Building Code adopted Branson’s equation and it first appeared in the 1971 edition of the publication and remains the recommended way of calculating the effective moment of inertia for the purpose of calculating the deflection of a reinforced concrete member. Since its adoption by the ACI Code in 1971, Branson’s model has been continually opposed. The reasons vary, but center around the accuracy of the model. Design engineers argue that the cumbersome calculation of Icr, especially for flanged sections, is complex and time consuming (Grossman 1981). They also argue that the effort required is not justified by the final product. Grossman (1981) states that the estimated deflection obtained by using Branson’s model is, at best, within ±20 % of experimental deflections obtained in a controlled lab setting. Another argument against Branson’s model is that its empirical nature can produce gross errors when applied to beams that are heavily or lightly reinforced and/or Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 110 subjected to non-uniform loads. Researchers have shown, that in some instances, Branson’s model can produce values that are 100% in error (Fikry and Thomas 1998). The arguments and concerns prompted various researchers to study the validity of Branson’s equation. The subsequent research produced numerous simplifications and enhancements to the Branson model. Modifications to the Ie method 1 Method (1) In 1991 scholars from King Saud University in Riyadh, Saudi Arabia published findings from research they conducted to determine if non-uniform load configurations are accurately accounted for by Branson’s effective moment of inertia model (Al-Zaid, Al-Shaikh, and Abu- Hussein 1991). The research compared theoretical moment of inertia values to experimental moment of inertia values obtained from subjecting reinforced concrete members of rectangular cross-section to a uniform load, a mid-span concentrated load, a third-point load, and a mid-span concentrated load combined with a uniform load. The service load moment applied to the member was the same for each load configuration. It was observed that the experimental moment of inertia values for a member subjected to a mid-span concentrated load was 12% greater than that experienced by a member subjected to a third-point load and 20% greater than the experimental moment of inertia exhibited by a member subjected to a uniform load. The experimental values proved that Branson’s model can not be accurate for all loading cases. Equation (1) returns a value comparable to the experimental value for the uniform loading case, which means that if the member is loaded with a concentrated load at mid-span the stiffness of the member would be significantly underestimated. The researchers addressed the discrepancy by suggesting that Branson’s model be generalized by modifying it to the form of Equation (2). cr m a cr g m a cr e I M M 1I M M I ×               −+×      = (2) where: m : experimentally determined exponent. In their report the researchers showed that by generalizing Equation (1) and in-turn solving for m (Equation 3) for each load case that the discrepancy could be eliminated. M M log I-I I-I logm a cr crg crexp               = (3) where: Iexp : Experimental moment of inertia. Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 111 2 Method (2) The researchers argued that the discrepancy revealed in Branson’s model was caused by the different lengths over which a beam cracks due to a specific load condition (Al-Zaid, et. al. 1991). Therefore, the authors suggested a model (Equation 4), similar in form to Branson’s model, that incorporated the ratio of cracked length to overall length which inherently accounted for the variation in the effective moment of inertia caused by different cracked lengths (Equation 4). g *m cr cr *m cr e I L L 1I L L I ×               −+×      = (4) where: m * : experimentally determined exponent. Lcr : cracked length of the member. L = length of member The proposed model is bounded by Ie = Ig when Lcr = 0, and Ie = Icr when the cracked length covers nearly the entire length of the member. The exponent m’ is calculated using Equation (3). In theory, the exponent m * is solely a function of the reinforcement ratio. This theory was later expanded on by the same researchers (Al-Shaikh and Al-Zaid 1993). L L log I-I I-I logm cr crg expg*               = (5) where: Iexp : experimental moment of inertia. The researchers exhibited that the “modified” form of Branson’s model and the proposed model incorporating cracked length both produce effective moment of inertia values relatively close to experimental moment of inertia values when the proper exponent is employed. As a continuation of the aforementioned study, two of the authors later executed an experimental program to study the effect that reinforcement ratio (ρ) plays on a reinforced concrete member’s effective moment of inertia (Al-Shaikh and Al-Zaid 1993). The experimental program was conducted by applying a mid-span concentrated load to reinforced concrete beams, of rectangular cross-section, containing varying amounts of reinforcement. The test specimen labels and reinforcement quantities were: Reinforcement Label Reinforcement Ratio Lightly 0.8 Normally 1.4 Heavily 2.0 Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 112 The study revealed that Branson’s model underestimated the effective moment of inertia of all test specimens. The underestimation of Ie was approximately 30% in the case of a heavily reinforced member and 12 % for a lightly reinforced specimen. Beyond the previously observed behavior of a reinforced concrete member subjected to a mid-span concentrated load (Al-Shaikh and Al-Zaid 1993), it is obvious that reinforcement ratio affects the accuracy of Branson’s model especially when the member is heavily reinforced. Therefore, by curve fitting, the authors derived an expression (Equation 6) to calculate the exponent m for use in Equation (3) which was introduced in the aforementioned study by the same authors. ρ×−= 8.00.3m (6) where: m = experimentally determined exponent ρ = reinforcement ratio. The authors also applied the more general Equation (4), introduced in their earlier research, and to the values obtained from this experiment. The experimental values were used to develop Equation (7) to determine the exponent m’ for Equation (4) a cr* M M m ×= β (7) where: m * = experimentally determined exponent and β = 0.8 ρ where: ρ = reinforcement ratio The use of Equation (4) may be better suited when considering the affects of reinforcement ratio on the effective moment of inertia, because the discrepancy created by load configuration is already taken into account by the cracked length term of the equation, which leaves the exponent m’ dependent only on the reinforcement ratio. 3 Method (3) In 1998, a new model was proposed by Fikry and Thomas were derived an effective moment of inertia model from basic concrete flexural response theory. Their focus was developing an effective moment of inertia model that eliminated the laborious Icr calculation associated with Branson’s model and more accurately accounted for variations in reinforcement ratio as well as load configuration. The derivation of the new model was based on an approximation for Icr, which the authors called Icre. The authors began their derivation with a cracked, singly reinforced, rectangular cross-section concrete member. They then derived Icr as a function of two variables (η and ρ) and represented it in the form of Equation (8), ( )         ×+= 12 bd I 3 cre βηρα (8) where, Icre = approximate moment of inertia, α = constant (given in literature), β = constant (given in literature), η = modular ratio, ρ = reinforcement ratio, b = width of member, and d = effective depth of reinforcement, this derivation achieved their first goal Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 113 (((( )))) (((( )))) (((( )))) (((( )))) 2 f 2fw2 3 fw2 2 f gfw2 2 f 3 fw1 2 gw 3 w g 2 h yhbb 12 hbb 2 h yhbb 2 h -c 12 hbb y- 2 h hb 12 hb I 2 2 2 2 2 11         −−−−−−−−++++ −−−− ++++         −−−−−−−−++++        ++++ −−−− ++++      ++++==== (eliminating the Icr calculation) and the approximation was within 6% of the cracked moment of inertia of all test specimens. The cracked moment of inertia approximation was then expanded to flanged cross-sections and doubly reinforced, rectangular and flanged cross- sections. 4 Proposed Model The deflections caused not only by changes of curvature but also by changes of shear deformations those are not always negligible, especially in the case of beams with span/depth ratio(L/H< 10) and in the case of amore pronounced of shear forces. The proposed model for the effective moment of inertia takes into account several effects such as (1. Type of loading, 2.(Span/depth) ratio, 3. Reinforcement ratio, 4. Ratio of (compression/tension) reinforcement, 5. (Effective depth/web width) ratio). The proposed model takes the following form: (9) where (10) The cross and cracked moment of inertia for beam with deferent cross sections as shown in figure below can be calculated as (11) )(I M M )(I M M I cr )( a cr g )( a cr e βξγξ γξγξ +××               −++×      = ++ ( ) )(III L H n f I I b d 525.026.5 creg l cr g w βξ ξ η αρρ γ η ρ ρ β α +≤≤ = +′ −= = ′ =       −= Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 114 ( ) ( ) ( ) ( ) ( )3s 2 1s 2f fw1 3 fw1 3 w cr cdAn dcA1n) 2 h -(chbb 12 hbb 3 cb I 2 1 1 1 1 −×+ −−+−+ − += g2 y-Hy , =         ++ = 0 10 2 11 a2 )a a4a(a- c, 2 0 wba ==== (((( )))) 111 fwss hbbnA1)A-(na 21 −−−−++++++++==== (((( )))) 2 212 fwss hbbdnAd1)A-(na 21 −−−−++++++++==== (12) where, fl= factor depend on loading type such as: 1. Distributed load =1.25 2. Two point load =1.0 3. Concentrated load =0.75 Properties and Abilities of the Program The computer program (EMIRCM)(Effective Moment of Inertia for Reinforced Concrete Members) is designed to deal with reinforced concrete members with many types of cross section with inclusion of transverse shear deformation effect and subjected to many types of loads. The computer program is coded in FORTRAN 90 language executed by PC Pentium IV 2800 MHz full cache Intel processor compatible computer with 2.0 GB RAM. The properties and abilities of this program may be summarized as follows: 1- Many types of loading such as (distributed loads, two point loads, and concentrated load). 2- Many types of cross section of members. 3- Using three different types of methods. Numerical Examples In order to verify the reliability of the adopted proposed method, some case studies reported by other researchers are utilized and compared with experimental and Branson' model, and Al-Zaid et al. model. 1 Comparison with experimental investigations of reinforced concrete members under concentrated loading at mid span a- Reinforced concrete simply supported beam under concentrated loading (with L/H=12.5) The accuracy of the results of the present analysis of real panels is checked through comparing with the experimental and numerical results studied by Al-Zaid et al. [1991] on g gr cr y If M = Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 115 simply supported reinforced concrete members and with Branson's model. The dimensions and material properties of the beams, as shown in Figure (2). Figure (3) shows a comparison with the experimental and the numerical results for the deflection at mid span. The results obtained from the present study give good agreement with the experimental results obtained by Al-Zaid, et al. [1991] with difference not more than (0.5%) with the experimental investigation while the difference between the Branson's model with the experimental results more than (17%) and so the difference between Al-Zaid et al. model with the experimental results more than (20%) at ultimate load stage. The load- deflection results are listed in Table (1). b- Reinforced concrete simply supported beam under concentrated loading (with L/H=٦.٥٣) one in a series of beams tested by (Bresler and Scordelis) was also examined the beam is simply supported and subjected to a concentrated load at mid span, The dimensions and material properties of the beams as shown in Figure (4). In the present study, this beam is analyzed using the proposed method with factor for typing of loading (fl=0.75). Figure (5) shows a comparison with the experimental and the numerical results for the deflection at mid span. The results obtained from the present study give good agreement with the experimental obtained by Bresler and Scordelis with difference about than (12%) with the experimental investigation while the difference between the Branson's model with the experimental results more than (42%) and so the difference between Al-Zaid et al. model with the experimental results more than (47%) at ultimate load stage. The load-deflection results are listed in Table (2). c- Reinforced concrete simply supported beam under concentrated loading (with L/H=٦.٥5) one in a series of beams analyzed by kreshna was also examined the beam is simply supported and subjected to a concentrated load at mid span, The dimensions and material properties of the beams as shown in Figure (6). Figure (7) shows a comparison with the experimental and the numerical results for the deflection at mid span. The results obtained from the present study give good agreement with the experimental obtained by kreshna with difference not more than (12%) with the experimental investigation. This test shows the effect of area of steel at compression region on the behavior of reinforced concrete members. The load-deflection results are listed in Table (3). Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 116 d- Reinforced concrete simply supported beam under concentrated loading (with L/H=7.5) Nurnbergerova et al. tested several reinforced concrete beams with I-cross section. The dimensions and materials properties of the beams as shown in Figure (8) Figure (9) shows a comparison with the experimental and the numerical results for the deflection at mid span. The results obtained from the present study give good agreement with the experimental obtained by Nurnbergerova et al. with difference not more than (4%) with the experimental investigation while the difference between the Branson's model with the experimental results more than (42%) and so the difference between Al-Zaid et al. model with the experimental results more than (44%) at ultimate load stage. The load-deflection results are listed in Table(4). 2 Comparison with experimental investigations of reinforced concrete members under Distributed loading a- Reinforced concrete simply supported beam under distributed loading (with L/H=12.5) The accuracy of the results of the present analysis of real panels is checked through comparing with the experimental and numerical results studied by Al-Zaid et al. [1991] on simply supported reinforced concrete members and with Branson's model. The dimensions and material properties of the beams, as shown in Figure (10). Figure (11) shows a comparison with the experimental and the numerical results for the deflection at mid span. The results obtained from the present study give good agreement with the experimental obtained by Al-Zaid, et al. with difference not more than (8%) with the experimental investigation while the difference between the Branson's model with the experimental results more than (15%) and so the difference between Al-Zaid et al. model with the experimental results more than (6%) at ultimate load stage. The load-deflection results are listed in Table (5). 3 Comparison with experimental investigations of reinforced concrete members under two point loading a- Reinforced concrete simply supported beam under two point loading (with L/H=12.5) The accuracy of the results of the present analysis of real panels is checked through comparing with the experimental and numerical results studied by Al-Zaid et al. [1991] on simply supported reinforced concrete members and with Branson's model. The dimensions and material properties of the beams, as shown in Figure (12). Figure (13) shows a comparison with the experimental and the numerical results for the deflection at mid span. The results obtained from the present study give good agreement with the experimental obtained by Al-Zaid, et al. with difference not more than (3%) with the experimental investigation while the difference between the Branson's model with the experimental results more than (3%) and so the difference between Al-Zaid et al. model with Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 117 the experimental results more than (7%) at ultimate load stage. The load-deflection results are listed in Table(6). 4 Parametric Study a- Effect of tension reinforcement steel ratio of flexural on the effective moment of inertia A simply supported rectangular cross section beam subjected to concentrated loading at mid span was analyzed with a range of (ρρρρ) from (0.5-3.0%). Figure (14) shows the effective moment of inertia ratio-applied moment ratio curve for the reinforced concrete member with a range of steel ratio (0.5-3.0%). The following properties of the beam are (H=200 m, b=200 mm, Ec=29.634×10 6 kN/m 2 , L=2500 mm,fy=153 MPa, fc'=38.12 MPa). b- Effect of slenderness ratio on the effective moment of inertia A two simply supported reinforced concrete beams subjected to concentrated loading at mid span were analyzed with a range of slenderness ratio (L/H) (5-20). Figure (15) and (16) show the effective moment of inertia ratio-applied moment ratio curve for the reinforced concrete member with a range of slenderness ratio (5-20). The following properties of the rectangular cross section beam are (H=200 m, b=200 mm, Ec=29.634×10 6 kN/m 2 , L=2500 mm,fy=413 MPa, fc'=38.12 MPa) and the properties of beam with I-section were mentioned at Figure(7). From these figures can be noticed that the effective moment of inertia is reduced by about 27% for the range of slenderness ratio (20-5) which can be attributed to shear deformations effect which increase with decreasing of slenderness ratio where this effect was neglected by the other models. d- Effect of compression tension reinforcement steel ratio on the effective moment of inertia A simply supported rectangular cross section beam subjected to concentrated loading at mid span was analyzed with a range of (As'/As) from (0.0-0.5%). Figure (17) and (18) shows the effective moment of inertia ratio-applied moment ratio curve for the reinforced concrete member with a range of (As'/As) from (0.0-0.5%). The following properties of the beam are (H=200 m, b=200 mm, Ec=29.634×10 6 kN/m 2 , L=2500 mm, fy=413 MPa, fc'=38.12 MPa). From these figures can be noticed that the effective moment of inertia is increased by about 21% for the range of ((As'/As)) (0.0-0.5) which can be attributed to compression reinforcement. The increase compression reinforcement lead to reduce the deformations of reinforced concrete members. Where this effect was neglected by the other models. Conclusions The research was presenting a new form of the effective moment of inertia by enhancement Branson's model taking into account the effect of several factors such as type of loading, shear deformations, reinforcement ratio. the results of the presented model were compared Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 118 with (experimental results, Branson's model results, and results of other models). The results of the present model give best agreement with experimental results than Branson's and the other models. The following conclusions are drawn with regard to the results obtained for the present study such as: 1. The results showed that the effective moment of inertia reduced by about 27% for span to depth ratio of (20 to 5) due to shear deformation effects. 2. The present model gives good agreement with the experimental results for all types of cross section. References 1. ACI Committee 318 "Building Code Requirement for Reinforced Concrete and Commentary (ACI 318-89 /ACI 318R-89),American Concrete Institute ,Detroit, 1989, 353 pp. 2. Al-Shaikh, A.H. and Al-Zaid, R.Z., “Effect of Reinforcement Ratio on The Effective Moment of Inertia of Reinforced Concrete Beams,” ACI, Struct. J.,Vol. 90, No.2, March-April,1993, pp.144-148. 3. Al-Zaid, R. Z. and Al-Shaikh, A.H. and Abu-Hussein, M., “Effect of Loading Type on The Effective Moment of Inertia of Reinforced Concrete Beams,” ACI, Struct.J.,Vol.88, No.2, March April 1991, pp.184-190. 4. Branson,D.E., "Instantaneous and Time Dependent Deflection of Simple and Continoues reinforced Concrete Beams" HPR Report No.7,Part 1,Albama,Highway Department /Us Bureau of Public Roads,1965,pp.1-78 5. Fikry, A.M. and Thomas, C., “Development of a Model for the Effective Moment of Inertia of One-Way Reinforced Concrete Elements,” ACI, Struct. J.,Vol.95,No.4, July-August 1998, pp.444-455. 6. Grossman, J.S., “Simplified Computations for Effective Moment of Inertia and Minimum Thickness to Avoid Deflection Computations,” ACI, Struct. J., Vol.78, No. 6,November-December 1981, pp. 423-434. 7. Joseph E. Wickline “A Study of Effective Moment of Inertia Models for Full-Scale Reinforced Concrete T-Beams Subjected to A Tandem-Axle Load Configuration”, M.Sc. Thesis, Dept. of Civil and Envir. Eng. Faculty of the Virginia Polytechnic Institute and State University. 8. Nurnbergerova, T., Krizma, M., and Hajek, J., “Theoretical Model of The Determination of The Deformation Rates of R/C Beams” J. Const. and Build. Mat., Vol.15, 2001, pp.169-176. 9. Muhaisin, M. H. “Nonlinear Analysis of Reinforced Concrete Plane Frames under Moving Loads”, M.Sc. Thesis, Dept. of Civil Eng., College of Eng., University of Babylon, (2003). Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 119 Deflection (mm) Load (kN) Experimental results Branson method Al-Zaid,et al. method Present Study 0.0 0.00 0.00 0.00 0.00 50.6 0.60 0.49 0.49 0.51 101.2 1.40 1.59 1.24 1.41 149.5 2.25 2.55 2.04 2.48 200.1 3.20 3.48 2.90 3.78 250.7 4.20 4.40 3.76 5.23 299.0 5.50 5.26 4.60 6.73 349.6 7.20 6.17 5.49 8.41 400.2 9.20 7.06 6.37 10.20 450.8 12.40 7.96 7.26 12.07 460.0 14.20 8.13 7.42 12.41 Deflection (mm) Load (kN) Experimental results Branson method Al-Zaid,et al. method Present Study 0.00 0.00 0.00 0.00 0.00 9.88 1.05 1.44 1.00 1.03 11.82 1.50 2.17 1.35 1.39 13.76 1.95 2.915 1.71 1.80 15.70 2.50 3.64 2.10 2.24 17.65 3.05 4.34 2.51 2.73 19.59 3.63 5.02 2.94 3.24 21.53 4.23 5.67 3.38 3.79 23.47 4.77 6.31 3.82 4.37 25.59 5.40 6.98 4.32 5.04 27.71 6.13 7.65 4.83 5.74 31.77 7.31 8.90 5.84 7.17 35.30 8.47 9.97 6.73 8.51 Table (1): Load-deflection results of reinforced concrete simply supported beam under concentrated load at mid span Table (2): Load-deflection results of reinforced concrete simply supported beam under concentrated load at mid span Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 120 Deflection (mm) Load (kN) Experimental results Branson method Al-Zaid,et al. method Present Study 0.00 0.00 0.00 0.00 0.00 44.83 0.52 0.81 0.75 0.65 89.65 1.3 2.07 1.78 1.69 134.48 2.25 3.17 2.82 2.90 179.31 3.65 4.27 3.90 4.32 224.14 5.2 5.37 4.98 5.85 260.00 6.35 6.21 5.82 7.12 Deflection (mm) Load (kN) Experimental results Branson method Al-Zaid,et al. method Present Study 0.0 0.00 0.00 0 0 25.0 0.40 0.39 0.35 0.38 50.0 0.85 1.04 0.85 0.96 75.0 1.40 1.60 1.37 1.66 100.0 2.10 2.16 1.91 2.45 125.0 3.00 2.70 2.44 3.29 150.0 4.10 3.25 2.98 4.21 175.0 5.10 3.80 3.53 5.18 200.0 6.10 4.35 4.07 6.19 225.0 7.25 4.89 4.61 7.24 250.0 8.50 5.43 5.15 8.33 275.0 9.50 5.16 4.87 7.79 300.0 10.70 6.52 6.23 10.63 325.0 11.7 7.07 6.77 11.82 350.0 12.70 7.61 7.32 13.04 375.0 13.85 8.15 7.86 14.29 400.0 15.00 8.69 8.40 15.56 Table (3): Load-deflection results of reinforced concrete simply supported beam under concentrated load at mid span Table (4): Load-deflection results of reinforced concrete simply supported beam under concentrated load at mid span Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 121 Deflection (mm) Total Load (kN) Experimental results Branson method Al-Zaid,et al. method Present Study 0.00 0.00 0.00 0.00 0.00 17.95 1.21 1.23 1.40 1.15 20.30 1.67 1.61 1.92 1.49 22.36 2.10 1.97 2.40 1.81 24.41 2.50 2.34 2.90 2.17 26.18 2.90 2.67 3.32 2.50 28.53 3.43 3.13 3.88 2.97 31.77 4.16 3.77 4.63 3.68 35.60 5.10 4.54 5.50 4.60 39.42 6.00 5.318 6.32 5.60 43.24 6.95 6.093 7.13 6.67 47.36 7.90 6.92 7.97 7.90 50.90 8.85 7.63 8.68 9.01 54.72 9.83 8.39 9.43 10.28 58.84 10.84 9.21 10.22 11.70 Deflection (mm) Total Load (kN) Experimental results Branson method Al-Zaid,et al. method Present Study 0.00 0.00 0.00 0.00 0.00 11.73 0.91 0.95 0.89 0.88 14.21 1.48 1.62 1.39 1.28 15.79 1.87 2.10 1.74 1.55 18.04 2.45 2.81 2.27 2.00 19.85 2.91 3.38 2.73 2.39 21.65 3.43 3.96 3.20 2.80 25.48 4.51 5.14 4.21 3.79 29.54 5.55 6.32 5.31 4.96 33.38 6.63 7.39 6.35 6.18 37.21 7.68 8.42 7.37 7.50 41.27 8.89 9.49 8.46 9.00 45.11 10.18 10.48 9.47 10.50 Table (5): Load-deflection results of reinforced concrete simply supported beam under distributed load Table (6): Load-deflection results of reinforced concrete simply supported beam under two point loads Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 122 2Φ16 1Φ10 MPayf MParf GPacE MPacf 413 393 63429 238 ==== ==== ==== ====′′′′ . . . L=2500 mm b=200 mm H=200 mm Figure (2): Details of a reinforced concrete simply supported beam under concentrated load at mid span with (L/H=12.5)(Al-Zaid, et al.) Figure (3): Load-deflection curve of reinforced concrete simply supported beam under concentrated load at mid span (Al-Zaid, et al.) 0 1 2 3 4 5 6 7 8 9 10 Central deflectin 0 5 10 15 20 25 30 35 40 L o a d ( k N ) Experimental work Branson method Al-Zaid, et.al. method Present method Figure (1): Details of uncracked and cracked cross section As2 As1 b1 b2 hf1 hf2 bw d d1 H (a) uncracked cross section nAs (n-1)As1 b1 N.A (b) Transformed cracked section Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 123 MPayf MParf GPacE MPacf 345 043 03723 124 ==== ==== ==== ====′′′′ . . . L=3660 mm b=305 mm H=560 mm Figure (4): Details of a reinforced concrete simply supported beam under concentrated load at mid span with (L/H=6.53)( Bresler and Scordelis) 4Φ28 2Φ12 Figure (5): Load-deflection curve of reinforced concrete simply supported beam under concentrated load at mid span (Bresler and Scordelis) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Central deflection (mm) 0 50 100 150 200 250 300 350 400 450 500 L o a d ( k N ) Experimental work Branson method Al-Zaid, et.al. method Present method MPayf MParf GPacE MPacf 345 003 03723 124 ==== ==== ==== ====′′′′ . . . L=3657 mm b=225 mm H=558 mm Figure (6): Details of a reinforced concrete simply supported beam under concentrated load at mid span with (L/H=6.55)( kreshna) 4Φ28 Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 124 0 1 2 3 4 5 6 7 8 Central deflection (mm) 0 50 100 150 200 250 300 L o a d ( k N ) Experimental work Branson method Al-Zaid et al method Present method Figure (7): Load-deflection curve of reinforced concrete simply supported beam under concentrated load at mid span (kreshna) 3600 11Φ16 2Φ10 MPayf MParf GPacE MPacf 345 262 8537 124 ==== ==== ==== ====′′′′ . . . L=3660 mm b=305 mm H=560 mm Figure (8): Details of a reinforced concrete simply supported beam under concentrated load at mid span with (L/H=7.5)( Nurnbergerova et al.) Figure (9): Load-deflection curve of reinforced concrete simply supported beam under concentrated load at mid span (Nurnbergerova, et al.) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Central deflection (mm) 0 50 100 150 200 250 300 350 400 L o a d ( k N ) Experimetal work Branson method Al-Zaid et al Present method Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 125 MPayf MParf GPacE MPacf 413 393 63429 238 ==== ==== ==== ====′′′′ . . . L=2500 mm b=200 mm H=200 mm Figure (10): Details of a reinforced concrete simply supported beam under distributed load with (L/H=12.5)(Al-Zaid, et al.) 2Φ16 1Φ10 0 1 2 3 4 5 6 7 8 9 10 11 12 Central deflection (mm) 0 5 10 15 20 25 30 35 40 45 50 55 60 L o a d ( k N ) Experimental work Branson method Al-Zaid et al Present method Figure (11): Load-deflection curve of reinforced concrete simply supported beam under distributed load (Al-Zaid et al.) MPayf MParf GPacE MPacf 413 393 63429 238 ==== ==== ==== ====′′′′ . . . L=2500 mm b=200 mm H=200 mm 2Φ16 1Φ10 Figure (12): Details of a reinforced concrete simply supported beam under two point load with (L/H=12.5)(Al-Zaid, et al.) Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 126 0 1 2 3 4 5 6 7 8 9 10 11 Central deflection (mm) 0 5 10 15 20 25 30 35 40 45 50 L o a d ( k N ) Experimental work Branson method AL- Zaid et al Present method Figure (13): Load-deflection curve of reinforced concrete simply supported beam under two point load (Al-Zaid et al.) Figure (14): Effect of reinforcement steel ratio of flexural on the effective moment of inertia of simply supported concrete beam under concentrated load by present study 0 1 2 3 4 5 Ma/Mcr 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Ie /I g ρ∗100 0.5 1.0 1.5 2.0 2.0 3.0 Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 127 Figure (15): Effect of (span/depth) ratio of simply supported concrete beam under concentrated load (rectangular section) 0 1 2 3 4 5 6 Ma/Mcr 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Ie /I g L/H 20.0 15.0 12.5 10.0 7.5 5.0 Figure (16): Effect of (span/depth) ratio of simply supported concrete beam under concentrated load (I-section) 0 2 4 6 8 10 12 14 16 18 20 Ma/Mcr 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Ie /I g L/H 20.0 15.0 12.5 10.0 7.5 5.0 Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 128 0 1 2 3 4 5 Ma/Mcr 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Ie /I g As'/As 0.0 0.1 0.2 0.3 0.4 0.5 Figure (18): Effect of compression reinforcement steel ratio on the effective moment of inertia of simply supported concrete beam under concentrated load by present study 0 1 2 3 4 5 Ma/Mcr 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Ie /I g As'/As 0.0 0.1 0.2 0.3 0.4 0.5 Figure (1٧): Effect of compression reinforcement steel ratio on the effective moment of inertia of simply supported concrete beam under concentrated load by Branson method