Residual deflection analysis in negative moment regions of steel-concrete composite beams under fatigue loading

Construction and Building Materials 158 (2018) 50–60

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Residual deflection analysis in negative moment regions of steel-concrete composite beams under fatigue loading Aiming Song, Shui Wan ⇑, Zhengwen Jiang, Jie Xu School of Transportation, Southeast University, Nanjing 210096, China

h i g h l i g h t s
The fatigue deformation behavior of steel-concrete composite beams under negative moment was investigated.
A prediction model of residual deflection related to pre-cracked loading was proposed.
An analytical model of residual deflection related to fatigue loading was proposed.

a r t i c l e

i n f o

Article history: Received 6 July 2017 Received in revised form 9 September 2017 Accepted 14 September 2017

Keywords: Steel-concrete composite beam Negative moment Fatigue tests Residual deflection Analytical model

a b s t r a c t This paper presents the results of an experimental and analytical study on the residual deflection behavior in the negative moment regions of steel-concrete composite beams under fatigue loading. Firstly, fatigue tests with different load amplitudes were performed on two steel-concrete composite plate beam specimens subjected to negative moment. The residual values of slip and deflection at the interval time of the loading procedure were measured by laser displacement sensors. To study the development law of residual deflection under fatigue loading, the total residual deformation fN is then regarded as the superposition of the plastic deflection f1 related to pre-cracking loading and the residual deflection fi related to fatigue loading. The analytical models of f1 and fi are presented based on the existing research results of previous researchers. Finally, the applicability and accuracy of the proposed analytical model are validated through the comparison between modeling results and the data of the experimental beams performed in this study and reported in the earlier companion paper. The fatigue design recommendations for the steel-concrete composite beams subjected to negative moment were given based on the fatigue test and model analysis. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Steel-concrete composite beams have been widely used in buildings and bridges because of the benefits of combining the advantages of the two component materials [1,2]. Served as efficiently lightweight and economical structural members, steel and concrete composite continuous beams are very attractive solutions for short and medium span bridges. However, the concrete slab is in tension and the lower flange of the steel part is subjected to compression in the negative moment regions of continuous composite beams [3]. As a result, the strongly nonlinear mechanical behavior of slip at the beam-slab interface and cracking in the concrete slab even under a low stress level generally has shortcomings in view of durability and strength. Over the last several decades, a ⇑ Corresponding author. E-mail addresses: [email protected] (A. Song), [email protected] (S. Wan). https://doi.org/10.1016/j.conbuildmat.2017.09.075 0950-0618/Ó 2017 Elsevier Ltd. All rights reserved.

lot of researchers have focused on experimental and theoretical studies on the crack control of concrete slab [4,5], as well as the inelastic mechanical behavior of composite beams under static loading [6,7]. Furthermore, in the actual conditions, composite bridges are mostly subjected to vibrating or oscillating forces generated by the live vehicles or the winds in addition to static load. The fatigue effect on mechanical behavior of structures under such repeated loading conditions generally causes the problems of structural serviceability and safety. A large amount of studies have investigated fatigue behavior of shear studs in push-out specimens [8,9] and composite beam specimens under positive bending moment [10,11]. For the fatigue behavior of continuous steel-concrete composite beams in the support region, very limited studies were reported. Zong and Che [12] analyzed the fatigue strength of the prestressed steel-concrete composite beams through experimental tests. The results showed that the repeated load cycles can lead to

A. Song et al. / Construction and Building Materials 158 (2018) 50–60

the gradual increase of slip at the beam-slab interface. Meanwhile, the serious cracking of concrete slab under hogging moment was the main reason for the failure of continuous composite beams. Ryu et al. [13] conducted an experimental test on a full-scale model of a two-span continuous composite bridge with prefabricated slabs to study the crack control method under static and fatigue loads. The results showed that crack widths can be controlled appropriately within an allowable crack width in the decks and transverse joints of the composite bridge with prefabricated slabs on an interior support under service loads. Zhou [14] performed an experimental study on fatigue behavior of composite girders with steel plate-concrete composite bridge decks. No fatigue failure was found for composite girders under negative bending moment according to the test results. Meanwhile, fatigue test with two million cycles did not show obvious effects on the static loading behavior. Lin et al. [15] investigated the fatigue performance of composite steel-concrete beams under hogging moment. The test results indicated that fatigue test with certain cycles could decrease the section stiffness and the ultimate load carrying capacity, while the repeated load was larger than the initial cracking load. Though as an important assessment index for mechanical properties, few studies have performed evaluation for the fatigue deflection of steel-concrete composite beams, especially in the negative moment regions. Wang et al. [11] studied the fatigue behavior of studs in steel-concrete composite beams through a fatigue test on seven specimens. Based on the experimental and theoretical analyses, a calculation method for the deflection of steel-concrete composite beams under positive bending was proposed and validated. In the fatigue tests conducted by Zhou [14], the residual deflection of composite beams under negative moment was measured. But the development laws of residual deflection were not analyzed deeply. Moreover, according to the foregoing research results on the fatigue behavior of composite structures, no analytical solution was presented for the residual deflection under fatigue loading of steel-concrete composite beams subjected to hogging moment till now. This paper is to understand the residual deflection behavior of steel-concrete composite plate beams subjected to negative moment under fatigue loading. The experimentally investigation concerning fatigue deformation of the tested specimens under cyclic loading with different load amplitudes was performed. The failure mode, fatigue life, and residual static test results were recorded and analyzed. Then an analytical model for the estimation of the total residual deformation fN is presented, which can be regarded as the superposition of the plastic deflection f1 in the first cycle and the residual deflection fi related to fatigue loading. Finally, through comparing with the existing experimental works conducted in this study and reported in the earlier companion paper, the applicability and limitations of the proposed analytical model are analyzed and discussed.

2. Fatigue tests on steel-concrete composite beams under negative moment 2.1. Details and materials of test specimens In order to investigate the residual deflection behavior in negative moment regions of steel-concrete composite beams under fatigue loading, three specimens SCB1-1, SCB1-2 and SCB1-3 were tested in this study. Each of the specimens was 3900 mm in length and was simply supported at a span of 3500 mm, as shown in Fig. 1 (a). The uniform thicknesses of steel plates were 12 mm, 12 mm, and 14 mm for the top flange, web and bottom flange respectively. The concrete slab had the thickness of 150 mm and the width of

51

600 mm (see Fig. 1(b)). The longitudinal and transverse reinforcement bars, with the diameters of 16 mm and 10 mm respectively, were arranged in both the top layer and the bottom layer of the concrete slab (see Fig. 1(c)). The longitudinal reinforcement ratio was 4.0%. The specimens were designed with studs as shear connectors, and the diameter and height were 16 mm and 90 mm respectively (see Fig. 1(d)). Two rows shear studs with the longitudinal and transverse spacings of 100 mm were welded on the top flange. In addition, the vertical stiffeners with the thickness of 12 mm were welded at the supports and loading points to prevent shear buckling failure and crippling of the web plate. Strength grade of the concrete was designed to be C50, and the average compressive cube strength was approximately 51.2 MPa at 28 days. The tensile reinforcement bars and steel plates used HRB400 and Q345 respectively of the same factory batch. The measured average yield strength and tensile strength of tensile reinforcement bars were 592 MPa and 718 MPa respectively. The measured average yield strengths of steel plates with the thicknesses of 12 mm and 14 mm were 443 MPa and 391 MPa, and the average tensile strengths were 608 MPa and 520 MPa respectively. 2.2. Loading and measurements All of the beam specimens were inverted to simulate the hogging moment region adjacent to the internal support of a continuous composite bridge (see Fig. 1(a)). First, the beam specimen SCB1-1 was initially tested under monotonic loading in order to determine the ultimate load-bearing capacity Fu and used as a reference beam for the fatigue tests. A single concentrated load by a hydraulic jack with the loading capacity of 2000 kN was applied monotonically downward on the bottom flange plate, as shown in Fig. 2. The monotonic test was conducted under force control until the applied load reaching 80% Fu and the loading rate was set to about 10 kN/min. After that, force control was switched to displacement control. The other two specimens SCB1-2 and SCB1-3 were tested under fatigue loading conditions with different load amplitudes. Fatigue loads were applied using the computer controlled twochannel electro-hydraulic servo static and dynamic loading test system (model: JAW-500K/4) with the maximum loading capacity of 500 kN, as shown in Fig. 3. The cyclic loading frequency was of 2 cycles/s, the stress ratio was 0.1, and sine waveform was used. The maximum load Fmax was determined by the ultimate load-bearing capacity Fu according to the monotonic test on specimen SCB1-1, and 25% Fu was for SCB1-2 and 40% Fu was for SCB1-3. The design details of fatigue tests are characterized in Table 1. Finally, residual static loading test was performed on specimen SCB1-2 which had not suffered a fatigue failure after 250  104 repeated cycles. The loading procedures and equipment were the same with the monotonic loading test on specimen SCB1-1 (see Fig. 2). During the static test, an unloading-reloading cycle was performed on specimen SCB1-1 when the load reached 50% Fu and 70% Fu. The residual deflection of SCB1-1 was recorded during the unloading-reloading process for the further modeling analysis in the later section. For the fatigue conditioned specimens, prior to starting the application of fatigue loading, 10–20 kN static load was pre-loaded for 2–3 times repeatedly. After the good bearing contact was ensured, a pre-cracked loading with the upper limit value Fmax was applied statically. Then fatigue loading started after the force of specimen was adjusted to the fatigue median. During the fatigue loading test, the repeated loading was periodically paused and the specimen was unloaded to zero after typical load cycles of 1  104, 5  104, 10  104, 50  104, 100  104, 150  104, 200  104 and 250  104. Then after the residual deformation was stable, the specimen was loaded to the maximum load Fmax monotonically. At the interval time of the loading procedure,

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A. Song et al. / Construction and Building Materials 158 (2018) 50–60

(a) Front elevation

(b) Side elevation

(c) Layout of reinforcements

(d) Layout of studs Fig. 1. Dimension details of the test specimens (unit: mm).

Hydraulic actuator

Force sensor

Hydraulic jack

Fig. 2. Static testing setup.

the residual slip of beam-slab interface at the support location and the residual deformation at the mid-span were measured by using laser displacement sensors, as shown in Fig. 3. The complete fatigue loading histories for the tested beams are given in Fig. 4. 2.3. Analysis of experimental results 2.3.1. Failure modes Fig. 5(a) shows the failure mode of specimen SCB1-1 under monotonic loading. It can be seen that the compressive bottom flange of SCB1-1 under the ultimate state was buckled locally near the loading point, where no vertical stiffeners were welded. For specimen SCB1-2 after 250  104 cycles, no fatigue failures of the

Laser displacement sensor at support location

Laser displacement sensor at mid-span

Fig. 3. Fatigue testing setup.

steel beam, shear connectors or reinforcement bars in the concrete slab had occurred under load amplitude of 250 kN. A final static test was then performed on SCB1-2 to study the residual static loading behavior of composite beam under hogging moment after the fatigue test. And a similar failure mode was also found for the final static test, which can be described in Fig.5(a). However, when load amplitude was equal to 360 kN for specimen SCB1-3, after 150  104 cycles, a fatigue crack occurred at the location of tensile top flange where the shear studs were welded, as shown in Fig. 5 (b). And the fatigue cracking of top flange can be considered as

53

A. Song et al. / Construction and Building Materials 158 (2018) 50–60 Table 1 Load conditions and characteristic experimental results. Specimens

Load mode

Fatigue failure

Ultimate load (kN)

SCB1-1 SCB1-2 SCB1-3

Static Fatigue Fatigue

/ No Yes

1033 973 /

Fatigue life (104)

Fatigue loads (kN) Upper limit

Lower limit

Load range

Experimental result

Predicted result

/ 250 400

/ 25 40

/ 225 360

/ / 152

/ 516 122

F F max Cycle

Cycle

Cycle to failure or Residual static loading

F min Pre-cracking loading

Fatigue loading

Intermediate static loading

Fatigue loading

Fig. 4. Fatigue loading procedure.

the result of stress concentration and local weakening generated by the studs. Thereafter, the fracture of longitudinal reinforcement occurred suddenly followed by the fatigue cracks of top flange and concrete slab developed rapidly after the next 2  104 repeated cycles.

2.3.2. Fatigue life The fatigue life of steel-concrete composite beams under hogging moment are mainly determined by the design details and stress ranges of the shear stud, tensile top plate and longitudinal reinforcement. In this research, the existing methods according to Eurocode 4-2 [16], TB 10002.2-2005 [17] and the empirical solution presented by Pan [18] were employed to estimate the fatigue life of shear stud, tensile top plate and longitudinal reinforcement respectively. And the three equations are shown respectively as follows:

lg Nf 1 þ 8 lgðDss Þ ¼ 21:935

ð1Þ

lg Nf 2 þ 3 lgðDrs Þ ¼ 12:02

ð2Þ

lg Nf 3 þ 5:3163 lgðDrr Þ ¼ 18:1547

ð3Þ

where, Nf1, Nf2 and Nf3 are the number of stress-range cycles of the shear stud, tensile top plate and longitudinal reinforcement respectively; 4ss, 4rs and 4rr are the fatigue strength related to the typical design details of shear stud, tensile top plate and longitudinal reinforcement respectively. Then the fatigue life Nfd of the test specimens as follow can be predicted by the minimum value of the above three equations:

 Nfd ¼ min Nf 1 ; N f 2 ; Nf 3

ð4Þ

Table 1 provides the experimental and predicted results of fatigue life of specimens SCB1-2 and SCB1-3. Coincided well with the tested phenomenon, both the fatigue failure modes are the fatigue cracking of top flange plate followed by the fracture of tensile reinforcement. It can be seen that the predicted equation may obtain a conservative result compared with the tested value of SCB1-3, as shown in Table 1.

2.3.3. Residual static test results The load-deflection responses at the mid-span of specimens SCB1-1 and SCB1-2 are shown in Fig. 6, in which SCB1-1 is the tested beam without fatigue test. It is found that the curvature of load-deflection curve of SCB1-1 becomes smaller when the applied load reached the initial cracking loading of 70 kN, which indicates that the section rigidity was decreased by the cracking of concrete slab. Different from specimen SCB1-1, the deflection of SCB1-2 behaved almost linearly before the yielding of longitudinal reinforcement. When the applied load was between the initial cracking loading and the yielding of reinforcement, the load-deflection relationships of the two specimens are almost the same. However, in the yielding stage of tested beam, the rigidity of specimen SCB11 became larger than SCB1-2 gradually, which may be ascribed to the fatigue test on the specimen. The ultimate load of specimen SCB1-1 was 1033 kN, which is 6.2% greater than the ultimate capacity of specimen SCB1-2 (973 kN). It is worth noting that the specimen SCB1-2 still had good ductility after 250  104 repeated cycles, when the upper limit was smaller than 250 kN. 3. Method for predicting the residual deflection of negative moment regions under fatigue loading When the upper limit of repeated loading is larger than the initial cracking loading, the fatigue tested specimens will be pre-cracked after the monotonic loading process which can be regarded as the first repeated cycle. After unloading, the nonrecoverable slip would occur at the beam-slab interface [8], as well as the residual crack width in the concrete slab [15]. Then the residual deflection behavior can be characterized by a specific non-recoverable deformation related to pre-cracked loading and an increasing plastic deflection related to fatigue loading cycles. On this base, an analytical model is proposed to estimate the residual deflection of negative moment regions of steel-concrete composite beams under fatigue loading in this section. 3.1. Analysis of residual deflection related to pre-cracked loading According to a series of experimental analyses on partially prestressed concrete beams, the residual deflection related to

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A. Song et al. / Construction and Building Materials 158 (2018) 50–60

Local bucking of bottom flange

(a) Static failure mode of SCB1-1 and SCB1-2

3

2

Cracking location of top flange

Fatigue cracking of top flange Fatigue crack developed rapidly 1

4

Cracking location of concrete slab

Static cracking of concrete slab

Fracture of reinforcement

(b) Fatigue failure mode of SCB1-3 Fig. 5. Failure modes of test specimens.

f 1 ¼ a1 þ a2

LMmax n0 qHM cr

ð5Þ

where a1 and a2 are the undetermined parameters; Mmax is the maximum applied moment corresponding to upper limit of repeated loading; Mcr is the cracking bending moment; q is the tensile reinforcement ratio; L and H are the span and height of the beam; n0 = Es/Ec; Ec and Es are the young’s modulus of concrete and steel respectively. As for the steel-concrete composite beams under negative moment, the cracking bending moment Mcr taking the combination effect into consideration can be determined using the following equation [20]:

Mcr ¼ Fig. 6. Load-deflection response of specimens.

pre-cracked loading was depended on the corresponding bending moment, the span and height of beam, and the material properties [19], which can be defined as the following form:

f tk I0 n0 esh Es As I0  þ esh Es As ys0 z0 A0 z 0

ð6Þ

where, ftk is the tensile strength of concrete; esh is the shrinkage strain of concrete; I0 is the moment inertia of composite section in the cracked section; z0 is the distance from the centroid of the concrete slab to the centroid of the composite section; ys0 is the distance from the centroid of the steel beam to the centroid of the composite section; A0 is the transformed steel area, and A0 = As + -

55

A. Song et al. / Construction and Building Materials 158 (2018) 50–60

Ar + Ac/n0; As, Ar and Ac are the areas of steel beam, longitudinal reinforcement and concrete slab respectively. The shrinkage strain of concrete is affected by various factors. In this work, a simple calculation method suggested in China Code GB 50917-2013 [21] was employed. According to the actual conditions of the test site, the annual average relative humidity (RH) was 76%, the nominal shrinkage coefficient of concrete (ecs0) was 310m, and the shrinkage reduction coefficient of cast-in-place concrete (ce) was 1.00. Thus shrinkage strain of concrete can be obtained as:

LMmax n0 qHMcr

Proportion Mmax/Mu

Residual deflection (mm)

SCB1-2 SCB1-3 SCB1-1 SCB1-1

175.0 473.5 350.0 612.5

20% 40% 50% 70%

0.50 1.10 1.38 2.80

3 Calculation = Test values Present study Zhou (2011)

2.5

Load proportion of 70%

2

1.5

1

0.5

0

0

1

0.5

2

1.5

2.5

3

Calculation (mm) Fig. 7. Comparison between calculated and tested values of residual deflection related to pre-cracked loading.

ð8Þ

To verify the validity and applicability of the proposed formula, existing experiments conducted by this study and Zhou [14] were selected and analyzed. Fig. 7 illustrates the comparison of residual deflection between measured results and prediction of Eq. (8). It is observed from this figure that the results of the proposed formula are in good agreement with the experimental results when the load proportion (Mmax/Mu) is smaller than 70%. As for the proportion of 70%, a large deviation occurred due to the yielding of tested beam (see Fig. 6). Thus, when 50% of the ultimate loading is defined as the service load of composite beams [22], it is accurate enough to predict the residual deflection related to pre-cracked loading in negative moment regions at the service limit states. 3.2. Analysis of residual deflection related to fatigue loading With the increasing number of cycles, the plastic slip at the beam-slab interface increases gradually, resulting in an increase of the residual deflection. Based on the existing formulation of plastic slip of shear studs (dstd,N) [23], a method for calculating the residual deflection (fr) of steel-concrete composite beams under positive moment was proposed by Wang et al. [11]:

fr ¼ k

Maximum applied moment Mmax (kNm)

ð7Þ

Table 2 shows the comparison of the cracking bending moment between the test results and the calculation values of Eq. (6). It can be seen that the shrinkage strain of concrete showed obvious effect on the cracking bending moment in the negative moment regions of composite beams. However, when the shrinkage strain was given by Eq. (7), the maximum error was controlled within 10%. Hence, it is effective to use Eqs. (6) and (7) to study the residual deflection of composite beams as expressed in Eq. (5). The residual deflections of tested beams produced by the precracked loading were characterized in Table 3. It is observed that the relationship between the maximum value of applied loading and the residual deflection were nearly linear, when the maximum applied moment (Mmax) was smaller than 70% of the ultimate bending moment (Mu). Based on the measured results, the parameters a1 and a2 in Eq. (5) were achieved by fitting the selected data which were corresponding to the proportions (Mmax/Mu) of 20%, 40% and 50% respectively. The equation of residual deflection related to pre-cracked loading was then obtained as follow:

f 1 ¼ 0:089 þ 0:0071

Specimen

Test values (mm)

esh ¼ ecs0 ce ¼ 310l

Table 3 The residual deflection produced by pre-cracked loading.

dstd;N L 12H

ð9Þ

and the plastic slip under fatigue loading can be expressed as follows:



dstd;N ¼ C 1  C 2 ln

 1  1 P 0; 0 < n=N < 0:9 n=N

ð10Þ

dstd;N ¼ 0; n=N ¼ 0

ð11Þ

where the coefficients C1 and C2 are given below:

C 1 ¼ 0:104e3:95Pmax =Pu;0

ð12Þ

C 2 ¼ 0:664Pmin =Pu;0 þ 0:029

ð13Þ

In the above equations, k = 10.11 is the dimensionless parameter; n is the number of loading cycles; N is the fatigue life; Pu,0 is the ultimate static strength of the shear stud; Pmax and Pmin are the upper limit and lower limit of fatigue loading of the shear stud respectively. However, the parameter k in the formula of residual deflection was achieved by fitting the limited test data, and it may not achieve reasonable results for other conditions especially when the composite beams subjected to negative moment. Meanwhile, the ultimate static strength of the shear stud was underestimated when the method was employed from China Code GB 50017-2003 [24]. In addition, when the plastic neutral axis was designed in the

Table 2 Comparison of cracking bending moment between test and calculation results. Specimen

SCB1-1 SCB1-2 SCB1-3

Calculation results (kNm) Test results Mcr,exp

esh=0m

esh=150m

esh=310m

esh=450m

Mcr,0

Mcr,exp/Mcr,0

Mcr,150

Mcr,exp/Mcr,150

Mcr,310

Mcr,exp/Mcr,310

Mcr,450

Mcr,exp/Mcr,450

61.25 59.50 58.63

113.15 113.15 113.15

0.54 0.53 0.52

85.62 85.62 85.62

0.72 0.69 0.68

56.26 56.26 56.26

1.09 1.06 1.04

30.57 30.57 30.57

2.00 1.95 1.92

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A. Song et al. / Construction and Building Materials 158 (2018) 50–60

concrete slab, the residual deformation related to pre-cracked loading should be taken into consideration separately. Therefore, an improved modified model based on theoretical analysis to predict the residual deflection in negative moment regions of composite beams under fatigue loading is presented in this section. 3.2.1. Plastic slip at the beam-slab interface under fatigue loading The mechanical behavior of shear studs in the standard pushout specimens is different from that in composite beams [25], and the ultimate shear strength obtained from the former is always larger than the latter. The main reason for this phenomenon is that the shear studs in the standard push-out specimens are close to the ideal pure shear state, while the studs in composite beams are affected by both bending moment and shear force. As for a simply supported composite beam, in the support regions, the shear force is equal to the maximum value, while the bending moment reaches zero, as shown in Fig. 8. Then a pure shear state is assumed at the support location in this study. Thus the mechanical behavior of shear studs in composite beams can be equivalent to that in the standard push-out specimens. Namely, the method for the plastic slip of studs in Eqs. (10) and (11) can be employed to the analysis of composite beams under negative moment. It should be noted that initial cracking of concrete or nonrecoverable deformation of stud will occur in the static loading process for the first time, i.e. in the first cycle of the repeated loading [8]. Thus the residual slip related to the pre-cracking loading should be separated from the total values. By refitting the measured data of push-out tests performed by Hanswille et al. [23], the new forms of Eqs. (12) and (13) can be then obtained without consideration of the pre-cracking effect which can be expressed as follows:

C 1 ¼ 0:116e3:309Pmax =Pu;0

ð14Þ

C 2 ¼ 0:384P min =Pu;0 þ 0:044

ð15Þ

The formula of the ultimate static strength for the shear studs based on a series of experiments and analyses is employed in this work [26], as can be expressed as follow:

 
1:7 0:80:15 lnðds 10Þ Pu;0 ¼ 0:2ds  10 f cu 0:002f y þ 0:24

ð16Þ

where ds is the stud diameter; fcu is the compressive strength of cubic concrete block; fy is the yield strength of the stud. As for the fatigue upper limit or lower limit of a stud at the support, the following method can be used:

Steel beam

Pmax=min ¼

V max=min S0c p I 0 ns

ð17Þ

where Vmax/min is the vertical shear force acted on the support section corresponding to the upper limit (Pmax) or the lower limit (Pmin); S0c is the area moment of the concrete slab with regard to the neutral axis of composite section; I0 is the moment inertia of composite section in the uncracked section; p is the distance between the studs; ns is the number of shear studs per row. The comparison of plastic slips (dstd,i) at the support location between measured values and calculated results is plotted with respect to normalized loading cycles (n/N) in Fig. 9. As no fatigue failures occurred in the tested beam SCB1-2, the fatigue life (N) was obtained by the method in Eq. (4). It is observed from the figure that during fatigue loading, the evolution of residual slip can be characterized by three stages: in the stage I, i.e. 0 < n/N  0.1, the residual slip increases rapidly with increment of loading cycles; in the stage II, i.e. 0.1 < n/N < 0.9, the growth rate of the residual slip becomes slow and stable gradually, and this stage accounts for about 80% of the fatigue life; and in the stage III, i.e. 0.9  n/ N < 1, the residual slip increases rapidly again until the fatigue failure occurred. It should be noticed that there was a good agreement between the calculation model and experimental results in the stage II, with the maximum error was controlled within 15%, as shown in Fig. 10. As for engineering application, the accuracy was satisfactory. However, a larger deviation occurred in the stage I and stage III (see Fig. 9), which may be ascribed to the discreteness of concrete strength and development of initial microcracks at the beginning of fatigue loading, as well as the different failure modes of components and the sharp growth of measured values at the end of fatigue loading. 3.2.2. Plastic slip at the reinforcement-concrete interface under fatigue loading The bond stress-slip relationship between reinforcing bars and concrete is complicated, especially when the reinforced concrete structures are subjected to repeated loadings [27]. Based on the experimental results, Jiang and Qiu [28] proposed an empirical formula for the plastic slip at the reinforcement-concrete interface under fatigue loading, which can be expressed as follow:

dr;N ¼ dr;1 nb

ð18Þ

where dr,N is the total plastic slip corresponding to the repeated cycles of n; dr,1 is the plastic slip produced in the first cycle; b = 0.057 is the dimensionless parameter.

Applied load

Concrete slab

Negative moment

Pure shear state at the support location

Shear force

Fig. 8. Mechanical behavior of simply supported composite under negative moment.

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A. Song et al. / Construction and Building Materials 158 (2018) 50–60

where dr,i is the plastic slip related to fatigue loading without consideration of the pre-cracking effect. Through a series of theoretical analyses for the stiffness and deflection of steel-concrete composite beams under negative bending, Nie et al. [29] found that the slip between reinforcement and concrete was small enough compared with the total slip and can thus be neglected at the service limit states. In this study, the fatigue loading level is smaller than the service loads (i.e. 50% of the ultimate loading), and then an assumption for the relationship of the residual slips at the beam-slab interface and the reinforcement-concrete interface can be made as follow:

0.4 Prediction Test values Residual slip related to pre-cracked loading

0.35 0.3

Stage

Stage

δ std,N (mm)

0.25

Stage

0.2 0.15 0.1

dstd;1  dr;1

δstd,i

0.05

δstd,1

0.015 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n/N

Prediction Test values Residual slip related to pre-cracked loading

δ std,N (mm)

Stage

3.2.3. Analytical model of residual deflection under fatigue loading The slip effect at the beam-slab interface is always not sensitive to the loading type [30]. Meanwhile, it can be assumed that the development law of the residual slip under fatigue loading is similar to the slip under static loading. Then after considering the fact that an actual beam usually carries different type of loading simultaneously, a formula of residual slip under fatigue loading is proposed for general loading as:

Stage

0.25

Stage

0.2 0.15 0.1

0

di ¼ k ðL  xÞx=L2

δstd,i

0.05

δstd,1

0.02 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n/N

(b) Specimen SCB1-3 Fig. 9. Comparison between calculated and tested values of plastic slip at support location.

Calculation = Test values Test values

0

fi ¼

kL 12H

ð23Þ

di jx¼L=2 ¼ dstd;i þ dr;i

0.2

Test values (mm)

ð22Þ

where k0 is the unknown parameter related to repeated cycles; x is the distance from the calculation section to the mid-span; L is the span of the beam. Based on the curvature-deflection relation of composite beams, the residual deflection at the mid-span can be derived from Eq. (22):

The residual slip at the support can be regarded as the boundary condition of Eq. (22), which can be expressed as following equation:

0.25

ð24Þ

And the unknown parameter k’ can be obtained by solving Eq. (24):

0.15

0

k ¼ 4 dstd;i þ dr;i 0.1




ð25Þ

When taking no account of the residual slip at the reinforcement-concrete interface and substituting Eq. (25) into Eq. (23), the formula of the residual deflection related to fatigue loading can be expressed as follow:

0.05

0

ð21Þ

Namely, the plastic slip at the reinforcement-concrete interface can be neglected compared with the plastic slip at the beam-slab interface.

0.4

0.3

When the fatigue loading reached 200  10 cycles, dr,i=1.286 dr,1 can be obtained according to Eq. (19). Thus the following reformation can be given:

dstd;i  dr;i

(a) Specimen SCB1-2

0.35

ð20Þ 4

fi ¼ 0

0.05

0.1

0.15

0.2

0.25

Calculation (mm)

ð26Þ

3.3. Total residual deflection

Fig. 10. Comparison between calculated and tested values of plastic slip in stage II (0.1 < n/N < 0.9).

After eliminating the pre-cracking effect, Eq. (18) can be transformed into a new form expressed as follow:


dr;i ¼ dr;1 nb  1

dstd;i L 3H

ð19Þ

During the whole progress of fatigue loading, the total residual deflection (fN) can be divided into two components, i.e., the residual deflection related to pre-cracked loading (f1) and the residual deflection related to fatigue loading (fi). Thus

f N ¼ 0:089 þ 0:0071

LMmax dstd;i L þ 3H n0 qHMcr

ð27Þ

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A. Song et al. / Construction and Building Materials 158 (2018) 50–60

3 Prediction Test values Residual deflection related to pre-cracked loading

2.5

fN (mm)

2 1.5

Stage

Stage Stage

1 fi

0.648 0.5

f1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n/N

(a) Specimen SCB1-2

curves and experimental results of specimens SCB1-2 and SCB1-3 are shown in Fig. 11. Three stages can also be defined for the overall developing process of the residual deflection: in the stage I (0 < n/N  0.1), the residual deflection increases rapidly; in the stage II (0.1 < n/N < 0.9), the residual deflection increases slowly and steadily; and in the stage III (0.9  n/N < 1), the residual values increase rapidly again. It can be seen from Fig. 11 that there is a good agreement between the calculation model and experimental results in the stage II. For the further verification of the proposed model quantitatively, the existing data of fatigue tests conducted both in present study and by Zhou [14] were selected and analyzed, as characterized in Table 4. It can be found that the most of deviation values were smaller than 15%. And the results further indicate that the proposed model can be used to predict the residual deflection in the negative moment regions of steel-concrete composite beams under fatigue loading. Thus the proposed model in this study can be a design reference for engineering applications of composite bridges.

3 2.5 2

fN /mm

4.2. Suggestion for the fatigue design

Prediction Test values Residual deflection related to pre-cracked loading

Stage

Stage Stage

1.5 fi

1.1 1

f1

0.5 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n/N

(b) Specimen SCB1-3 Fig. 11. Comparison between calculated and tested values of residual deflection at mid-span.

4. Verification and suggestion for the analytical model 4.1. Test verification To verify the accuracy of the analytical model for estimating the residual deflection in the negative moment regions of steelconcrete composite beams under fatigue loading, the predicted

The fatigue failure mode of the composite beams under negative moment is determined by three components, i.e. the shear stud, tensile top plate and longitudinal reinforcement, which is different from the composite beams under positive moment. The larger stress amplitude of tensile top plate may result in the poor ductility at the end of fatigue loading, as observed from the fatigue test on SCB1-3. Hence, it is important to make a reasonable design for the negative moment regions, whose typical failure mode was the fatigue failure of studs, and then the composite beams may still maintain good ductility although the fatigue failure had already occurred. In addition, the residual deformation increased gradually under the repeated loading, as observed in the fatigue test. The residual deflection in the stage II listed in Table 4 accounted for about 20–30% of the static deflection caused by the fatigue upper limit under the monotonic loading (see Fig. 6). Thus the residual deformation caused by fatigue loading should not be ignored in order to improve the accuracy of the total deflection analysis. When the fatigue loading is smaller than the service loads, the following formula is suggested to calculate the deflection in the negative moment regions of steel-concrete composite beams under fatigue loading:

f ¼ f 0 þ f N ; ð0:1 < n=N < 0:9Þ

ð28Þ

where f0 is the elastic deflection of the composite beam under negative moment corresponding to the fatigue upper limit and

Table 4 Comparison between calculated and tested values of residual deflection in stage II (0.1 < n/N < 0.9). Fatigue life (104) N = Nf

Specimen

Present study

Zhou [14]

SCB1-2

516

SCB1-3

152

CBF-1200-2 CBF-1200-3 CBF-1200-4

1398 1715 455

n/N

Residual deflection (mm)

Deviation

Test results

Predicted values

0.10 0.19 0.29 0.39 0.48 0.33 0.66

0.848 0.948 0.988 1.018 1.028 1.577 1.607

0.755 0.843 0.901 0.949 0.993 1.430 1.586

11.0% 11.1% 8.8% 6.8% 3.5% 9.3% 1.3%

0.14 0.10 0.11 0.21 0.33 0.47

1.565 1.945 2.890 2.920 2.930 3.005

1.320 1.631 2.352 2.596 2.803 2.999

15.7% 16.1% 18.6% 11.1% 4.3% 0.2%

A. Song et al. / Construction and Building Materials 158 (2018) 50–60

can be obtained according to the calculation method for positive moment; fN is the total residual deflection at mid-span calculated by the proposed model in this paper; N is the fatigue life obtained by the Eq. (4). As for the stage I and stage III, the residual deflection can not be predicted accurately (see Fig. 11), due to the discreteness of the material property, the development of initial microcracks, and the nondeterminacy of damage degree at the end of fatigue loading. Then a simple method may be used to estimate the structure security under fatigue loading according to the proposed model in the stage II (see Eq. (28)): in the stage I (0 < n/N  0.1), the residual deflection can be regarded as a consistent value which is equal to the lower limit of proposed model; and in the stage III (0.9  n/ N < 1), the composite beams can be assumed to have reached the fatigue failure stage. And for engineering application, the suggestion is conservative. 5. Conclusions In this paper, the residual deflection behavior in the negative moment regions of steel-concrete composite beams was investigated through the static and fatigue tests conducted on three specimens. Through using laser displacement sensors, the plastic slip of beam-slab interface at the support and the residual deflection at the mid-span were cautiously measured and analyzed. Then an analytical model for the residual deflection was proposed and verified. Based on the presented results, the following conclusions can be drawn. (1) The fatigue life was significantly affected by the load amplitude of the tested beams under negative moment. When the fatigue upper limit was equal to 40% of the ultimate load, the fatigue failure mode of the specimen was the fatigue cracking of tensile top plate followed by the fracture of longitudinal reinforcement. However, when the fatigue upper limit was equal to 25% of the ultimate load, no fatigue failure occurred after 250  104 repeated cycles. Meanwhile, the specimen still showed good ductile failure characteristic in the residual static test. (2) The plastic slip at support and the residual deflection at midspan gradually increases with the increasing number of repeated cycles. When the fatigue upper limit was smaller than service load, the typical evolution of the residual values can be characterized by three stages: in the stage I (0 < n/ N  0.1), the residual values increased rapidly; in the stage II (0.1 < n/N < 0.9), the residual values increased slowly and steadily; and in the stage III (0.9  n/N < 1), the residual values increased rapidly again. Moreover, to obtain a reasonable result in the overall process of fatigue loading, the residual values produced by the pre-cracking effect were analyzed separately. (3) An analytical model was proposed for the prediction of residual deflection in negative moment regions of steelconcrete composite beams. The total residual deformation was regarded as the superposition of the residual deflections related to the pre-cracking loading and fatigue loading respectively. In the stage II (0.1 < n/N < 0.9), the good agreement can be observed through the comparison between the predicted values and the measured results of the experimental beams performed in this study and reported in the earlier companion paper. Based on the fatigue test and model analysis, the fatigue design recommendations for the steelconcrete composite plate beams subjected to negative moment were given. For engineering design and application,

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this work can provide reference for calculation and evaluation of the negative moment regions of steel-concrete composite beams under long-term live load. Acknowledgments This research is sponsored by the Major State Basic Research Development Program of China (973 Program) (No. 2012CB026200), and the Fundamental Research Funds for the Central Universities of China (No. KYLX15_0143). References [1] J. Liu, F.X. Ding, X.M. Liu, Z.W. Yu, Study on flexural capacity of simply supported steel-concrete composite beam, Steel Comp. Struct. 21 (4) (2016) 829–847. [2] J. Deng, M.M.K. Lee, S.Q. Li, Flexural strength of steel–concrete composite beams reinforced with a prestressed CFRP plate, Constr. Build. Mater. 25 (1) (2011) 379–384. [3] S.M. Chen, X.D. Wang, Y.L. Jia, A comparative study of continuous steel– concrete composite beams prestressed with external tendons: experimental investigation, J. Constr. Steel Res. 65 (7) (2009) 1480–1489. [4] C.S. Shim, S.P. Chang, Cracking of continuous composite beams with precast decks, J. Constr. Steel Res. 59 (2) (2003) 201–214. [5] H.K. Ryu, S.P. Chang, Y.J. Kim, B.S. Kim, Crack control of a steel and concrete composite plate girder with prefabricated slabs under hogging moments, Eng. Struct. 27 (11) (2005) 1613–1624. [6] H.K. Ryu, C.S. Shim, S.P. Chang, C.H. Chung, Inelastic behaviour of externally prestressed continuous composite box-girder bridge with prefabricated slabs, J. Constr. Steel Res. 60 (7) (2004) 989–1005. [7] Q.T. Su, G.T. Yang, C. Wu, Experimental investigation on inelastic behavior of composite box girder under negative moment, Int. J. Steel Struct. 12 (1) (2012) 71–84. [8] G. Hanswille, M. Porsch, C. Ustundag, Resistance of headed studs subjected to fatigue loading part I: experimental study, J. Constr. Steel Res. 63 (4) (2007) 475–484. [9] B. Wang, Q. Huang, X.L. Liu, Deterioration in strength of studs based on twoparameter fatigue failure criterion, Steel Comp. Struct. 23 (2) (2017) 239–250. [10] R. Seracino, D.J. Oehlers, M.F. Yeo, Partial-interaction fatigue assessment of stud shear connectors in composite bridge beams, Struct. Eng. Mech. 13 (4) (2002) 455–464. [11] Y.H. Wang, J.G. Nie, J.J. Li, Study on fatigue property of steel-concrete composite beams and studs, J. Constr. Steel Res. 94 (2014) 1–10. [12] Z.H. Zong, H.M. Che, Fatigue behavior of prestressed composite steel concrete beams. Journal of the China Railway Society, J. China Railway Soc. 22 (3) (2000) 92–95 (in Chinese). [13] H.K. Ryu, Y.J. Kim, S.P. Chang, Crack control of a continuous composite twogirder bridge with prefabricated slabs under static and fatigue loads, Eng. Struct. 29 (6) (2007) 851–864. [14] X.W. Zhou, Research on Mechanical Behavior of composite Beam with Steel Plate-Concrete Composite Decks Under Static and Fatigue Loads Master Dissertation, Xi’an University of Architecture and Technology, Xi’an, 2011 (in Chinese). [15] W.W. Lin, T. Yoda, N. Taniguchi, Fatigue tests on straight steel–concrete composite beams subjected to hogging moment, J. Constr. Steel Res. 80 (1) (2013) 42–56. [16] E.N. Bs 1994–2, Eurocode 4: Design of composite steel and concrete structures, Part 2: General rules and rules for bridges 2005 European Committee for Standardization; Brussels Belgium. [17] TB 10002.2-2005, Code for Design on Steel Structure of Railway Bridge, Ministry of Railways of China, Beijing, China, 2005 (in Chinese). [18] H. Pan, Experimental Study on Fatigue Behaviors of Concrete Flexural Components, Ph.D. Dissertation, Southeast University, Nanjing (2006) (in Chinese). [19] B. Lei, Experimental Research and Numerical Simulation on Mechanical Properties of P.P.C Beam Under Fatigue Loading, Master Dissertation, Dalian University of Technology, Dalian (2013) (in Chinese). [20] J.S. Fan, Experiments and Research on Continuous Composite Beams of Steel and Concrete, Ph.D. Dissertation, Tsinghua University, Beijing (in Chinese) (2003). [21] GB 50917-2013, Code for Design of Steel and Concrete Composite Bridges, Ministry of Construction of China, Beijing, China, 2013 (in Chinese). [22] J.G. Nie, J.S. Fan, C.S. Cai, Experimental study of partially shear-connected composite beams with profiled sheeting, Eng. Struct. 30 (1) (2008) 1–12. [23] G. Hanswille, M. Porsch, C. Ustundag, Resistance of headed studs subjected to fatigue loading part II: analytical study, J. Constr. Steel Res. 63 (4) (2007) 485– 493. [24] GB 50017-2003, Code for Design of Steel Structures, Ministry of Construction of China, 2003. Beijing, China(in Chinese).

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