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Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load

Data publikacji: 15 Jul 2022
Tom & Zeszyt: AHEAD OF PRINT
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Otrzymano: 04 Feb 2022
Przyjęty: 30 Mar 2022
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
2444-8656
Pierwsze wydanie
01 Jan 2016
Częstotliwość wydawania
2 razy w roku
Języki
Angielski
Abstract

With the help of fractional differential equations, this article studies the failure morphology, fatigue strength, and fatigue life of corroded reinforced concrete beams under fatigue loading. Studies have shown that the mid-span deflection of low-corrosion reinforced concrete beams is smaller than that of uncorroded reinforced concrete beams. The corrosion-fatigue coupling effect accelerates the fatigue crack growth rate of steel bars. This reduces the fatigue modulus of concrete and causes the stiffness of the beam to degrade. The research results provide a theoretical basis for the fatigue performance evaluation of corroded reinforced concrete beams.

Keywords

MSC 2010

Introduction

A certain degree of damage will inevitably occur to the structure due to the environment and load within the design service life. The most common ones are steel corrosion, concrete carbonation, freeze-thaw cycles, etc. These factors lead to a reduction in the seismic performance of the structure. The dynamic response of reinforced concrete structures under the action of corrosion and earthquake affects the comfort of occupants and directly affects the safety and durability of the structure. The structure is subjected to reciprocating loads in an earthquake, and the restoring force model is the basic element to describe the ability of reinforced concrete structures to resist deformation under this action. China is a country prone to earthquakes, and many structures are in a severely corrosive environment at the same time. Many tests and studies have shown that corroded reinforced concrete structures’ bearing capacity, stiffness, flexibility, and energy consumption have been greatly reduced. The shape of the skeleton curve of corroded reinforced concrete members is the same as that of intact members, but the various parameters of the skeleton curve are reduced [1]. The current research results are based on the experimental research of the specific force state or characteristic geometric specimens. The results are different. Therefore, to some extent, the existing research has certain limitations.

Low-cycle cyclic loading test of corroded reinforced concrete members

This experiment aims to obtain the restoring force model of corroded reinforced concrete members through experimental research [2]. The size, reinforcement, and loading form of the 6 specimens are the same, but the degree of steel corrosion is different. The size and reinforcement of the test piece are shown in Figure 1.

Figure 1

Appearance and dimensions of the test piece (mm)

The corrosion of steel bars adopts the electrochemical corrosion method. After pouring and curing, the test piece is soaked in 5% sodium chloride solution for 20 days, and then the DC power supply is applied to accelerate the corrosion electrochemically. The measured corrosion rates of specimens L1-L6 were: 0, 2.76%, 5.47%, 8.63%, 9.81%, 11.59%, respectively. Both ends of the test piece are hinged, and vertical repeated loads are applied to the end of the middle column.

When loading, the load control cycle is first used to load until the specimen enters the yielding state. After the specimen yields, the displacement control cycle loading is adopted according to the yield displacement. The load control cycle level is 0.5kN, the displacement control cycle level is the yield displacement Δy, and each level is cycled three times until the specimen fails. We divide the loading state of the specimen into cracking state, yield state, limit state, and failure state according to the pre-loading system of the test. The failure load should be less than 80% of the peak load [3]. After the specimen reaches the failure state, it can be determined that the loaded specimen has been destroyed. At this time, we will stop the experiment.

Selection of restoring force model of corroded reinforced concrete members

The internal force, deformation, concrete cracks, and slip deformation between reinforced concrete structures of corroded reinforced concrete structures undergo reciprocal changes under the action of an earthquake. It is necessary to have the constitutive relationship of the material or section performance under repeated loads. This is also called the resilience model [4]. The restoring force model is the basis of the nonlinear analysis of structures under earthquake action. It can be divided into two types: curve type and broken line type. The stiffness given by the curvilinear restoring force model is continuously changing, which is closer to the actual engineering. Still, it is insufficient in the determination of stiffness and the calculation method. At present, the steel structure mostly adopts the double-line type. The degenerate three-line restoring force model is often used in reinforced concrete structures [5].

Determination of seismic analysis parameters of corroded reinforced concrete members
The constitutive relationship of damaged concrete

When the steel bar begins to yield, the concrete has already passed the linear elastic stage (0 − 0.4 fc). Therefore, it is not accurate to calculate the elastic modulus of concrete as Ec before the steel bar yields. We construct a simplified concrete stress-strain calculation curve (Figure 2). Before the concrete strain reaches its peak, its stress and strain become a linear relationship.

σc={fcε0<εcεcu {\sigma _c} = \left\{{{f_c}\,{\varepsilon _0} < {\varepsilon _c}\, \le {\varepsilon _{cu}}} \right.

Figure 2

Constitutive relationship of damaged concrete

According to the assumption that the concrete strain of the normal section of the member conforms to the assumption of the plane section, we take the neutral axis height conversion factor β1 = 0.8. The article can obtain ε0 = 0.001456.

The deterioration of the mechanical properties of corroded steel bars and the constitutive relationship
Deterioration of the mechanical properties of corroded steel bars

According to the test results of corroded steel bars, it can be seen that the mechanical properties of the steel bars have changed after corrosion [6]. fyc=(10.151ρ)fy {f_{yc}} = \left({1 - 0.151\rho} \right){f_y} fuc=fu(10.0132ρ) {f_{uc}} = {f_u}\left({1 - 0.013\,2\rho} \right) Eyc=Ey(10.0117ρ) {E_{yc}} = {E_y}\left({1 - 0.011\,7\rho} \right)

In formulas (2)(4), fy, fu and Es are the nominal yield strength, nominal ultimate strength, and nominal elastic modulus of the uncorroded steel bars, respectively.

The constitutive relationship of corroded steel bars

We need to determine the critical point of steel corrosion rate when the yield platform of the corroded steel stress-strain curve is degraded [7]. After statistical analysis of the tensile load-displacement curve of corroded steel bars and the corresponding corrosion rate, the article concludes that the critical point of the cross-sectional loss rate of different types of steel bars is 20%. According to the test results, the change law of strengthening strain εshc and ultimate strain εsuc with the increase of steel corrosion rate ρ is obtained. εshc={fycEsc+(εsh0fy0E0)(1ρsρs,cr)(ρsρs,cr)εsyc=fycEsc(ρs>ρs,cr) {\varepsilon _{shc}} = \left\{{\matrix{{{{{f_{yc}}} \over {{E_{sc}}}} + \left({{\varepsilon _{sh0}} - {{{f_{y0}}} \over {{E_0}}}} \right)\left({1 - {{{\rho _s}} \over {{\rho _{s,cr}}}}} \right)} \hfill & {\left({{\rho _s} \le {\rho _{s,cr}}} \right)} \hfill \cr {{\varepsilon _{syc}} = {{{f_{yc}}} \over {{E_{sc}}}}\,\,\left({{\rho _s} > {\rho _{s,cr}}} \right)} \hfill & {} \hfill \cr}} \right.

In the formula, εsy0, εsh0 is the yield strain and strengthening strain of the uncorroded steel bars. εsuc=e3.519ρsεsh0 {\varepsilon _{suc}} = {e^{- 3.519{\rho _s}}}{\varepsilon _{sh0}}

The mathematical model of the stress-strain relationship of the corroded steel can be obtained by using the calculation formulas of the characteristic parameter change law. σsc={Escεsc(εscfyc/Esc)fyc(fyc/Esc<εscεshc)fyc+εscεshcεsucεshc(fucfyc)(εsc>εshc) {\sigma _{sc}} = \left\{{\matrix{{{E_{sc}}{\varepsilon _{sc}}\,\left({{\varepsilon _{sc}}\, \le {f_{yc}}/{E_{sc}}} \right)} \hfill \cr {{f_{yc}}\,\left({{f_{yc}}/{E_{sc}} < {\varepsilon _{sc}} \le {\varepsilon _{shc}}} \right)} \hfill \cr {{f_{yc}} + {{{\varepsilon _{sc}} - {\varepsilon _{shc}}} \over {{\varepsilon _{suc}} - {\varepsilon _{shc}}}}\left({{f_{uc}} - {f_{yc}}} \right)\,\left({{\varepsilon _{sc}} > {\varepsilon _{shc}}} \right)} \hfill \cr}} \right.

Stiffness degradation of corroded reinforced concrete structures

It can be seen from Fig. 3 that as the corrosion rate increases, the stiffness of the test piece gradually decreases as a whole [8]. Corroded specimens have relatively low initial stiffness due to rust damage and cracks, and stiffness degradation is relatively slow. With the continuous increase of displacement, the decay rate of the stiffness of the rusted specimen and the uncorroded specimen decreases and finally stabilizes.

Figure 3

Specimen stiffness

We perform a fitting analysis on the average value of the relative stiffness attenuation rate of the rusted specimen (see Figure 4) to obtain the relative decay rate of the loading and unloading stiffness of the rusted specimen. See equations (8) and (9) for details: y=1.67e0.515x+0.002 y = 1.67{e^{- 0.515x}} + 0.002 y=1.571e0.124x0.388 y = 1.571{e^{- 0.124x}} - 0.388

Figure 4

Relative stiffness decay rate of corroded specimen

It can be seen from Figure 4 that the stiffness of the specimen continues to decrease with the increase of displacement. The stiffness degrades more rapidly after the specimen is cracked, especially after reaching the yield load [9]. When the load reaches the peak load, the attenuation of stiffness tends to be gentle.

Determination of the characteristic points of the skeleton curve of corroded reinforced concrete members

First, assume that the cross-section of the structure remains flat after deformation. The concrete strain on the section is distributed in a straight line. The tensile strength of concrete is not considered, but the slippage of corroded steel bars and concrete should be considered.

Yield load and yield displacement

The relationship between the yield load of the test member and the yield moment of the section is: Fy=2My/l {F_y} = 2{M_y}/l

l is the simplified calculation of the length of the cantilever section. My is the yield bending moment of the section. Fy is the yield load. Corrosion of steel bars changes the structure of damage to a certain extent. A member with a low corrosion rate and a good design is a tensile failure. The steel bar yields first; however, the concrete first reaches the ultimate compressive strain when the member with a larger corrosion rate is destroyed, and the failure form is a brittle failure. However, there is no conclusion about the critical corrosion rate that affects the failure form of the component. Therefore, the following two situations are discussed:

1) Brittle failure and corroded reinforced concrete members have ΣM = 0 at the center of the steel bar in the compression zone. Therefore, we can obtain the yield bending moment of the section as: My=Ascσsc(h0ξh0)+Asc'σsc'(h0a)+bfcqc'ξ2h023 {M_y} = {A_{sc}}{\sigma _{sc}}\left({{h_0} - \xi {h_0}} \right) + A_{sc}^{'}\sigma _{sc}^{'}\left({{h_0} - a} \right) + bf_{cqc}^{'}{{{\xi ^2}{h_0}^2} \over 3}

From Figure 5, we can assume that the ultimate compressive strain of concrete is taken as εcc = 0.0033 at the time of brittle failure, and then from the geometric relationship: σsc=εscEsc=0.00331ξξηEsc {\sigma _{sc}} = {\varepsilon _{sc}}{E_{sc}} = 0.0033{{1 - \xi} \over \xi}\eta {E_{sc}} σsc'=εsc'Esc=0.00331a/(ξh0)ξηEsc \sigma _{sc}^{'} = \varepsilon _{sc}^{'}{E_{sc}} = 0.0033{{1 - a/\left({\xi {h_0}} \right)} \over \xi}\eta {E_{sc}}

Figure 5

Force analysis of member section under the assumption of non-planar section

2) The damage component of ductile failure has ΣM = 0 in the centroid of the steel bar in the compression zone. Therefore, we can obtain the yield bending moment of the section as: My=Ascfyc(h0ξh0)+Asc'σsc'(h0a)+bfcc'ξ2h023 {M_y} = {A_{sc}}{f_{yc}}\left({{h_0} - \xi {h_0}} \right) + A_{sc}^{'}\sigma _{sc}^{'}\left({{h_0} - a} \right) + bf_{cc}^{'}{{{\xi ^2}{h_0}^2} \over 3}

fcc' f_{cc}^{'} is the maximum compressive stress of concrete at yield. fyc is the yield strength of corroded steel bars.

The non-slip strain εsc' \varepsilon _{sc}^{'} of the steel bar complies with the assumption of a flat section. The bond performance of the rusted steel bar no longer meets the flat section's assumption after the bond performance's deterioration. The relationship between the actual strain εsc of the rusted steel bar and the non-slip strain εsc' \varepsilon _{sc}^{'} is εsc=ηεsc' {\varepsilon _{sc}} = \eta \varepsilon _{sc}^{'} . The geometric similarity of triangles assumed by the plane section is solved by transformation and reuse. Under the assumption of a non-planar section, it can be obtained according to the similar relationship and balance equation in Figure 5: fcc'=fycαEcηξ1ξ f_{cc}^{'} = {{{f_{yc}}} \over {{\alpha _{Ec}}\eta}}{\xi \over {1 - \xi}} σsc'=fyc(ξh0a)h0ξh0as \sigma _{sc}^{'} = {{{f_{yc}}\left({\xi {h_0} - a} \right)} \over {{h_0} - \xi {h_0} - {a_s}}} ξ=αE(ρsc+ρsc')+[αE(ρsc+ρsc')]2+2αEρsc \xi = - {\alpha _E}\left({{\rho _{sc}} + \rho _{sc}^{'}} \right) + \sqrt {{{\left[{{\alpha _E}\left({{\rho _{sc}} + \rho _{sc}^{'}} \right)} \right]}^2} + 2{\alpha _E}{\rho _{sc}}}

Where fyc = Escεscy,σc = Ecξ/(1 − ξ)εscy, σs'ξ/(1ξ) \sigma _s^{'} \approx \xi /\left({1 - \xi} \right)

Escεscy, αE = Esc / Ec, ρsc = Asc/(bh0), ρsc'=Asc'/(bh0) \rho _{sc}^{'} = A_{sc}^{'}/\left({b{h_0}} \right)

The test specimen can be simplified as a cantilever beam for calculation. The relationship between bending moment and curvature adopts an ideal two-fold line model. The yield curvature of the specimen is expressed as ϕyc = εscy/[(1 − ξ)h0]. According to its bending deformation distribution, it can be obtained: θyc=12ϕycl {\theta _{yc}} = {1 \over 2}{\phi _{yc}}l Δyc=12ϕycl23l=13ϕycl2 {\Delta _{yc}} = {1 \over 2}{\phi _{yc}}l{2 \over 3}l = {1 \over 3}{\phi _{yc}}{l^2}

Peak load and peak displacement

According to the test results of corroded reinforced concrete members, it is found that when the concrete is crushed, it is taken as the limit state of the structure. From Figure 8, the ultimate bending moment is: Mu=fckbx(h0x/2)εsc'EscAsc'(h0as') {M_u} = {f_{ck}}bx\left({{h_0} - x/2} \right)\varepsilon _{sc}^{'}{E_{sc}}A_{sc}^{'}\left({{h_0} - a_s^{'}} \right)

Considering the influence of rust on the bond-slip performance, the rotation angle in the limit state can be expressed as: θuc=12ϕycl+(ϕucϕyc)lp {\theta _{uc}} = {1 \over 2}{\phi _{yc}}l + \left({{\phi _{uc}} - {\phi _{yc}}} \right){l_p}

According to the balanced equation, the peak displacement can be solved: Δuc=13ϕycl2+(ϕucϕyc)lp(l0.5lp) {\Delta _{uc}} = {1 \over 3}{\phi _{yc}}{l^2} + \left({{\phi _{uc}} - {\phi _{yc}}} \right){l_p}\left({l - 0.5{l_p}} \right)

The experiment found that the corrosion degree of the stirrup in the reinforced concrete specimen is more serious than that of the longitudinal reinforcement. The stirrups are prone to severe corrosion or rust breakage, especially at the intersection of the stirrup and the longitudinal reinforcement. This is also very similar to the test results of many actual engineering structures. The severe corrosion of the stirrups leads to a decrease in the shear bearing capacity of the structure. We assume that the ductility reduction factor of the uncorroded specimen is 1. We perform regression analysis on the relationship between the rusted specimen's ductility reduction coefficient and the rust rate change to obtain equation (23). ηu=10.0196ρ {\eta _u} = 1 - 0.019\,6\rho Δuc'=ηuΔuc=(10.0196ρ)Δuc \Delta _{uc}^{'} = {\eta _u}{\Delta _{uc}} = \left({1 - 0.0196\,\rho} \right){\Delta _{uc}}

Failure shear and failure displacement

The failure load of the reinforced concrete structure is taken as the state. Where the peak load of the component is reduced by 15%, Fcu=0.85Fu {F_{cu}} = 0.85{F_u}

The failure displacement can be determined according to the peak displacement method, and the rotation angle in the failure state can be expressed as θpc=12ϕycl+(ϕpcϕyc)lp {\theta _{pc}} = {1 \over 2}{\phi _{yc}}l + \left({{\phi _{pc}} - {\phi _{yc}}} \right){l_p}

Section curvature in failure state: θpuc=ηεyc/(h0x/β1) {\theta _{puc}} = \eta {\varepsilon _{yc}}/\left({{h_0} - x/{\beta _1}} \right)

n is the ratio of steel bar strain to yield strain in the failure state. By calculating the peak displacement as: Δpc=13ϕycl2+(ϕpcϕyc)lp(l0.5lp) {\Delta _{pc}} = {1 \over 3}{\phi _{yc}}{l^2} + \left({{\phi _{pc}} - {\phi _{yc}}} \right){l_p}\left({l - 0.5{l_p}} \right)

We consider the influence of stirrup corrosion and introduce the ductility reduction factor to obtain: Δpc'=ηuΔpc=(10.0196ρ)Δpc \Delta _{pc}^{'} = {\eta _u}{\Delta _{pc}} = \left({1 - 0.019\,6\rho} \right){\Delta _{pc}}

Comparison of model calculation results and reference experiments

We compare and analyze the experimental value and the calculated value. The purpose is to verify the accuracy of the earthquake damage-based restoring force model of corroded reinforced concrete compression-bending members established in this study. The failure form of the specimen or structure with a corrosion rate of less than 10% shall be considered a ductile failure. The failure mode of the specimen or structure with a corrosion rate greater than 10% shall be considered as brittle failure. Figure 6 shows the comparison between the calculated value and the experimental value of the characteristic points of the skeleton curve of each specimen using the model in this article.

Figure 6

Comparison of experimental and calculated values of characteristic points of skeleton curve

It can be seen from Figure 6 that the calculated skeleton curve described by the research model in this paper is generally in good agreement with the experimental results of the references. Some calculated values are different from experimental values. This is due to the existence of test errors. As the number of repeated load cycles increases for each specimen, the hysteresis loop of the component gradually decreases, and the energy consumption capacity decreases. As the corrosion rate increases, the components’ bearing capacity, stiffness, ductility, and energy consumption gradually decrease. The above phenomenon is consistent with the test. This also shows that it is feasible to determine the restoration force model of corroded reinforced concrete compression-bending members based on earthquake damage according to the method in this paper.

Conclusion

(1) Rebar corrosion greatly influences the hysteretic performance of concrete members under repeated loads. As the corrosion degree of each specimen increases, the hysteresis performance of the specimen decreases. Especially the severely corroded specimens are more likely to be brittle failures in earthquakes. (2) Corrosion of steel bars changes the damage pattern of the structure to a certain extent. The failure mode of structures or components with a high corrosion rate under repeated loads should consider the influence of corroded steel bars and analyze without the assumption of a flat section.

Figure 1

Appearance and dimensions of the test piece (mm)
Appearance and dimensions of the test piece (mm)

Figure 2

Constitutive relationship of damaged concrete
Constitutive relationship of damaged concrete

Figure 3

Specimen stiffness
Specimen stiffness

Figure 4

Relative stiffness decay rate of corroded specimen
Relative stiffness decay rate of corroded specimen

Figure 5

Force analysis of member section under the assumption of non-planar section
Force analysis of member section under the assumption of non-planar section

Figure 6

Comparison of experimental and calculated values of characteristic points of skeleton curve
Comparison of experimental and calculated values of characteristic points of skeleton curve

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