Calculation methods of the crack width and deformation for concrete beams with high-strength steel bars

Abstract

Three groups of concrete beams reinforced with high-strength steel bars were tested, and the crack width and deformation of the specimens were observed and studied. To facilitate the predictions, two simplified formulations according to a theory developed by the first author were proposed. The advantages of the formulations were verified by the test data and compared with several formulas in different codes.

Keywords

concrete beam / high-strength steel bar / crack width / deformation

Introduction

The crack width and deflection constitute two important aspects in the design of concrete beams at the serviceability limit state. The first author has noted that the development of cracks and the flexural stiffness of concrete beams were both related to the bond slip at the interface between the reinforcement and the concrete [1,2]. Therefore, a unified method was established for predicting the slip, crack width and flexural stiffness of concrete beams. The method proved to be precise but complex in calculation. Furthermore, a higher reinforcement strength is required in the new Chinese concrete code GB 50010-2010 [3]. Thus, the suitability of the former method should be re-evaluated. Combined with recent tests of reinforced concrete (RC) and pre-stressed reinforced concrete (PRC) beams using high-strength steel bars, the corresponding simpler formulas are proposed for the calculation of the short-term crack width and deflection.

Test

Details of the test

The specimens were classified into three groups. The first two groups were RC beams loaded negatively with a load step in 0.1P_u increments with an initial duration of 10 min and a 0.05P_u increment with a 30 min duration after the loading value reached 0.9P_u (where the ultimate load, P_u, was applied when flexural failure occurred; the corresponding moment is denoted by M_u); the third group was a PRC beam loaded positively, as shown in Fig. 1, with a load step in 0.05P_u and 0.1P_u increments before and after concrete cracking, respectively, for 30 min. The details of each group are shown in Fig. 2 and Tables 1 and 2.

In Fig. 1, L = beam length; L” = edge distance= 200 mm for Group 1 and 150 mm for Groups 2 and 3; L' = length of the shear span (with stirrups) = (L-2L”)/4 for Groups 1 and 2 and 1200 mm (for L = 4500 mm) or 1650 mm (for L = 6000 mm) for Group 3; and (L-2L'-2L”) = length of the pure bending segment (no stirrups for Groups 1 and 2).

Observation

Initially, the strains of the reinforcement and concrete in the elastic beam were congruous, and the mid-span deflection developed linearly with an increase in the load. Subsequently, the concrete at the tensile edge reached its maximum tensile strain and failed to work, transferring the corresponding tensile stress to the reinforcement and causing slip at the interface in the form of bend cracks, which led to the nonlinearly accelerated development of deflection at the mid span. As the load level increased, the flexural cracks widened and increased in number, and diagonal cracks later emerged in the shear segment. When the load increased to 0.4-0.5 P_u for the RC beams (or 0.6P_u for the PRC beams), no more cracks developed. After that, the tensile steel bars in the RC beams and the unprestressed HRB500 bars in the PRC beams both yielded when the load increased to close to P_u, and the concrete was finally crushed. During the tests, it was observed that the average concrete strains along the height at the mid-span were in accordance with the plane cross-section assumption.

Introduction of the unified formulas

The average crack width for concrete beams consists of two parts:

(1){\omega }_m={\omega }_{\mathrm{1}}+{\omega }_{\mathrm{2}},

where {\omega }_{\mathrm{1}} = crack width caused by the strain difference between the reinforcement and concrete= {\alpha }^{\prime }\cdot c\cdot {ϵ}_s^{\mathrm{0}}, with c= concrete cover thickness and {ϵ}_s^{\mathrm{0}}=reinforcement strain at the cracked section; {\omega }_{\mathrm{2}}= crack width caused by the reinforcement slip and is usually calculated from traditional bond-slip theory:

(2){\displaystyle \frac{d^{\mathrm{2}}S(x)}{d{x}^{\mathrm{2}}}}=-C\tau (x).

One commonly used method of solving Eq. (2) is to assume that \tau (x) obeys some specific distribution first, i.e., to suppose that {ϵ}_s(x) is known before calculating {\omega }_{\mathrm{1}} explicitly through two integrals.

The average flexural stiffness of a concrete beam can be calculated from the following equation, based on the moment-curvature relationship and the plane strain hypothesis (see Fig. 3):

(3)B_s={\displaystyle \frac{M_k}{{\phi }_m}}={\displaystyle \frac{M_k}{{\overline{ϵ}}_s/\left(h_{\mathrm{0}}-\overline{x}\right)}}={\displaystyle \frac{M_k{h}_{\mathrm{0}}\left(\mathrm{1}-{\overline{\xi }}_c\right)}{{\overline{ϵ}}_s}}.

in which \overline{x} and {\overline{\xi }}_c are the average depth and the relative average depth of the concrete compression zone, respectively, and {\overline{\xi }}_c=\overline{x}/h_{\mathrm{0}} = a function of the average strain of the reinforcement, {\overline{ϵ}}_s.

Based on these analyses, the formulas for the crack width and flexural stiffness of concrete beams can be derived from the same preconditions, i.e., using the strain distribution of the reinforcement between cracks, {ϵ}_s(x).

The arguments mentioned above have been explained systematically by the first author Zhou et al. [2], and the corresponding unified formulas for RC and PRC beams are the following:

for crack width,

(4){\omega }_m\cong \left[\mathrm{0.2}\cdot c+\mathrm{0.85}{k}_{\mathrm{1}}{k}_{\mathrm{2}}\cdot \sqrt{{\displaystyle \frac{cd}{{\rho }_{te}}}}\right]\times \left[\mathrm{1}-{\displaystyle \frac{\beta f_t}{{\rho }_{te}{\sigma }_s}}\cdot (\mathrm{1}+\mathrm{2}{\alpha }_E{\rho }_{te})\right]\cdot {\displaystyle \frac{{\sigma }_s^{\mathrm{0}}}{E_s}}\cdot {\eta }_w

and for flexural stiffness,

(5)B_s={\displaystyle \frac{{\overline{E}}_s(A_s+A_p)h_{\mathrm{0}}^{\mathrm{2}}}{\mathrm{1.78}-\mathrm{2.1}{\lambda }_{\mathrm{0}}+\mathrm{4}{\alpha }_E\rho -(\mathrm{1.2}+\mathrm{2.9}{\lambda }_{\mathrm{0}}+\mathrm{4.7}{\alpha }_E\rho )(\beta f_{t}b{h}_{\mathrm{0}}^{\mathrm{2}}\rho /M_k{\rho }_{te})}}.

Equations (4) and (5) are in good agreement with the test results but are too complex to calculate efficiently.

Thus, the corresponding simplified formulas are derived here for easier application in engineering.

Suggested formula for the crack width

Equation (4) gives good predictions for the crack width of RC and PRC beams (see Table 3), and the average crack spacing calculated using Eq. (6) is also precise enough.

The average crack spacing [2] is the following:

(6)l_m=\mathrm{0.85}\cdot k_{\mathrm{1}}{k}_{\mathrm{2}}\sqrt{{\displaystyle \frac{cd}{{\rho }_{te}}}}.

In calculating the crack width, two different popular theories were used. One was the bond-slip theory, which attributed the development of the crack width to the discordance of the deformation between the reinforcement and concrete and mainly considered two parameters,d/{\rho }_{te} and {\sigma }_s; the other theory was the non-slip theory, which assumed that the deformation retraction difference in the concrete around steel led to cracks, and only the concrete cover,c,was highlighted. However, the observed crack width proved to be a combined result of these two theories; therefore, most codes adopt the combined theory, and the corresponding formulas can be classified into two kinds according to how they address the parameters mentioned above:

1) Linear superposition, e.g., the crack width formula in the Chinese concrete code, GB 50010-2010 [3]:

(7){\omega }_m={\alpha }_c\phi {\displaystyle \frac{{\sigma }_{sk}}{E_s}}\left(1.9c+0.08{\displaystyle \frac{d_{eq}}{{\rho }_{te}}}\right).

2) Nonlinear superposition, e.g., the Chinese Code for Design of Highway Reinforced Concrete and Prestressed Concrete Bridges and Culverts, JTG D62-2004 [4], specifies that the crack width can be calculated from:

(8){\omega }_{fk}=C_{\mathrm{1}}{C}_{\mathrm{3}}{\displaystyle \frac{{\sigma }_{ss}}{E_s}}\left({\displaystyle \frac{30+d}{0.28+10\rho }}\right).

The American code, ACI 318-08 [5], does not give specific formulas for calculating the crack width but sets limits on the maximum reinforcement spacing according to Frosch [7]:

(9){\omega }_s=\mathrm{2}{\displaystyle \frac{f_s}{E_s}}\sqrt{d_c^{\mathrm{2}}+d_s^{\mathrm{2}}}.

In the European code, EN 1992-1-1:2004 [6], the consideration of the parameters for the crack width appears to be of the first kind:

(10){\omega }_k=\left(k_{\mathrm{3}}c+{\displaystyle \frac{k_{\mathrm{1}}{k}_{\mathrm{2}}{k}_{\mathrm{4}}\varphi }{{\rho }_{p,eff}}}\right)({ϵ}_{sm}-{ϵ}_{cm}).

Combining the bond-slip and non-slip theories, \sqrt{cd/{\rho }_{te}} is supposed to be an important factor affecting the crack width. In Eq. (4), the first component, [\mathrm{0.2}c+\mathrm{0.85}{k}_{\mathrm{1}}{k}_{\mathrm{2}}\sqrt{cd/{\rho }_{te}}], contains two constants, k_{\mathrm{1}} and k_{\mathrm{2}}, and the second component, \left[\mathrm{1}-{\displaystyle \frac{\beta f_{tk}}{{\rho }_{te}{\sigma }_s}}(\mathrm{1}+\mathrm{2}{\alpha }_E{\rho }_{te})\right], is complex to calculate. Considering that most codes calculate the crack width by integrating the reinforcement strain, {\sigma }_s/E_s, through the average crack spacing, the same rule is obeyed, and {\omega }_m is supposed to be a function of \sqrt{cd/{\rho }_{te}} and {\sigma }_s/E_s:

(11){\omega }_m=\left[A^{\prime }+C^{\prime }\sqrt{{\displaystyle \frac{cd}{{\rho }_{te}}}}\right]\cdot {\displaystyle \frac{{\sigma }_s}{E_s}}.

where c denotes the concrete cover thickness; d and E_s denote the diameter and elastic modulus of the reinforcement, respectively; {\sigma }_s is the calculated stress of the tensile reinforcement; and {\rho }_{te}=A_s/A_{te}, where A_{te} is taken as 0.5bh in accordance with the current theory.

Rewriting Eq. (11) gives

(12){\displaystyle \frac{{\omega }_m{E}_s}{{\sigma }_s}}=A^{\prime }+C^{\prime }\sqrt{{\displaystyle \frac{cd}{{\rho }_{te}}}},

where the two constant parameters A^{\prime } and C^{\prime } can be estimated from a linear regression analysis of the test data, A^{\prime }=\mathrm{34} and C^{\prime }=\mathrm{0.24}. Thus,

(13){\omega }_m^{c1}=\left(\mathrm{34}+\mathrm{0.24}\sqrt{{\displaystyle \frac{cd}{{\rho }_{te}}}}\right){\displaystyle \frac{{\sigma }_s}{E_s}}.

A comparison of the calculation from Eq. (13) and the test results is shown in Fig. 4, which illustrates the linear relationship between {\omega }_m and \sqrt{cd/{\rho }_{te}}.

Suggested formula for deflection

Based on the test results, the development of the relation between the loading moment, M, and the deflection at the mid span, f, can be simplified into three stages, as shown in Fig. 5, the initial elastic linear stage, the subsequent inelastic (nearly linear) stage after concrete cracking and the final nonlinear stage, when the reinforcement in tension yields (usually, a load of 0.8M_u is reached).

In structural design, the yield of the reinforcement is one important control principle, and the calculation of the elastic stiffness is precise enough. Thus, the current research focuses on the second stage, and an equivalent flexural stiffness formula is usually established for convenience in practical applications. The existing methods of predicting the flexural stiffness for beams are widely used and can be summarized into the following two categories: one is a semi-empirical analytic method, emphasizing mathematical description of the tested M-f curve and applying the superimposed deflection or stiffness of the first two stages to derive the corresponding equivalent flexural stiffness (see equivalent line in Fig. 5), including the first and second subtypes; the other is an analytic method based on the plane cross-section hypothesis, including the third subtype.

Subtype 1: the deflection of beams at the second stage is written as

(14)f_s=f_{cr}+\mathrm{\Delta }f,

where f_{cr} denotes the total elastic deflection of the first stage, and \mathrm{\Delta }f is the deflection increment due to concrete cracking at the second stage.

Assuming a double linear M-f relationship, Eq. (14) is rewritten in the form of the flexural stiffness as

(15){\displaystyle \frac{M_k}{B_s}}={\displaystyle \frac{M_{cr}}{B_{\mathrm{0}}}}+{\displaystyle \frac{M_k-M_{cr}}{B_{cr}}},

(16)B_s={\displaystyle \frac{B_{\mathrm{0}}}{{\displaystyle \frac{M_{cr}}{M_k}}+\left(1-{\displaystyle \frac{M_{cr}}{M_k}}\right){\displaystyle \frac{B_0}{B_{cr}}}}},

where M_{cr}/M_k and \mathrm{1}-M_{cr}/M_k denote the two weighting coefficients for deflection interpolation. For general purposes, the short-term flexural stiffness is written as

(17)B_s={\displaystyle \frac{B_{\mathrm{0}}}{\alpha +(1-\alpha ){\displaystyle \frac{B_0}{B_{cr}}}}},

where \alpha and \mathrm{1}-\alpha are the corresponding weighting coefficients related to the assumed M-f mathematical form \alpha ={(M_{cr}/M_k)}^n (where, as mentioned above, Chinese concrete code GB 50010-2010 [3] on prestressed concrete beams and Recommendations on Partially Prestressed Concrete Structural Design take n as 1).

In the recommendations,

(18)B_s={\displaystyle \frac{\mathrm{0.85}{E}_c{I}_{\mathrm{0}}}{{\displaystyle \frac{M_{cr}}{M_k}}+{\displaystyle \frac{\mathrm{0.85}}{\beta }}\left(\mathrm{1}-{\displaystyle \frac{M_{cr}}{M_k}}\right)}},

where \mathrm{0.85}/\beta may be seen as the coefficient obtained from the linear regression analysis of B_{\mathrm{0}}/B_{cr} about {\alpha }_E\rho. In code JTG D62-2004 [4], a parabolic relationship is used, i.e., n=\mathrm{2}. Although European code EN 1992-1-1: 2004 [6] recommends calculating the maximum curvature directly (in a quadratic relation with the moment) rather than the flexural stiffness, the same conclusion can be drawn after transforming the curvature to the stiffness. According to the test results, the influence of n on the deflection, f, is that a higher value of n usually leads to an increasing f, which may be attributed to the enlarged effect of B_cr due to n.

Subtype 2: American code ACI 318-08 [5], using the effective moment of inertia to calculate the stiffness, gives

(19)I_e={\left({\displaystyle \frac{M_{cr}}{M_k}}\right)}^{\mathrm{3}}{I}_g+\left[\mathrm{1}-{\left({\displaystyle \frac{M_{cr}}{M_k}}\right)}^{\mathrm{3}}\right]I_{cr}.

where {(M_{cr}/M_k)}^{\mathrm{3}} and \mathrm{1}-{(M_{cr}/M_k)}^3 can be understood as the two weighting coefficients of the stiffness interpolation.

Subtype 3: Chinese code GB 50010-2010 [3] on concrete beams, which establishes the relationship of the curvature and cross-sectional average strain, is based on the plane cross-section assumption before introducing {\alpha }_E\rho and \phi =\mathrm{1.1}(\mathrm{1}-M_{cr}/M_k) to reflect the influence of the reinforcement content and crack.

The results of the comparison indicate the similarity of the first two methods in interpolating between two points, (M_{cr},f_{cr}) and (M_k,f_k). When M_k is close to M_{cr}, the influence of the parameter n is not obvious, but when load increases from M_{cr} to \mathrm{0.8}{M}_u, the precision of the interpolation seems to depend more on the form of the assumed mathematical function. Furthermore, in the denominator of Eq. (17), i.e., \alpha +(1-\alpha )B_0/B_{cr}, \alpha becomes less important asM_k increases. Given M_{cr}/M_k=\mathrm{0.25} (i.e., cracking happens at\mathrm{0.3}{M}_u, and M_k is taken as \mathrm{0.8}{M}_u) and B_{\mathrm{0}}/B_{cr}\ge \mathrm{2}, then \alpha (B_{\mathrm{0}}/B_{cr}-\mathrm{1}) approximates {\mathrm{0.25}}^n\times \mathrm{2}(or\mathrm{3}) and is less than 0.2 when n is larger than 2.

In principle, B_s is a function of B_{\mathrm{0}}, B_{cr}, M_{cr}/M_k and {\alpha }_E\rho. Because B_{cr} can be calculated from B_{\mathrm{0}} and {\alpha }_E\rho, B_s is only related to three independent parameters. B_s=B(B_{\mathrm{0}},{\alpha }_E\rho ,M_{cr}/M_k) is used here for its simplicity and clear concepts (which can also be drawn from Eq. (5)); B_{\mathrm{0}} denotes the elastic stiffness; {\alpha }_E\rho reflects the contribution of the reinforcement in tension to the stiffness; and the load’s negative effect on beams is considered in terms of M_{cr}/M_k.

Assuming that

(20)B_s=B_{\mathrm{0}}f({\alpha }_E\rho )\cdot g\left({\left({\displaystyle \frac{M_{cr}}{M_k}}\right)}^n\right),

the corresponding mid-span deflection is

(21)f_s=\beta {\displaystyle \frac{M_k}{B_{\mathrm{0}}f({\alpha }_E\rho )}}h\left({\left({\displaystyle \frac{M_{cr}}{M_k}}\right)}^n\right),

where \beta denotes a factor reflecting the influence of the load form and position and equals \mathrm{1}/\mathrm{8}-\mathrm{1}/\mathrm{6}{(L^{\prime }/L_{\mathrm{0}})}^{\mathrm{2}} for the two-point loading condition (where B_s^t/B_s^c and L^{\prime } denote the lengths of the support-support span and shear span, respectively); and h(x)=g^{-\mathrm{1}}(x).

Equation (21) should satisfy the following two preconditions:

(22a)f_s=\beta {\displaystyle \frac{M_k}{B_{\mathrm{0}}f({\alpha }_E\rho )}}h\left({\left({\displaystyle \frac{M_{cr}}{M_k}}\right)}^n\right)=\beta {\displaystyle \frac{M_k}{B_{\mathrm{0}}}},i.e.,h\left({\left({\displaystyle \frac{M_{cr}}{M_k}}\right)}^n\right)/f({\alpha }_E\rho )=\mathrm{1}when{\displaystyle \frac{M_{cr}}{M_k}}=\mathrm{1},

(22b)f_s=\beta {\displaystyle \frac{M_k}{B_{\mathrm{0}}f({\alpha }_E\rho )}}h\left({\left({\displaystyle \frac{M_{cr}}{M_k}}\right)}^n\right)=\beta {\displaystyle \frac{M_k}{B_{\mathrm{0}}f({\alpha }_E\rho )}}when{M}_k=\mathrm{0.8}{M}_u\left(i.e.,h\left({\left({\displaystyle \frac{M_{cr}}{M_k}}\right)}^n\right)=\mathrm{1}\right).

For simplicity, suppose that

(23)h\left({\left({\displaystyle \frac{M_{cr}}{M_k}}\right)}^n\right)=\mathrm{1}-(\mathrm{1}-f({\alpha }_E\rho )){\left({\displaystyle \frac{M_{cr}}{M_k}}\right)}^n,n\ge \mathrm{2.}

The regression analysis at the point (\mathrm{0.8}{M}_u,f_{s,lim}) indicates that f({\alpha }_E\rho )=\mathrm{0.234}+\mathrm{1.674}{\alpha }_E\rho.

After moderate adjustment,

(24)f({\alpha }_E\rho )=\mathrm{0.24}+\mathrm{1.67}{\alpha }_E\rho .

Substituting Eq. (24) into Eq. (23) yields

(25)h\left({\left({\displaystyle \frac{M_{cr}}{M_k}}\right)}^n\right)=\mathrm{1}-(\mathrm{1}-(\mathrm{0.}24+1.67{\alpha }_E\rho )){\left({\displaystyle \frac{M_{cr}}{M_k}}\right)}^n.

The final problem is the estimation of the unknown n. Obviously, a higher n is required by Eq. (22b) (which may make our calculation at the serviceability limit state more precise); on the other hand, a higher n may lead to a larger influence of the load. It therefore should be limited considering the difference of the tested and calculated cracking moments. A very low value of n, e.g., n=\mathrm{1}, is not desired and makes it hard to separate the influence of the load.

In this study, n is taken as 2 to reasonably predict the deflection at the serviceability states:

(26)h\left({\left({\displaystyle \frac{M_{cr}}{M_k}}\right)}^{\mathrm{2}}\right)=\mathrm{1}-(\mathrm{1}-(\mathrm{0.}24+1.67{\alpha }_E\rho )){\left({\displaystyle \frac{M_{cr}}{M_k}}\right)}^{\mathrm{2}}.

Thus, the suggested formula for a short-term mid-span deflection is

(27)f_s=\beta {\displaystyle \frac{M_k}{B_{\mathrm{0}}(\mathrm{0.24}+\mathrm{1.67}{\alpha }_E\rho )}}\left(\mathrm{1}-(\mathrm{0.76}-\mathrm{1.67}{\alpha }_E\rho ){\left({\displaystyle \frac{M_{cr}}{M_k}}\right)}^{\mathrm{2}}\right).

Verification

Besides the test introduced previously in the paper, other related test data were collected to verify the proposed equations, including, for the crack width, tests of RC beams from Tongji University [8,9], Zhejiang University [10], Tianjin University [11], Hunan University [12-14], Zhengzhou University [15-17], Huaqiao University [18], Dongnan University [19,20] and other related universities [21,22] and of PRC beams from Zhengzhou University [23-27]; for the flexural stiffness, tests of RC beams from Tongji University [28], Zhengzhou University [15-17], Huaqiao University [18], Dongnan University [19,29,30]and Tianjin University [31] and of PRC beams from Tongji University [32], Zhengzhou University [23-27] and the Dalian University of Technology [33].

The calculation results with the suggested equations are summarized in Tables 3 and 4. Relative values, i.e., the ratio of the calculated width to tested mean crack width ({\omega }_m^c/{\omega }_m^t) and the ratio of the tested deflection to the calculated deflection (f_s^t/f_s^c), are adopted for clarity, and two commonly used statistical characteristics, i.e., the average (μ) and coefficient of variation (δ), are applied. For comparison, calculations from the equations of the different codes (including GB 50010 [3], JTG D62 [4], ACI 318 [5], and EN 1992-1-1 [6]) as well as the unified formulas [2] are also listed in Tables 3 and 4. Detailed comparisons are shown in Figs. 7 to 8.

The results of the comparison indicate:

1) Average crack width. The calculated values from codes GB 50010-2010 [3] and ACI 318-08 [5] agree well with the test results. The average ratios of the calculated values to the tested values are both smaller than 1.15, but the high variability reflects their discrete precision for individual specimen. The former unified formula of the crack width is too complex, and a bigger deviation is obtained using the equations from codes JTG D62-2004 [4] and EN 1992-1-1 [6]. Based on part of the test results, the mean value of Eq. (13) is in line with observations.

2) Deflection. Both JTG D62-2004 [4] and EN 1992-1-1 2004 [6] overestimate the deflection significantly; in contrast, the predictions from code GB 50010-2010 [3], ACI 318-08 [5], Zhou [2] and Eq. (27) are more accurate, indicating their similar precision. However, the equations of the Chinese code GB 50010-2010 adopt two different forms for RC and PRC beams, and calculations from the equations of Zhou [2] are complex. The suggested formula is simple to calculate and gives precise predictions.

Conclusions

The crack width and deflection are two elements that most Chinese engineers face in design at the serviceability limit state. Unreasonably, however, different forms are adopted in the calculation of the flexural stiffness for RC and PRC beams, as RC beam is a special PRC beam with zero prestress. Because a higher strength grade of the reinforcement is specified in the new Chinese concrete code, it is necessary to verify the former formula of the crack width. According to the unified method, tests of RC and PRC beams have been conducted in Tongji University, and the corresponding simplified formulas were put forward for the crack width and deflection of concrete beams with high-strength steel bars. Combined with the collected test data, the suggested formulas were verified to have good predictions, and comparing them to formulas from different codes highlighted their effectiveness.

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