Design of slender reinforced concrete rectangular columns subjected to eccentric loads by approximate methods

Normas para projeto de estruturas de concreto armado estabelecem que a analise dos efeitos de 2a ordem em pilares esbeltos deve ser feita de modo a assegurar que para as combinacoes mais desfavoraveis das acoes de calculo, nao ocorra perda de estabilidade, nem esgotamento da capacidade resistente de calculo. Para isso e obrigatorio considerar a nao-linearidade fisica dos materiais, juntamente com a nao-linearidade geometrica. Os calculos dos efeitos de 2a ordem podem ser feitos por metodos aproximados ou pelo metodo geral. Neste trabalho o foco sera no calculo desses efeitos de 2a ordem com base nos metodos aproximados prescritos na NBR 6118: metodo do pilar-padrao com curvatura aproximada e metodo do pilar-padrao com rigidez aproximada. Dada a importância dos pilares e de sua estabilidade e importante que os metodos aproximados sejam capazes de apresentar solucoes seguras para o dimensionamento destes. Neste cenario, o objetivo desse trabalho e avaliar estes metodos aproximados em termos de seguranca, precisao e economia com relacao a resultados de ensaios de pilares encontrados na literatura. A analise deste estudo comparativo revela que o emprego do metodo da rigidez aproximada apresenta resultados mais proximos dos encontrados nos ensaios de pilares.


Introduction
Reinforced concrete (RC) columns are linear structural elements, usually cast vertically, responsible for carrying the loads from floors to the foundations.The stability of any given structure is directly linked to the stiffness and strength of the columns.Thus, the design of reinforced concrete columns must include local checks as well as global analysis of the structural system.Concrete codes worldwide require that columns must be designed to resist not only axial and bending moments computed from an ordinary first order frame analysis including allowances for construction imperfections but also moments due to internal force effects resulting from deflections (second-order effects).Thus the design of concrete columns must be based on the factored forces and moments from a second-order analysis considering material nonlinearity and cracking, as well as the effects of member curvature and lateral drift, duration of the loads, shrinkage and creep, and interaction with the supporting foundation.Column sections, under the above conditions, must then be designed to ensure that there is neither instability nor material failures.Use of such an analysis to determine column compressive axial forces and bending moments for section design is the most rational approach.But this analysis is very complex.Thus concrete codes worldwide allow the use of approximate design methods for slender columns.The simplified methods shown in NBR 6118 [1] are approximate curvature method and approximate stiffness procedure.Due to the responsibility of columns in the stability and strength of concrete structures, these approximate approaches must provide adequate safety in the design.In this scenario, the goal of this paper is to analyze NBR 6118 [1] approximate design criteria for slender rectangular columns subjected to eccentric loads with respect to safety, precision and economy, by comparing code based calculations with respect to experimental results of columns built with conventional concrete (f c ≤ 55 MPa).The investigation also includes columns with concrete having compressive strength above 55 MPa since NBR 6118 procedures are being changed to allow the use of concrete with compressive strength above 55 MPa.

NBR 6118 approximate design approaches for slender columns
The NBR 6118 [1] provides two simplified design procedures for the evaluation of second order effects in slender columns: approximate curvature method and approximate stiffness procedure.The simplifications in both procedures are related to geometric and material nonlinearities.

Approximate curvature method
The approximate curvature method is applicable to RC columns with slenderness ratios less or equal to 90 and symmetrical location of the reinforcement.It can only be used in columns subjected to axial loads and bending on one axis.The geometrically nonlinear behavior is simplified by assuming a deformed shape represented by a sine curve.The material nonlinearity is taken into account by an approximate equation for the curvature at the critical column cross-section.The lateral displacement 2 e due to the second order effects is given by: (1) The curvature is calculated from: (2) The column maximum bending moment is then equal to: These symbols used are explained in the notation.

Approximate stiffness procedure
The approximate stiffness procedure is applicable to RC columns subjected to combined flexure and axial loads with unsupported length to radius of gyration ratio less or equal to 90 and symmetrical location of the reinforcement.It can only be used in columns with rectangular cross-sections.The deformed shape in this case is also represented by a sine curve.The material nonlinearity is taken into account by an approximate equation for column flexural stiffness.The maximum design bending moment in a column is equal to: (5) where the dimensionless stiffness k is calculated from: Design of slender reinforced concrete rectangular columns subjected to eccentric loads by approximate methods have longitudinal and transversal (stirrups) reinforcement, and were tested in one axis bending with axial compressive loads.Columns fabricated with concrete containing light aggregates or fibers are not included.The most important characteristics of the columns in the databank are shown in Tables 1 and 2. Each table corresponds to a range of concrete compressive strength: up to 55 MPa and above 55 MPa.Each table also presents the number of columns tested by each research team, their respective geometrical characteristics, the longitudinal reinforcement ratio as well as the concrete compressive strength.The indicated concrete compressive strength were obtained from 15 cm x 30 cm cylinder specimens.

Comparative Study
The comparative study between the actual and predicted second order effects can be quantified based on the ratio of the test failure moment M test to the predicted one.The test failure moment M test provides a close estimate of the true capacity.It can be determined from the loads and the total lateral displacements measured at the column failure.The design moment M pred is calculated using equations 4 and 7 respectively.For the evaluation of the predicted moment M pred , all material resistance factors were set equal to one.Further, the measured concrete compressive strength c f of each test specimen was used in determining M pred .For each column the ratio M test /M pred was calculated.Statistical analyses of this ratio include its average m, the median m d , the standard deviation SD, the coefficient of variation CV as well as the maximum and minimum values.The average of M test /M pred is used as a measure of the conservative bias of the procedure while the coefficient of variation is taken as an indication of accuracy.With the objective of evaluating the reliability and of comparing the performance of shear design code equations for reinforced concrete beams, COLLINS [4] developed a demerit point scale methodology.Considering safety, precision and economy, a score is attributed for each range of M test /M pred ratio: these values are The symbols used are also explained in the notation.The dimensionless stiffness κ is needed to calculate M d,tot and is also a function of M d,tot .Thus an iterative process must be used.However, Scadelai [2] has shown that an iterative process is not necessary and M d,tot can be calculated from:

Slender reinforced concrete columns database
The database for this analysis was originally assembled by Souza [3] as part of his final undergraduate report.It consists of slender columns tested up to failure in laboratories worldwide.All columns

Columns made with concrete of f c ≤ 55 MPa
The results of the M test /M pred ratio for columns made with concrete of f c ≤ 55 MPa are presented in the Table 4.They were obtained from the comparison with test results of 51 columns.The analysis of these results reveals that NBR 6118 [1] approximate stiffness criterion is more conservative than the approximate curvature method: average and median values are larger than the unity.In terms of accuracy, the approximate stiffness procedure is also more precise, since the coefficient of variation is 2% smaller.With respect to safety, the demerit scale (shown in the part B of Table 4) reveals that the results obtained with the approximate curvature method are of concern: 22% of the M test /M pred values  The effects of concrete compressive strength f c , of the column slenderness ratio l and of the dimensionless axial force u on the ratio M test /M pred were verified for both the approximate curvature method and the approximate stiffness procedure.For both design methods, this analysis revealed a reduction in the M test /M pred ratio for larger values of the column slenderness l as shown in Figure 1.For the other two parameters no trend was found.
Overall, the analysis shows that NBR 6118 [1] approximate stiffness criterion provides better predicting results in terms of safety, precision and economy for columns made with concrete of f c ≤ 55 MPa .

Columns made with concrete of f c > 55 MPa
Since NBR 6118 [1] procedures are being changed to allow the use of concrete with compressive strength above 55 MPa, a comparative analysis of the current approximate design approaches for slender columns is presented next.The results of the M test /M pred ratio for columns made with concrete of f c > 55 MPa are presented in the Table 5.They were obtained from the comparison with test results of 65 columns.The analysis of these results reveals that NBR 6118 [1] approximate stiffness criterion and approximate curvature method have average and median values for the M test /M pred ratio larger than the unity.The approximate stiffness criterion is more conservative since its average and median are larger.In terms of accuracy, the approximate stiffness procedure is also more precise, since the coefficient of variation is 3% smaller.With respect to safety, the demerit scale (shown in the part B of Table 5) shows that the results obtained with the approximate stiffness criterion are safer with only 4,6% of the M test /M pred values below the appropriate safety range (smaller than 0.85) and all of them are in the low safety range.This fact is also reflected in the total demerit point score of this method, which is much smaller.The approximate curvature method, on the other hand, has larger percentage of results in the low safety and dangerous range with no values in the extremely dangerous zone.Analysis of the effects of concrete compressive strength f c , of the column slenderness ratio l and of the dimensionless axial force u on the ratio M test /M pred were also done in this case for both approximate design procedures.For both design methods, the results (Figure 2) showed a slight reduction in the M test /M pred ratio with increasing values of the dimensionless axial force u.For the other two parameters no trend was found.The analysis of these results also indicates that NBR 6118 [1] approximate stiffness criterion provides better predicting results in terms of safety, precision and economy for columns made with concrete of f c > 55 MPa.

Concluding remarks
The

A c =
gross area of column cross-section.b = side dimension of column cross section.f c = concrete cylindrical compressive strength at time of tests.f cd = design concrete compressive strength.h = overall column height.l e = column unsupported length.M d,tot = total bending moment in midheight section.M 1d,A = column maximum factored end moment.N d = factored axial force.a b =factor relating actual moment diagram to an equivalent uniform moment diagram.l = slenderness ratio.n = dimensionless axial force.r l =longitudinal steel ratio.

Figure
Figure 2 -M /M versus dimensionless axial force n test pred

Table 1 -
Column database (f ≤ 55 MPa) are below the appropriate safety range (smaller than 0.85) with 2% in the extremely dangerous zone (smaller than 0.5).This fact shown in Table3.A score value smaller than 0,5 is worse in terms of safety than one greater than 2. On the other hand, a score value equal 2, corresponded to the low safety zone is attributed to an extremely conservative ratio for being not economical.The total demerit point score of each design code equation is calculated by summing the products of the percentage of M test /M pred obtained in each range times the demerit value attributed to that range.The largest the total demerit point score is the worst is the performance of the design equation.

of slender reinforced concrete rectangular columns subjected to eccentric loads by approximate methods is
also reflected in the total demerit point score of this method, which is much larger.The approximate stiffness procedure, on the other hand, has smaller percentage of results in the low safety and dangerous range with no values in the extremely dangerous zone.

Table 5
goal of this paper was to analyze with respect to safety, precision and economy NBR 6118 [1] approximate design criteria for slender rectangular RC columns subjected to eccentric loads by comparing to experimental results.The study includes columns built with conventional concrete (f c ≤ 55 MPa) and with concrete having compressive strength above 55 MPa.The simplified methods prescribed by NBR 6118 [1] analyzed herein were -approximate curvature method and approximate stiffness procedure.For the comparative study a database was created containing the results of columns tested in the laboratory.The overall analysis of the comparative study indicates that the approximate stiffness procedure provides better results than the approximate curvature method in relation to the experimental results of columns built with conventional concrete (f c ≤ 55 MPa) and fabricated with concrete having compressive strength above 55 MPa.