Shear strength evaluation in the existance of axial compressive loads for reinforced concrete beams

ABSTRACT Response-2000 is a program developed to calculate the shear capacities of beams and columns under any combinations of shear, moment, and axial loads. It is based on the modified compression field theory. The capabilities of the program Response-2000 to obtain shear strength capacities in the existence of axial compressive loads are demonstrated. The comparison with the experimental tests proved that the program gives good estimate for the failure shear in this case. The shear provisions of the Egyptian reinforced concrete code (ECP203-2017) are also evaluated against the experimental results. It is found that the accuracy of the obtained results using Rseponse-2000 is better than those obtained using the shear equations of the Egyptian reinforced concrete code (ECP203-2017). The program Response-2000 is then used to generate 336 results for 7 different cross-sections considering different concrete compressive strengths, transverse reinforcement, and values of axial loads. These generated results along with 33 experimental results for other 7 cross-sections are used to develop a new formula for the shear strength capacity in the existence of axial compressive loads. This formula is found to give better results when verified against experimental data.


Introduction
Beams subjected to combined bending, shear, axial force, and torsion are encountered for example in the cases of buildings and frames which resist wind or seismic forces. The analysis of these beams has attracted lots of research. Halim et al. (2011) [1] used experimental data and finite elements package to evaluate the ACI318 and AASHTO shear and combined shear and torsion provisions for shear-critical beams. They indicated the possibility of deriving shear-torsion interaction equations and reported that they plotted the interaction envelope points. A MathCAD design program employing the modified compression field theory and truss analogy was developed for sections under combined shear and torsion or shear only. Metwally (2012) [2] evaluated the program Response-2000 [3] in terms of its capability for predicting the shear strength of reinforced and prestressed concrete members under moment and shear. He found that it can predict the experimental tests accurately. Esfandiari and Adebar (2009) [4] presented a procedure for evaluating the shear capacity of reinforced concrete girders which is similar to the 2008 AASHTO LRFD method. They compared their method with the modified compression field procedure for the case of an element under uniform shear. They also compared their method with the program Response-2000 for beams under shear and bending moment. A comparison with test results for reinforced and prestressed beams under shear and moment and a comparison with the predictions of the AASHTO LRFD and ACI318 codes were made. Esfandiari and Adebar (2009) [4] found that their method gives good shear strength predictions. Their equations start by defining the nominal shear resistance as follows Nominal shear resistance (V n ) = part from concrete (V c ) + part from stirrups The modified compression field theory (MCFT) is used to define θ and β assuming no normal stress in z-direction. This way the relationship between shear stress and strain can be obtained. The work of Esfandiari and Adebar (2009) [4] calculates the axial strain ε x from the amount of tensile reinforcement and applied forces. There are three shear failure modes. These are: 1-longitudinal reinforcement yielding; two-transverse reinforcement yielding; and 3-concrete crushing after transverse reinforcement yielding. The θ is related to the major compressive strain, which in turn is a function of the shear stress.v/f ' c . As the shear stress ratio is not determined during evaluation, ρ z f y /f ' c is used in the following relation For the case when the stirrups are yielding and at crushing of concrete The authors provide figures, which can be reviewed in their work, for β in terms of axial strain ε x . For the case of concrete crushing, they provide the following equation  [3] to 149 experimental tests for beams with minimum stirrups. They found that the average measured-to-predicted shear capacities are 1.02 for reinforced beams and 1.11 for prestressed beams. Collins et al. (1996) [6] developed a unified method to design reinforced concrete beams under shear and axial tension or compression. Their method adopts the concept of modified compression field theory. They indicated that the maximum longitudinal strain ε x in the beam web can be taken as the reinforcement tension strain as follows The major tensile strain ε 1 is related to the axial strain ε x , and the major compressive strain ε 2 by the following relation The nominal shear strength V n is given by The largest tensile strain ε 1 may be found from ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi To use the equation of V n to calculate the needed shear reinforcement, the engineer shall define values for β and θ. The authors give The shear stiffness, S, for the case of single core is the shear modulus of the core, G c , times the cross-section of the core, A c . For the case of two cores, the authors use the energy method to expand the model to this case and the derived shear stiffness is as follows where d = distance between the axis of the face sheets b = width of beam h c2 , h c1 = second and first core thicknesses, respectively G c1 , G c2 = shear moduli of cores 1 and 2, respectively The axial stresses in the core and face sheets of sandwich beams, assuming that the distribution of strain is plane, are given by where M z = bending moment at the considered section of the beam y = depth from the neutral axis The shear stresses in the core and face sheets are given by S c , S f = first moment of area of the core and the face sheet V y = shear force at the considered section of the beam Ou and Nguyen (2016) [9] proposed flexure-shear-axial interaction approach which accounts for corrosion of reinforcement. The shear effect on confinement of concrete and buckling of reinforcement, and the softening effect of transverse reinforcement on concrete flexural compression zone are considered. The proposed approach uses corroded models for bond, steel, and concrete. The method was generally able to capture the behavior of uncorroded and corroded beams. Rajapakse et al. (2019) [10] developed a finite element approach based on fiber force formulation to evaluate the response of frames or walls considering shear force-axial force-bending moment interaction. Their approach calculates the stresses corresponding to certain strain using the modified compression field theory. The proposed element proved its capability when compared to experimental results. Kirkland et al. (2015) [11] studied the axial force-shear-moment interaction for composite beams and developed a finite element formulation which gave good predictions compared with experimental tests. They developed a design approach for shear and flexure interaction of axially loaded beams. Lai et al. (2019) [12] assessed axial force-moment interaction of concrete encased steel columns made of a range of concrete and steel grades. They collected from the literature experimental tests covering steel yield strength ranging from 280 MPa to 913 MPa and concrete strength ranging from 20 MPa to 104 MPa. They used these data to evaluate the approach of EN 1994-1-1 for calculating the ultimate strength of these columns. Then, they performed an analytical study to obtain the interaction curves for axial-moment capacities of cross-sections of concrete encased steel columns. They reported that the EC4 method yields un-conservative values. Lai et al. (2019) [12] also presented a simple method to calculate the strength of composite columns under axial compression and bending. This general method is given here for comparison purpose. To plot the interaction curve, four points are calculated. These points are the pure compression, point close to the pure bending, and points for some strain states. The pure compression point is obtained using the following relation concrete area, steel section area, and longitudinal reinforcement area, respectively. fc = compressive strength of concrete f ' ys & f ' yr = steel section effective yield strength, and longitudinal reinforcement effective yield strength, respectively (f ' y = min(E s ε co ,f y )) α c = factor which accounts for difference between concrete in cylinder and concrete in column, size effect, and the change in the strength of concrete over the length of the column = 0.85 E = elastic modulus of steel ε co = peak strain f y = f ys or f yr = steel section yield strength, and longitudinal reinforcement yield strength, respectively The bending and compression points are obtained for three strain cases. The ultimate strain of concrete, which is 0.003, is assumed for the three points. One of the points is when the geometric centroid of the composite cross-section lies on the neutral axis. In this case, they assume that the steel section is in the elastic range and the stress in concrete is nonlinear and is represented by the rectangular stress block. The effective height coefficient β and the effective strength coefficient α are calculated using the following equations.
The strain at the centroid of the flange of steel section, at the tip of web of steel section, and at reinforcing bar is calculated as follows: where d f , d w , and d r = distance from the centroid of flange of steel section, tip of web of steel section, and centroid of reinforcing bars, respectively, to concrete surface x = neutral axis depth ε cu = ultimate strain of concrete = 0.003 From the longitudinal strains calculated above, the stresses can be found assuming elastic-perfectly plastic behavior. Thus, the bending moment and axial force may be obtained for the cross-section.
Xu and Zhang (2012) [13] suggested a hysteretic model for the interaction of flexural-shear-axial strengths of reinforced concrete columns. The model was used in a finite element program which was verified with experimental tests for columns under different axial loads. They also obtained the seismic response of a prototype bridge. Kocer et al. (2019) [14] calculated the shear and moment strengths of reinforced concrete columns and their displacements based on artificial neural network. They reported that the success of this method for calculating the shear and moment strengths was good while for displacements it was not sufficient. Popovic (2012) [15] reported that the shear design equations of the FIB model code (2010) [16] are simplifications of the modified compression field theory. He presented a method for the bending moment-shear force-axial force interaction curves to be used in the design of reinforced concrete beams. He also studied the effect of the axial forces on the shapes of the interaction diagrams. Popovic (2012) [15] reported that the longitudinal strain can be obtained by the following relation after some approximations: where M Ed = applied design moment V Ed = applied design shear force on the section shear area N Ed = applied design axial force (positive for tension and negative for compression) d v = effective shear depth (flexural lever arm) A s = area of tensile flexural reinforcement E s = elastic modulus of tensile flexural reinforcement The shear capacity of the beam is calculated from where V Rd = design shear capacity V Rd,c = concrete deign shear capacity V Rd,s = design shear capacity of shear reinforcement The design shear capacity of concrete for the case where minimum shear reinforcement exists is given by ffi ffi ffi ffi ffi f ck p � 8MPa) γ c = concrete partial safety factor b w = width of beam k v = coefficient which considers aggregate interlock and shear reinforcement greater than the minimum The design shear capacity of vertical stirrups is given by Lodhi and Sezen (2012) [17] reported that reinforced concrete columns with transverse reinforcement not designed for earthquake codes may suffer shear failure in the event of strong seismic load. The authors applied two models for displacement-based study of reinforced concrete columns. Their proposed method gave results consistent with experimental data. Saritas and Filippou (2009) [18] presented a beam element which considers shear-flexure-axial coupling. The coupling is attained using numerical integration of the axial model for the material over the depth. The examples proved the quality of their beam element.
Mullapudi and Ayoub (2013) [19] presented a three dimensional concrete model for fiber analysis of reinforced concrete. The model considers the interaction between shear, bending, axial force, and torsion. The shear behavior is considered using Timoshenko approach. The capability of the model has been evaluated. Zhang et al. (2017) [20] suggested a mechanics approach to model reinforced concrete beams under shear. They also extended their approach to consider axial load and developed a closed form solution. They validated their method using experimental results. Their closed-form solution has the following form: where b = width of the reinforced concrete beam d NA = neutral axis depth for the case of reinforced concrete beam with axial load A = shear friction material property before sliding P stpi = tensile force by the legs of stirrup number i n = number of stirrups crossed by diagonal crack C = coefficient which is a function in β CDC and B β CDC = critical diagonal plane angle B = shear friction material property before sliding Jd = lever arm between P rt and P conc P rt = tensile reinforcement force P conc = concrete compression force d stpi = the distance from the crack at tensile reinforcement to the i-th stirrup P a = axial force on the cross-section d = effective depth of beam h = overall depth of beam M a = bending moment at the considered section V a = shear force at the considered section M a = V a * a a = the effective shear span Rasheed and Abouelleil (2015) [21] addressed the analysis of columns subjected to axial-moment-shear interaction. They compared different procedures for calculating the shear strength and paid more attention to the AASHTO LRFD 2014 procedure. They validated with experimental tests and improved the accuracy of AASHTO LRFD method for calculating the shear strength. The authors used the AASHTO LRFD equations with the following two modifications. The first modification is an equation for calculating the inclination angle (α) of spiral reinforcement for circular columns in relation to the longitudinal axis. The equation has the following form ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where s = spiral spacing (pitch) D r = diameter of circular helix The second modification is the equation for shear strength of transverse steel (V s ) which has the following form for the case of hoops perpendicular on the center line of circular column: is a two-dimensional nonlinear program for sectional analysis according to the modified compression field theory. It considers beams and columns subjected mainly to shear. The program can integrate the behavior of the section to beam segments. Response-2000 assumes that there is no clamping stress over the beam depth and that plane sections remain plane after deformation. Response-2000 gives diagrams over the depth of the cross-section. It gives the shear-strain versus shear and the curvature versus moments which allow noting the flexural failures versus shear failures. Also, the program gives information about cracking of concrete and yielding of reinforcement. Longitudinal strains and transverse strains are also supplied. Among other produced information is the load-deformation curve.
The aim of this paper is to evaluate the Response-2000 analysis results and the results from the Egyptian reinforced concrete code (ECP203-2017) [22] model for the case of reinforced concrete beams under shear and axial compressive load. The evaluation is to be made comparing with experimental results. Further, the work uses the program to develop extra data which are to be used to obtain new formula for the shear strength in the existence of axial compressive load. This formula is then tested.

Shear model of the Egyptian code for reinforced concrete (ECP203-2017) [22]
The Egyptian code for reinforced concrete (ECP203-2017) [22] defines the shear strength of cracked concrete when the shear is resisted by concrete and steel as V cu ¼ 0:12 ffi ffi ffi ffi f cu γ c q MPa where f cu = standard cube 28-day compressive strength, in MPa γ c = concrete strength reduction factor = 1.5 Beams with web reinforcement develop the flexural capacity after shear-inclined cracking. The shear is carried by the uncracked concrete before flexural cracking. Between flexural and inclined cracking this shear is taken by concrete and steel. At last, the stirrups crossing the crack yield and the inclined crack open rapidly and a splitting dowel failure occurs, the compression zone crushes or the web crushes. The last two components present a brittle failure and are lumped by V cu which is taken equal to the inclined cracking shear. Axial compressive forces increase the inclined cracking load. This is because they delay the flexural cracking and reduce its penetration into the beam (Wight and MacGregor, 2012 [23]).
In the case where there is compressive axial load, the above equation is to be multiplied by the following factor δ c ¼ 1 þ 0:07 P u A c � 1:5 where P u = the ultimate axial compressive load substituted for as positive, in N A c = the area of cross-section, in mm 2

Added results using Response-2000 and the developed shear strength equation
The program Response-2000 [3] is used to generate 336 results for the specimens shown in Table 3 Table 3 presents different values of the axial load for the same cross-section. The obtained values of concrete ultimate shear strengths (V cu ) are shown in Table 4 for the cross-sections of no moment and Table 5 for the cross-sections with shear and moment. The last column of both tables gives the ultimate concrete shear strength corresponding to each axial load in Table 3.
All the test results and the generated results are used to develop an empirical relationship for the ultimate shear strength of concrete (V cu ) in the existence of axial compressive load. The procedure adopted for developing this equation consists of plotting relation between the axial compressive      form of the equation of the Egyptian code is adopted. The change is in using different coefficients as follows:  [25] as shown in Tables 8 and 9, respectively. It is clear that the degree of conservatism for this equation is less than that of the Egyptian code. Figures 3 and 4 show the variation of concrete shear strength calculated from the Response-2000 [3] and the proposed     equation with the axial load and the concrete compressive strength, respectively. It is very clear that the increase in either the axial normal force or the concrete compressive strength results in increase in the ultimate concrete shear capacity. The increase in shear capacity with the increase in the axial compression is due to the reduction in the major tensile stress in concrete.

Disclosure of potential conflicts of interest
No potential conflict of interest was reported by the author(s).