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Numerical study on the deformation behavior of steel concrete composite girders considering partial shear interaction
Structures 23 (2020) 437–446
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Numerical study on the deformation behavior of steel concrete composite girders considering partial shear interaction ⁎
Madhusudan G. Kalibhat , Akhil Upadhyay
T
⁎
Department of Civil Engineering, Indian Institute of Technology, Roorkee 247667, India
A R T I C LE I N FO
A B S T R A C T
Keywords: Steel concrete composite girder Deformations Partial shear interaction Parametric study Numerical model
The present paper focuses on the study of deformation behavior of steel concrete composite (SCC) girders with partial shear interaction. The main objective of this work is to bring out the relative significance of the partial shear interaction with respect to full interaction, which has been addressed with the help of parametric study considering various influencing parameters such as, degree of shear connection, span length, type of loading, distribution of shear stud connectors, geometry of steel girder, yield strength of structural steel and concrete grade. The parametric study has been carried out using a simplified numerical modelling technique with the help of commercially available finite element software SAP2000. The numerical modelling involves simulation of composite action between the concrete deck and the steel girder in the SCC girder, which has been carried out by using various elements such as, beam elements, rigid link elements and spring elements. Further, a comparative study has been carried out to study the accuracy of numerical model with the available analytical models. The results of the numerical model are also compared with the results of the equations given in several national codes to calculate deflections, to study the efficacy of the codal provisions. It is seen that the various aforementioned parameters have considerable influence on the deformations of SCC girders. It is also seen that the change in distribution type of shear stud connectors considerably reduce their number without significantly affecting the deformations of SCC girder.
1. Introduction The use of SCC girder as an alternative to the RC girder or the steel girder has become ubiquitous in the recent decades because of the fact that it is advantageous in terms of strength and stiffness when compared to RC girder and steel girder alone [1,2]. As shown in the Fig. 1 a typical SCC girder consists of a concrete deck component connected to the steel girder component with the help of shear stud connectors at their interface and the composite action is achieved by these shear stud connectors [3]. However, the extent of this composite action is characterized by the behavior of the shear stud connectors [4–7]. If the shear connectors are infinitely rigid, the full composite action or the full interaction is achieved. In this case, the analysis of the SCC girder is performed by transformed section approach (i.e., by transforming one of the components with respect to the other, using appropriate modular ratio). Conversely, these shear connectors tend to slip at the interface, regardless of the magnitude of load, because of their finite stiffness [8,9] and consequently the interaction between the concrete deck and the steel girder is partial. Hence, the transformed section approach cannot be used for the analysis of this partially connected composite
⁎
girder. Therefore, the slip at the interface resulting because of the finite stiffness of the connector should be taken into consideration during the design so as to accurately predict the deflections of composite girder which is generally omitted by the designers. In the past researchers across the globe have shown keen interest in effects of slip on the deflections of SCC girders. The very first work on SCC beams considering slip effects dates back to mid twentieth century by Newmark et al. [10], which is a highly cited and acclaimed work. Johnson and May [11] derived design rules are for SCC beams with partial-interaction for determining the flexural strength at ultimate limit state, and for checking deflections at serviceability limit state. Since then different researchers have developed different analytical models by solving differential equations considering the relative displacement at the interface of composite section of SCC girders [1,12–19]. These analytical models can be used to compute the deformations in SCC girders. However, these analytical models are based on the simplifying assumption like uniform distribution of shear connectors which limits the practical applicability. Further, the analytical models are complex to use in day-to-day designs. Alternatively, numerical methods [20–30] have been developed and are widely used.
Corresponding authors. E-mail addresses: [email protected] (M.G. Kalibhat), [email protected] (A. Upadhyay).
https://doi.org/10.1016/j.istruc.2019.10.007 Received 20 May 2019; Received in revised form 12 October 2019; Accepted 16 October 2019 2352-0124/ © 2019 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
Structures 23 (2020) 437–446
M.G. Kalibhat and A. Upadhyay
Nomenclature Ac= A s= hc= h s= dsh= Ec= EIeff= EIf= Es =
f′c= fck= h= Ic= Is= K= L= m= Ns= p= Q u= V L= Δf= Δpi=
Area of cross section of concrete deck Area of cross section of steel girder Centroidal distance of concrete deck from the interface of SCC girder Centroidal distance of steel girder from the interface of SCC girder Shank diameter of the shear connector Modulus of elasticity of concrete Effective flexural rigidity of SCC girder for any degree of interaction Flexural rigidity of transformed cross section of SCC girder for full interaction Modulus of elasticity of steel girder
These also include rigorous finite element modeling techniques [20,23–26] and stiffness matric approach [28–30] for the analysis of SCC girders with partial shear interaction. Numerical modelling is more suitable than analytical models because of the ease with which different components of composite beam can be modelled and composite action between the connecting components can be established. Further, additional degrees of freedom because of the slip of the shear connector also can be easily incorporated in the numerical model. A lot of experimental and numerical studies have also been conducted in the past on SCC girders considering the slip of the stud connectors [21,31–34]. All these studies showed that the slip of the shear stud connectors increase the deformations in SCC girders when compared to the SCC girders analyzed using transformed section approach. A few national codes [35,36] recommend considering partial shear interaction in the design and provide equations to calculate the deflections of SCC girders. In this paper, the effect of partial shear connection on the deformations of SCC girders has been discussed with the help of parametric study considering various design parameters such as, degree of shear connection, span length, type of loading, distribution of shear stud connectors, geometry of steel girder, yield strength of steel and concrete grade. For this purpose, a simplified numerical modelling technique has been adopted because of the following advantages: (1) Easiness in modeling at design offices when compared to analytical and other rigorous numerical methods which cannot be used in day-to-day design because of their complexity and the computational effort involved in them. (2) Flexibility and generality in dealing with different types of loading and boundary conditions. (3) Practicality in terms of ease with which the distribution of the shear connectors can be varied along the span, which is not possible in analytical models because of the limitations imposed by the assumptions made. Besides these, the modeling approach is simple when compared to rigorous finite element methods and can also be extended to the nonlinear domain (though it is
Characteristic strength of concrete cylinder Characteristic strength of concrete cube Total height of SCC girder Second moment of area of concrete deck about its centroid Second moment of area of steel girder about its centroid Shear connector’s stiffness in load per unit slip Span length of SCC girder Modular ratio Number of shear connectors provided Pitch of the shear connectors Ultimate strength of the shear connector Longitudinal shear per unit length (i.e. Shear flow) Deflection of SCC girder with full interaction Deflection of SCC girder with partial shear interaction
out of the scope of the present work). The design engineer usually assumes full interaction during the design of SCC girders which is not the case in reality. Therefore, this type of study not only gives an in-depth understanding about the deformations due to the partial shear interaction in SCC girders, but also gives the usefulness of SCC girders when compared to its counterparts such as RC and steel girders, to the design engineer in developing countries where SCC as a structural system is sparsely adopted. 2. Numerical modelling In the present study, a simplified numerical modeling technique has been adopted to study the behavior of SCC girders incorporating partial interaction. Fig. 2 shows a schematic representation of typical 2D numerical model which is based on linear elastic finite element approach. The numerical modeling has been carried out using finite element software SAP2000 [37]. Different types of elements used to imitate partial composite action in a SCC girder are described as follows: A. Element 1 as shown in the Fig. 2 depicts the concrete deck element and is modelled as a beam element. The properties of element 1 are same as of the concrete deck at its centroid. B. Element 2 as shown in the Fig. 2 depicts the steel girder element and is modelled as a beam element. The properties of element 2 are same as of the steel girder at its centroid. C. Element 3 represents a spring element that is used to simulate shear connector’s behavior at the interface. This spring element has a finite stiffness in longitudinal direction of the SCC girder and is equal to the stiffness of the shear connector, whilst it has infinite stiffness in other directions. Eq. (1) gives stiffness of the spring element K as given in [1] and is as follows.
K=
Qu d sh (0.16 − 0.0017f ′c)
(1)
Concrete cylinder strength f′c is obtained by multiplying 0.8 to fck (as per recommendations of IS 516: 1959) [39]. A. Elements 4 represents rigid link element used to connect the nodes of centroid of concrete deck (i.e. element1) with the spring element (i.e. element 3) at the interface. B. Elements 5 represents rigid link element used to connect the nodes of centroid of steel girder (i.e. element 2) with the spring element (i.e. element 3) at the interface. The element 3, element 4 and element 5 put together bring about the composite action in the SCC girder. The required degree of shear connection can be established, merely by varying the spacing of the spring element 3, element 4 and element 5.
Fig. 1. Typical cross section of SCC girder. 438
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Fig. 2. A schematic representation of numerical model of SCC girder.
3. Comparative study
A1 =
In this section a comparative study has been carried out which involves the comparison of the results of the numerical model against the results of available analytical models for different degree of shear connection. The results of the numerical models are also compared with the results of several national codes. Note that the details about the degree of shear connection is explained in the “Section 5.1” of this paper.
A0 I0 + A 0 (h c + h s )2
3.1.3. Girhammar’s model Eq. (4) simplified analytical model as proposed by Girhammar [18] for the determination of the deflection of the SCC girder considering partial shear connection and is as follows
Δpi =
wL4 X X3 2X 2 . ⎛ − + 1⎞ 24EIeff L ⎝ 3L3 L2 ⎠ ⎜
⎟
3.1. Analytical models EI
f − 1⎤ ⎡ ∑ EI Where: EIeff ≈ ⎢1 + ⎥ (α L)2 ⎢ 1+ 2 ⎥ π ⎣ ⎦
Here the various analytical models proposed by different researchers considered in the present study are explained.
Δf
=1+
∑ EI = Ec Ic + Es Is
24(C − 1) 1 1 ⎡ − (1 − cosh( R ) + tanh( R )sinh( R ) ⎤ 5R R ⎦ ⎣2 (2)
3.2. National codes In the following sub sections, the provisions of various national code with regard to partial shear connection are explained.
A (h + h s)2 ⎤ Where: C = ⎡1 + 0 c ⎥ ⎢ I0 ⎦ ⎣
R=
3.2.1. AISC provisions AISC 360 [35] provides the equation for the determination of effective flexural rigidity of the SCC girder and is defined in the equation (5).
Kj 16
Kj = G1L3
EIeq = EIs + (Ns /Nf )1/2 ∗ (EIf − EIs)
K ⎡ A (h + h s)2 ⎤ G1 = 1+ 0 c ⎥ XEs A 0 ⎢ I0 ⎦ ⎣ A0 =
I0 =
IC + IS m
3.2.2. Eurocode provisions Eurocode 4 [36] also provides the equation for the determination of deflection of SCC girders due to partial shear interaction and is given in Eq. (6).
3.1.2. Nie’s model Eq. (3) gives analytical model as proposed by Nie and Cai [15] for the determination of the deflection of the SCC girder considering partial shear connection and is as follows
⎛ L2 2e( ) − 1 − e μL ⎞ + Δpi = + βw⎜ 384EIf μ2 h(1 + e μL) ⎟⎟ ⎜ 8h ⎝ ⎠
β=
Δpi
N Δ = 1 + C ⎛1 − s ⎞ ⎛ s − 1⎞ N f ⎠ ⎝ Δf ⎠ ⎝ ⎜
Δf
μL 2
5wL4
(5)
where, EIeq is the equivalent flexural rigidity of the cross section of SCC girder and Nf is the number of shear stud connectors required to achieve the full interaction. Merely by varying the ratio Ns/Nf, (Ns/Nf < 1), the required equivalent flexural rigidity for any degree of shear connection is obtained.
ASA C mAS + A C
Where: μ2 =
∑ EI
1 1 (h + hS )2 ⎤ + + C α2 = K ⎡ ⎢ EC A C ES AS ∑ EI ⎥ ⎣ ⎦
3.1.1. Jasim’s model Eq. (2) gives the analytical model proposed by Jasim [14] for the determination of the deflection of the SCC girder considering partial shear connection and is as follows
Δpi
(4)
−1
⎟⎜
⎟
(6)
where C is a coefficient, taken as 0.3 and 0.5 for unpropped and propped construction respectively, and Δs is the deflection of the steel girder for the same load.
(3)
K A1ES I0 p
3.3. Model verification
A1 (hC + hS )p K
For the purpose of comparison, a SCC girder of span 20 m, subjected to uniformly distributed load of 55kN/m has been considered, with 439
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three different degree of shear interaction (i.e., full interaction, full shear connection and 0.5 degree of shear connection). The concrete deck has width of 2500 mm and depth of 250 mm. The geometric dimensions of the steel girder is given in table 4. The spacing of shear connectors for full shear connection and for 0.5 degree of shear connection is 119 mm and 240 mm respectively, and the connector’s stiffness is 54600 N/mm. In case of full interaction the shear connectors are modelled by providing rigid properties to the spring element. The mid-span deflection of the SCC girder obtained by the numerical model, different analytical models and several national codes are tabulated in the Table 1. It is observed from the Table 1 that the results of the numerical model closely match with that of different analytical models, which validates the accuracy of the numerical model. It can be seen from the results of the codal provisions that, they do not distinguish between the full interaction and the full shear connection, which is also observed from Eqs. (5) and (6). However, for partial shear connection cases, the codal predictions are good.
Table 2 Design parameters and their values considered in the study.
Degree of shear connection Span length (m) Ratio of bottom flange area to the top flange area of the steel girder Yield strength of steel (fy) in N/mm2 Concrete Grade (fck) in N/mm2
1, 0.75, 0.5 10, 15, 20, 25, 30, 35, 40 1, 1.5, 2, 3
Span Length (m)
10 15 20 25 30 35 40
To study the effects of various aforementioned parameters on the deformations of SCC girder a parametric study has been carried out. The pertinent values of various parameters considered in this work are tabulated in Table 2. The parametric study carried out here involves modeling and analysis of SCC girders, which are designed according to Indian standards IRC: 22 (2008) [38]. Two different loading types viz, uniformly distributed load (UDL), and the point load (PL) at the mid-span are considered, and magnitude of the UDL and PL at mid-span for different spans are given in Table 3. Here the magnitude of the PL at mid-span considered is such that it produces same magnitude of bending moment as produced by UDL. In this study three different types of distribution of shear connectors have been considered viz, uniform distribution, two steps distribution and three steps distribution. In case of uniform distribution of shear connectors the number of shear connectors is obtained from Eq. (7) as per IRC recommendations.
VL*L/2 ∑Q
Considered Values
250, 300, 350, 450 25, 30, 40, 50
Table 3 Loading type details for various span length.
4. Parametric study
Ns=
Design Parameters
Loading Type UDL (kN/m)
Point Load (kN)
50 50 55 60 65 70 70
250 375 550 750 975 1225 1400
respectively obtained by calculating the shear force and the corresponding shear flow at the support section, at a distance of L/6 from the support section and at a distance of L/3 from the support section. The number of shear connectors so obtained in three different portions are uniformly distributed in respective portions. Fig. 3 shows the qualitative representation of distribution of shear connectors in the shear span for various distribution types. To study the influence of geometry of steel girder on the deformations of the SCC girder four different types of variation in cross section of the steel girder have been considered. However, for all the variations the second moment of area of the cross section of SCC girder is kept constant. The variation is realized by only increasing the area of bottom flange there by keeping symmetry of the steel girder about its vertical axis. The variations are, cross section having equal flange area, cross section having bottom flange area equal to 1.5 times the top flange area, cross section having bottom flange area equal to twice the top flange area and cross section having bottom flange area equal to thrice the top flange area. These variations are respectively designated as S1, S2, S3 and S4 as shown in the Fig. 4. The cross section dimension of the steel girder for different spans is designed as per IRC recommendations and are given in the Table 4. In the Table 4, hs, btf, ttf, bbf, tbf and tw, respectively refer to the total height of the steel section, breadth of the top flange, thickness of the top flange, breadth of the bottom flange, thickness of the bottom flange and thickness of the web. The concrete deck is 2500 mm wide and 250 mm deep. The flow of design of SCC girder and calculation of deflection are explained with the help flow chart as given in Fig. 5. In Fig. 5, np and nf respectively stand for the number of connectors for any partial shear
(7)
The number of shear stud connectors so obtained are distributed uniformly over the shear span i.e. from the point of zero moment to the point of maximum moment (i.e. from support to the mid span in case of simply supported girders). In case of two steps distribution the shear span is divided into two equal portions. The number of shear connectors in the first portion and the second portion of the shear span is respectively obtained by calculating the shear force and the corresponding shear flow at the support section and at a distance of L/4 from the support section. The number of shear connectors so obtained in two different portions are uniformly distributed in respective portions. Similarly, in case of three steps distribution the shear span is divided into three equal portions and the number of shear connectors in the first portion, second portion and third portion of the shear span is Table 1 Comparison of results of numerical and various analytical models. Model's Name
Numerical Jasim’s Nie’s Girhammar’s AISC Eurocode
Full interaction
Full shear connection
0.5 shear connection degree
Mid-span deflection (mm)
Percentage difference w.r.t. numerical model
Mid-span deflection (mm)
Percentage difference w.r.t. numerical model
Mid-span deflection (mm)
Percentage difference w.r.t. numerical model
20.61 19.84 19.84 19.84 19.84 19.84
– 3.88 3.88 3.88 3.88 3.88
23.62 22.88 22.46 22.88 19.84 19.84
– 3.23 5.16 3.23 19.05 19.05
26.00 25.34 24.58 25.34 24.07 24.31
– 2.60 5.78 2.60 8.02 6.95
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Fig. 3. Distribution of shear connectors.
interaction and for full shear interaction.
Table 4 Details of steel cross section for different span length.
5. Results and discussion The effects of various aforementioned parameters on the deflection of SCC girder are discussed here. 5.1. Effect of degree of shear connection The two limits of the interaction in a composite girder are the full interaction (i.e. shear connectors have infinite stiffness and slip at the interface of the composite girder is zero) and no interaction between the two connecting elements. The analysis of a SCC girder with full interaction is carried out by transforming the cross section using appropriate modular ratio. In reality, the interaction exists between the full interaction and no interaction. The degree of shear connection is defined as the ratio of number of shear connectors provided to the number of shear connectors required for the full shear connection. The number of shear connectors required for the full shear connection is obtained by considering the minimum of the strengths of the concrete deck element and the steel girder element. In the present study, the slip at the interface and the deflection of SCC girder has been studied for various degree of shear connection. Fig. 6a and 6b show the slip profile at the interface of a SCC girder subjected to UDL and PL respectively. The graph is plotted, for varying degree of shear connection, between the interface slip of the composite girder and the span of 20 m, keeping other parameters constant. It can be observed that the slip at the interface increases with decrease in the degree of shear connection. It can also be observed that even for the full shear connection (i.e. degree of shear connection = 1) the shear connector tends to slip because of its finite stiffness. Fig. 7a and 7b show the deflection profile of a SCC girder subjected to UDL and PL at mid-span respectively. It is seen from Fig. 7a and 7b that the deflection increases with decrease in degree of shear connection. It is also observed that even for the full shear connection there is an increase in the deflection with respect to full interaction. This shows that deflection is sensitive to the slip of the shear connector because of its finite stiffness. From Fig. 8 it can be seen that the percentage increase in deflection varies from 15% to 28% for a composite girder
Span Length (m)
hs (mm)
btf (mm)
ttf (mm)
bbf (mm)
tbf (mm)
tw (mm)
10 15 20 25 30 35 40
765 1010 1295 1530 1705 1950 2200
200 300 350 400 500 600 600
20 30 35 40 40 50 50
200 300 350 400 500 600 600
20 30 35 40 40 50 50
8 8 10 12 12 14 14
subjected to both UDL and PL at mid-span respectively, when the degree of shear connection is varied from 1 to 0.5. Here the percentage increase in deflection has been calculated with respect to the deflection corresponding to the full interaction. 5.2. Effect of span length Here the effect of length of span on the deflection of SCC girder is presented. Keeping the concrete deck area and the material properties constant, the span length has been varied. Fig. 9 shows the plot of mid span deflection verses different span lengths of SCC girder subjected to both UDL and PL at mid span for full interaction and full shear connection. Fig. 10 shows the percentage increase in deflection with respect to full interaction for different span lengths. It is rather obvious from the Fig. 9 that the mid span deflection increases with increase in span length. However, the percentage increase in deflection decreases with the increase in span length as shown in the Fig. 10. It can also be seen that the effect of degree of shear connection is more sensitive for SCC girders having shorter span length. 5.3. Effect of geometry of steel girder In this section, the influence of the geometry of the steel girder on the deflections of SCC girder is presented. Keeping material properties and the cross section area of the concrete deck constant, the geometry of the cross section of the steel girder has been varied. Here, it should be noted that for all the variations considered in the study the second
Fig. 4. Variation in the cross section of the steel girder. 441
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Fig. 5. Flow chart showing design of SCC girder and calculation of deflection.
Fig. 11a and 11b respectively show the variation of mid span deflection with the change in the steel flange area ratio, for full interaction and partial shear interaction subjected to both UDL and PL at mid span, for a span length of 20 m. Here it can be observed that, for a SCC girder with full interaction subjected to both UDL and PL at the mid span, deflection remains approximately the same when steel girder geometry ratio is changed. However, with partial shear connection (i.e., with same shear connection stiffness) the mid span deflection increases
moment of area of the cross section of SCC girder and the stiffness of the shear connections are kept same. The variation in the steel girder geometry is brought about by only increasing the area of the bottom flange of the steel girder, keeping the area of the top flange of the steel girder constant. This variation is named as “steel flange area ratio” and is defined as the ratio of area of the bottom flange of the steel girder to the area of the top flange of the steel girder. In this study, as mentioned above four different variations are considered.
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Fig. 6. Slip profile at the interface of SCC girder for various degree of shear connection.
with increase in steel flange area ratio. From Fig. 12, it is also observed that the percentage increase in deflection with respect to full interaction increases when the steel flange area ratio is varied from 1 to 3. Similar observations are made for other spans as well. Fig. 13 shows that slip at the interface of the SCC girder increases with the increase in the steel flange area ratio. This conveys the designer that there is an increase in slip demand on the shear connector with the increase in the steel flange area ratio. 5.4. Effect of type of shear connector’s distribution Table 5 shows the effect of type of distribution of connectors on the deflection of the SCC girder. It can be seen from the Table 5 that there is substantial reduction in number of shear connectors when the type of distribution of shear connectors is changed from uniform to two steps distribution and to three steps distribution. The increase in deflection because of change in the distribution type of shear connectors is so small that it can be ignored. It is to be noted that for the spans 25 m and above the number of shear connectors obtained are same. This is because of the fact that the use of concrete deck having same cross section for all span lengths.
Fig. 8. Effect of degree of shear connection on percentage increase in deflection of SCC girder.
strength of the steel is tabulated in Table 6. Keeping, the span, the grade and deck dimension of the concrete constant, the yield strength of the steel girder has been varied. It can be observed from Fig. 14 that the mid span deflection of the SCC girder increases when the yield strength of the steel is varied from 250 N/mm2 to 450 N/mm2. This increase in deflection can be attributed to the fact that the cross sectional area of the steel girder gets reduced with increase in the yield strength of the steel which eventually reduces the second moment of area of the SCC girder. It is interesting to see from Fig. 15 that, though the mid span
5.5. Effect of yield strength of steel In this section, the effect of yield strength of steel on the deflection of SCC has been studied. The various yield strength of steel considered in the study are; fy = 250 N/mm2, 300 N/mm2, 350 N/mm2, 450 N/ mm2. The change in the girder geometry because of the change in yield
Fig. 7. Effect of degree of shear connection on deflection of SCC girder. 443
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from 25 N/mm2 to 50 N/mm2. It is observed from the Fig. 16 that the mid span deflection of SCC girder decreases with increase in the grade of the concrete. Fig. 17 shows percentage increase in deflection decreases with increase in concrete grade. Here the percentage increase in deflection is with respect to the full interaction. 6. Conclusions In this work, a parametric study has been carried out considering various influencing parameters such as degree of shear connection, span length, type of loading, distribution of shear stud connectors, geometry of steel girder, yield strength of steel and concrete grade, to bring out the relative significance of partial shear interaction with respect to full interaction in SCC girders. A simplified numerical method has been adopted to model SCC girders considering the partial shear interaction. Following are the outcomes of the present study.
Fig. 9. Mid span deflection of SCC girder for various span length.
i) It is evident from the study that, there is always slip at the interface of the SCC girder because of the finite stiffness of the shear connector regardless of the magnitude of the load. Consequently, deformations of the SCC girder increase with decrease in degree of shear connection. For an instance, for a span length of 20 m the percentage increase in deflection varies from 15% to 27%, when the degree of shear connection is varied from 1 to 0.5. ii) The results obtained by the equations given in several national codes underestimate the deflections. Further, these equations do not distinguish between the full interaction and the full shear connection, which is not the case in reality. iii) The effect of partial shear interaction is more pronounced for shorter spans. It is observed that the percentage increase in deflection due to partial interaction is about 33% for 10 m span and 8% for 40 m span. iv) The deformations of the SCC girder are significantly influenced by the geometry of cross section of the steel girder. It can be seen that, for the same second moment of area of SCC girder with partial shear interaction the mid-span deflection increases with increase in the steel flange area ratio. However, for the same composite girder with full interaction the mid-span deflection remains the same with increase in the steel flange area ratio. v) It is also interesting to note that the slip at the interface of the SCC girder increases with increase in the steel flange area ratio. This conveys that there is an increase in slip demand on the shear connectors with the increase in steel flange area ratio. vi) There is considerable savings in number of shear connectors when the type of distribution of stud shear connectors is changed, at the same time the increase in deflection of SCC girders is negligibly small. For an instance up to 25% and 50%, savings in number of
Fig. 10. Percentage increase in deflection w.r.t. full interaction for various span length.
deflection of the SCC girder increases with increase in yield strength of the steel the percentage increase in deflection with respect to full interaction decreases. This is attributed to the fact that the distance of the centroidal axis of the SCC girder with respect to the interface of the connecting elements decreases. 5.6. Effect of concrete grade Here the effect of grade of concrete on the deflections of the SCC has been studied. Keeping, the span, the yield strength of steel and deck dimension of the concrete constant, the grade of the concrete is varied
Fig. 11. Variation of mid-span deflection of SCC girder with change in steel flange area ratio . 444
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Table 6 Details of steel girder cross section for different yield strength of steel. Yield strength (m)
hs (mm)
btf (mm)
ttf (mm)
bbf (mm)
tbf (mm)
tw (mm)
250 300 350 450
1295 1195 1135 1035
350 350 330 300
35 35 30 30
350 350 300 300
35 35 30 30
10 10 10 8
Fig. 12. Effect of steel flange area ratio on the percentage increase in deflection of SCC girder.
Fig. 14. Effect of steel strength on deflection of SCC girder.
Fig. 13. Effect of variation in geometry of steel girder on slip at the interface of SCC girder.
Table 5 Effect of distribution of shear connectors. Span (m)
10 15 20 25 30 35 40
Description
Number of Connectors Deflection (mm) Number of Connectors Deflection (mm) Number of Connectors Deflection (mm) Number of Connectors Deflection (mm) Number of Connectors Deflection (mm) Number of Connectors Deflection (mm) Number of Connectors Deflection (mm)
Distribution Type Uniform
2 Steps
3 Steps
65 8.6 118 14.8 169 23.6 228 35.5 228 55.0 228 63.0 228 82.9
49 (24.62%)* 8.77 (1.86%)** 90 (23.73%) 15.04 (1.62%) 127 (24.85%) 24.02 (1.68%) 172 (24.56%) 36.0 (1.27%) 172 (24.56%) 55.6 (1.16%) 172 (24.56%) 64.0 (1.57%) 172 (24.56%) 84.2 (1.57%)
33 (49.23%)* 8.86 (2.90%)** 80 (32.20%) 15.2 (2.84%) 116 (31.36%) 24.2 (2.58%) 154 (32.46%) 36.2 (1.97%) 154 (32.46%) 56.0 (1.91%) 154 (32.46%) 64.4 (2.20%) 154 (32.46%) 84.7 (2.17)
Fig. 15. Percentage increase in deflection with variation in strength of steel.
* Number in parentheses indicates percentage decrease in number of connectors w.r.t. uniform distribution. ** Number in parentheses indicates percentage difference in deflection w.r.t. uniform distribution.
shear connectors is observed, when the distribution is changed from uniform distribution to two steps and three steps distribution respectively. This leads to considerable savings in the cost of shear connectors. vii) With the increase in the yield strength of steel or the grade of the concrete, the effect of partial shear interaction on the deflection
Fig. 16. Effect of concrete grade on deflection of SCC girder.
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approach. J Constr Steel Res 1999;49(3):291–301. [8] Seracino R, Oehlers DJ, Yeo MF. Partial-interaction flexural stresses in composite steel and concrete bridge beams. Eng Struct 2001;23(9):1186–93. [9] Zeng X, Jiang SF, Zhou D. Effect of shear connector layout on the behavior of steelconcrete composite beams with interface slip. Appl Sci 2019;9(1):207. [10] Newmark NM, Siess CP, Viest IM. Tests and analysis of composite beams with incomplete interaction. Proc Soc Exp Stress Anal 1951;9(1):75–92. [11] Johnson RP, May IM. Partial-interaction design of composite beams. Struct Eng 1975;8(53):305–11. [12] Girhammar UA, Gopu VK. Composite beam-columns with interlayer slip—exact analysis. J Struct Eng 1993;119(4):1265–82. [13] Jasim NA, Mohamad Ali AA. Deflections of composite beams with partial shear connection. Struct Eng 1997;75(4). [14] Jasim NA. Deflections of partially composite beams with linear connector density. J Constr Steel Res 1999;49(3):241–54. [15] Nie J, Cai CS. Steel–concrete composite beams considering shear slip effects. J Struct Eng 2003;129(4):495–506. [16] Ranzi G, Bradford MA, Uy B. A general method of analysis of composite beams with partial interaction. Steel Compos Struct 2003;3(3):169–84. [17] Schnabl S, Planinc I, Saje M, Cas B, Turk G. An analytical model of layered continuous beams with partial interaction. Struct Eng Mech 2006;22(3):263–78. [18] Girhammar UA. A simplified analysis method for composite beams with interlayer slip. Int J Mech Sci 2009;51(7):515–30. [19] Foraboschi P. Analytical solution of two-layer beam taking into account nonlinear interlayer slip. J Eng Mech 2009;135(10):1129–46. [20] Wang YC. Deflection of steel-concrete composite beams with partial shear interaction. J Struct Eng 1998;124(10):1159–65. [21] Leaf David, Laman Jeffrey A. Testing and Analysis of Composite Steel-Concrete Beam Flexural Strength. Int J Struct Civ Eng Res 2013;2(3):90–103. [22] Faella C, Martinelli E, Nigro E. Shear connection nonlinearity and deflections of steel–concrete composite beams: a simplified method. J Struct Eng 2003;129(1):12–20. [23] Martinelli E, Nguyen QH, Hjiaj M. Dimensionless formulation and comparative study of analytical models for composite beams in partial interaction. J Constr Steel Res 2012;75:21–31. [24] Zona A, Ranzi G. Shear connection slip demand in composite steel–concrete beams with solid slabs. J Constr Steel Res 2014;102:266–81. [25] Queiroz FD, Vellasco PCGS, Nethercot DA. Finite element modelling of composite beams with full and partial shear connection. J Constr Steel Res 2007;63(4):505–21. [26] Marconcin LR, Machado RD, Marino MA. Numerical modeling of steel-concrete composite beams. Revista IBRACON de Estruturas e Materiais 2010;3(4):449–76. [27] Turmo J, Lozano-Galant JA, Mirambell E, Xu D. Modeling composite beams with partial interaction. J Constr Steel Res 2015;114:380–93. [28] Faella C, Martinelli E, Nigro E. Steel and concrete composite beams with flexible shear connection: “exact” analytical expression of the stiffness matrix and applications. Comput Struct 2002;80(11):1001–9. [29] Ranzi G, Bradford MA, Uy B. A direct stiffness analysis of a composite beam with partial interaction. Int J Numer Meth Eng 2004;61(5):657–72. [30] Nguyen Quang-Huy, Martinelli Enzo, Hjiaj Mohammed. Derivation of the exact stiffness matrix for a two-layer Timoshenko beam element with partial interaction. Eng Struct 2011;33(2):298–307. [31] Nie J, Fan J, Cai CS. Experimental study of partially shear-connected composite beams with profiled sheeting. Eng Struct 2008;30(1):1–12. [32] Al-deen S, Ranzi G, Vrcelj Z. Shrinkage effects on the flexural stiffness of composite beams with solid concrete slabs: an experimental study. Eng Struct 2011;33(4):1302–15. [33] Nie J, Xiao Y, Chen L. Experimental studies on shear strength of steel–concrete composite beams. J Struct Eng 2004;130(8):1206–13. [34] Shanmugam NE, Baskar K. Steel–concrete composite plate girders subject to shear loading. J Struct Eng 2003;129(9):1230–42. [35] American Institute for Steel Construction (AISC). Load and resistance factor design specification for structural steel buildings. Chicago: AISC; 2016. [36] Eurocode 4: Design of composite steel and concrete structures part 1-1: General rules and rules for buildings. European Standard BS EN (1994): 1-1. [37] SAP2000, CSI. (2018). v20 Integrated Finite Element Analysis and Design of Structures. Computers and Structures Inc.: Berkely, CA, USA. [38] IRC: 22 (2008) Standard specification and code of practice for road bridges, Section VI, Composite construction. Indian Road Congress, New Delhi. [39] IS 516:1959 Method of tests for strength of Concrete, Bureau of Indian Standards, New Delhi.
Fig. 17. Percentage increase in deflection with variation in concrete grade.
reduces. viii) The deformations of a SCC girder subjected to point load at mid-span are marginally higher when compared to the composite girder subjected to UDL for the same mid-span moment. ix) In a partially connected SCC girder, the deformations depend on the distance of the centroidal axis of the SCC girder with respect to the interface of the connecting components. As this distance increases the deformations also increase. From this study, it can be said that the interaction between the concrete element and the steel element is always partial which is generally ignored in the designs. This eventually leads to serviceability issues if the partial shear interaction is not accounted for during the design of SCC girders. The parametric study carried out here gives not only an in-depth understanding about the deformations due to the partial shear interaction, but also gives usefulness of SCC girders to the designer in developing countries where SCC as a structural system is sparsely adopted. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Oehlers DJ, Bradford MA. Steel and Concrete Composite Structural Members: Fundamental Behaviour. Oxford: Pergamon Press; 1995. [2] Johnson RP. Composite Structures of Steel and Concrete. Oxford: Blackwell Scientific Publishers; 1994. [3] Ranzi G, Leoni G, Zandonini R. State of the art on the time-dependent behaviour of composite steel–concrete structures. J Constr Steel Res 2013;80:252–63. [4] Bradford MA, Gilbert RI. Composite beams with partial interaction under sustained loads. J Struct Eng 1992;118(7):1871–83. [5] Dezi L, Ianni C, Tarantino AM. Simplified creep analysis of composite beams with flexible connectors. J Struct Eng 1993;119(5):1484–97. [6] Dezi L, Leoni G, Tarantino AM. Algebraic methods for creep analysis of continuous composite beams. J Struct Eng 1996;122(4):423–30. [7] Jasim NA, Atalla A. Deflections of partially composite continuous beams: a simple
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Numerical study on the deformation behavior of steel concrete composite girders considering partial shear interaction ⁎
Madhusudan G. Kalibhat , Akhil Upadhyay
T
⁎
Department of Civil Engineering, Indian Institute of Technology, Roorkee 247667, India
A R T I C LE I N FO
A B S T R A C T
Keywords: Steel concrete composite girder Deformations Partial shear interaction Parametric study Numerical model
The present paper focuses on the study of deformation behavior of steel concrete composite (SCC) girders with partial shear interaction. The main objective of this work is to bring out the relative significance of the partial shear interaction with respect to full interaction, which has been addressed with the help of parametric study considering various influencing parameters such as, degree of shear connection, span length, type of loading, distribution of shear stud connectors, geometry of steel girder, yield strength of structural steel and concrete grade. The parametric study has been carried out using a simplified numerical modelling technique with the help of commercially available finite element software SAP2000. The numerical modelling involves simulation of composite action between the concrete deck and the steel girder in the SCC girder, which has been carried out by using various elements such as, beam elements, rigid link elements and spring elements. Further, a comparative study has been carried out to study the accuracy of numerical model with the available analytical models. The results of the numerical model are also compared with the results of the equations given in several national codes to calculate deflections, to study the efficacy of the codal provisions. It is seen that the various aforementioned parameters have considerable influence on the deformations of SCC girders. It is also seen that the change in distribution type of shear stud connectors considerably reduce their number without significantly affecting the deformations of SCC girder.
1. Introduction The use of SCC girder as an alternative to the RC girder or the steel girder has become ubiquitous in the recent decades because of the fact that it is advantageous in terms of strength and stiffness when compared to RC girder and steel girder alone [1,2]. As shown in the Fig. 1 a typical SCC girder consists of a concrete deck component connected to the steel girder component with the help of shear stud connectors at their interface and the composite action is achieved by these shear stud connectors [3]. However, the extent of this composite action is characterized by the behavior of the shear stud connectors [4–7]. If the shear connectors are infinitely rigid, the full composite action or the full interaction is achieved. In this case, the analysis of the SCC girder is performed by transformed section approach (i.e., by transforming one of the components with respect to the other, using appropriate modular ratio). Conversely, these shear connectors tend to slip at the interface, regardless of the magnitude of load, because of their finite stiffness [8,9] and consequently the interaction between the concrete deck and the steel girder is partial. Hence, the transformed section approach cannot be used for the analysis of this partially connected composite
⁎
girder. Therefore, the slip at the interface resulting because of the finite stiffness of the connector should be taken into consideration during the design so as to accurately predict the deflections of composite girder which is generally omitted by the designers. In the past researchers across the globe have shown keen interest in effects of slip on the deflections of SCC girders. The very first work on SCC beams considering slip effects dates back to mid twentieth century by Newmark et al. [10], which is a highly cited and acclaimed work. Johnson and May [11] derived design rules are for SCC beams with partial-interaction for determining the flexural strength at ultimate limit state, and for checking deflections at serviceability limit state. Since then different researchers have developed different analytical models by solving differential equations considering the relative displacement at the interface of composite section of SCC girders [1,12–19]. These analytical models can be used to compute the deformations in SCC girders. However, these analytical models are based on the simplifying assumption like uniform distribution of shear connectors which limits the practical applicability. Further, the analytical models are complex to use in day-to-day designs. Alternatively, numerical methods [20–30] have been developed and are widely used.
Corresponding authors. E-mail addresses: [email protected] (M.G. Kalibhat), [email protected] (A. Upadhyay).
https://doi.org/10.1016/j.istruc.2019.10.007 Received 20 May 2019; Received in revised form 12 October 2019; Accepted 16 October 2019 2352-0124/ © 2019 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
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Nomenclature Ac= A s= hc= h s= dsh= Ec= EIeff= EIf= Es =
f′c= fck= h= Ic= Is= K= L= m= Ns= p= Q u= V L= Δf= Δpi=
Area of cross section of concrete deck Area of cross section of steel girder Centroidal distance of concrete deck from the interface of SCC girder Centroidal distance of steel girder from the interface of SCC girder Shank diameter of the shear connector Modulus of elasticity of concrete Effective flexural rigidity of SCC girder for any degree of interaction Flexural rigidity of transformed cross section of SCC girder for full interaction Modulus of elasticity of steel girder
These also include rigorous finite element modeling techniques [20,23–26] and stiffness matric approach [28–30] for the analysis of SCC girders with partial shear interaction. Numerical modelling is more suitable than analytical models because of the ease with which different components of composite beam can be modelled and composite action between the connecting components can be established. Further, additional degrees of freedom because of the slip of the shear connector also can be easily incorporated in the numerical model. A lot of experimental and numerical studies have also been conducted in the past on SCC girders considering the slip of the stud connectors [21,31–34]. All these studies showed that the slip of the shear stud connectors increase the deformations in SCC girders when compared to the SCC girders analyzed using transformed section approach. A few national codes [35,36] recommend considering partial shear interaction in the design and provide equations to calculate the deflections of SCC girders. In this paper, the effect of partial shear connection on the deformations of SCC girders has been discussed with the help of parametric study considering various design parameters such as, degree of shear connection, span length, type of loading, distribution of shear stud connectors, geometry of steel girder, yield strength of steel and concrete grade. For this purpose, a simplified numerical modelling technique has been adopted because of the following advantages: (1) Easiness in modeling at design offices when compared to analytical and other rigorous numerical methods which cannot be used in day-to-day design because of their complexity and the computational effort involved in them. (2) Flexibility and generality in dealing with different types of loading and boundary conditions. (3) Practicality in terms of ease with which the distribution of the shear connectors can be varied along the span, which is not possible in analytical models because of the limitations imposed by the assumptions made. Besides these, the modeling approach is simple when compared to rigorous finite element methods and can also be extended to the nonlinear domain (though it is
Characteristic strength of concrete cylinder Characteristic strength of concrete cube Total height of SCC girder Second moment of area of concrete deck about its centroid Second moment of area of steel girder about its centroid Shear connector’s stiffness in load per unit slip Span length of SCC girder Modular ratio Number of shear connectors provided Pitch of the shear connectors Ultimate strength of the shear connector Longitudinal shear per unit length (i.e. Shear flow) Deflection of SCC girder with full interaction Deflection of SCC girder with partial shear interaction
out of the scope of the present work). The design engineer usually assumes full interaction during the design of SCC girders which is not the case in reality. Therefore, this type of study not only gives an in-depth understanding about the deformations due to the partial shear interaction in SCC girders, but also gives the usefulness of SCC girders when compared to its counterparts such as RC and steel girders, to the design engineer in developing countries where SCC as a structural system is sparsely adopted. 2. Numerical modelling In the present study, a simplified numerical modeling technique has been adopted to study the behavior of SCC girders incorporating partial interaction. Fig. 2 shows a schematic representation of typical 2D numerical model which is based on linear elastic finite element approach. The numerical modeling has been carried out using finite element software SAP2000 [37]. Different types of elements used to imitate partial composite action in a SCC girder are described as follows: A. Element 1 as shown in the Fig. 2 depicts the concrete deck element and is modelled as a beam element. The properties of element 1 are same as of the concrete deck at its centroid. B. Element 2 as shown in the Fig. 2 depicts the steel girder element and is modelled as a beam element. The properties of element 2 are same as of the steel girder at its centroid. C. Element 3 represents a spring element that is used to simulate shear connector’s behavior at the interface. This spring element has a finite stiffness in longitudinal direction of the SCC girder and is equal to the stiffness of the shear connector, whilst it has infinite stiffness in other directions. Eq. (1) gives stiffness of the spring element K as given in [1] and is as follows.
K=
Qu d sh (0.16 − 0.0017f ′c)
(1)
Concrete cylinder strength f′c is obtained by multiplying 0.8 to fck (as per recommendations of IS 516: 1959) [39]. A. Elements 4 represents rigid link element used to connect the nodes of centroid of concrete deck (i.e. element1) with the spring element (i.e. element 3) at the interface. B. Elements 5 represents rigid link element used to connect the nodes of centroid of steel girder (i.e. element 2) with the spring element (i.e. element 3) at the interface. The element 3, element 4 and element 5 put together bring about the composite action in the SCC girder. The required degree of shear connection can be established, merely by varying the spacing of the spring element 3, element 4 and element 5.
Fig. 1. Typical cross section of SCC girder. 438
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Fig. 2. A schematic representation of numerical model of SCC girder.
3. Comparative study
A1 =
In this section a comparative study has been carried out which involves the comparison of the results of the numerical model against the results of available analytical models for different degree of shear connection. The results of the numerical models are also compared with the results of several national codes. Note that the details about the degree of shear connection is explained in the “Section 5.1” of this paper.
A0 I0 + A 0 (h c + h s )2
3.1.3. Girhammar’s model Eq. (4) simplified analytical model as proposed by Girhammar [18] for the determination of the deflection of the SCC girder considering partial shear connection and is as follows
Δpi =
wL4 X X3 2X 2 . ⎛ − + 1⎞ 24EIeff L ⎝ 3L3 L2 ⎠ ⎜
⎟
3.1. Analytical models EI
f − 1⎤ ⎡ ∑ EI Where: EIeff ≈ ⎢1 + ⎥ (α L)2 ⎢ 1+ 2 ⎥ π ⎣ ⎦
Here the various analytical models proposed by different researchers considered in the present study are explained.
Δf
=1+
∑ EI = Ec Ic + Es Is
24(C − 1) 1 1 ⎡ − (1 − cosh( R ) + tanh( R )sinh( R ) ⎤ 5R R ⎦ ⎣2 (2)
3.2. National codes In the following sub sections, the provisions of various national code with regard to partial shear connection are explained.
A (h + h s)2 ⎤ Where: C = ⎡1 + 0 c ⎥ ⎢ I0 ⎦ ⎣
R=
3.2.1. AISC provisions AISC 360 [35] provides the equation for the determination of effective flexural rigidity of the SCC girder and is defined in the equation (5).
Kj 16
Kj = G1L3
EIeq = EIs + (Ns /Nf )1/2 ∗ (EIf − EIs)
K ⎡ A (h + h s)2 ⎤ G1 = 1+ 0 c ⎥ XEs A 0 ⎢ I0 ⎦ ⎣ A0 =
I0 =
IC + IS m
3.2.2. Eurocode provisions Eurocode 4 [36] also provides the equation for the determination of deflection of SCC girders due to partial shear interaction and is given in Eq. (6).
3.1.2. Nie’s model Eq. (3) gives analytical model as proposed by Nie and Cai [15] for the determination of the deflection of the SCC girder considering partial shear connection and is as follows
⎛ L2 2e( ) − 1 − e μL ⎞ + Δpi = + βw⎜ 384EIf μ2 h(1 + e μL) ⎟⎟ ⎜ 8h ⎝ ⎠
β=
Δpi
N Δ = 1 + C ⎛1 − s ⎞ ⎛ s − 1⎞ N f ⎠ ⎝ Δf ⎠ ⎝ ⎜
Δf
μL 2
5wL4
(5)
where, EIeq is the equivalent flexural rigidity of the cross section of SCC girder and Nf is the number of shear stud connectors required to achieve the full interaction. Merely by varying the ratio Ns/Nf, (Ns/Nf < 1), the required equivalent flexural rigidity for any degree of shear connection is obtained.
ASA C mAS + A C
Where: μ2 =
∑ EI
1 1 (h + hS )2 ⎤ + + C α2 = K ⎡ ⎢ EC A C ES AS ∑ EI ⎥ ⎣ ⎦
3.1.1. Jasim’s model Eq. (2) gives the analytical model proposed by Jasim [14] for the determination of the deflection of the SCC girder considering partial shear connection and is as follows
Δpi
(4)
−1
⎟⎜
⎟
(6)
where C is a coefficient, taken as 0.3 and 0.5 for unpropped and propped construction respectively, and Δs is the deflection of the steel girder for the same load.
(3)
K A1ES I0 p
3.3. Model verification
A1 (hC + hS )p K
For the purpose of comparison, a SCC girder of span 20 m, subjected to uniformly distributed load of 55kN/m has been considered, with 439
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three different degree of shear interaction (i.e., full interaction, full shear connection and 0.5 degree of shear connection). The concrete deck has width of 2500 mm and depth of 250 mm. The geometric dimensions of the steel girder is given in table 4. The spacing of shear connectors for full shear connection and for 0.5 degree of shear connection is 119 mm and 240 mm respectively, and the connector’s stiffness is 54600 N/mm. In case of full interaction the shear connectors are modelled by providing rigid properties to the spring element. The mid-span deflection of the SCC girder obtained by the numerical model, different analytical models and several national codes are tabulated in the Table 1. It is observed from the Table 1 that the results of the numerical model closely match with that of different analytical models, which validates the accuracy of the numerical model. It can be seen from the results of the codal provisions that, they do not distinguish between the full interaction and the full shear connection, which is also observed from Eqs. (5) and (6). However, for partial shear connection cases, the codal predictions are good.
Table 2 Design parameters and their values considered in the study.
Degree of shear connection Span length (m) Ratio of bottom flange area to the top flange area of the steel girder Yield strength of steel (fy) in N/mm2 Concrete Grade (fck) in N/mm2
1, 0.75, 0.5 10, 15, 20, 25, 30, 35, 40 1, 1.5, 2, 3
Span Length (m)
10 15 20 25 30 35 40
To study the effects of various aforementioned parameters on the deformations of SCC girder a parametric study has been carried out. The pertinent values of various parameters considered in this work are tabulated in Table 2. The parametric study carried out here involves modeling and analysis of SCC girders, which are designed according to Indian standards IRC: 22 (2008) [38]. Two different loading types viz, uniformly distributed load (UDL), and the point load (PL) at the mid-span are considered, and magnitude of the UDL and PL at mid-span for different spans are given in Table 3. Here the magnitude of the PL at mid-span considered is such that it produces same magnitude of bending moment as produced by UDL. In this study three different types of distribution of shear connectors have been considered viz, uniform distribution, two steps distribution and three steps distribution. In case of uniform distribution of shear connectors the number of shear connectors is obtained from Eq. (7) as per IRC recommendations.
VL*L/2 ∑Q
Considered Values
250, 300, 350, 450 25, 30, 40, 50
Table 3 Loading type details for various span length.
4. Parametric study
Ns=
Design Parameters
Loading Type UDL (kN/m)
Point Load (kN)
50 50 55 60 65 70 70
250 375 550 750 975 1225 1400
respectively obtained by calculating the shear force and the corresponding shear flow at the support section, at a distance of L/6 from the support section and at a distance of L/3 from the support section. The number of shear connectors so obtained in three different portions are uniformly distributed in respective portions. Fig. 3 shows the qualitative representation of distribution of shear connectors in the shear span for various distribution types. To study the influence of geometry of steel girder on the deformations of the SCC girder four different types of variation in cross section of the steel girder have been considered. However, for all the variations the second moment of area of the cross section of SCC girder is kept constant. The variation is realized by only increasing the area of bottom flange there by keeping symmetry of the steel girder about its vertical axis. The variations are, cross section having equal flange area, cross section having bottom flange area equal to 1.5 times the top flange area, cross section having bottom flange area equal to twice the top flange area and cross section having bottom flange area equal to thrice the top flange area. These variations are respectively designated as S1, S2, S3 and S4 as shown in the Fig. 4. The cross section dimension of the steel girder for different spans is designed as per IRC recommendations and are given in the Table 4. In the Table 4, hs, btf, ttf, bbf, tbf and tw, respectively refer to the total height of the steel section, breadth of the top flange, thickness of the top flange, breadth of the bottom flange, thickness of the bottom flange and thickness of the web. The concrete deck is 2500 mm wide and 250 mm deep. The flow of design of SCC girder and calculation of deflection are explained with the help flow chart as given in Fig. 5. In Fig. 5, np and nf respectively stand for the number of connectors for any partial shear
(7)
The number of shear stud connectors so obtained are distributed uniformly over the shear span i.e. from the point of zero moment to the point of maximum moment (i.e. from support to the mid span in case of simply supported girders). In case of two steps distribution the shear span is divided into two equal portions. The number of shear connectors in the first portion and the second portion of the shear span is respectively obtained by calculating the shear force and the corresponding shear flow at the support section and at a distance of L/4 from the support section. The number of shear connectors so obtained in two different portions are uniformly distributed in respective portions. Similarly, in case of three steps distribution the shear span is divided into three equal portions and the number of shear connectors in the first portion, second portion and third portion of the shear span is Table 1 Comparison of results of numerical and various analytical models. Model's Name
Numerical Jasim’s Nie’s Girhammar’s AISC Eurocode
Full interaction
Full shear connection
0.5 shear connection degree
Mid-span deflection (mm)
Percentage difference w.r.t. numerical model
Mid-span deflection (mm)
Percentage difference w.r.t. numerical model
Mid-span deflection (mm)
Percentage difference w.r.t. numerical model
20.61 19.84 19.84 19.84 19.84 19.84
– 3.88 3.88 3.88 3.88 3.88
23.62 22.88 22.46 22.88 19.84 19.84
– 3.23 5.16 3.23 19.05 19.05
26.00 25.34 24.58 25.34 24.07 24.31
– 2.60 5.78 2.60 8.02 6.95
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Fig. 3. Distribution of shear connectors.
interaction and for full shear interaction.
Table 4 Details of steel cross section for different span length.
5. Results and discussion The effects of various aforementioned parameters on the deflection of SCC girder are discussed here. 5.1. Effect of degree of shear connection The two limits of the interaction in a composite girder are the full interaction (i.e. shear connectors have infinite stiffness and slip at the interface of the composite girder is zero) and no interaction between the two connecting elements. The analysis of a SCC girder with full interaction is carried out by transforming the cross section using appropriate modular ratio. In reality, the interaction exists between the full interaction and no interaction. The degree of shear connection is defined as the ratio of number of shear connectors provided to the number of shear connectors required for the full shear connection. The number of shear connectors required for the full shear connection is obtained by considering the minimum of the strengths of the concrete deck element and the steel girder element. In the present study, the slip at the interface and the deflection of SCC girder has been studied for various degree of shear connection. Fig. 6a and 6b show the slip profile at the interface of a SCC girder subjected to UDL and PL respectively. The graph is plotted, for varying degree of shear connection, between the interface slip of the composite girder and the span of 20 m, keeping other parameters constant. It can be observed that the slip at the interface increases with decrease in the degree of shear connection. It can also be observed that even for the full shear connection (i.e. degree of shear connection = 1) the shear connector tends to slip because of its finite stiffness. Fig. 7a and 7b show the deflection profile of a SCC girder subjected to UDL and PL at mid-span respectively. It is seen from Fig. 7a and 7b that the deflection increases with decrease in degree of shear connection. It is also observed that even for the full shear connection there is an increase in the deflection with respect to full interaction. This shows that deflection is sensitive to the slip of the shear connector because of its finite stiffness. From Fig. 8 it can be seen that the percentage increase in deflection varies from 15% to 28% for a composite girder
Span Length (m)
hs (mm)
btf (mm)
ttf (mm)
bbf (mm)
tbf (mm)
tw (mm)
10 15 20 25 30 35 40
765 1010 1295 1530 1705 1950 2200
200 300 350 400 500 600 600
20 30 35 40 40 50 50
200 300 350 400 500 600 600
20 30 35 40 40 50 50
8 8 10 12 12 14 14
subjected to both UDL and PL at mid-span respectively, when the degree of shear connection is varied from 1 to 0.5. Here the percentage increase in deflection has been calculated with respect to the deflection corresponding to the full interaction. 5.2. Effect of span length Here the effect of length of span on the deflection of SCC girder is presented. Keeping the concrete deck area and the material properties constant, the span length has been varied. Fig. 9 shows the plot of mid span deflection verses different span lengths of SCC girder subjected to both UDL and PL at mid span for full interaction and full shear connection. Fig. 10 shows the percentage increase in deflection with respect to full interaction for different span lengths. It is rather obvious from the Fig. 9 that the mid span deflection increases with increase in span length. However, the percentage increase in deflection decreases with the increase in span length as shown in the Fig. 10. It can also be seen that the effect of degree of shear connection is more sensitive for SCC girders having shorter span length. 5.3. Effect of geometry of steel girder In this section, the influence of the geometry of the steel girder on the deflections of SCC girder is presented. Keeping material properties and the cross section area of the concrete deck constant, the geometry of the cross section of the steel girder has been varied. Here, it should be noted that for all the variations considered in the study the second
Fig. 4. Variation in the cross section of the steel girder. 441
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Fig. 5. Flow chart showing design of SCC girder and calculation of deflection.
Fig. 11a and 11b respectively show the variation of mid span deflection with the change in the steel flange area ratio, for full interaction and partial shear interaction subjected to both UDL and PL at mid span, for a span length of 20 m. Here it can be observed that, for a SCC girder with full interaction subjected to both UDL and PL at the mid span, deflection remains approximately the same when steel girder geometry ratio is changed. However, with partial shear connection (i.e., with same shear connection stiffness) the mid span deflection increases
moment of area of the cross section of SCC girder and the stiffness of the shear connections are kept same. The variation in the steel girder geometry is brought about by only increasing the area of the bottom flange of the steel girder, keeping the area of the top flange of the steel girder constant. This variation is named as “steel flange area ratio” and is defined as the ratio of area of the bottom flange of the steel girder to the area of the top flange of the steel girder. In this study, as mentioned above four different variations are considered.
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Fig. 6. Slip profile at the interface of SCC girder for various degree of shear connection.
with increase in steel flange area ratio. From Fig. 12, it is also observed that the percentage increase in deflection with respect to full interaction increases when the steel flange area ratio is varied from 1 to 3. Similar observations are made for other spans as well. Fig. 13 shows that slip at the interface of the SCC girder increases with the increase in the steel flange area ratio. This conveys the designer that there is an increase in slip demand on the shear connector with the increase in the steel flange area ratio. 5.4. Effect of type of shear connector’s distribution Table 5 shows the effect of type of distribution of connectors on the deflection of the SCC girder. It can be seen from the Table 5 that there is substantial reduction in number of shear connectors when the type of distribution of shear connectors is changed from uniform to two steps distribution and to three steps distribution. The increase in deflection because of change in the distribution type of shear connectors is so small that it can be ignored. It is to be noted that for the spans 25 m and above the number of shear connectors obtained are same. This is because of the fact that the use of concrete deck having same cross section for all span lengths.
Fig. 8. Effect of degree of shear connection on percentage increase in deflection of SCC girder.
strength of the steel is tabulated in Table 6. Keeping, the span, the grade and deck dimension of the concrete constant, the yield strength of the steel girder has been varied. It can be observed from Fig. 14 that the mid span deflection of the SCC girder increases when the yield strength of the steel is varied from 250 N/mm2 to 450 N/mm2. This increase in deflection can be attributed to the fact that the cross sectional area of the steel girder gets reduced with increase in the yield strength of the steel which eventually reduces the second moment of area of the SCC girder. It is interesting to see from Fig. 15 that, though the mid span
5.5. Effect of yield strength of steel In this section, the effect of yield strength of steel on the deflection of SCC has been studied. The various yield strength of steel considered in the study are; fy = 250 N/mm2, 300 N/mm2, 350 N/mm2, 450 N/ mm2. The change in the girder geometry because of the change in yield
Fig. 7. Effect of degree of shear connection on deflection of SCC girder. 443
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from 25 N/mm2 to 50 N/mm2. It is observed from the Fig. 16 that the mid span deflection of SCC girder decreases with increase in the grade of the concrete. Fig. 17 shows percentage increase in deflection decreases with increase in concrete grade. Here the percentage increase in deflection is with respect to the full interaction. 6. Conclusions In this work, a parametric study has been carried out considering various influencing parameters such as degree of shear connection, span length, type of loading, distribution of shear stud connectors, geometry of steel girder, yield strength of steel and concrete grade, to bring out the relative significance of partial shear interaction with respect to full interaction in SCC girders. A simplified numerical method has been adopted to model SCC girders considering the partial shear interaction. Following are the outcomes of the present study.
Fig. 9. Mid span deflection of SCC girder for various span length.
i) It is evident from the study that, there is always slip at the interface of the SCC girder because of the finite stiffness of the shear connector regardless of the magnitude of the load. Consequently, deformations of the SCC girder increase with decrease in degree of shear connection. For an instance, for a span length of 20 m the percentage increase in deflection varies from 15% to 27%, when the degree of shear connection is varied from 1 to 0.5. ii) The results obtained by the equations given in several national codes underestimate the deflections. Further, these equations do not distinguish between the full interaction and the full shear connection, which is not the case in reality. iii) The effect of partial shear interaction is more pronounced for shorter spans. It is observed that the percentage increase in deflection due to partial interaction is about 33% for 10 m span and 8% for 40 m span. iv) The deformations of the SCC girder are significantly influenced by the geometry of cross section of the steel girder. It can be seen that, for the same second moment of area of SCC girder with partial shear interaction the mid-span deflection increases with increase in the steel flange area ratio. However, for the same composite girder with full interaction the mid-span deflection remains the same with increase in the steel flange area ratio. v) It is also interesting to note that the slip at the interface of the SCC girder increases with increase in the steel flange area ratio. This conveys that there is an increase in slip demand on the shear connectors with the increase in steel flange area ratio. vi) There is considerable savings in number of shear connectors when the type of distribution of stud shear connectors is changed, at the same time the increase in deflection of SCC girders is negligibly small. For an instance up to 25% and 50%, savings in number of
Fig. 10. Percentage increase in deflection w.r.t. full interaction for various span length.
deflection of the SCC girder increases with increase in yield strength of the steel the percentage increase in deflection with respect to full interaction decreases. This is attributed to the fact that the distance of the centroidal axis of the SCC girder with respect to the interface of the connecting elements decreases. 5.6. Effect of concrete grade Here the effect of grade of concrete on the deflections of the SCC has been studied. Keeping, the span, the yield strength of steel and deck dimension of the concrete constant, the grade of the concrete is varied
Fig. 11. Variation of mid-span deflection of SCC girder with change in steel flange area ratio . 444
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Table 6 Details of steel girder cross section for different yield strength of steel. Yield strength (m)
hs (mm)
btf (mm)
ttf (mm)
bbf (mm)
tbf (mm)
tw (mm)
250 300 350 450
1295 1195 1135 1035
350 350 330 300
35 35 30 30
350 350 300 300
35 35 30 30
10 10 10 8
Fig. 12. Effect of steel flange area ratio on the percentage increase in deflection of SCC girder.
Fig. 14. Effect of steel strength on deflection of SCC girder.
Fig. 13. Effect of variation in geometry of steel girder on slip at the interface of SCC girder.
Table 5 Effect of distribution of shear connectors. Span (m)
10 15 20 25 30 35 40
Description
Number of Connectors Deflection (mm) Number of Connectors Deflection (mm) Number of Connectors Deflection (mm) Number of Connectors Deflection (mm) Number of Connectors Deflection (mm) Number of Connectors Deflection (mm) Number of Connectors Deflection (mm)
Distribution Type Uniform
2 Steps
3 Steps
65 8.6 118 14.8 169 23.6 228 35.5 228 55.0 228 63.0 228 82.9
49 (24.62%)* 8.77 (1.86%)** 90 (23.73%) 15.04 (1.62%) 127 (24.85%) 24.02 (1.68%) 172 (24.56%) 36.0 (1.27%) 172 (24.56%) 55.6 (1.16%) 172 (24.56%) 64.0 (1.57%) 172 (24.56%) 84.2 (1.57%)
33 (49.23%)* 8.86 (2.90%)** 80 (32.20%) 15.2 (2.84%) 116 (31.36%) 24.2 (2.58%) 154 (32.46%) 36.2 (1.97%) 154 (32.46%) 56.0 (1.91%) 154 (32.46%) 64.4 (2.20%) 154 (32.46%) 84.7 (2.17)
Fig. 15. Percentage increase in deflection with variation in strength of steel.
* Number in parentheses indicates percentage decrease in number of connectors w.r.t. uniform distribution. ** Number in parentheses indicates percentage difference in deflection w.r.t. uniform distribution.
shear connectors is observed, when the distribution is changed from uniform distribution to two steps and three steps distribution respectively. This leads to considerable savings in the cost of shear connectors. vii) With the increase in the yield strength of steel or the grade of the concrete, the effect of partial shear interaction on the deflection
Fig. 16. Effect of concrete grade on deflection of SCC girder.
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Fig. 17. Percentage increase in deflection with variation in concrete grade.
reduces. viii) The deformations of a SCC girder subjected to point load at mid-span are marginally higher when compared to the composite girder subjected to UDL for the same mid-span moment. ix) In a partially connected SCC girder, the deformations depend on the distance of the centroidal axis of the SCC girder with respect to the interface of the connecting components. As this distance increases the deformations also increase. From this study, it can be said that the interaction between the concrete element and the steel element is always partial which is generally ignored in the designs. This eventually leads to serviceability issues if the partial shear interaction is not accounted for during the design of SCC girders. The parametric study carried out here gives not only an in-depth understanding about the deformations due to the partial shear interaction, but also gives usefulness of SCC girders to the designer in developing countries where SCC as a structural system is sparsely adopted. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Oehlers DJ, Bradford MA. Steel and Concrete Composite Structural Members: Fundamental Behaviour. Oxford: Pergamon Press; 1995. [2] Johnson RP. Composite Structures of Steel and Concrete. Oxford: Blackwell Scientific Publishers; 1994. [3] Ranzi G, Leoni G, Zandonini R. State of the art on the time-dependent behaviour of composite steel–concrete structures. J Constr Steel Res 2013;80:252–63. [4] Bradford MA, Gilbert RI. Composite beams with partial interaction under sustained loads. J Struct Eng 1992;118(7):1871–83. [5] Dezi L, Ianni C, Tarantino AM. Simplified creep analysis of composite beams with flexible connectors. J Struct Eng 1993;119(5):1484–97. [6] Dezi L, Leoni G, Tarantino AM. Algebraic methods for creep analysis of continuous composite beams. J Struct Eng 1996;122(4):423–30. [7] Jasim NA, Atalla A. Deflections of partially composite continuous beams: a simple
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