Reinforcement design of concrete sections based on the arclength method 1

Resumo The reinforcement design of concrete cross-sections with the parabola-rectangle diagram is a well-established model. A global limit analysis, considering geometrical and material nonlinear behavior, demands a constitutive relationship that better describes concrete behavior. The Sargin curve from the CEB-FIP model code, which is defined from the modulus of elasticity at the origin and the peak point, represents the descending branch of the stress-strain relationship. This research presents a numerical method for the reinforcement design of concrete cross-sections based on the arc length process. This method is numerically efficient in the descending branch of the Sargin curve, where other processes present convergence problems. The examples discuss the reinforcement design of concrete sections based on the parabola-rectangle diagram and the Sargin curve using the design parameters of the local and global models, respectively.


Introduction
Different constitutive relations have been used for the reinforcement design of concrete beams and columns.Mörsch [1] considered linear-elastic material behavior in allowable stress design.Several authors contributed to the flexural model that is nowadays used in ultimate limit state design.In the 1950s, Bernoulli's plane section hypothesis, equilibrium conditions, and nonlinear constitutive relationships for concrete and steel provided the basis for the development of reinforcement design theories.The literature review on concrete stress distribution presented by Hognestad [2] includes the contributions of Whitney [3] and Bittner [4] for rectangular and parabolarectangle diagrams, respectively.Simplified theories for ultimate strength under combined bending and normal force were consolidated in the early 1960s using approximate constitutive relations for concrete without any significant loss of precision.Mattock, Kriz, and Hognestad [5] adopted the rectangular diagram, while Rüsch, Grasser, and Rao [6] used the parabola-rectangle diagram.Concrete stress distribution is currently approximated by a rectangular stress block in ACI 318-14 [7].CEN Eurocode 2: 2004 [8], FIB Model Code 2010 [9], and ABNT NBR 6118: 2014 [10] all use the parabola-rectangle diagram.Such simplified stress diagrams require limiting strain states for reinforcing steel and concrete to ensure valid results under combined axial and bending effects.The approximated diagrams simplify numerical design procedures and design graphs, but do not represent the characteristics of concrete, such as the initial modulus of elasticity and the descending branch of the stress-strain relationship.Physical and geometric nonlinear analyses of reinforced concrete framed structures require stress-strain relationships that better describe the behavior of concrete.The Sargin model [11] represents several characteristics of the uniaxial behavior of concrete.The Sargin curve presented in the CEB-FIP Model Code 1990 [12] is defined by the initial modulus of elasticity, minimum compression stress, and critical strain.This curve also represents the descending branch of the stress-strain relationship.The convergence of the Newton-Raphson method is not stable in descending branches of stress-strain curves.This study presents a numerical method for the reinforcement design of concrete sections under combined bending and normal forces that is suitable for the Sargin curve.It is based on the arc-length technique, which is stable for negative derivatives of the stress-strain diagram.The numerical procedure automatically identifies the strain distribution in the ultimate limit state without having to consider a variable strain limit in compression (domain 5).Concrete and steel strain limits are not required but can be included to avoid excessive deformations.The examples given of reinforcement design apply both the parabola-rectangle and the Sargin curve.Design stress-strain diagrams are based on characteristic curves and code provisions for local and global analysis.

Simplifying assumptions
The following assumptions are considered at the outset: 1.There is no relative displacement between the steel and the surrounding concrete (steel and concrete have the same mean strain).
2. Cross-sections remain plane after deformation (Bernoulli's hypothesis).In the interests of simplifying the formulation, steel area is not deducted from concrete area.The influence of the type of aggregate is not discussed in the present investigation.

Constitutive relations
Compression stresses and strains are negative.The constitutive stress-strain relationship of steel is defined by where steel stress σ s is a function of steel strain ε s .The yield strength and modulus of elasticity of the steel are f y and E s , respectively.The corresponding yield strain ε sy is: The steel stress-strain curve is divided into three regions (Figure 1), which are respectively defined by: (3) The convergence of the Newton-Raphson process in the yielding range is stabilized by the reduced tangent modulus K s E s .The arclength method uses K s =0.The steel tangent modulus E s (ε s ) is defined by the derivative: Expressions and yield: (5) Concrete stress σ c is a function of concrete strain ε c , i.e., (6) Figure 1 Stress-strain relationship of steel Reinforcement design of concrete sections based on the arc-length method CEB-FIP Model Code 1990 [12] defines the Sargin curve from the minimum compression stress σ c1 , the critical strain ε c1 , and the initial modulus of elasticity E c0 (Figure 2).Concrete stress is defined by: (7) where ε c lim is the strain that separates the first two branches of the curve.The secant modulus of elasticity E c1 at the critical point is: Coefficient k 1 , variable η, and strain limit ε c lim are respectively defined by: (9) (10) (11) where Parameters b and c of equation (7) are respectively expressed by: (14) (15) where (16) The tangent modulus of elasticity of concrete, E c (ε c ), is defined by the derivative: Expressions and yield: The initial modulus of elasticity can be ascertained from equations and , i.e.,

(19)
The provisions of item 5.8.6 from CEN Eurocode 2:2004 [8] are also considered.The critical strain and initial elasticity modulus are, respectively, The partial factor for the elasticity modulus of concrete is γ cE =1.2 and the effect of the aggregate type is not discussed in this investigation.The mean compressive strength of the concrete is estimated by f cm = f ck + 8 MPa, where f ck is the characteristic compressive strength of concrete.The partial safety factors for concrete and steel are γ c = 1.4 and γ s =1.15, respectively, as recommended in ABNT NBR 6118:2014 [10].The effect of long-term sustained loads on the ultimate strength of concrete (Rüsch [13]) is considered by using α c = 0.85 in: The reinforcement design examples apply both the Sargin and the parabola-rectangle curve.The reinforcement design with the parabola-rectangle diagram assumes the constitutive relation, the limit strains, and the ultimate limit-state domains provided in ABNT NBR 6118:2014 [10].The numerical procedure proposed for the Sargin curve automatically identifies the strain distribution in the ultimate limit state without having to consider a variable strain limit in compression (domain 5).Concrete and steel strain limits are not required, but they are included to avoid excessive deformations.Steel strain is limited by: Concrete strain is limited by: (24) CEN Eurocode 2:2004 [8] provides the following expression: Figure 2 Stress-strain relationship of concrete Since ε c u1 > ε c lim (Figure 2), the branch of the Sargin curve defined by ε c ≤ ε c lim is not used in the reinforcement design.

Equilibrium and compatibility equations
Figure 3 shows the coordinate system of the cross-section.The concrete section is discretized into area elements dA c .The position of each element centroid is defined by the coordinates y c and z c .The position of each steel reinforcing bar, whose area is denoted as A s , is defined by the coordinates y s and z s (Figure 4).The stress resultants are presented in Figure 5. Positive normal forces N x are tension forces.Positive bending moments M y and M z correspond to tension stresses at the positive y and z faces, respectively.According to assumption 1, there is no slip between the steel and the surrounding concrete.Concrete and steel strains, which are respectively denoted as ε c and ε s , have the same value, i.e., (26) where ε is the strain at a point in the cross-section.Cross-sections remain plane after deformation (assumption 2).Strain ε at a point is expressed as: where k x is the strain at the origin.Parameters k y and k z are the curvatures with inverted signs.The compatibility equation ( 27) is rewritten as: where p=[1 y z] T is a position vector and k=[k x k y k z ] T is the generalized strain vector.The following expressions are obtained from the equilibrium conditions of the cross section: The equilibrium equations ( 29), (30), and (31) are rewritten as: where σ(ε) is the stress at a point and S = [N x M y M z ] T is the stress resultant vector.The following incremental equation is obtained from (32):  where the tangent matrix E is expressed by: (35)

Numerical methods for section analysis and reinforcement design
Figure 6 shows the solution for a nonlinear structural system of a single degree of freedom based on the Newton-Raphson process.The arc-length process is a variant of the Newton-Raphson method that controls the progress of the iterative process (Figure 7).The arc-length and load factor are denoted as l and λ, respectively.The incremental process is capable of passing through critical points.
The section analysis and reinforcement design methods are applicable, but not limited, to the Sargin stress-strain relationship.

Arc-length method
The arc-length method presented by Crisfield [14] is an alternative formulation of the method originally proposed by Riks [15].The stress resultant vector is defined as , where is a load factor and is the stress resultant vector that is established as a reference.The term ΔS i is defined as: where S i = [N x,i M y,i M z,i ] is the stress resultant vector associated with the generalized strain vector k i = [k x,i k y,i k z,i ] T at iteration i. Equation is rewritten as: where E i is a tangent matrix and Dk i is the increment of the generalized strain vector at iteration i. Equations ( 36) and (37) yield: where The arc-length l is expressed by: (41) The substitution of (38) into (41) yields:

Section analysis
The parameters required for section analysis are the steel and concrete properties, the geometric characteristics of the cross-section, the position and area of the reinforcing steel bars, the reference stress resultant vector , and the arc-length l.The maximum load factor found throughout the incremental process defines the cross-section strength.A brief summary of the iterative process is presented next.

I. Generalized strains k i at iteration i
Iteration i starts with vector k i .The first iteration can start with k 1 =0.

Figure 7
Arc-length method

II. Generalized stresses S i and tangent matrix E i
The strains ε = p T k i , stresses σ(ε), and tangent moduli of elasticity E(ε) are determined for each area element of the steel and concrete.Expressions (32) and (35) yield the generalized stresses S i and tangent matrices E i , respectively.

III. Load factors λ A and λ B
Equations ( 39) and (40) yield the auxiliary vectors and , respectively.Load factors λ A and λ B are the solutions of the quadratic equation established by ( 43) and (44).

IV. Load factor λ
The root of (43) that pushes forward the incremental process is selected.The first iteration elects λ 1 = max (λ A , λ B ).For iteration i>1, equation (38) yields: where Δk A and Δk B are the strain vector increments of roots λ A and λ B , respectively.The slopes θ A and θ B of roots λ A and λ B are respectively defined as: The load factor λ associated with the maximum slope θ = max (θ A , θ B ) is selected.The corresponding increment Δk A or Δk B is denoted Δk i .strain vector k i+1 of the next iteration is: The procedure returns to step II to start a new iteration.The process terminates when steel or concrete strains reach their limit values.Section strength is defined by , where is the maximum load factor found throughout the incremental process.

Reinforcement design
The parameters required for reinforcement design are the steel and concrete properties, the geometric characteristics of the crosssection, the position and relative area of each reinforcing steel bar, the minimum and maximum steel ratios, the reference stress resultant vector S ̅ , and the arc-length l.The design stress resultants are defined by λ d S ̅ , where λ d is the corresponding load factor.A brief summary of the iterative process is presented next.

I. Stress analysis for minimum reinforcement
The procedure in item 5.2 yields the maximum load factor for minimum reinforcement A s min .If , the required reinforcement is A s min and the process is terminated.Otherwise, and A INF = A s min .

II. Stress analysis for maximum reinforcement
The procedure in item 5.2 yields the maximum load factor λ s A max for the maximum reinforcement s max , the crosssection is not adequate and the process is terminated.Otherwise, .

III. Iterative process
The required reinforcement is estimated by linear interpolation The procedure in item 5.2 yields the maximum load factor for A s .If , the new limit is defined by and A s SUP = A s .Otherwise, and A S INF = A s .A new iteration restarts when A s SUP -A s INF >TOL d , where TOL d is the tolerance for the reinforcement design.The iterative process ends when A s SUP -A s INF ≤ TOL d .The required reinforcement is conservatively assumed to be A s SUP .This study considers TOL d = 1 × 10 -7 m².

Examples and numerical results
The reinforcement design procedure based on the arc-length method is implemented in two Fortran programs, which use parabola-rectangle and Sargin curves, respectively.Programs Fx4 and Fx5 are presented in Kabenjabu [16].The typical rectangular cross-section is defined by b y = 0.25 m and b z = 0.80 m (Figure 8).The rebar edge distances in y and z directions are d' y = 0.05 m and d' z = 0.05 m, respectively.The concrete section is discretized in 25×80 area elements.The section is considered doubly reinforced in most examples, but it is also studied as singly reinforced for pure bending.The characteristic yield strength of steel is f yk = 500MPa.

Reinforcement design of concrete sections based on the arc-length method
The examples investigate concrete grades C15, C30, and C45.The corresponding compressive strengths are 15 MPa, 30 MPa and 45MPa, respectively.Although C15 concrete is no longer in use, it is included in the study because of its widespread use in the past.The partial safety factors for concrete and steel are γ c = 1.4 and γ s = 1.15, respectively, as recommended in ABNT NBR 6118:2014 [10].N x , M y , and M z are the design values of the stress resultants.The examples are summarized in Tables 1 to 9, where A s tot is the required total reinforcement, ε c min is the minimum concrete strain, and ε s max is the maximum steel strain.The relative difference ΔA s tot ⁄ A s tot is defined as: where A s tot,PAR-RECT and A s tot,SARGIN are the required total reinforcement for parabola-rectangle and Sargin curves, respectively.The section is subjected to pure compression in Table 1.The Sargin curve yields lower reinforcement values than the parabola-rectangle diagram.The limit strain ε cu2 = -0.002 of the parabola-rectangle  diagram is smaller in modulus than the design value of the yield strain of the steel (ε syd = 0.00207).Steel stresses are higher with the Sargin curve since they reach the yield point.The differences between the two models are small and less than 5% in required reinforcement.Tables 2 and 3

Table 5
Doubly-reinforced cross-section subjected to compression and biaxial bending (e y = b y ⁄2 and e z = b z ⁄2)

Conclusion
The reinforcement design of concrete sections based on the parabola-rectangle diagram is a practical and well-established model.However, the initial modulus of elasticity and plastic range of the parabolarectangle diagram do not represent the actual behavior of concrete.Stress-strain relationships that better characterize concrete properties are needed for global limit analyses of concrete structures that consider their physical and geometric non-linear behavior.The Sargin curve is selected because it is a function of the peak point and initial modulus of elasticity and represents the descending branch of the stress-strain relationship.This research proposes a numerical procedure for the reinforcement design of concrete sections that uses an arc-length method and yields good convergence in the descending branch of the Sargin curve, without having to consider the distributions of strain limits around pivot C in domain 5. Strain limits for concrete and steel are not required, but they are included in order to avoid excessive deformation.The parabola-rectangle and Sargin curves are considered by using the code provisions for cross-sections and global limit analyses, respectively.The reinforcement design using the parabola-rectangle diagram is based on the section model in ABNT NBR 6118: 2014 [10].The Sargin curve is implemented according to the global nonlinear model in CEN Eurocode 2: 2004 [8].
The examples consider characteristic concrete strength values of 15, 30, and 45 MPa.The typical 0.25 m × 0.85 m rectangular cross-section is subjected to several loading cases which include pure compression and pure bending.Eccentricities in each direction of 1/4 and 1/2 of the corresponding dimension are considered in uniaxial and biaxial bending.
The required reinforcement shows a good correspondence in pure compression, pure bending of doubly-reinforced cross-sections, and uniaxial and biaxial bending with the highest relative eccentricity.The results also show good correspondence in pure bending of singly-reinforced cross-sections when reinforcing steel reaches the yield point.The comparison of the results shows that the use of compression reinforcement in beams to avoid the neutral axis in domain 4 also improves the correspondence between the results of the parabola-rectangle and Sargin curves.
More significant differences are observed in uniaxial and biaxial bending with the lowest relative eccentricity.The parabola-rectangle diagram is more conservative for C15 concrete, which shows a relative difference of -9.0%.The Sargin curve yields more reinforcement for C30 and C45, which present relative differences of 13.9% and 28.7%, respectively.The relative differences are higher for the lower reinforcement ratios, since the absolute differences are small and limited to -1.4 cm 2 , 1.4 cm 2 , and 3.6 cm 2 for C15, C30, and C45, respectively.Despite the good correspondence observed in most examples, the investigation shows that the results of the Sargin curve are not necessarily conservative when compared to the parabola-rectangle diagram.For this reason, a global limit analysis using the Sargin curve still requires the analysis of all cross-sections with the parabola-rectangle diagram.
The proposed reinforcement design method is efficient, numerically robust, and capable of considering other stress-strain relationships with or without descending branches.The examples use local and global analysis parameters for the parabola-rectangle and Sargin curves, respectively.The validation of a single calculation model for section and global limit analyses could motivate future investigations.

Figure 4
Figure 3 Cross-section

Figure 5
Figure 5 Stress resultants One of the roots of equation (43) corresponds to the factor λ of the next iteration.The appropriate root is discussed in the next item.
consider combined compression and uniaxial bending with eccentricities of e z = b z ⁄ 4 and e z = b z ⁄ 2, respectively, where e z = |M z ⁄ N x |.In Table 4, the section is subjected to com-pression and biaxial bending with e y = b y ⁄ 4 and e z = b z ⁄ 4, where e y = |M y ⁄ N x |.Table 5 discusses compression and biaxial bending with e y = b y ⁄ 2 and e z = b z ⁄ 2. Tables6 and 7consider compression and uniaxial bending with e y = b y ⁄ 4 and e y = b y ⁄ 2, respectively.Table 8 investigates pure bending with compression reinforcement.The relative differences are always less than 5% in Tables 3, 5, 7 and 8.

Figure 10 investigates
an example in pure flexion without compression reinforcement (Table 9, M z = 1050 kNm).The resultants of the compressive stresses in concrete are obtained by numerically integrating the parabola-rectangle and Sargin curves.The required reinforcements are A s PAR-RECT = 70.54cm² and A s SARGIN = 74.56cm², respectively.The corresponding level arms are z s PAR-RECT = 0.524 m and z s SARGIN = 0.510 m, respectively.Reinforcing bars do not reach the yield point in either case.The parabolarectangle diagram and the Sargin curve yield σ c topo = σ c min and |σ c top | < |σ c min |, respectively, where σ c top is the stress at the top of the section and σ c min is the minimum compressive stress in the concrete.The concrete and steel force resultants are R c = R s = 2003.73kN and R c = R s = 2057.53kN for the parabola-rectangle and Sargin curves, respectively.

1
Doubly-reinforced cross-section subjected to pure compression

Table 2
Doubly-reinforced cross-section subjected to compression and uniaxial bending (e z = b z ⁄ 4)

Table 3
Doubly-reinforced cross-section subjected to compression and uniaxial bending (e z = b z ⁄ 2)

Table 4
Doubly-reinforced cross-section subjected to compression and biaxial bending (e y = b y ⁄4 and e z = b z ⁄4)

Table 6
Doubly-reinforced cross-section subjected to compression and uniaxial bending (e y = b y ⁄4)

Table 8
Doubly-reinforced cross-section subjected to pure bending