Influence of steel–concrete bond damage on the dynamic stiffness of cracked reinforced concrete beams

Abstract

In this article, experiments focusing at the influence of steel–concrete bond damage on the dynamic stiffness of cracked reinforced concrete beams are reported. In these experiments, the bond between concrete and reinforcing bar was damaged using appreciate flexural loads. The static stiffness of cracked reinforced concrete beam was assessed using the measured load–deflection response under cycles of loading and unloading, and the dynamic stiffness was analyzed using the measured natural frequencies with and without sustained loading. Average moment of inertia model (Castel et al. model) for cracked reinforced beams by taking into account the respective effect of bending cracks (primary cracks) and the steel–concrete bond damage (interfacial microcracks) was adopted to calculate the static load–deflection response and the natural frequencies of the tested beams. The experimental results and the comparison between measured and calculated natural frequencies show that localized steel–concrete bond damage does not influence remarkably the dynamic stiffness and the natural frequencies both with and without sustained loading applied. Castel et al. model can be used to calculate the dynamic stiffness of cracked reinforced concrete beam by neglecting the effect of interfacial microcracks.

Keywords

cracked reinforced concrete steel–concrete bond stiffness vibration

Introduction

Considering the low tensile strength of concrete, cracking may appear in reinforced concrete structures due to in-service loading, early age restrained shrinkage, and so on. Cracking-induced damages lead to a reduction in the stiffness of reinforced concrete members, which influences both static behavior and dynamic response of the structures (Deng et al., 2014; Zhao et al., 2016).

The stiffness $EI$ is a fundamental property that governs the behavior of cracked beams. Comprehensive experimental, analytical, and simulated studies have been reported on dynamic behavior of cracked structural members (Dimarogonas, 1996). It was found that the static stiffness $E I_{s}$ can be used to calculate the static deformations but underestimates the stiffness of cracked concrete beams under dynamic excitation $E I_{d}$ (Jerath and Shibani, 1985). Similar results were reported by several authors showing that the dynamic stiffness $E I_{d}$ is remarkably larger than the static stiffness $E I_{s}$ for cracked reinforced concrete beam (Eccles, 1999; Hamad et al., 2015; Maeck et al., 2000; Musial, 2012, 2017; Neild et al., 2003; Pešić et al., 2015; Tan, 2003; Xu and Castel, 2016). Pešić et al. (2015) investigated the influence of damage on the dynamic properties of the cracked reinforced concrete beam experimentally and proposed an updated finite element analysis model with static measured elastic modulus and recommended tensile strength (5% of the characteristic compressive strength). Moreover, Musial (2017) studied the effect of the load level on the natural frequency of cracked reinforced concrete beams and observed that the dynamic stiffness of the unloaded cracked beam was higher than that of the loaded cracked beams with ballast masses. The explanation of the difference in stiffness was related to the nonlinear vibration mechanisms, including the breathing crack mechanism, nonlinear behavior of concrete in compression, bond damage between the reinforcing bar and the concrete, and matrix-aggregate interaction over the crack surface. The effects of both breathing cracks and matrix-aggregate interaction on the nonlinear dynamic stiffness of cracked reinforced concrete beams were studied experimentally. It was found that breathing cracks and matrix-aggregate interaction do not influence significantly the nonlinear vibration as once a primary crack has been opened during loading, loose aggregate prevents the crack from closing when unloading (Neild et al., 2002). However, the effect of the steel–concrete bond damage on the dynamic stiffness was not considered, and more experimental work are required.

For static loading, when the applied bending moment is larger than the cracking bending moment, primary cracks form in the tensile zone of the beam weakening the moment of inertia at the cracked section resulting in the reduction of the overall stiffness of the beam. Due to the bond between the reinforcing steel bars and the concrete, the tensile concrete between the primary cracks continue to carry a part of the tensile stress and, as a result, contribute to the stiffness of the beam. This phenomenon is known as tension stiffening. When the applied bending moment is increased further and becomes larger than the so-called cover-controlled cracking moment, microcracks develop at the interface between tensile reinforcements and concrete leading to a steel–concrete bond damage and reducing the tension stiffening effect (Wu and Gilbert, 2009). Castel and François (2011) experimentally observed steel–concrete interfacial microcracks under static loading and measured the reduction in static stiffness $E I_{s}$ of cracked beams and then proposed a cover-controlled cracking criteria and a model allowing assessing the average beam stiffness $E I_{a}$ under static cycles of loading and unloading, which is proportional to the so-called average moment of inertia $I_{a}$ (Castel et al., 2014). This model can capture the respective effects of primary cracks and steel–concrete interfacial microcracks that develop between the primary cracks.

Using Castel et al. model, Xu and Castel (2016) studied the effect of steel–concrete bond damage on the dynamic stiffness of cracked reinforced concrete beams under low-amplitude vibration. It was found that bond damage appears not to be relevant to the dynamic response. The level of stress in the steel was very low under low-amplitude vibration. As a result, the dynamic stiffness $E I_{d}$ was always assessed experimentally larger than the static stiffness $E I_{s}$ . However, in practice, when a cracked concrete member is vibrating in service (e.g. high-speed train on a concrete bridge), the structural members are subjected to both static permanent loads and dynamic actions due to the live loads. The applied bending moment can be much higher than the one considered for low-amplitude vibration. In such cases, whether the bond damage can affect the dynamic behavior remains a question unanswered.

In this article, experiments focusing at static and dynamic stiffness of cracked reinforced concrete beams are reported. A total of two cracking damage levels were considered: low and high. For low damaged level, only primary cracks were formed due to the applied bending moment. For heavy damage stage case, steel–concrete bond damage between the primary cracks was created. The static stiffness $E I_{s}$ of cracked reinforced concrete beam was assessed using the measured load–deflection response under cycles of loading and unloading. For each damaged level considered, the beams were subjected to a sustained load and then the fundamental natural frequency of the beams was measured to assess the dynamic stiffness $E I_{d}$ . Furthermore, the static load–deflection response and natural frequencies were analyzed using Castel et al. model in order to assess the effects of the steel–concrete bond damage on the dynamic stiffness.

Experiments

The experiment includes two reinforced concrete beams labeled B1 and B2. Both beams were priorly subjected to cycles of loading and unloading to create cracks and bond damage. Compared to previous study by the same authors (Xu and Castel, 2016), the natural frequency of the beams was measured under sustained loading instead of testing the beams unloaded. The sustained load applied was equal to the one used to create cracking and bond damage during the preliminary cycles of loading and unloading. Hence, the bond damage was expected to be sufficiently simulated during the dynamic test.

Test specimens

As shown in Figure 1, B1 and B2 had the same dimensions (length of 3500 mm and 400 × 300 mm² rectangular cross section). The only difference between B1 and B2 is the tensile reinforcement. The main longitudinal reinforcing bars in B2 are ribbed bars of 16 mm diameter with a reinforcement ratio of 0.56%. B2 is a typical lightly reinforced concrete beam, for which the tension stiffening effect and the bond between steel and concrete play a more important role in the stiffness after cracking compared to the beams with high reinforcement ratio (Gilbert, 2007). The diameter of the tensile reinforcement of B1 is 18 mm, leading to a reinforcement ratio of 0.71%. The cover to the main reinforcing bars was 35 mm for both beams. The density of the beams was 2570 kg/m³. The compressive strength and elastic moduli of the concrete at 28, 63, and 80 days are given in Table 1. The average yield and ultimate stress of reinforcing bars were 450 and 620 MPa, respectively. The average elastic modulus of the reinforcing bars was 205.8 GPa. The beams were demolded 24 h after casting and moist cured until the load tests commenced. Beam B1 was tested at 63 days, and beam B2 was tested at 80 days.

Figure 1.

Static loading test: (a) cross section (mm), (b) loading arrangements, (c) loading setup, and (d) damage cases.

Table 1.

Average strength and elastic moduli of concrete (MPa).

	28 days	63 days	80 days
Compressive strength	48.67	63.30	70.52
Tensile strength	−	3.45	3.15
Elastic modulus	33,500	38,000	43,000

Experimental program

The two pinned supports were set with a distance of 3300 mm for the static tests as shown in Figure 1. The experiments consisted of the static cycles of loading and unloading, the natural frequency test, and finally the failure loading. As shown in Figure 2, the experimental program was as follows:

Step 1: Undamaged beams were supported by cables in a quasi-free–free configuration as shown in Figure 3(a). The supporting points were located at the bottom of the beam with the distance of 82.5 cm from the end of beam, which are the inflection points of the first bending mode. A point excitation was impacted at the top side at mid-span of the beam. The response of the beam was measured using an accelerometer located at the top of the beam as shown in Figure 3(a). Using fast Fourier transform (FFT), the natural frequency of the beam was calculated as shown in Figure 4.

Step 2: Both beams were loaded in four-point bending configuration using a hydraulic actuator as shown in Figure 1. For the lightly damaged level, the maximum applied load $P_{a}$ was equal to 50 kN, and the maximum applied bending moment, including the weight of the steel spreader beam (5 kN), was 30.25 kN m. According to the cover-controlled cracking criterion proposed by Castel and François (2011), no steel–concrete bond damage was created. A total of five cycles of loading and unloading were then applied. During the loading and unloading cycles, the deflection at mid-span was measured to assess the overall beam stiffness, and the cracking patterns were recorded.

Step 3—low damage: After completing the static cycles of loading and unloading, beams were subjected to a sustained load equivalent to 78 kN using two steel rails. By moving the steel rail along the span of the beam, the maximum applied moment could be precisely adjusted. To ensure no extra cracking or steel–concrete interface damage were created due to the additional mass added, only two levels of mid-span bending moments were adopted: 17.55 and 29.25 kN m corresponding to the two load cases named S1 and S2, respectively. In each load case, the natural frequency of the cracked reinforced concrete beam carrying the additional mass loading was measured as described in step 4.

Step 4: In this stage, a point excitation was impacted at the bottom side of the beam. The time histories of the impact excitations were recorded, and the response of the beam was measured using an accelerometer located at the top of the beam as shown in Figure 3(b). Using FFT, the natural frequency of the beam under static loads was calculated as shown in Figure 4.

Step 5—heavy damage: The beams were loaded further using the same setup described in step 2 to create the heavy damage. The maximum applied load $P_{a}$ was equal to 95 kN, and the maximum applied bending moment, including the weight of the steel spreader beam (5 kN), was 55.00 kN m. According to the cover-controlled cracking criterion (40.13 and 33.64 kN m for B1 and B2, respectively), the steel–concrete bond was damaged. A total of five cycles of loading and unloading were carried out. During the loading and unloading, the deflection at mid-span was measured to assess the overall beam static stiffness, and the cracking patterns were recorded.

Step 6: Repeating the natural frequency test as described in steps 3 and 4. For the heavily damaged beams, the sustained applied bending moments were 40.95 and 52.65 kN m, corresponding to the load cases named S3 and S4, respectively. The bending moment in case S4 was very close to the maximum applied bending moment used to create the steel–concrete bond damage in step 5.

Step 7: Finally, the beams were loaded up to failure using the same four-point bending system as shown in Figure 1.

Figure 2.

Experimental program.

Figure 3.

Natural frequency test on (a) the beam supported by cables and (b) the simply supported beam.

Figure 4.

Natural frequency analysis.

Experimental results

Static cycles of loading and unloading

The load–deflection responses for two beams including the cycles and the failure are plotted in Figure 5. To show the response of the beams under the cycles of loading and unloading clearly, only the displacements up to 12 mm were plotted in Figure 5. The ultimate loads of B1 and B2 were 282 and 263 kN, respectively. For both beams, the cycle history was typical of in-service load levels (i.e. less than 50% of the ultimate load (Castel et al., 2014).

Figure 5.

Load–deflection response under cycles of loading and unloading: (a) B1 and (b) B2.

The cracking patterns of both B1 and B2 are illustrated in Figure 6. In Figure 6, the cracks of lightly and heavily damaged beams are plotted in solid lines and dotted lines, respectively. Additionally, the average spacing of the cracks $l_{s, av}$ and the average width of the residual cracks after unloading $w_{r, av}$ are listed in Table 2. From light to heavy damage, more visible cracks can be found in the surface of the beam as shown in Figure 6 leading to a decrease in the average spacing of cracks as listed in Table 2. Meanwhile, the average width of the residual cracks increases remarkably between the lightly and heavily damaged cases as listed in Table 2 due to the steel–concrete bond damage created in the heavily damaged beams (Xu et al., 2017).

Figure 6.

Surface cracking patterns of (a) beam B1 and (b) beam B2.

Table 2.

Surface cracking information of tested beams.

	Damage	$l_{s, av}$ (m)	$w_{r, av}$ (mm)
B1	Light	0.18	0.048
	Heavy	0.17	0.061
B2	Light	0.22	0.051
	Heavy	0.19	0.072

Natural frequency test

The natural frequencies measured for all loading levels and all beams are listed in Table 3. It can be seen that the frequencies of both beams are considerably decreasing after cracking with an average reduction of 32.42%. As shown in Figure 6, visible surface primary cracks were observed on the lightly damaged beams leading to the stiffness reduction. However, although more visible surface primary cracks were created and existing cracks propagated (Figure 6), only a marginal reduction in the natural frequency was observed between the lightly and heavily damaged beams in cases free–free supported, S1, and S2. The average reduction ratio was about 7.69%.

Table 3.

Experimental natural frequency of B1 and B2 (Hz).

	Damage	Free–free	S1	S2	S3	S4
B1	No	121.30	−	−	−	−
	Light	81.57	21.30	17.40	−	−
	Heavy	75.25	20.05	14.66	12.42	11.04
B2	No	120.90	−	−	−	−
	Light	82.10	22.10	16.80	−	−
	Heavy	78.10	20.00	16.40	13.50	11.80

As shown in Table 3, the natural frequencies of both beams decreased from cases S1–S4. One of the main reasons is that the location of the attached mass vibrating with the beam was close to the mid-span. Another possible reason is the influence of the steel–concrete bond damage. Analysis of the results using Castel et al. (2014) model is presented in the next section, allowing to identify the influencing parameters leading to the reduction in both static and dynamic stiffness.

Modeling and discussion

Modeling the moment of inertia distribution $I (s)$

The distribution of stiffness along the beam $EI (s)$ can be used in either analytical or numerical methods to calculate the static and dynamic behaviors of cracked reinforced concrete beams, where $s$ is the location of the cross section along the beam. For the in-service performance of concrete structures, the compressive concrete behavior can be considered linear elastic, that is, the elastic modulus $E_{c}$ of compressive concrete is assumed constant. According to the experimental results by Pešić et al. (2015), the value of the elastic modulus of concrete $E_{c}$ can be considered as equal to the static modulus obtained from the standard cylinder compression tests as shown in Table 1. Hence, the moment of inertia distribution $I (s)$ along the cracked beam becomes the key parameter in the stiffness $EI (s)$ governing the bending problem.

In this article, along the uncracked parts of the beam, the overall stiffness is assessed using the gross moment of inertia $I_{g}$ as shown in Figure 7(a). For the cracked parts, the average moment of inertia $I_{a}$ proposed by Castel et al. (2014) is used. In Castel et al. model, the effects of primary cracks and steel–concrete interfacial microcracks between the primary cracks are taken into account.

Figure 7.

Simulation models: (a) assembling the moment of inertia and (b) vibration model with additional mass.

Castel et al. (2014) model is based on a linear steel–concrete bond between the cracks. The linear distribution of the bond stress $τ (x)$ proposed by Yankelevsky et al. (2008) is adopted

τ (x) = τ_{\max} (1 - \frac{x}{l_{s, \max}})

(1)

where $τ_{\max}$ is the maximum shear stress close to the exact crack locations; $x$ $(0 < x < l_{s, \max})$ is the distance from the cracks, and $l_{s, \max}$ is the length over which slip between concrete and steel occurs as recommended by CEB-FIB Model (2010).

According to the equilibrium between the axial force and the shear force, the tensile steel and concrete strains can be written as

ε_{s} (x) = [1 - g (x)] ε_{s 0} + g (x) ε_{snc}

(2a)

ε_{tc} (x) = g (x) ε_{tc, \max}

(2b)

g (x) = 2 \frac{x}{l_{s, \max}} - {(\frac{x}{l_{s, \max}})}^{2}

(2c)

where $ε_{s 0}$ is the tensile strain of the steel bars at the cracked section, $ε_{snc}$ is the steel strain at $x = l_{s, \max}$ , and $ε_{tc, \max}$ is the tensile strain of the concrete along the reinforcement at $x = l_{s, \max}$ . According to the definition of $l_{s, \max}$ , for the bond damage–free beams, the strains of concrete and tensile steel bars are equal at $x = l_{s, \max}$ as

ε_{tc, \max} = ε_{snc}

(3)

For excessive service loads, cover-controlled cracks occur at the steel–concrete interface leading to a bond damage (Wu and Gilbert, 2009). A scalar variable $D_{ccc}$ was introduced to evaluate the bond damage by calculating the difference of the strain between the concrete and steel at $x = l_{s, \max}$ in Castel et al. (2014)

ε_{tc, \max} = (1 - D_{ccc}) ε_{snc}

(4)

The bending moment $M (x)$ applied can be calculated as

M (x) = [ε_{s} (x) E_{s} A_{s} + ε_{tc} (x) E_{c} A_{tc, ef}] Z (x)

(5)

where $Z (x)$ is the distribution of the lever arm of the internal forces along $l_{s, \max}$ .

When a beam is at the stabilized cracking stage, compared to the whole span of the beam, $l_{s, \max}$ is small allowing to assume that the bending moment at all cross sections located along $l_{s, \max}$ is constant. Considering the bending moment at the cracked section $x = 0$ and at $x = l_{s, \max}$ , $M (x = 0) = M (x = l_{s, \max})$

ε_{s 0} E_{s} A_{s} z_{c} = (ε_{snc} E_{s} A_{s} + ε_{tc, \max} E_{c} A_{tc, ef}) z_{nc}

(6)

where $z_{c}$ and $z_{nc}$ are the lever arms of the internal forces on the cracked and uncracked cross sections, respectively; $A_{s}$ is the cross section area of tensile reinforcement; $E_{s}$ and $E_{c}$ are the elastic moduli of steel and concrete, respectively; and $A_{tc, eff}$ is the effective area of active tensile concrete, deduced by assuming the stress distribution in the concrete presented in Figure 8 (Castel et al., 2006) and given by

A_{tc, eff} = b {\frac{d - y_{onc} - a / 2}{2} + a + (h - d - a / 2) [1 - \frac{h - d - a / 2}{2 (d - y_{onc} - a / 2)}]} - A_{s}

(7)

where $b$ is the width of cross section; $d$ stands for the effective depth of the tensile reinforcement; $h$ is the height of the cross section; $a$ is the thickness of the reinforcement layers in the cross section as indicated in Figure 8; and $y_{oc}$ and $y_{onc}$ are the depths of neutral axis at the cracked and uncracked sections, respectively.

Figure 8.

Stress distribution in the concrete in uncracked cross section.

Substituting equation (4) into equation (6) yields

\frac{ε_{snc}}{ε_{s 0}} = \frac{z_{c}}{z_{nc}} \frac{1}{1 + (1 - D_{ccc}) \frac{A_{tc, eff}}{n A_{s}}}

(8)

where $n$ is the modular ratio of steel and concrete ( $E_{s} / E_{c}$ ).

Assuming that the bending moment is constant along $l_{s, \max}$ , $M (x)$ can be expressed as

M (x) = M (x = 0)

(9)

Substituting equations (2), (5), and (8) into equation (9), $Z (x)$ can be obtained

Z (x) = \frac{Z_{c} Z_{nc}}{Z_{c} g (x) + (1 - g (x)) Z_{nc}}

(10)

For a rectangular section, the depth to the neutral axis $y_{0} (x)$ along $l_{s, \max}$ can be derived as (Castel et al., 2014)

y_{0} (x) = 3 [d - Z (x)]

(11)

Using equation (11), the moment of inertia distribution $I (x)$ along the $l_{s, \max}$ can be calculated. As recommended by CEB-FIB Model (2010), the average crack spacing is equal to $1.5 l_{s, \max}$ . Considering the symmetry between half cracks spacing, the average moment of inertia $I_{a}$ between cracks can be calculated as follows

I_{a} = \frac{\int_{0}^{0.75 l_{s, \max}} I (x) dx}{0.75 l_{s, \max}}

(12)

The analytical solution of equation (12) for the rectangle beam was proposed by Xu and Castel (2016). It can be seen that the variable $l_{s, \max}$ appears in both numerator and denominator in equation (12). As a result, the equation of $I_{a}$ does not depend on the flexural crack location. When assembling the moment of inertia in the calculation as shown in Figure 7(a), it is not necessary to define a node at exact crack locations.

Bond damage variable $D_{ccc}$

In Castel et al. (2014) model, $D_{ccc}$ is the key variable governing the steel–concrete bond damage. Castel and François (2011) proposed a bond damage criterion (i.e. $σ_{s 0} > σ_{s, ccc}$ ) as follows

σ_{s, ccc} (f_{tc}) = \frac{z_{nc}}{0.9 z_{c}} [n + \frac{A_{tc, eff}}{A_{s}}] f_{tc}

(13)

where $σ_{s, ccc}$ is the critical axial steel stress at the crack location leading to cover-controlled cracking and $f_{tc}$ is the tensile strength of concrete. $D_{ccc}$ was calibrated using experimental results (Castel et al., 2014; Murray et al., 2016; Xu et al., 2016)

D_{ccc} = {\bar{σ}}^{0.05}

(14a)

\bar{σ} = \frac{σ_{s 0} - σ_{s, ccc}}{f_{y} - σ_{s, ccc}}

(14b)

where $f_{y}$ is the yield strength of reinforcement.

When $D_{ccc} = 0$ , there is no steel–concrete bond damage; when $D_{ccc} = 1$ , the steel–concrete bond is totally lost.

Static stiffness analysis

According to equation (5), the bending moment at the cracked cross section can be calculated as

M (x = 0) = σ_{s 0} A_{s} z_{c}

(15)

Implementing equation (13) into equation (15), the bending moment $M_{s, ccc}$ leading to cover-controlled cracking is derived as follows

M_{s, ccc} = \frac{z_{nc}}{0.9 z_{c}} [n + \frac{A_{tc, eff}}{A_{s}}] f_{tc} A_{s} z_{c}

(16)

According to equation (16), the criteria for cover-controlled cracking of beams B1 and B2 are $M_{s, ccc} = 40.13 kN m$ and $M_{s, ccc} = 35.64 kN m$ , corresponding to the criteria loads $P_{s, ccc} = 65 kN$ and $P_{s, ccc} = 63 kN$ , respectively.

As shown in Figure 5, for the light damage loading level, the maximum applied load (equal to 50 kN) is less than the cover-controlled cracking criteria for both beams. Meanwhile, for the heavy damage case, the maximum applied load (equal to 95 kN) is much larger than the cover-controlled cracking criteria for both beams. Hence, in the simulation, the steel–concrete bond damage had to be considered for the heavily damaged beam only. According to the model mentioned above, the distributions of the average moment of inertia were calculated and are plotted in Figure 9. For the sake of comparison, the calculated average moment of inertia distributions without considering the steel–concrete bond damage for the heavily damaged beam are plotted in Figure 9 as well. It can be seen in Figure 9(a) and (c) that the average moment of inertia for the cracked zone was reduced by approximately 60% due to cracking damage, compared to the gross moment of inertia $I_{g}$ at the uncracked cross section. And, as illustrated in Figure 9(b) and (d), due to the steel–concrete bond damage, the average moment of inertia continued decreasing and approaching the moment of inertia at the fully cracked cross section $I_{c}$ (about $0.2 I_{g}$ ) as a result of the reduction in tension stiffening. If neglecting the steel–concrete bond damage, the value of the average moment of inertia for heavily damaged beam is the same as that for lightly damaged beam, and only the cracked zone is expanded as shown in Figure 9.

Figure 9.

Distribution of the average moment of inertia $I_{a} (s)$ of B1 and B2: (a) light damage (B1), (b) heavy damage (B1), (c) light damage (B2), and (d) heavy damage (B2).

The distribution of the average moment of inertia is shown in Figure 7(a). The responses of beams B1 and B2 under static cycles of loading and unloading were simulated and plotted in Figure 5. The static stiffness of the beams were deduced from the load–deflection results and compared to the model predictions (Figure 10). A good agreement can be found between the experimental and simulated results using Castel et al. model in Figures 5 and 10 except if neglecting the steel–concrete bond damage in the heavily damaged cases (inertia $I_{a 1}$ ) where the static stiffness was always overestimated.

Figure 10.

Comparison between measured and calculated static stiffness of the damaged beams ( $I_{a}^{1}$ : bond damage not considered; $I_{a}^{2}$ : with bond damage): (a) B1 and (b) B2.

Dynamic stiffness analysis

Free–free supports

The natural frequency of the damaged beams supported in quasi-free–free system were calculated using the average moment of inertia $I_{a}$ considering or not the effect of steel–concrete bond damage. The simulated results are plotted in Figure 11 showing that the simulations without considering the effect of steel–concrete bond damage provide a good prediction. For the heavily damaged cases, the calculated results taking the effect of bond damage into account underestimated the natural frequency by about 14% and 19% for B1 and B2, respectively. It is consistent with the results reported by Xu and Castel (2016) for cracked reinforced concrete beams under low-amplitude vibration.

Figure 11.

Comparison between measured and calculated natural frequencies of the damaged beams in the quasi-free–free boundary configuration.

Simple supports with sustained loads

In the previous natural frequency test (free–free supports), the amplitude of the excitation and applied load were low. As a result, it was assumed that the interfacial microcracks were closed, and the bond deterioration was not relevant to the vibration response (Xu and Castel, 2016). In this test, similar amplitude of the excitation was applied to the beam. However, the natural frequencies were measured under sustained loading. The maximum sustained loading applied led to a bending moment $M_{a}$ of 17.55, 29.25, 40.95, and 52.65 kN m, successively, for load cases S1–S4. In S3 and S4, the applied bending moments due to the sustained loadings are larger than the cover-controlled cracking criteria $M_{s, ccc}$ . Hence, it was expected that the bond damage of the beam was simulated sufficiently.

The simulation model shown in Figure 7(b) was adopted to calculate the natural frequency of the beams including the additional mass. The simulated results for the lightly damaged beam are plotted in Figure 12. The average error ratio is only about 4%.

Figure 12.

Comparison between measured and calculated natural frequencies of the lightly damaged beams under sustained loading.

For heavily damaged beams, the natural frequencies were calculated using the average moment of inertia $I_{a}$ with and without considering the bond damage and are plotted in Figure 13. Figure 13 shows that the calculation without considering the bond damage performed better. The average error ratio are about 5% (no bond damage considered) and 16% (with bond damage), respectively. The steel–concrete bond damage has only marginal effect on the natural frequency more often. As a result, the reduction in the natural frequencies from the cases S1–S4 observed experimentally is only due to the locations of the attached mass. The moderate difference in the natural frequency of the lightly and heavily damaged beams is due to the propagation of the bending cracks along the beam. Hence, the experimental results imply that the effects of steel–concrete bond do not affect significantly the dynamic stiffness $E I_{d}$ even if the bond damage is exposed sufficiently by applying a sustained loading to the beams during the dynamic test.

Figure 13.

Comparison between measured and calculated natural frequencies of the heavily damaged beams under sustained loading: (a) B1 and (b) B2.

Conclusion

The presented work explored the influence of steel–concrete bond damage on the dynamic stiffness of cracked reinforced concrete beams in free vibration configuration using both experiment and analytical models and led to the following conclusions as follows:

The steel–concrete bond damage affects the static stiffness of the cracked reinforced concrete beam. In this experiment, compared to the cracked reinforced concrete beam without bond damage between primary cracks, the heavily loaded beam (with bond damage between primary cracks) showed up to 30% decrease in the static stiffness.

The measured natural frequencies of the nonloaded reinforced concrete beams decreased considerably after cracking compared to before cracking with an average reduction of 32.5%. However, only a marginal reduction was observed between the lightly (without bond damage) and heavily damaged (with bond damage) nonloaded beams.

The measured natural frequencies of the cracked reinforced concrete beam with sustained loading decreased along with the increment of the applied load due to the location of the attached mass and the propagation of the bending cracks along the beam.

The experimental results and the comparison between measured and calculated natural frequencies of the cracked beams showed that localized steel–concrete bond damage does not influence remarkably the dynamic stiffness and the natural frequency with or without sustained loading applied.

Castel et al. model can be used to calculate the dynamic stiffness of cracked reinforced concrete beam by neglecting the effect of interfacial microcracks induced bond loss.

Footnotes

Acknowledgements

T. Xu wishes to thank the Key Laboratory of High-Speed Railway Engineering, Ministry of Education, Southwest Jiaotong University, People’s Republic of China, for its support.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Key Research and Development Program of China (Grant Nos 2016YFC0802205 and 2016YFB1200401), National Natural Science Foundation of China (projects 51778535, 51778531, and 51778537), and Australian Research Council (projects DP110103028 and DP140100529).

ORCID iD

Tengfei Xu

References

Castel

François

(2011) Modeling of steel and concrete strains between primary cracks for the prediction of cover-controlled cracking in RC-beams. Engineering Structures 33(12): 3668–3675.

Castel

Gilbert

Ranzi

(2014) Instantaneous stiffness of cracked reinforced concrete including steel-concrete interface damage and long-term effects. Journal of Structural Engineering 140(6) 04014021.

Castel

Vidal

François

(2006) Effective tension active cross-section of reinforced concrete beams after cracking. Materials and Structures 39(1): 115–126.

CEB-FIB Model (2010) CEB-FIB Model Code 2010–Final Draft. London: Thomas Telford Ltd.

Deng

Pan

Lam

et al . (2014) A simplified model for analysis of high-rise buildings equipped with hysteresis damped outriggers. Structural Design of Tall and Special Buildings 23(15): 1158–1170.

Dimarogonas

(1996) Vibration of cracked structures: a state of the art review. Engineering Fracture Mechanics 55(5): 831–857.

Eccles

(1999) The use of non-linear vibrations in the health monitoring of reinforced concrete structures. PhD Thesis, University of Nottingham, Nottingham.

Gilbert

(2007) Tension stiffening in lightly reinforced concrete slabs. Journal of Structural Engineering 133(6): 899–903.

Hamad

Owen

Hussein

(2015) Modelling the degradation of vibration characteristics of reinforced concrete beams due to flexural damage. Structural Control and Health Monitoring 22(6): 939–967.

10.

Jerath

Shibani

(1985) Dynamic stiffness and vibration of reinforced concrete beams. ACI Journal Proceedings 82(2): 196–202.

11.

Maeck

Wahab

Peeters

et al . (2000) Damage identification in reinforced concrete structures by dynamic stiffness determination. Engineering Structures 22(10): 1339–1349.

12.

Murray

Castel

Gilbert

et al . (2016) Time-dependent changes in the instantaneous stiffness of reinforced concrete beams. Engineering Structures 126: 641–651.

13.

Musial

(2012) Static and dynamic stiffness of reinforced concrete beams. Archives of Civil and Mechanical Engineering 12(2): 186–191.

14.

Musial

(2017) Dynamic stiffness of non-loaded and loaded beams. In: Proceedings of the IOP conference series: materials science and engineering, Seoul, Korea, 26–28 May, vol. 216, pp. 012020. Bristol: IOP Publishing.

15.

Neild

Williams

McFadden

(2002) Non-linear behaviour of reinforced concrete beams under low-amplitude cyclic and vibration loads. Engineering Structures 24(6): 707–718.

16.

Neild

Williams

McFadden

(2003) Nonlinear vibration characteristics of damaged concrete beams. Journal of Structural Engineering 129(2): 260–268.

17.

Pešić

Živanović

Dennis

et al . (2015) Experimental and finite element dynamic analysis of incrementally loaded reinforced concrete structures. Engineering Structures 103: 15–27.

18.

Tan

(2003) Nonlinear vibrations of cracked reinforced concrete beams. PhD Thesis, University of Nottingham, Nottingham.

19.

Gilbert

(2009) Modeling short-term tension stiffening in reinforced concrete prisms using a continuum-based finite element model. Engineering Structures 31(10): 2380–2391.

20.

Castel

(2016) Modeling the dynamic stiffness of cracked reinforced concrete beams under low-amplitude vibration loads. Journal of Sound and Vibration 368: 135–147.

21.

Castel

Gilbert

(2017) Assessing immediate and time-dependent instantaneous stiffness of cracked reinforced concrete beams using residual cracks. Journal of Structural Engineering 114. pp. 04018022.

22.

Castel

Gilbert

et al . (2016) Modeling the tensile steel reinforcement strain in RC-beams subjected to cycles of loading and unloading. Engineering Structures 126: 92–105.

23.

Yankelevsky

Jabareen

Abutbul

(2008) One-dimensional analysis of tension stiffening in reinforced concrete with discrete cracks. Engineering Structures 30(1): 206–217.

24.

Zhao

Gao

et al . (2016) Green’s functions for the forced vibrations of cracked Euler–Bernoulli beams. Mechanical Systems and Signal Processing 68: 155–175.

Influence of steel–concrete bond damage on the dynamic stiffness of cracked reinforced concrete beams

Abstract

Keywords

Introduction

Experiments

Test specimens

Experimental program

Experimental results

Static cycles of loading and unloading

Natural frequency test

Modeling and discussion

Modeling the moment of inertia distribution I ( s )

Bond damage variable D ccc

Static stiffness analysis

Dynamic stiffness analysis

Free–free supports

Simple supports with sustained loads

Conclusion

Footnotes

Acknowledgements

Declaration of Conflicting Interests

Funding

ORCID iD

References

Modeling the moment of inertia distribution $I (s)$

Bond damage variable $D_{ccc}$