Sensitivity Analysis and Optimum Design Curves in Concrete Beams Reinforced with FRP Bars

Sensitivity analysis and Optimum design curves in concrete beams reinforced with FRP bars

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Abstract: Optimization of structural elements is one of the most important topics in structural engineering with a wide range of application such as Concrete beams reinforced with fiber-reinforced polymer (FRP) bars. Thus, the objective of this paper is to employ Lagrangian Multiplier Method (LMM) to acquire minimum cost design of singly reinforced concrete beams with rectangular shape. Concrete and FRP material costs are used as objective cost functions to be minimized, and the ultimate flexural strength of the beam is considered to be the main constraint. Optimum designs are obtained by using the LMM and are presented in closed form solution. The application of the proposed relations and curves are demonstrated through one real life example of SRB design situation and it is shown that the minimum cost design is actually reached using the proposed method.

Keywords: composite reinforced concrete beam, optimization, design curves, FRP bar,  Lagrangian Multiplier Method, ACI

1.  Introduction

Plain concrete members are strong in compression, but they cannot bear tension stresses due to their natural characteristics. Thus, at the very beginning it was used for simple, massive structures. Later designers and builders developed techniques to embed bars into concrete members so as to provide additional capacity to resist tensile stresses. This revolutionary effort ended with what is known as reinforced concrete (RC). Steel bars were the only option for reinforcing concrete structures. Later, with the advent of Fiber reinforced polymer (FRP) instead of steel bars to achieve higher performance in reinforced concrete members revealed that use of FRP is remarkably beneficial [1]. Corrosion of steel reinforcements in intensive environments can cause serious damage in RC structures [2]. As a solution to prevent the damage, the use of FRP bars as flexural reinforcement was presented. Due to, non-corrosive property, the use of FRP bars which can reduce protection and rehabilitation costs[3], [4]. Researchers in concrete engineering have recently realized that it is time to present design optimization methods for such applications [5], [6]. The goal of structural optimization is to make the best or most effective use of materials for structural elements that can satisfy design requirements which are usually to reduce design costs of materials. With accurate implementation to detailing and quality control; concrete structures have much more overtaking position in weight, strength and ductility compared to metal-framed structures[7]. Optimization design plays a key role in economic design of reinforced concrete elements. The optimal section economic designs carried out low reinforcement rates owing to high cost of reinforcing bars[8], [9]. Beams as structural components are of the main parameters that implementation on their optimized design can reduce the overall cost of the structure[10]. Beam component costs significantly are controlled based on their dimensions, ratio of reinforcements and costs per unit [11]. The objective of optimization (e.g. minimizing the cost, weight, etc.) thus, the range of design variables and constraints are widely presented in recent literatures with different optimization methods to acquire an optimized design [12]–[15]

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Designing structural elements requires adjudication, insight, experience; furthermore, the ability to design structural elements at safe levels concerning serviceability and economy. Design codes and codes of practice do not implement on aforementioned parameters. so as the designers have to perform multiple design cycles to meet the best solution. It is relatively difficult to achieve minimum design cost of RC beams using conventional methods while there are large number of design solutions that yield equal bending moment capacity[16]. Thus, implementing to gain a numerical optimization technique becomes necessary to develop a cost effective design approach [17], [18]. The optimization includes selecting the design parameters concerning the overall cost of the beam to be minimized by which performance and geometrical constraints are satisfied. The prerequisite to design of reinforced concrete beams are to limit geometrical dimensions because of architectural considerations and the load applied to the beam which is greater than limited moment capacity in beams with doubly reinforced system compared with singly reinforced concrete beams[8]. Attempt to minimization of weight of the structures by controlling the compression steel in concrete structures has been done by researchers [19] whereas most researchers are acquiring techniques to reach optimized cost of the structures[20]. It is obvious that the weight of the structure is proportional to the cost of materials; thus minimizing the use of materials should be the main objective in designing of RC structures. Many researchers implemented on using ultimate load method to design concrete elements[21]–[24], which not many employed limit state method. To meet constraint satisfaction criterion, researchers implemented to satisfy moment capacity constraints[10], [18], others considered weight of the structure in their analyses [16], [21]–[23] and likewise few researchers have given design equivalents related to the deflection by considering factored loads[5], [6].

Optimization techniques can be categorized into three main fields: mathematical programming, methods based on optimality criteria and heuristic search algorithms[16] . A few researchers have applied heuristic search algorithms such as artificial bee colony, simulated annealing and artificial neural networks for acquiring the optimum design of reinforced concrete beams[25]–[27]. Lagrangian multiplier methods has been successfully applied in optimizing constrained problems in engineering[28]. These methods perform direct transformation of constrained problems to unconstrained problems resulting in a solution through a system of sequential unconstrained optimization subordinate problems. This approach has been employed for minimizing cost design of singly reinforced concrete beams with rectangular shapes to resist external action of flexural bending based on British Standard [10]. Creativity of researchers in optimization engineering has led to combination of Lagrangian Multiplier Method with other optimization approaches. For instance, an application of the Continuum-type Optimality Criteria (COC) method to the design of RC beams where the conditions of minimality are derived using the augmented Lagrangian method has been proposed which the cost that is minimized consists of concrete, reinforcement and formwork costs with active constraints on maximum deflection, bending and shear strength[5].

Thus, the objective of this paper is to employ LMM to acquire minimum cost design of singly reinforced concrete beams with rectangular shapes based on ACI-318-14 and ACI-440-15 codes. Concrete and FRP material costs are used as objective cost functions to be minimized, and the ultimate flexural strength of the beam is considered to be the main constraint. The optimum design curves achieved in this study can be used for minimum cost design of the beams without prior knowledge of optimization. The results were presented in closed form equations which can be easily used for scientists and engineers and without the need for performing design process.

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Concrete properties

Several models representing the behavior of concrete in compression are available. The compressive stress–strain diagram for normal strength concrete proposed by Todeschini, Bianchini, and Kesler[29] is represented in Figure 1  f′c is the design concrete compressive strength and εcu is the maximum usable concrete compressive strain and is assumed equal to 0.003.

Figure.1

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For ultimate strength calculations which controlled by concrete crushing, ACI 318-14 [30]allows the approximation of the stress–strain curve to an equivalent rectangular stress, or “stress block,” distribution as discussed in Section 4.5. In the design examples discussed in the chapters of Part 3, the stress–strain curve proposed by Todeschini is adopted when concrete crushing does not control failure.

Modulus of elasticity of concrete. The modulus of elasticity of concrete is dependent to concrete compressive strength (f′c), concrete age, properties of cement and aggregates, and rate of loading. Based on statistical analysis of experi­mental data available for concrete with unit weights, w, varying between 90 and 155 pcf (1442 and 2483 kg/m3), ACI 318-14[30] provides the follow­ing empirical equation for computing the modulus of elasticity:

Econcrete=0.43 w1.5fc’

For normal-weight concrete weighing 145 pcf (2323 kg/m3), the following simplified equation is suggested by ACI 318-14:

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Econcrete=4700fc’

4.5.3 Minimum FRP reinforcement

ACI 440.1R-15 prescribes that at every section of a flexural member where tensile reinforcement is required by analysis,

Afprovided should not be less than the area given by:

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Afmin=0∙41 f’cffubwd>2.3ffubwd  →   ρmin=0∙41 f’cffu>2.3ffu

(1)

Where

bwand d are the cross-section web width and the distance from the extreme com­pression fiber to the centroid of tension reinforcement, respectively.

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It has been shown that the requirements of Eq. (1) may become unrealistic for large concrete cross sections. It is therefore suggested that Eq. (1) need not be applied in a member where at every cross sec­tion, the area of tensile reinforcement provided is at least one-third greater than that required by analysis.

Maximum FRP reinforcement

Provision 10.3.5 in ACI 318-14 limits the minimum tensile strain at failure in the longitudinal steel reinforcement of flexural members to a value of 0.004, which corresponds to roughly twice the yield stain of a Grade 60 steel (420 MPa). This strain limit is to ensure that the failure of the steel-RC structural member will always be ductile. Even though this limit loses rel­evance in the case of FRP reinforcement, it may be argued that irrespective of the fact that FRP bars do not yield, this strain threshold would at least ensure some visible level of distress in terms of deflection and crack width for a flexural member approaching failure. According to this provision, the maximum reinforcement ratio, ρ_max, for an FRP-RC member in flexure would be obtained as Eq.(14)

ρmax=0.85β1f’c0.004Ef0.0030.003+0.004=91.1β1f’cEf (16)

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Optimization Design

1.1. Assumptions

As it has been demonstrated experimentally, irrespectively of the reinforcing material used (FRP), the basic assumptions for the flexural theory of RC beams reinforced with FRP bars can be summarized as follows:

1. Plane sections remain plane; this means that shear deformations can be disregarded (Euler–Bernoulli beam theory)

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2. Perfect bond exists between reinforcing bars and the surrounding concrete; in other words, the strain in the reinforcement is equal to the strain in the concrete at the same level.

3. Stresses in both concrete and reinforcement are presented based on the strain level reached in each material using the appropriate constitutive relations for concrete and reinforcing bars. In this study, the case of concrete, up to the serviceability limit state, a linear–elastic relationship will be used; past the linear elastic point up to crushing, either the Todeschini model or the equivalent stress block can be used. Regardless of the limit state considered FRP reinforcing bars is considered linear–elastic.

4. The tensile strength of the concrete is neglected.

5. The concrete is considered to fail when it reaches a maximum specified compressive strain.

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The basic safety relationship at the ultimate limit state can be written as:

∅Mn≥Mu

2.2. Optimization method

LMM method is employed to obtain optimum design of RC beam element. The problem’s goal is to minimize the objective function presented in the following form;

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s=fx1.x2.x3.….xn                      (1)

Subjected Constraints

hix1.x2.x3.….xn=0                  i=1.2.…. p                   (2)

Where n is the number of independent variables x_i and p is the number of constraints.

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To solve the optimization problem based on Eqs. (1) and (2), the unconstrained Lagrangian function L is presented in the following form:

Lx1.x2.x3.….xn.λ1 .λ2.λ3.….λp=fx1.x2.x3.….xn+∑i=1pλihix1.x2.x3.….xn    (3)

where λ_p parameters are Lagrangian multipliers. The necessary conditions for Lagrangian function are as follows:

∂L∂xk=∂f∂xk+∑i=1pλi∂hi∂xk=0             k=1.2.3. …. n           (4)

∂L∂λi=hi=0                                  i=1.2.3.….p        (5)

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Expressions of (4) and (5) show that we have a system of n + p equalities with n + p unknowns. Therefore, their solution produces stationary values for x₁, x₂, x₃, …, x_n and λ₁, λ₂, λ₃, . . ., λ_p  in which an optimized solution would be achieved.

Applying the Lagrangian Multiplier Method

Reinforced concrete beams with FRP bars are primarily designed to bear the applied loading which are mentioned in ACI 440-1R-2015 for singly reinforced concrete beams[3]. In this study minimizing the cost of the beam which is subjected to nominal bending moment Mn is the objective which means the cost of the beam is considered to be the objective function and the nominal flexural strength as the active constraint.

Singly reinforced concrete beam 

The cost function is formulated as objective function to be minimized afterwards the nominal flexural strength of RC beam is obtained and proposed as constraint function. At last the Lagrangian function is presented and solved according to Eqs. (3) , (4), (5), respectively. Setting the ratio of the material costs to q= C_f/C_c where C_{f and} Cc are cost per unit volume, respectively. The total cost objective function per unit length which is based on reinforcement area of the beam, geometry and material costs is as follows:

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C=Ccbρfrpqd+1+rd                 (6)

where ρ_frp is the reinforcement ratio As/bd, As it is the area of tensile rebar, b and d are the breadth and effective depth of the section, respectively and r is the ratio of reinforcement cover to effective depth d. b is assumed to be constant and concrete cover does not change in the design procedure. Based upon these simplification Eq. (6) can be re-written in the following form:

C=ρfrpqd+1+rd                           (7)

Flexural strength design at the section of a member, refereed as nominal flexural strength, of the member multiplied by the strength reduction factor (φ). Based on ACI 440-1R-2015 relation between nominal and ultimate flexural strength is:

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φMn≥Mu       (8)

In Eq. (8), ϕMn is the factored bending moment capacity of the member and is based on the member geometrical parameters, where the reinforcement is located, and the mechanical properties of the materials; the safety factors associated with the materials or the failure mode depending upon the calculation procedures followed. The second term of Equation (8), Mu, is the factored bending moment resulting from the analysis of the member and is a function of the member. The nominal flexural strength at a section can be expressed in terms of FRP reinforcement ratio mentioned in ACI 440-1R-2015 is presented in following equation:

Mnbd2=ρfrpffrp1-0.59ffrpρfrpfc’     (9)

Where

Mnis the nominal design moment,

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ffrpis the reduced strength of FRP bar ,

∅ ffrp, and

f’cis the reduced compressive strength of concrete,

∅fc, where

∅frp,

∅c, are the strength reduction factors of FRP and Concrete, respectively.

Figure.2 Strength reduction factor as a function of reinforcement ratio based on ACI 440.1 R 2015

The geometry of singly reinforced beam with stress block presented by Todeschini, Bianchini, and Kesler[29] is shown in Figure 3

Figure 3

The formulation of Lagrangian function and then solving the problem to achieve the minimized cost design is presented according to the proposed method, the unconstrained problem is formed using the Lagrangian function L according to the Eq.(3) as follows:

L=ρfrpqd+1+rd+λ ρfrpffrp1-0.59ffrpρfrpfc’bd2-Mn         (10)

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The Eq. (10) is the Lagrangian function for singly reinforced concrete beam formed by applying partial derivatives of the Lagrangian function according to Eq.(10). By simplifying and equating the partial derivatives to zero. Then substituting the equations, the optimum tensile reinforcement ratio of concreate beam is achieved as follows:

ρ frp  opt= 50fc’1+r59ffrp1+r+50fc’q=11.18ffrpfc’+(q1+r)                  (11)

Eq.(11) is used to obtain the optimum tensile reinforcement ratio for singly reinforced concrete beam . Also, the optimum effective depth for singly reinforced concrete beam is then derived by substituting Eqs. (9) and (11):

dopt=Mnbρfrp opt  ffrp1-0.59ρfrpoptfc’         12

Since Eqs.(11), and (12) are for singly reinforced concrete beam, it is essential to  calculate upper bond

ρ frp  opt. According to ACI 318-14, tensile reinforcement ratio must be bonded by the allowable maximum reinforcement ratio presented in Eq.(16)  which is the bond of limiting reinforcement ratio for which both FRP and concrete gain their ultimate strength assuming Euler-Bernoulli’s principle and linear behavior.  The compressive reinforcement ratio must be bounded by the allowable minimum reinforcement ratio as in ACI regulations.

ρ frpb= 0.85 β1fc’ffrpEfεcuEfεcu+ffrp   (13)

ρ frpb=0.85 β1fc’0.004Efrp0.0030.003+0.004=91.1 β1fc’ffrp Efrp  (14)

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It should not be neglected that the value of maximum reinforcement ratio cannot be practically applied for GFRP reinforced concrete flexural members since it is impossible to fit may bars in the cross section.

Figure.4 represents the optimum tensile reinforcement ratio for singly reinforced concrete beam given by Eq.(11). A group of lines have been drawn for various material cost ratio q for a constant value of r = 0.15. The graphs are restricted by the maximum and minimum limitations of

ρ maxand

ρ minon the reinforcement ratios given by ACI 318-14 and ACI 440-1R-15. It must be noted that these limitations are as the boundary constraints in the process of optimization analyses and the nominal flexural strength as the main constraint. Thus, in the process it is considered to be active.

a)

fc’=20 MPa

b)

fc’=30 MPa

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Figure.4: Optimum tensile reinforcement ratio for singly reinforced concrete beams

The figure 4 represents different compression values of  20 to 30 Mpa, q= 10,25,50,75,100,150,200 and r=0.15 which covers all possible ranges based on ACI. For other values of q , it is possible to mathematically derive the valid stress ratio range for each design type.

Sensitivity analysis

In this section, the optimum solutions for singly reinforced concrete beam sections for different values of material stresses are compared and various practical design solutions are presented. The reinforcement ratio

ρ frp  optis restricted by bonding value

ρ frp  balanced. Through which the following inequality expression is obtained:

91.1 β1fc’ffrp Efrp≤ 50fc’1+r59ffrp1+r+50fc’q           (15)

Based on Eq.(15)