Corner cracking model for non-uniform corrosion-caused deterioration of concrete covers

Construction and Building Materials 234 (2020) 117410

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Corner cracking model for non-uniform corrosion-caused deterioration of concrete covers Yanlong Zhang, Ray Kai Leung Su ⇑ Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China

h i g h l i g h t s
A corner cracking model for simulating corrosion-caused deterioration is proposed.
The non-uniform distribution of rust and the cover boundary are considered.
The mechanisms of the initiation and propagation of corner cracking are analyzed.
The cover surface crack opening and the budging of cover are predicted.
This model has been verified by numerical models.

a r t i c l e

i n f o

Article history: Received 4 July 2019 Received in revised form 24 October 2019 Accepted 26 October 2019

Keywords: Concrete cover Non-uniform corrosion Cover cracking Corner rebar Analytical model

a b s t r a c t The failure of concrete covers due to corrosion of the corner rebars is different from that of the interior rebars. To date, there is no specific analytical model for simulating corner cracking in the literature. In response, this paper proposes a novel analytical model, which considers both the non-uniform distribution of rust and the boundary conditions of the cover surface, to study the process and propagation of corner cracking. This nondestructive evaluation method is capable of obtaining the volume of corroded steel by measuring the bulging of the cover surface and/or the crack opening. This model has been verified by comparing the analytical results with the numerical results. Using the verified analytical model, the effects of the tensile strength of concrete, cover thickness, rebar diameter and diagonal crack angle on the bulging of the cover surface and crack opening are studied, which provide useful information for the design and prediction of the service life and crack width in reinforced concrete structures. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Reinforced concrete (RC) is the most prevalently used construction material worldwide. Corrosion-caused cover cracking, spalling and delamination not only cause aesthetic and water leakage problems, but also reduce the strength of concrete covers, accelerate the deterioration process and reduce the bonding strength between steel and concrete [1–4], which would adversely affect the durability, serviceability, strength and ductility of RC structures [5–10]. Under chloride environments, the steel bars in proximity to a concrete cover normally depassivates before those that are located further away [11,12], which leads to macrocell corrosion around the perimeter of the steel bars and causes the non-uniform corrosion of the rebars [13,14]. A number of non-uniform corrosion

⇑ Corresponding author. E-mail address: [email protected] (R.K.L. Su). https://doi.org/10.1016/j.conbuildmat.2019.117410 0950-0618/Ó 2019 Elsevier Ltd. All rights reserved.

models [11,12,15–23] have been proposed in the literature to describe the concrete cover failures from such corrosion. They assume that the rust is more distributed and corrosion-induced expansion pressure is higher on the side near the concrete cover [12,19–22,24,25]. A review of the literature shows that rust is distributed non-uniformly around the cross section area of steel [11,15–17,19,20,22,23,26] and along the rebars themselves [23,27]. In this study, non-uniform corrosion is defined as the non-uniform distribution of rust as cracking of the concrete cover is examined as a two dimensional problem for simplicity purposes and computational efficiency. Extensive studies have been conducted to study concrete cover cracking caused by rebar corrosion [11,12,15,17–24,26,28,29]. However, they have only studied the interior rebars and not the corner rebars as the failure modes of concrete covers caused by these two types of rebars differ [15,20,30–32] due to the following reasons. First, if the steel rebar is exposed to an environment with chloride ions, the ions can only penetrate into the interior rebar

Y. Zhang, R.K.L. Su / Construction and Building Materials 234 (2020) 117410

concrete

restrained side (a) Corrosion of interior rebar

Fig. 2. Typical cracking modes of interior and corner rebars.

2. Proposed corner cracking model In this section, the corner cracking model will be discussed in detail. Based on the results obtained from an FE model which will be discussed in Section 3, the entire progression of corner cracking can be observed as three main stages: (1) the elastic stage of concrete, (2) the partial cracking stage and (3) the stage of crack widening. 2.1. Distribution of rust The penetration of chloride ions into the surface of the concrete cover first causes corrosion of the side of the rebar that is facing the cover as shown in Fig. 1a and 1b in which ds,max1 and ds,max2 are the maximum thickness of the corroded steel of two different locations on the concrete cover, c1 and c2, respectively. The rust distribution caused by the non-uniform corrosion of the interior rebars is assumed to be a crescent shape, which has been widely observed in other experimental studies and adopted in numerical models [19,22,39]. For the corner of the cover, the chloride ions penetrate into the cover surface in both directions. Thus, the shape of the distribution of the rust as shown in Fig. 1c can be obtained by the superposition of the distributed rust in Fig. 1a and b [33,35]. When c1 and c2 are the same, i.e. c1 = c2 = c, ds,max1 and ds,max2 could also be equal, i.e. ds,max1 = ds,max2 = ds,max. Meanwhile, the maximum thickness of the rust on the two sides of the rebar near the cover dr,max1 and dr,max2 is equal, i.e., dr,max1 = dr,max2 = dr,max, as shown in Fig. 1c. Therefore, the total volume of corroded steel, Vsteel, can be written as:

V steel ¼ 2V steel;1 ¼ 2 

+

ds,max2

c2

"
# 1 pD2 1 pD D pDds;max  ds;max  ¼ 2 4 2 2 2 2

ð1Þ

dr,max2 rebar

concrete

=

restrained side (b) Corrosion of interior rebar Fig. 1. Corrosion of interior and corner rebars.

diffusion of chloride ions rust c1

c2

ds,max

dr,max rebar concrete

restrained side (c) Corrosion of corner rebar

restrained side

ds,max1

concrete

diffusion of chloride ions

rebar

c1

restrained side

dr,max1

corner rebar

interior rebar

diffusion of chloride ions

restrained side restrained side

diffusion of chloride ions

from one direction, and more rust would be produced on the side of the rebar that is in proximity to the cover as shown in Fig. 1a and b. However, the chloride ions can penetrate into concrete from two orthogonal directions and more rust is deposited on the two sides of the rebar near the cover as shown in Fig. 1c [22,33]. Second, while the expansion of rust caused by the interior rebar is mainly limited by the concrete on the three sides of the rebar with only one side without concrete as shown in Fig. 1a and 1b, the expansion of rust in a corner rebar is mainly limited by concrete on two sides of the rebar. However, there are still two sides of the rebar near the concrete, as shown in Fig. 1c. As a result, the configurations for cover cracking caused by these two kinds of rebars would be different as illustrated in Fig. 2. Therefore, research that specifically focuses on cover cracking caused by corner rebars is warranted. Nevertheless, very little research has been carried out on the corner cracking of concrete covers due to corrosion [34,35]. Ye et al. [34] and Zhao et al. [18,19] experimentally found that the rust distribution and cover cracking patterns caused by the corrosion of the corner rebars are different from those caused by the interior rebars. Finite element (FE) models have also been proposed to study the behavior of corner cracks that are caused by the corrosion of rebars [16,20,21,26,32,33,35–38]. However, they have mainly compared various parameters, such as the volume of rust or the expansion pressure on cover cracking in different regions of the concrete, but did not consider the deterioration process or progression of corner cracking, except [38]. Furthermore, the proposed numerical methods are relatively complex and time consuming, making them rather difficult for engineers and designers to implement in practice. To the best of the knowledge of the authors, there is currently no specific analytical model that simulates corner cracking in the literature. Therefore, it is imperative to develop a simple analytical model that would describe the process behind crack initiation, propagation and widening, as well as their progression, and predict the volume of corroded steel caused by corrosion of the corner rebars. In this study, a simplified corner cracking model is proposed by considering the non-uniform distribution of rust at the steel/concrete interface and on a flat cover surface. Furthermore, an FE analysis is conducted with ATENA to verify the proposed model by comparing the analytical and numerical results. Lastly, a parametric study is conducted to study the effects of the tensile strength of concrete, cover thickness, rebar diameter and diagonal crack angle on the bulging at the cover surface and crack opening.

restrained side

2

3

Y. Zhang, R.K.L. Su / Construction and Building Materials 234 (2020) 117410

where Vsteel,1 is the volume of corroded steel caused by the corrosion of an interior rebar. 2.2. Corrosion induced expansion The total amount of corrosion-caused rust, Vrust, originates from four sources [3,5,40], including the rust that: (1) occupies the original volume of the corroded steel, Vsteel; (2) penetrates into the porous zone at the steel/concrete interface, Vporous, (3) causes expansion pressure on the surrounding concrete, Vnet, and (4) is deposited into the corrosion-caused cracks, Vcracks. Thus, the total volume of rust can be written as:

V rust ¼ V steel þ V porous þ V net þ V cracks

ð5Þ

As all rust is produced from the corrosion of steel rebars, the relationship between Vrust, and Vsteel can be written as:

V rust ¼ bV steel

ð6Þ

where b is the ratio of the volume of rust to that of the original corroded steel. Therefore, the thickness of the rust dr,max can be written as follows:

dr;max ¼ bds;max

ð7Þ

WP,h,E WP,v,E deformed cover surface

deformed cover surface WA,v,E

V porous ¼

M

N c1

WM,h,E initial cover surface

ð8Þ

where d0 is the thickness of the porous zone. It has been found that rust penetrates into the surrounding concrete and causes expansion pressure on the surrounding concrete at the same time [40,45]. However, to simplify the calculation process, it is assumed that rust first fills the porous zone and then exerts pressure onto the surrounding concrete in this study

deformed cover surface C WA,v A P c2

WP,v

c2 B

3pd0 ðD þ d0 Þ 4

WP,h

A

P

The parameter b has been experimentally determined to be about 3 for rust mixtures [41,42], although it could largely range from 1.7 to 6.15 for different rust components (FeO = 1.7, Fe3O4 = 2, Fe2O3 = 2.1, a-FeO(OH) = 2.95, b-FeO(OH) = 3.53, Fe (OH)2 = 3.6, c-FeO(OH) = 3.07, Fe(OH)3 = 4.0, Fe(OH)33H2O = 6.15) [2,43]. Some rust may penetrate into the porous zone at the steel/concrete interface, which can reduce the service life of RC structures [2,3,44]. In this study, it is assumed that only the porous zone which is adjacent to the corroded steel would be filled with rust [16]. For the corner rebar, the left and top sides and the side facing the corner as shown in Fig. 1c or Fig. 3, are corroded (about 3/4 of the rebar perimeter), which means that only the side opposite to the corner (1/4 of the rebar perimeter) is not corroded. Therefore, the volume of rust in the porous zone can be expressed as:

initial cover surface

O

deformed cover surface

E N

M

concrete

F rebar initial cover surface

Point D

(a) Elastic stage of concrete

O

c1 WM,h

rebar

(b) Partial cracking stage

deformed cover CMODA,R surface WP,h A WA,v WP,v P C c2 initial cover B surface E WM,h M N O c1 deformed F cover rebar surface Point D

initial cover surface

B

initial cover surface

concrete

(c) Stage of crack widening Fig. 3. Progression of corner cracking.

concrete

4

Y. Zhang, R.K.L. Su / Construction and Building Materials 234 (2020) 117410

[2,46]. The thickness of the porous zone has been found to be between 12.5 mm and 120 mm [2,44,46,47]. In this analysis, the volume of rust in the cracks is taken as zero as there is no consensus in the literature yet on how much rust is deposited into corrosion-induced cracks [1,23,24,40,48–51]. The volume of rust in the cracks is conservatively neglected in the analysis. The maximum thickness of the corroded steel that fills up the porous zone, ds,0,max, can be obtained by substituting Eqs. (1), (6) and (8) into Eq. (5),

ds;0;max ¼

3d0 2ð b  1Þ

ð9Þ

The effect of the uncertainty of b on the uncertainty of ds,0,max is analyzed here in. As b ranges from 1.7 to 6.15, ds,0,max could vary from 0.29d0 to 2.14d0 according to Eq. (9). Thus, the ratio of the maximum b to the minimum b is 6.15/1.7 = 3.62, while the ratio of the maximum ds,0,max to the minimum ds,0,max is 2.14/0.29 = 7.38 which is about 2 times of that of b. This means that the uncertainty of ds,0,max is larger than that of b. When rust fills up the porous zone around the corroded steel, the higher volume of rust Vnet would cause expansion pressure in the surrounding concrete. Thus, the net maximum thickness of the corroded steel df,max which corresponds to Vnet can be expressed as:

df;max ¼ ðb  1Þðds;max  ds;0;max Þ

ð10Þ

2.3. Properties of materials The linear elastic stress-strain relationship of concrete under tension is:

et ¼

rt Ec;ef

ð11Þ

where et is the tensile strain of concrete, rt is the tensile stress of concrete, and Ec,ef is the effective elastic modulus of concrete and can be written as:

Ec;ef ¼ Ec =ð1 þ /ct Þ

ð12Þ

where Ec is the elastic modulus of concrete and /ct is a creep coefficient of concrete. Once the tensile stress reaches the tensile strength of the concrete, the corresponding strain of concrete ect can be expressed as:

ect ¼

ft Ec;ef

ð13Þ

where ft is the tensile strength of the concrete. In this study, rebars are treated as rigid bodies as the Young’s modulus of steel (about 200 GPa) is much greater than that of concrete (about 20–30 GPa) [46]. Moreover, the deformation of rust can be neglected [3,46,52] when the Young’s modulus of rust Erust or the bulk modulus of rust Krust is larger than an adequately sufficient value, for example, Erust  1 GPa [46], Erust  500 MPa [3], Krust  300 MPa [52] or Krust  4 GPa [53]. In this analysis, rust is also treated as a rigid body for simplicity. 2.4. Process and progression of corner cracking The typical progression of corner cracking caused by the nonuniform corrosion of the rebars is shown in Fig. 3 in which c1 and c2 are the thickness of two different locations on the corner of the concrete cover. After the porous zone around the corroded steel is filled with rust, the progression of cover cracking has three

main stages - the elastic stage of concrete, partial cracking stage and stage of crack widening. The elastic stage of concrete begins with rust that causes expansion pressure on the surrounding concrete and results with the initiation of cracks at the steel/concrete interface as shown in Fig. 3a in which Points A and M are on the cover surface, Points B and N are located at the steel/concrete interface, Point P is just at the corner of the cover and Point O is at the center of the initial rebar. The definition of crack initiation here is the point when the tensile stress of the concrete equals the tensile strength of the concrete ft. During this stage, the concrete of the cover is linearly elastic. The corrosion-caused expansion pressure on the two sides of the rebar near the cover pushes against the corner of the cover, causing the elastic concrete body ABNMP to move outward. As the expansion of rust is confined by the surrounding concrete, the bulging of the cover surface at Points A and M, WA,v,E and WM,h,E, respectively, are less than df,max at Points B and N. Furthermore, the deformation at Point P in the vertical and horizontal directions, WP,h,E and WP,v,E respectively, is also less than df,max. The partial cracking stage as shown in Fig. 3b starts with crack initiation at the steel/concrete interface and ends with the failure of the cover at c2. The cover failure is defined as the moment when there is no tensile capacity of the concrete along the cover, which causes df,max to be just equal to the thickness of the bulging at both Points A (in the vertical direction) and M (in the horizontal direction). During this stage, the cracks gradually penetrate along Points A to B and Points M to N and their width progressively increases with an increase in the volume of rust. It should be noted that the tensile cracking along Points A to B is mainly caused by the expansion of the rust that faces c1, and that along Points M to N is mainly caused by the expansion of the rust that faces c2. Theoretically, when c1 = c2, the cracking at c1 and c2 would propagate simultaneously at the same speed. However, as concrete is not a homogeneous material, the cracking in one area of the cover would propagate faster even if they are equal; that is, even if c1 is equal to c2. In this analysis, it is assumed that the area of c2 is more brittle and the cracks in this area propagate slightly faster. Therefore, with an increase in the volume of rust, c2 would first reach its maximum tensile capacity. After that, the tensile capacity along Points A to B would decrease. Meanwhile, the tensile stress of the concrete along Points M to N is unloaded, which causes a reduction in the width of cracks in that area. As a result, the expansion of the rust that faces c1 would be confined by the concrete along Points F to D, thus causing ABENFDP (the concrete body) to gradually rotate about the pivot point (Point D), like a cantilever beam. This causes the formation of a diagonal crack along Points F to D, and at the same time, increases the width of the crack along Points A to B. At the end of this stage, the crack along Points A to B is wide enough so that the concrete there can no longer resist the corrosion-caused expansion pressure. This leads to the complete rotation of ABENFDP about Point D, which results in df,max equal to the bulging at Points A and M. Meanwhile, the expansion of the rust that faces c2 causes the rotation of ABEP (concrete body) about the pivot point (Point C) as shown in Fig. 3b. During this stage, the bulging at Points A and M is still less than that at df,max as the concrete cover still has a confinement effect against the expansion of rust. The bulging along Points D to P is determined by both the rotation and the outward movement of ABENFDP. Meanwhile, the deformation along Points A to P is determined by both the rotation and the outward movement of ABEP. Furthermore, the effects of the rotation of both ABENFDP and ABEP gradually increase with crack growth, while their outward movements are reduced. The third stage is the stage of crack widening in the corner of the cover. In this stage, the rotation of ABENFDP and ABEP mainly determine the deformation along Points D to P and Points A to P, respectively, as shown in Fig. 3c. Meanwhile, the bulging at Points

5

Y. Zhang, R.K.L. Su / Construction and Building Materials 234 (2020) 117410

A and M is equal to df,max, and the bulging along both Points D to P and Points P to A varies almost linearly.

The horizontal deformation at Point A, WA,h, can be obtained by considering an elastic thick-walled cylinder with an internal diameter of D/2 and external diameter of D/2 + c2, and expressed as [54]:

2.5. Elastic stage of concrete In the elastic stage of concrete, the corner of the cover is elastic and has no cracks. Thus, the linear elastic process can be used to study the deformation of the concrete cover. Then, the bulging at Points M, A and P, can be obtained by simulating an elastic thickwalled cylinder usually with an internal diameter of D/2 but different external diameters of D/2 + c1, D/2 + c2, and D/2 + c3, respectively, and can be expressed as [54]:

W M;h;E ¼

W A;v;E ¼

2df;max DðD þ 2c1 Þ D2 ð1  tÞ þ ð1 þ tÞðD þ 2c1 Þ2 2df;max DðD þ 2c2 Þ D ð1  tÞ þ ð1 þ tÞðD þ 2c2 Þ2 2

W P;h;E ¼ W P;v;E ¼

2df;max DðD þ 2c3 Þ D2 ð1  tÞ þ ð1 þ tÞðD þ 2c3 Þ2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D ðc1 þ D=2Þ2 þ ðc1 þ D=2Þ2  2

2df;max D=2ðD=2 þ c2 Þ

W A;h ¼

ðD=2Þ2 ð1  tÞ þ ð1 þ tÞðD=2 þ c2 Þ2 2df;max DðD þ 2c2 Þ

¼

D2 ð1  tÞ þ ð1 þ tÞðD þ 2c2 Þ2

ð24Þ

ð14Þ

According to the crack band model [55,56], the crack band width wcb is a material property of concrete and approximately 2da  4da with uniform strain ect, where da is the maximum aggregate size. When c2 has just failed, the horizontal deformation of the concrete cover at Point A, WA,h,u, can be written as:

ð15Þ

W A;h;u ¼ wcb ect

ð16Þ

ð17Þ

The bulging along Points M to P and Points A to P, dcs1(x) (Point M is the coordinate origin) and dcs2(x) (Point A is the coordinate origin) respectively, is assumed to decrease linearly from Points M to P and Points A to P, respectively. Thus, they can be expressed as:

ð25Þ

By substituting Eq. (24) into (25), the net maximum thickness of the rust that causes the failure of c2, df,max,u, can be obtained by:

df;max;u ¼

where WM,h,E, and W A,v,E are the bulging at Points M and A respectively, WP,h,E and WP,v,E are the horizontal and vertical bulging at Point P respectively, m is the Poisson’s ratio of the concrete, and c3 is the length of PO minus D/2 which can be expressed as:

c3 ¼

2.6. Failure of c2

h i wcb ect D2 ð1  tÞ þ ð1 þ tÞðD þ 2c2 Þ2 2DðD þ 2c2 Þ

ð26Þ

2.7. Crack widening After df,max,u < df,max, the bulging at Point M, WM,h,R, is equal to df,max, i.e.,

W M;h;R ¼ df;max

ð27Þ

Due to the rotation of ABENFDP, the bulging increases almost linearly from Points D to P. Therefore, the horizontal deformation at Point P, WP,h,R, and the bulging along Points D to P, dcs1(x) (Point D is the origin of the coordinates), can be written as:

dcs1 ðxÞ ¼ W P;h;E þ

 LMP  x  W M;h;E  W P;h;E LMP

ð18Þ

W P;h;R ¼

df;max LDP LDM

ð28Þ

dcs2 ðxÞ ¼ W P;v;E þ

LAP  x ðW A;v;E  W P;v;E Þ LAP

ð19Þ

dcs1 ðxÞ ¼

xW P;h;R LDP

ð29Þ

where LMP and LAP are the length of Points M to P and Points A to P respectively, and are:

LMP ¼ c2 þ

D 2

ð20Þ

LAP ¼ c1 þ

D 2

ð21Þ

The concrete at the steel-concrete interface will stretch due to the expansion of the rust and the elongation of the concrete WE can be calculated as follows:

" 

WE ¼

p 3 D þ df;max 2

2

 

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
# D D pD pdf;max  þ df;max þ 2 2 2 2

ð22Þ

In this stage, the tangential strain of concrete around the part of corroded rebar (about 3/4 of the rebar perimeter as shown in Fig. 3) is assumed to be uniform [5,44,46,48]. Cracks initiate when WE reaches the critical tensile deformation rate of concrete WEc, i.e.:

W E ¼ W Ec ¼

3pDect 4

ð23Þ

At the end of this stage, cracks initiate at the steel-concrete interface and the corresponding net maximum thickness of the corroded steel, df,max,Ec, and the bulging at Points M, A and P, WM, h,Ec, WA,v,Ec, WP,h,Ec and WP,v,Ec, can be obtained with Eqs. (22)– (23) and Eqs. (14)–(16), respectively.

where LDM and LDP are the length of Points D to M and Points D to P, respectively, which are:

LDM ¼


 D þ c1 tan u1 2


LDP ¼

 D D þ c1 tan u1 þ þ c2 2 2

ð30Þ ð31Þ

where u1 is the angle of Points M, O, and D, which is about 45° based on the results obtained from the FE model developed in Section 3. During this stage, the bulging at Point A, WA,v,R, can be expressed as:

W A;v;R ¼ df;max

ð32Þ

As the rotation of ABEP could be affected by the rotation of ABENFDP, the final pivot point of ABEP is at Point C as shown in Fig. 2c. Thus, the bulging along Points P to A (coordinate origin at Point A) can be expressed as:

dcs2 ðxÞ ¼

LAC  x df;max LAC

ð33Þ

where LAC is the length of Points A to C and can be expressed as:




LAC ¼

 D þ c2 tan u2 2

ð34Þ

6

Y. Zhang, R.K.L. Su / Construction and Building Materials 234 (2020) 117410

where u2 is the angle of Points A, O, and C which is about 40° based on the results obtained from the FE model developed in Section 3. Thus, the vertical deformation at Point P can be expressed as:

W P;v;R ¼

LAC  c1  D=2 df;max LAC

ð35Þ

After c2 fails, the cover crack opening at Point A, CMODA,R, is mainly determined by the rotation of ABENFDP, and would be approximately equal to WP,h,R, i.e.,

df;max LDP LDM

CMODA;R ¼ W P;h;R ¼

ð36Þ

The rotation of ABENFDP gradually increases with the growth of rust, and eventually, the concrete along Points F to D is not able to resist the expansion of the rust, thus causing ABENFDP to peel off. 2.8. Partial cracking stage

¼

2df;max DðD þ 2c1 Þð1  qÞ

D ð1  tÞ þ ð1 þ tÞðD þ 2c1 Þ2 2

df;max  df;max;Ec df;max;u  df;max;Ec

ð37Þ

Since the bulging at Point M, WM,h is determined by both the rotation and the outward movement of ABNDP, WM,h has two parts: (1) (1-q)df,max which can be obtained from the elastic process shown in Section 2.5, and (2) qdf,max which can be obtained from the rotation of the concrete cover shown in Section 2.7. Thus, WM,h can be expressed as:

ð38Þ

Similarly, the bulging at Points A and P (in both the horizontal and vertical directions), WA,v, WP,h, and WP,v, respectively, can be expressed as:

W A;v ¼ ð1  qÞW A;v;E þ qW A;v;R ¼

2df;max DðD þ 2c2 Þð1  qÞ

D2 ð1  tÞ þ ð1 þ tÞðD þ 2c2 Þ2

þ qdf;max

ð39Þ

W P;h ¼ ð1  qÞW P;h;E þ qW P;h;R ¼

2df;max DðD þ 2c3 Þð1  qÞ

D2 ð1  tÞ þ ð1 þ tÞðD þ 2c3 Þ2

¼

2df;max DðD þ 2c3 Þð1  qÞ

D2 ð1  tÞ þ ð1 þ tÞðD þ 2c3 Þ2

þ

qdf;max LDP

ð40Þ

LDM

200 mm

fixed support

rebar D

þ

qdf;max ðLAC  c1  D=2Þ LAC ð41Þ

In this stage, the bulging along Points D to P, dcs1(x), is assumed to linearly decrease, respectively, from Points M to D and Points M to Point P as shown in Fig. 2b. Thus, dcs1(x) (Point D is the coordinate origin) can be expressed as:

dcs1 ðxÞ ¼

8 xW < L M;h

0  x  LDM

DM

:W

M;h

þ

ðxLDM ÞðW P;h W M;h Þ LMP

LDM  x  LDP

ð42Þ

Meanwhile, the bulging along Points A to P, dcs2(x), is assumed to linearly vary from Points P to A. Thus, dcs2(x) (Point A is the coordinate origin) can be expressed as:

dcs2 ðxÞ ¼ W A;v 

c1

þ qdf;max

W P;v ¼ ð1  qÞW P;v;E þ qW P;v;R

After cracking initiates at the steel/concrete interface, the cracks gradually penetrate toward the cover surface with increase in rust. During this stage, the amount of bulging caused by the rotation of the concrete bodies, q, gradually increases from 0 to 1 as df,max increases from df,max,Ec to df,max,u. Thus, q can be expressed as:

q¼

W M;h ¼ ð1  qÞW M;h;E þ qW M;h;R

xðW A;v  W P;v Þ LAP

ð43Þ

It should be noted that this analytical model is a nondestructive evaluation method, which can be used to predict the volume of corroded steel by measuring the bulging of the cover surface and/or the cover crack opening for all stages of the progression of corner cracking. To achieve that, the largest cover bulging and/ or crack opening are recommended to measure in practice. They are WA,v,E and WM,h,E at Points A and M in the elastic stage of concrete as shown in Fig. 3a, WA,v and WP,h in the partial cracking stage as shown in Fig. 3b, and WP,h and CMODA,R in the stage of crack widening as shown in Fig. 3c. Furthermore, the predicted volume of corroded steel can be used to evaluate the bonding strength and residual service life of corroded RC structures.

concrete 3. Finite element model

c2

A nonlinear FE analysis was conducted with ATENA software to verify the analytical model developed in Section 2. The schematic and boundary conditions of the concrete sample are shown in Fig. 4. The rebar diameter, cover thickness and concrete properties (modeled by using a SBETA material model with an exponential

200 mm Fig. 4. Schematic and boundary conditions of concrete sample.

Table 1 Rebar diameter, cover thickness and concrete properties. Rebar diameter D (mm)

Cover thickness c1 (mm)

Cover thickness c2 (mm)

Tensile strength of concrete ft (MPa)

Young’s modulus of concrete Ec,ef (GPa)

Fracture energy Gf (N/ m)

12 16 20

25 35 45

25 35 45

2.317 3.678 4.82

30.23 39.27 43.69

57.93 91.96 120.5

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1.6

Non-uniform temperature assigned to each part on the side facing cover

Numerically derived results (ATENA) Analytically modeled results

1.4 1.2

df,max,Ec (μm)

assigned temperature Non-uniform temperature assigned to each part on the side facing cover

1.0 0.8 0.6 0.4

rebar

Fig. 5. Cross section and assigned temperature of segments of rebar.

tension softening curve) used in the simulation are shown in Table 1. ATENA software and the material model have been used for the FEM analysis in [28,29]. The cross section of the rebar is evenly divided into 24 segments as shown in Fig. 5. The thermal analogy method is used to simulate the rust expansion by providing an expansion with ther-

0.2 0.0

2

3

ft (MPa)

4

5

Fig. 7. Comparison of df,max,Ec obtained from FE and analytical models for different ft (m = 0.2, D = 16 mm, b = 3, d0 = 12.5 lm, c1 = c2 = 25 mm, u1 = 45°, u2 = 39°, wcb = 80 mm).

mal effects. Thus, the increase in the radius of the rebar, df,max, can be expressed as:

Fig. 6. Bulging of cover surface based on FE and analytical models: (a) and (c) in partial cracking stage (df,max = 12 lm), and (b) and (d) in stage of crack widening (df,max = 100 lm) (ft = 2.317 MPa, Ec,ef = 30.23 GPa, m = 0.2, D = 16 mm, b = 3, d0 = 12.5 lm, c1 = c2 = 25 mm, u1 = 45°, u2 = 39°, wcb = 80 mm).

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240 200

Numerically derived results (ATENA) Analytically modeled results

Numerically derived results (ATENA) Analytically modeled results

200

160

WM,h (μm)

WA,v (μm)

160 120

120 3

3

80

80 2

2

40

1 0

0

0

40

80

1

40 0

1

120

df,max (μm)

2

3

4

160

0

5

200

0

0

40

80

0

120

1

df,max (μm)

(a)

2

160

3

4

200

5

240

(b)

400

Numerically derived results (ATENA) Analytically modeled results

WP,h (μm)

300

200

100

0

0

40

80

120

df,max (μm)

160

200

(c) Fig. 8. Comparison of bulging of cover surface obtained from analytical and FE models: (a) Point M, (b) Point A, and (c) Point P (ft = 2.317 MPa, Ec,ef = 30.23 GPa, m = 0.2, D = 16 mm, b = 3, d0 = 12.5 lm, c1 = c2 = 25 mm, u1 = 45°, u2 = 39°, wcb = 80 mm).

df;max ¼ al DTD=2

ð44Þ

where DT is the temperature increase and al is the thermal expansion coefficient of steel which is 1.2  105 m/(mK). In order to model a crescent shape to simulate the rust distribution, the temperature assigned to every segment of the corroded rebar linearly decreases with an increase in the distance from each part of the steel to the cover surface as shown in Fig. 5. It should be noted that the stresses provided by surrounding concrete may confine the elastic expansion of the rust, however, in the FE simulation, the Young’s modulus of rust Erust is equal to that of the rebar, i.e. Erust = 200 GPa. Thus, the deformation of rust caused by the confined effect could be ignored and the rust can be treated as a rigid body. 4. Verification and discussion The deformation of the concrete cover surface, df,max,Ec, WM,h, WA,v, WP,h, and CMODA,R obtained from analytical and FE models is compared in this section to verify the proposed analytical model.

4.1. Deformation of cover surface The deformation of the cover surface obtained from the analytical and FE models are shown in Fig. 6. It can be found that both the shape and the value of the deformation obtained are in very good agreement in the partial cracking stage (df,max = 12 lm) and the stage of crack widening (df,max = 100 lm). Furthermore, the regions of cracking and tensile stresses obtained from the FE simulation as shown in Fig. 6c and d agree well with those reported in Fig. 3b and c. 4.2. df,max,Ec Fig. 7 shows a comparison between the df,max,Ec obtained from the FE and analytical models for different ft. It can be found that the results obtained from the latter agree well with those from the FE method. A higher ft means a higher df,max,Ec. This is reasonable as a higher ft requires a larger df,max,Ec to initiate concrete cracking in the concrete/steel interface.

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4.3. WM,h, WA,v, WP,h

400

Numerically derived results (ATENA) Analytically modeled results

Fig. 8 shows the variations in the WM,h, WA,v, and WP,h with an increase in the df,max. It can be found that all of the results obtained from the analytical model are in agreement with those from the FE model.

CMODA,R (μm)

300

4.4. CMODA,R 200

Fig. 9 compares the CMODA,R obtained from the analytical and the FE models. It can be observed that the two sets of results are in very good agreement.

100

5. Parametric study

0

0

40

80

120

160

In this section, the effects of ft, c, D and u1 on WM,h, WA,v, WP,h, df,max,u and CMODA,R are studied, which is useful information for designing and predicting the service life of RC structures and width of cracks in such structures.

200

df,max (μm) Fig. 9. Comparison of CMODA,R obtained from analytical and FE models: (ft = 2.317 MPa, Ec,ef = 30.23 GPa, m = 0.2, D = 16 mm, b = 3, d0 = 12.5 lm, c1 = c2 = 25 mm, u1 = 45°, u2 = 39°, wcb = 80 mm).

5.1. Effect of ft The effects of ft on WM,h, WA,v, and WP,h are shown in Fig. 10. It can be found that in the elastic and the partial cracking stages, WM,

200

200

ft = 2.3 MPa

ft = 2.3 MPa ft = 3.7 MPa

ft = 3.7 MPa

ft = 4.8 MPa

ft = 4.8 MPa

WA,v (μm)

160

WM,h (μm)

160

120

120

the initiation of the stage of crack widening

80

15

the initiation of the stage of crack widening

80

12

15 12

9

9

6

40

6

40

3 0

0

0

40

3 0

80

3

120

6

9

12

0

15

160

0

200

0

40

0

80

3

120

df,max (μm)

df,max (μm)

(a)

(b)

400

6

9

160

12

15

200

ft = 2.3 MPa ft = 3.7 MPa ft = 4.8 MPa

WP,h (μm)

300

200

the initiation of the stage of crack widening

15 12 9

100

6 3 0

0

0

40

80

0

3

120

df,max (μm)

6

9

160

12

200

(c) Fig. 10. Effects of ft on (a) WM,h, (b) WA,v, and (c) WP,h (m = 0.2, D = 16 mm, b = 3, d0 = 12.5 lm, c1 = c2 = 25 mm, u1 = 45°, wcb = 80 mm).

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30

CMODA,R is shown in Fig. 18. It can be found that D has no effect on CMODA,R.

25

5.4. Effect of diagonal crack angle The diagonal crack angle (u1) may vary in practice as the aggregate can affect the development of the crack direction. Therefore, the effects of u1 on WP,h and CMODA,R are studied as shown in Fig. 19 with the developed analytical model. It can be found that both WP,h and CMODA,R increase with an increase in u1. Among all of the parameters in Section 5, u1 has the most significant influence on WP,h and CMODA,R. Therefore, more field and experimental studies should be conducted to investigate the factors that could affect u1.

df,max,u (μm)

20

15

10

5

6. Conclusion 2

3

ft (MPa)

Fig. 11. Effect of ft on df,max,u (m = 0.2, c1 = c2 = 25 mm, u1 = 45°, wcb = 80 mm).

4

D = 16 mm,

5

b = 3,

d0 = 12.5 lm,

h,

WA,v, and WP,h decrease with an increase in the ft while in the stage of crack widening, ft has no influence on all deformations of the cover surface. This is reasonable as a higher ft means higher confinement effect of the concrete cover, which results in less deformation of the cover surface. However, in the stage of crack widening, the crack development of the cover is determined by the rotation of the concrete body as explained in Section 2.4, which would not be affected by ft. The effect of ft on df,max,u is shown in Fig. 11. It can be seen that df,max,u increases with an increase in ft. Fig. 12 shows the effect of ft on CMODA,R. It can be observed that ft has no influence on CMODA,R. The explanation for the effects of ft on WM,h, WA,v, and WP,h is also applicable for this effect. 5.2. Effect of c Fig. 13 shows the effects of c on WM,h, WA,v, and WP,h. It can be seen that a thicker c results in a smaller WM,h, WA,v, and WP,h in the elastic stage and partial cracking stage. This is because a thicker c will have a greater confinement effect on the expansion of rust, thus leading to less deformation of the cover surface. However, in the stage of crack widening, c has no effect on WM,h, WA,v, and WP,h. The explanation for the effects of ft on WM,h, WA,v, and WP,h can also be applicable for this effect. Fig. 14 shows the effect of c on df,max,u. It can be observed that df,max,u increases with an increase in c. Fig. 15 shows the effect of c on CMODA,R. It can be seen that c has no influence on CMODA,R. The explanation for the effect of ft on CMODA,R is applicable for this effect. 5.3. Effect of D Fig. 16 is a comparison among WM,h, WA,v, and WP,h for different diameters (D). It is observed that in both the elastic and partial cracking stages, WM,h, WA,v, and WP,h increase with an increase of D, while in the stage of crack widening, D has no influence on WM,h, WA,v, and WP,h. It should be noted that when df,max is the same for different D, the total volume of rust is different as the D is different. Fig. 17 compares the influence of D on df,max,u. It can be seen that df,max,u decreases with an increase in D. The effect of D on

In this paper, a novel analytical model is proposed to study the process and progression of corrosion-induced corner cracking by considering the non-uniform distribution of rust at the steel/concrete interface and on a flat cover surface. This model can be applied for the nondestructive evaluation of the degree of corrosion by measuring the bulging of the cover surface and/or the cover crack opening for the entire progression of the cover cracking process: the elastic stage of concrete, partial cracking stage and stage of crack widening. It is found that the relationship between the bulging of the cover surface and the net maximum thickness of the rust differ in the different cracking stages. In the elastic stage of the concrete and the partial cracking stage, the bulging of the cover surface is less than the net maximum thickness of the rust due to the confinement effect of the concrete cover. However, in the stage of crack widening, the bulging at the points of cover surface vertical to the rebar are equal to the net maximum thickness of the rust, and the bulging along the cover surface to the corner location increase almost linearly. Furthermore, the crack opening increases linearly with an increase in the net maximum thickness of the rust. By using the thermal analogy method, a nonlinear FE analysis has been conducted with ATENA to simulate the non-uniform expansion of rust. It is found that the results obtained by using the FE method agree very well with those obtained from the analytical model.

400

ft = 2.3 MPa ft = 3.7 MPa ft = 4.8 MPa

300

CMODA,R (μm)

0

200

100

0

0

40

80

120

df,max (μm)

160

200

Fig. 12. Effect of ft on CMODA,R (m = 0.2, D = 16 mm, b = 3, d0 = 12.5 lm, c1 = c2 = 25 mm, u1 = 45°, wcb = 80 mm).

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c1 = c2 = 25 mm

200

c1 = c2 = 25 mm

200

c1 = c2 = 35 mm

c1 = c2 = 35 mm

c1 = c2 = 45 mm

c1 = c2 = 45 mm

160

WA,v (μm)

WM,h (μm)

150

120

100

the initiation of the stage of crack widening

the initiation of the stage of crack widening

80

15

15 12

12

9

9

50

6

6

40

3

3 0

0

0

50

0

3

6

100

9

12

150

df,max (μm)

0

15

0

200

0

40

80

0

3

6

9

120

160

df,max (μm)

12

15

200

(b)

(a) 400

c1 = c2 = 25 mm c1 = c2 = 35 mm c1 = c2 = 45 mm

WP,h (μm)

300

200

the initiation of the stage of crack widening

15 12 9

100

6 3 0

0

0

50

0

3

6

100

9

12

150

df,max (μm)

15

200

(c) Fig. 13. Effects of c on (a) WM,h, (b) WA,v and (c) WP,h (ft = 2.317 MPa, Ec,ef = 30.23 GPa, m = 0.2, D = 16 mm, b = 3, d0 = 12.5 lm, u1 = 45°, wcb = 80 mm).

30

400

c1 = c2 = 25 mm c1 = c2 = 35 mm c1 = c2 = 45 mm

25 300

CMODA,R (μm)

df,max,u (μm)

20

15

10

200

100

5

0

25

30

35

c (mm)

40

45

Fig. 14. Effect of c on df,max,u (ft = 2.317 MPa, Ec,ef = 30.23 GPa, m = 0.2, D = 16 mm, b = 3, d0 = 12.5 lm, u1 = 45°, wcb = 80 mm).

0

0

40

80

120

df,max (μm)

160

200

Fig. 15. Effect of c on CMODA,R (ft = 2.317 MPa, Ec,ef = 30.23 GPa, m = 0.2, D = 16 mm, b = 3, d0 = 12.5 lm, u1 = 45°, wcb = 80 mm).

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Y. Zhang, R.K.L. Su / Construction and Building Materials 234 (2020) 117410

200

200

D = 12 mm D = 16 mm D = 20 mm

D = 12 mm D = 16 mm D = 20 mm

160

WM,h (μm)

WA,v (μm)

150

120

100

the initiation of the stage of crack widening

15

the initiation of the stage of crack widening

80

12

15 12 9

9

50

6

6

40

3

3 0

0

0

50

0

3

6

100

9

12

150

df,max (μm)

0

15

0

200

0

40

80

(a)

0

3

6

120

df,max (μm)

9

12

15

160

200

(b)

400

D = 12 mm D = 16 mm D = 20 mm

WP,h (μm)

300

200

the initiation of the stage of crack widening

30 20

100 10 0

0

0

40

80

0

3

6

120

9

160

df,max (μm)

12

15

200

(c) Fig. 16. Effects of D on (a) WM,h, (b) WA,v, and (c) WP,h (ft = 2.317 MPa, Ec,ef = 30.23 GPa, m = 0.2,b = 3, d0 = 12.5 lm, c1 = c2 = 25 mm, u1 = 45°, wcb = 80 mm).

30

400

D = 12 mm D = 16 mm D = 20 mm

25 300

CMODA,R (μm)

df,max,u (μm)

20

15

200

10 100

5

0

12

14

16

D (mm)

18

20

Fig. 17. Effect of D on df,max,u (ft = 2.317 MPa, Ec,ef = 30.23 GPa, m = 0.2,b = 3, d0 = 12.5 lm, c1 = c2 = 25 mm, u1 = 45°, wcb = 80 mm).

0

0

50

100

df,max (μm)

150

200

Fig. 18. Effect of D on CMODA,R (ft = 2.317 MPa, Ec,ef = 30.23 GPa, m = 0.2,b = 3, d0 = 12.5 lm, c1 = c2 = 25 mm, u1 = 45°, wcb = 80 mm).

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500

ϕ1 = 35°

500

ϕ1 = 35°

ϕ1 = 45°

ϕ1 = 45°

ϕ1 = 55°

ϕ1 = 55°

400

CMODA,R (μm)

400

WP,h (μm)

300

200

30

300

200

20

100

0

0

100

10

0

40

80

120

0

df,max (μm)

3

6

160

9

12

15

200

(a)

0

0

40

80

120

df,max (μm)

160

200

(b)

Fig. 19. Effects of diagonal crack angle u1 on (a) WP,h and (b) CMODA,R (ft = 2.317 MPa, Ec,ef = 30.23 GPa, m = 0.2, D = 16 mm, b = 3, d0 = 12.5 lm, c1 = c2 = 25 mm, wcb = 80 mm).

A parametric study is conducted to study the effects of the tensile strength of the concrete, cover thickness, rebar diameter and diagonal crack angle on the bulging of the cover, crack opening and net maximum thickness of rust for a crack opening on one of the corners of the concrete cover. It is found that in the elastic stage of the concrete and the partial cracking stage, the bulging of the cover surface decreases with an increase in the tensile strength of concrete and cover thickness, but increases with a larger rebar diameter that has the same net maximum thickness of rust. However, in the stage of crack widening, the bulging of the cover surface and the crack opening are not affected by the tensile strength of the concrete, cover thickness and rebar diameter, but decrease with an increase in the diagonal crack angle. The work in this study is useful for designing and predicting the service life of RC structures and widths of the cracks in such structures. However, to the authors’ best knowledge, there is no available research about the diagonal crack angles u1 and u2, and therefore, further research to establish such analytical relationship is highly recommended. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement This research did not receive any specific grant from any funding agency in the public, commercial, or not-for-profit sector. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.conbuildmat.2019.117410. References [1] K. Bhargava, A.K. Ghosh, Y. Mori, S. Ramanujam, Model for cover cracking due to rebar corrosion in RC structures, Eng. Struct. 28 (8) (2006) 1093–1109.

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