An analysis model accounting for time-dependent shear lag effects in assembled composite bridge decks

Abstract

The concrete deck of steel-concrete composite bridges made via prefabricated construction consists of precast slabs and construction joints between the slabs. When the age difference between the longitudinal joint concrete and precast slab concrete (new-old concrete) across a beam section is considered, the time-dependent characteristic of the assembled deck is different from that of the whole prefabricated deck. Based on the virtual work principle, differential equations were established to analyse the influence of the time-dependent characteristics of new-old concrete across a beam section on the shear lag effect of the composite beams. The analysis of examples showed that the proposed model was more accurate than the traditional model, which ignores the concrete age difference across a beam section, especially when analysing the long-term stress of joint concrete. The long-term stress of the joint concrete obtained by the proposed model decreased faster than that obtained by the traditional model with time, which increased the non-uniformity of the distribution of stress along the concrete flange. Overall, the proposed model could reasonably forecast the long-term performance of composite beams with assembled decks, which is beneficial to the life-cycle design and service behaviour analysis of composite bridges.

Keywords

composite beam assembled long-term shear lag effective flange width

Introduction

Steel-concrete composite beams have been widely used in urban bridges and deck structures of long-span bridges such as cable-stayed bridges and suspension bridges. The steel part of the composite beams will restrain the deck deformation induced by the shrinkage and creep of concrete. The deformation constraint easily causes concrete cracking in the composite beams and thus affects the durability of the bridges. To reduce concrete cracking in the large-span composite bridges, the concrete slabs of the bridges are generally prefabricated and spliced into an integral deck through the construction joints upon steel girders, shown in Figure 1.

Figure 1.

Assembled deck in composite bridges.

Unfortunately, the cracking of construction joints is common (Nie 2011). Kwak et al. (2000) and Tong et al. (2018) pointed out that the concrete shrinkage difference of assembled decks could cause concrete cracking in the composite beams. Furthermore, low-shrinkage concrete was used to cast construction joints in some bridge projects (Gara et al. 2013), but only as a passive engineering countermeasure. A considerable amount of attention should be paid to studying how the normal stress redistribution of assembled decks will change at the life-cycle service of the composite beams.

When the time-dependent analysis of the composite beams with assembled decks was carried out, engineering designers generally neglected the difference in shrinkage and creep between the longitudinal joint concrete and precast slab concrete. The concrete age of deck sections was considered the same as that of the precast slab in the corresponding cross-section. It is difficult to assess the stress state of longitudinal joint concrete in the long term by this traditional analytic strategy (Huang et al. 2019a). Based on the deformation coordination between precast slabs and longitudinal joints of the composite beams, Huang et al. (2018, 2019b) proposed a one-dimensional time-dependent analysis model that considers the age difference of new-old concrete across beam sections. However, because of the assumption that the plane section of assembled decks remains plane, the above model is accurate only when analysing composite bridges with a small width-span ratio. For composite bridges with a large width-span ratio, the shear lag effect of the concrete flange should be considered in the mechanical performance analysis (Dezi et al. 2003). Also, considerable attention should be paid to the shear lag effect when composite beam prestressing is loaded (Dezi et al. 2006). Dezi et al. (2006) and Gara et al. (2009) defined a warping displacement function to describe the time-dependent shear lag effect of the bridge flanges. The results showed that the shrinkage and creep of the concrete would change the effective flange width of concrete decks where concentrated axial loads acted. Afterward, Gara et al. (2010, 2011a, b) further extended the time-dependent shear lag analysis to cable-stayed composite bridges using finite beam elements. The research results showed that the effective flange width of the concrete decks gradually changed with time under a sustainable loading. It is suggested that the effective flange width could be equal to the entire width of the bridge deck in the long-term performance analysis of composite beams. In addition, Nguyen and Hjiaj (2016) and Zhu and Su (2017) proposed some high-precision methods for solving the closed-form solution of the time-dependent shear lag differential equations. Further, the variation regularity of long-term stress of composite beam deck was discussed.

Although the researchers mentioned above carried out some theoretical researches on the time-dependent shear lag effect of composite beams, the age difference of concrete across the deck section of composite beams is not considered. Therefore, this work will propose a time-dependent shear lag model considering the age difference between precast slab concrete and longitudinal joint concrete to study the influence of the age difference on the shear lag effect of composite beams with assembled decks.

Time-dependent shear lag model

Principle assumptions

The compressive stress of concrete in engineering structures is always less than 50% of the ultimate compressive strength of the concrete (Xue et al. 2008). Therefore, it is assumed that the concrete deck exhibits linear viscoelastic behaviour under tensile and compressive stresses and that the steel girder is a linear elastic material. Moreover, it is assumed that the tensile stress of concrete is lower than its failure stress. Neville et al. (1983) showed that the tensile creep strain of concrete is approximately equal to that of the compressive creep of concrete under the same absolute value of concrete stress. Therefore, it is further assumed that the creep strain of concrete follows the same regularity under tensile stress and compressive stress.

Based on the connection assumption of the continuous distribution between the steel girders and concrete decks along the axial direction of a composite beam, the horizontal shear flow $τ_{s l i p}$ is proportional to the slip $s_{t}$ at the steel-concrete interface at any time. Namely, $τ_{s l i p} = k s_{t}$ , where k denotes the average shear stiffness of the shear connectors. According to the Chinese specification (China JTG/T, 2015), $k = K \cdot n_{s} / ρ_{s u}$ , where $K = 13 d_{s} \sqrt{E_{c} f_{c k}}$ . Moreover, only steel-concrete interface slip is permitted, while the uplift displacement of the concrete deck is negligible in the following analysis. Additionally, the interface slip between the longitudinal joints and precast slabs in the assembled deck is ignored.

Generalized coordinates

It is generally believed that the longitudinal displacement of concrete can be divided into two parts in the composite beam. The first is caused by longitudinal bending and axial length changes based on the assumption of the Euler–Bernoulli beam. The second is caused by longitudinal warping related to the shear lag effect of the concrete flange. The longitudinal displacement diagram of the assembled composite beam is assumed to be Figure 2.

Figure 2.

Displacement of composite beams.

The local coordinate systems of the concrete deck and steel girder of the composite beam are constructed, shown in Figure 2. The corresponding coordinate origins (o′′ and o′) are on the neutral axis of the concrete deck and steel girder, respectively. The $Z_{c}$ and $Z_{s}$ are the vertical coordinate within the corresponding local coordinate systems, respectively. The $x_{c}$ and $x_{s}$ are the longitudinal coordinate within the corresponding local coordinate systems, respectively. Moreover, $u_{c} (x, t)$ and $u_{s} (x, t)$ denote the longitudinal displacement of the concrete and steel caused by the change in the axial length at time step t, respectively, abbreviated as $u_{c t}$ and $u_{s t}$ . $w (x, t)$ refers to the vertical deflection of the composite beam at time step t, abbreviated as $w_{t}$ . $s (x, t)$ stands for the slip of the steel-concrete interface of the composite beam at time step t, abbreviated as $s_{t}$ . The longitudinal displacement of concrete is

U_{c} (x, y, z, t) = u_{c} (x, t) - Z_{c} w^{'} (x, t) + ω_{c ζ} (y) ζ_{c} (x, t)

(1)

where

ζ_{c} (x, t)

is the longitudinal warping displacement of the concrete caused by the shear lag effect, abbreviated as

ζ_{c t}

; and

ω_{c ζ} (y)

denotes the shape function related to the longitudinal warping, abbreviated as

ω_{c ζ}

. The superscript ′ represents the derivative of the corresponding function with respect to x.

Huang et al. (2019b) had validated that the assumption of deformation coordination between longitudinal joints and precast slabs across beam sections is feasible for analysing the long-term performance of composite beams. Therefore, the warping displacement function proposed by Dezi et al. (2006) could still be used to analyse the shear lag effect

ω_{c ζ} (y) = {\begin{cases} {(\frac{y}{b})}^{2} + 2 \frac{y}{b} + \frac{b_{1}}{b} (2 - \frac{b_{1}}{b}), - b \leq y \leq - b_{1} \\ {(\frac{y}{b})}^{2} - {(\frac{b_{1}}{b})}^{2}, \begin{matrix} - b_{1} \leq y \leq b_{1} \end{matrix} \\ {(\frac{y}{b})}^{2} - 2 \frac{y}{b} + \frac{b_{1}}{b} (2 - \frac{b_{1}}{b}), b_{1} \leq y \leq b \end{cases}

(2)

where b₁ denotes the half spacing between the two steel girders across the beam section, b is the half-width of the assembled deck and y indicates the transverse coordinate in Figure 3.

Figure 3.

Warping displacement function of concrete.

Figure 3 shows the distribution of the longitudinal warping displacement function of concrete. h_c stands for the height of the concrete deck. b_w denotes the width of the construction joint upon one side steel girder. b_sl is the half-width of the construction joint upon the small steel beam, which is arranged between the two longitudinal steel girders. Due to the width-span ratio of the steel girder being generally less than 0.1, the warping deformation of the steel roof plate caused by the shear lag effect could be ignored (Dezi et al., 2003). Based on the assumption that the plane section of the steel girder remains planar, the longitudinal displacement of steel is

U_{s} (x, z, t) = u_{s} (x, t) - Z_{s} w^{'} (x, t)

(3)

It can be seen from Figure 3 and equation (2) that the longitudinal warping displacement of the concrete caused by the shear lag effect is 0 at the position of the steel girder, where the concrete deck is supported. Therefore, the slip of the steel-concrete interface can be expressed as

s (x, t) = u_{s} (x, t) - u_{c} (x, t) - \frac{h}{2} w^{'} (x, t)

(4)

where h is the height of the composite beam.

Generalized internal force

After the derivative of equations (1) and (3) with respect to x, the expressions of the strain of the concrete and steel can be obtained

ε_{c} (x, y, z, t) = u_{c}^{'} (x, t) - Z_{c} w^{″} (x, t) + ω_{c ζ} (y) ζ_{c}^{'} (x, t)

(5)

ε_{s} (x, y, z, t) = u_{s}^{'} (x, t) - Z_{s} w^{″} (x, t)

(6)

The shear strain of the concrete can be obtained by the derivative of equation (1) with respect to y

γ_{c x y} (x, y, t) = \frac{d ω_{c ζ} (y)}{d y} ζ (x, t)

(7)

According to the age-adjusted effective modulus method (Bazant 1972), it is easy to obtain the long-term normal stress and shear stress of the concrete

σ_{c i} (x, y, z, t) = E_{c t i} {\begin{cases} ε_{c} (x, y, z, t) - ε_{c s i} (t, t_{s}) - \\ ε_{c} (x, y, z, t_{0}) ϕ_{i} (t, t_{0}) [1 - χ_{i} (t)] \end{cases}}

(8)

τ_{c i} (x, y, t) = \frac{E_{c t i}}{2 (1 + ν_{c})} {\begin{cases} γ_{c x y} (x, y, t) \\ - γ_{c x y} (x, y, t_{0}) ϕ_{i} (t, t_{0}) [1 - χ_{i} (t)] \end{cases}}

(9)

where the subscript i denotes one batch of concrete, namely, i = 1 and 2 indicate the precast slab concrete and longitudinal joint concrete, respectively;

E_{c t i}

is the age-adjusted effective modulus of the concrete;

ε_{c s i} (t, t_{s})

refers to the shrinkage of the joint concrete or precast slab concrete after casting the joint concrete, abbreviated as

ε_{c s i}

;

ϕ_{i} (t, t_{0})

and

χ_{i} (t)

stand for the creep coefficient and ageing coefficient of the concrete, respectively; and

ν_{c}

indicates the Poisson’s ratio of the concrete.

According to Hooke’s law, the long-term normal stress of the steel can be obtained

σ_{s} (x, y, z, t) = E_{s} ε_{s} (x, y, z, t)

(10)

where

E_{s}

denotes the modulus of elasticity of the steel.

Based on the assumption of the Euler-Bernoulli beam, the axial force of the assembled deck can be deduced

N_{c t} = \int_{A_{c}} σ_{c t i} d A = \sum_{i = 1}^{2} E_{c t i} [\begin{array}{l} (u_{c t}^{'} - ε_{c s i}) A_{c i} + ζ_{c t}^{'} S_{c ζ i}^{*} - \\ ϕ_{i} (1 - χ_{i}) (u_{c 0}^{'} A_{c i} + ζ_{c 0}^{'} S_{c ζ i}^{*}) \end{array}]

(11)

where

σ_{c t i}

is the abbreviation of

σ_{c i} (x, y, z, t)

;

A_{c i}

denotes the cross-sectional area of the precast slab (i = 1) or longitudinal joint (i = 2);

S_{c ζ i}^{*}

refers to the moment of area related to the cross-section warping of the precast slab (i = 1) or longitudinal joint (i = 2); and

S_{c ζ i}^{*} = \int_{A_{c i}} ω_{c ζ} d A

The bending moment of the assembled deck can be expressed as

M_{c t} = - \int_{A_{c}} σ_{c t i} Z_{c} d A = \sum_{i = 1}^{2} E_{c t i} I_{c i} [w_{t}^{″} - ϕ_{i} (1 - χ_{i}) w_{0}^{″}]

(12)

where

I_{c i}

denotes the moment of inertia of the precast slab (i = 1) or longitudinal joint (i = 2), and

I_{c i} = \int_{A_{c i}} Z_{c}^{2} d A

;

w_{0}

is the instantaneous deflection.

Similarly, the axial force $N_{s t}$ and bending moment $M_{s t}$ of the steel girder are

N_{s t} = \int_{A_{s}} σ_{s t} d A = E_{s} u_{s t}^{'} A_{s}

(13)

M_{s t} = - \int_{A_{s}} σ_{s t} Z_{s} d A = E_{s} I_{s} w_{t}^{″}

(14)

where

σ_{s t}

is the abbreviation of

M_{c ζ t} = \int_{A_{c}} σ_{c t i} ω_{ζ} d A = \sum_{i = 1}^{2} E_{c t i} [\begin{array}{l} I_{c ζ i} ζ_{c t}^{'} + S_{c ζ i}^{*} (u_{c t}^{'} - ε_{c s i}) \\ - ϕ_{i} (1 - χ_{i}) (I_{c ζ i} ζ_{c 0}^{'} + S_{c ζ i}^{*} u_{c 0}^{'}) \end{array}]

(15)

σ_{s} (x, y, z, t)

A_{s}

denotes the cross-sectional area of the steel girder,

I_{s}

stands for the moment of inertia of the steel girder, and

I_{s} = \int_{A_{s}} Z_{s}^{2} d A

Moreover, the internal forces related to the warping of the concrete deck (Zhang 2012), namely, warping bi-moment $M_{c ζ t}$ and warping bi-moment $T_{c ζ t}$ , are given by

T_{c ζ t} = \int_{A_{c}} τ_{c t i} \frac{d ω_{ζ}}{d y} d A = \sum_{i = 1}^{2} \frac{E_{c t i}}{2 (1 + ν_{c})} I_{d ω i} [ζ_{c t} - ϕ_{i} (1 - χ_{i}) ζ_{c 0}]

(16)

where

I_{c ζ i}

is the warping moment of inertia of the precast slab (i = 1) or longitudinal joint (i = 2), and

I_{c ζ i} = \int_{A ci} ω_{ζ}^{2} d A

; similarly,

I_{d ω i}

denotes the warping product of inertia of the precast slab (i = 1) or longitudinal joint (i = 2), and

I_{d ω i} = \int_{A_{c i}} {(\frac{d ω_{ζ}}{d y})}^{2} d A

Equilibrium equations

According to the principle of virtual work

\begin{array}{l} \int_{0}^{L} [\begin{array}{l} N_{c t} δ u_{c t}^{'} + N_{s t} δ u_{s t}^{'} + (M_{c t} + M_{s t}) δ w_{t}^{″} + M_{c ζ t} δ ζ_{c t}^{'} + T_{c ζ t} δ ζ_{c t} \\ + τ_{s l i p} (δ u_{s t} - δ u_{c t} - \frac{h}{2} δ w_{t}^{'}) \end{array}] d x = \\ - \int_{0}^{L} q δ w_{t} d x + {[{\bar{N}}_{p c} δ u_{c t} + {\bar{N}}_{p s} δ u_{s t} + \bar{M} δ w_{t}' - \bar{V} δ w_{t} + {\bar{M}}_{c ζ} δ ζ_{c t}]}_{0}^{L} \end{array}

(17)

where

{\bar{N}}_{p c}

{\bar{N}}_{p s}

\bar{M}

\bar{V}

and

{\bar{M}}_{c ζ}

are the resultants of the surface forces applied to the beam end;

{\bar{N}}_{p c}

and

{\bar{N}}_{p s}

denote the axial force on the concrete deck and steel girder, respectively;

\bar{M}

and

\bar{V}

indicate the bending moment and shear force, respectively;

{\bar{M}}_{c ζ}

refers to the warping bi-moment; q is the uniformly distributed load.

The integral-differential system will be transformed into a simpler differential system through the step-by-step integration of equation (17) and further separation of displacement variables. Equations (11)–(16) are further substituted into the differential system

{\begin{cases} \sum_{i = 1}^{2} E_{c t i} [u_{c t}^{″} A_{c i} + ζ_{c t}^{″} S_{c ζ i}^{*} - ϕ_{i} (1 - χ_{i}) (u_{c 0}^{″} A_{c i} + ζ_{c 0}^{″} S_{c ζ i}^{*})] = - k (u_{s t} - u_{c t} - \frac{h}{2} w_{t}^{'}) \\ E_{s} u_{s t}^{″} A_{s} = k (u_{s t} - u_{c t} - \frac{h}{2} w_{t}^{'}) \\ \sum_{i = 1}^{2} E_{c t i} I_{c i} [w_{t}^{‴} - ϕ_{i} (1 - χ_{i}) w_{0}^{‴}] + E_{s} I_{s} w_{t}^{‴} = \\ - q - \frac{h}{2} k (u_{s t}^{'} - u_{c t}^{'} - \frac{h}{2} w_{t}^{″}) \\ \sum_{i = 1}^{2} E_{c t i} {\begin{cases} I_{c ζ i} ζ_{c t}^{″} + S_{c ζ i}^{*} u_{c t}^{″} - ϕ_{i} (1 - χ_{i}) (I_{c ζ i} ζ_{c 0}^{″} + S_{c ζ i}^{*} u_{c 0}^{″}) \\ - \frac{I_{d ω i}}{2 (1 + ν_{c})} [ζ_{c t} - ϕ_{i} (1 - χ_{i}) ζ_{c 0}] \end{cases}} = 0 \end{cases}

(18)

Meanwhile, the boundary conditions are

{\begin{cases} {\sum_{i = 1}^{2} E_{c t i} [(u_{c t}^{'} - ε_{c s i}) A_{c i} + ζ_{c t}^{'} S_{c ζ i}^{*} - ϕ_{i} (1 - χ_{i}) (u_{c 0}^{'} A_{c i} + ζ_{c 0}^{'} S_{c ζ i}^{*})] - {\bar{N}}_{p c}} δ u_{c t} = 0 \\ (E_{s} u_{s t}^{'} A_{s} - {\bar{N}}_{p s}) δ u_{s t} = 0 \\ {\sum_{i = 1}^{2} E_{c t i} I_{c i} [w_{t}^{‴} - ϕ_{i} (1 - χ_{i}) w_{0}^{‴}] + E_{s} I_{s} w_{t}^{‴} + \frac{h}{2} k (u_{s t} - u_{c t} - \frac{h}{2} w_{t}^{'}) - \bar{V}} δ w_{t} = 0 \\ {\sum_{i = 1}^{2} E_{c t i} I_{c i} [w_{t}^{″} - ϕ_{i} (1 - χ_{i}) w_{0}^{″}] + E_{s} I_{s} w_{t}^{″} - \bar{M}} δ w_{t}^{'} = 0 \\ {\sum_{i = 1}^{2} E_{c t i} [I_{c ζ i} ζ_{c t}^{'} + S_{c ζ i}^{*} (u_{c t}^{'} - ε_{c s i}) - ϕ_{i} (1 - χ_{i}) (I_{c ζ i} ζ_{c 0}^{'} + S_{c ζ i}^{*} u_{c 0}^{'}) - {\bar{M}}_{c ζ}]} δ ζ_{c t} = 0 \end{cases}} \begin{matrix} (x = 0, L) \end{matrix}

(19)

Numerical solution

Generally, it is difficult to solve a closed differential system (equation (18)). It is common to numerically solve the problem by approximating the derivatives with algebraic expressions using the finite difference method. By introducing the discretization of the x axis into n equally spaced parts, the derivatives of a generic function $Γ$ can be expressed as

{\begin{cases} Γ^{'} \approx \frac{Γ_{j + 1} - Γ_{j - 1}}{2 d} \\ Γ^{″} \approx \frac{Γ_{j + 1} - 2 Γ_{j} + Γ_{j - 1}}{d^{2}} \\ Γ^{‴} \approx \frac{Γ_{j + 2} - 2 Γ_{j + 1} + 2 Γ_{j - 1} - Γ_{j - 2}}{2 d^{3}} \\ Γ^{‴} \approx \frac{Γ_{j + 2} - 4 Γ_{j + 1} + 6 Γ_{j} - 4 Γ_{j - 1} - Γ_{j - 2}}{d^{4}} \end{cases}

(20)

where d is the length of the partition element and the subscript j denotes the node number of the partition element.

The differential system (equations. (18) and (19)) contains four variables, namely, $u_{c t}$ , $u_{s t}$ , $w_{t}$ and $ζ_{c t}$ . If the composite beam is divided into n fibre elements along its length, that is, n+1 nodes, there will be 4×(n+1) variables related to the node. It is worth noting that $u_{c 0}$ , $u_{s 0}$ , $w_{0}$ and $ζ_{c 0}$ correspond to the instantaneous values of $u_{c t}$ , $u_{s t}$ , $w_{t}$ and $ζ_{c t}$ , respectively. The variable $w_{t}$ of equations (18) and (19) has a fourth derivative, and it can be seen from equation (20) that the extra four variables ( $w_{t, - 2}$ , $w_{t, - 1}$ , $w_{t, n + 1}$ and $w_{t, n + 2}$ ) need to be considered to replace $w_{t}^{‴}$ at the end nodes j=0 and n. Similarly, the variables $u_{c t}$ , $u_{s t}$ and $ζ_{c t}$ all have a second derivative in equations (18) and (19) and require an extra 2 × 3=6 variables to replace the corresponding second derivative variable. In summary, there are 4+6=10 variables to consider, excluding the 4 × (n+1) variables directly related to the n+1 nodes. According to the node equilibrium condition of j=0, 1, …, n, one can obtain 4 × (n+1) equations. The differential system equations (18) and (19) could be solved with ten boundary condition equations and the 4 × (n+1) node equilibrium equations.

According to assuming t = 0, $u_{c 0}$ , $u_{s 0}$ , $w_{0}$ and $ζ_{c 0}$ can be firstly solved by using the finite difference method. Then, $u_{c 0}$ , $u_{s 0}$ , $w_{0}$ and $ζ_{c 0}$ will be brought into equations. (18)-(19) as constants, and $u_{c t}$ , $u_{s t}$ , $w_{t}$ and $ζ_{c t}$ could be obtained.

Validation of analytical model

According to the comparison with the long-term experimental data from the mock-up test by Huang et al. (2019a), the feasibility of the proposed model will be verified. The mock-up test was conducted for 390 days in an indoor environment with an average annual temperature of 17.5°C and average annual humidity of 67.1%. The mock-up was a continuous composite beam with a length of 7426 mm and a width of 5000 mm and included a lattice steel girder with a height of 354 mm and a concrete deck with a thickness of 55 mm. The shear connection and composite action of the concrete deck and steel girder depended on headed studs. There were five rows of headed studs along the roof flange of one side steel girder, and the spacing of the studs in the axial direction of the corresponding steel girder was 55 mm. The dimensional drawing of the mock-up is given in Figure 4.

Figure 4.

Dimensional drawing of the mock-up (Huang et al. 2019a).

As shown in Figure 4, the concrete deck was made of prefabricated and assembled components, and the plane dimensions of precast slabs include two types: 1910 mm × 586 mm and 586 mm × 270 mm. The precast slab parts were first made and stored outside 6 months, and then the lattice steel girder was lifted and laid on the supports shown in Figure 4. Before casting the construction joints, all precast slabs were laid on the lattice steel girder for 1 month. The cubic strengths of the precast slab concrete and joint concrete were measured to be 45.6 MPa and 49.7 MPa, respectively, and the corresponding moduli of elasticity were 33.9 GPa and 34.8 GPa, respectively. The steel girder was made with Chinese grade Q370qD steel plates, and its modulus of elasticity was 206 GPa, following the Chinese specification (China JTG/T, 2015). The mock-up was subjected to a uniformly distributed load (2 × 7.918 kN/m) and external prestressing (2 × 288.6 kN) for 390 days. The prestressing tendons were placed symmetrically in the cavity at the elevation of 40 mm under the top flange of every side steel girder.

Calculation for mock-up test

According to the shrinkage and creep test of precast slab concrete and joint concrete (Huang et al., 2019a), the shrinkage strain and creep coefficient of the concrete are

{\begin{cases} ε_{c s 1} = 0.399 ε_{c s o} β_{s} {(t - t_{s})}^{0.664} \\ ε_{c s 2} = 0.978 ε_{c s o} β_{s} {(t - t_{s})}^{0.450} \end{cases}

(21)

ϕ_{1} = ϕ_{2} = 1.287 ϕ_{0} β_{c} {(t - t_{0})}^{0.466}

(22)

where

ε_{c s o}

and

ϕ_{0}

denote the nominal coefficient of shrinkage and creep in CEB-FIP (1993), respectively.

β_{s} (t - t_{s})

and

β_{c} (t - t_{0})

refer to the coefficient describing the development of shrinkage and creep with time in CEB-FIP (1993), respectively. As the reinforcement ratio of the concrete deck in the mock-up is only 1.57%, the effect of reinforcement on the shrinkage and creep of concrete is ignored. According to equations (21) and (22), the shrinkage strain and creep coefficient of the precast slab concrete and joint concrete for analysing the long-term performance of mock-up is shown in Table 1.

Table 1.

Shrinkage strain and creep coefficient of the concrete at the final test period.

Concrete batch	Cube strength/MPa	Age/days	Shrinkage/uε	Creep coefficient
Precast slab	45.6	602	−146	1.486
Construction joint	49.7	390	−358	2.129

To solve equations (18) and (19) by the finite difference method, the mock-up is divided into 44 units along the length direction, shown in Figure 5. The corresponding deflection at the nodes 4, 16, 28 and 40 is 0, namely $w_{t, j} = 0$ (j = 4, 16, 28 and 40).

Figure 5.

Beam units for finite differential calculation.

The numerical solutions of equations (18) and (19) could be solved according to the process shown in Figure 6. The stress state and deformation of the mock-up in instantaneous and long-term could be obtained.

Figure 6.

Process of numerical calculation.

Comparison of long-term strain

The longitudinal strain distribution of the concrete along the concrete flange is shown in Figure 7. A comparison of the strain between the proposed model and experimental data at the initial loading and final test is shown in Figure 7. Furthermore, The longitudinal strain of the mock-up, which is lonely subjected to the effect of creep or shrinkage of concrete, is added in Figure 7. It is worth noting that this part of the long-term strain only related to creep or shrinkage includes the instantaneous response.

Figure 7.

Longitudinal strain distributions of concrete along the concrete flange: a) Cross-section Ⅰ-Ⅰ; b) Cross-section Ⅱ-Ⅱ; c) Cross-section Ⅲ-Ⅲ; and d) Cross-section Ⅳ-Ⅳ

Figure 7 shows that the longitudinal strain of the concrete obtained by the proposed model at the initial loading agrees well with the corresponding results of the test. Moreover, the instantaneous strain distribution of the concrete along the concrete flange shows the positive shear lag phenomenon in all cross-sections. Namely, the concrete strain upon the steel girder is greater than that at other places along the concrete flange in the same cross-section. As shown in Figure 4, the mock-up is subjected to compression and bending, and its concrete deck is mainly under compression due to the axial prestressing in the steel girder. Therefore, the concrete stress distribution far away from the axial loading section is more uniform. Namely, the strain distribution along the concrete flange is gradually uniform from the cross-section Ⅰ-Ⅰ to cross-section Ⅳ-Ⅳ.

Under the sustainable load, the long-term strain of the concrete affected by shrinkage and creep increases with time. As shown in Figure 7, the long-term strain of concrete caused by the new-old concrete shrinkage effect is greater than that of the new-old concrete creep effect. Moreover, the new-old concrete shrinkage hardly increases the shear lag effect of the assembled concrete deck. Namely, the new-old concrete shrinkage effect does not increase the non-uniformity of normal stress distribution along the concrete flange. As for the cross-sections I-I and Ⅱ-Ⅱ near the axial loading position, the increase of the shear lag effect with time is caused by the creep effect of the new-old concrete.

The degree of agreement between the proposed model and the long-term test is slightly lower than that under the instantaneous response. The reason is that the low level of instantaneous strain resulted in the substantial interference of the accumulated error of the sensor in the long-term test. In summary, the proposed model could well characterize the long-term strain distribution along the concrete flange, which verifies the accuracy of the proposed model.

Time-dependent analysis of the shear lag

To analyse the time-dependent shear lag effect of the composite beams under bending load and axial compression load separately, the example beams are designed as a simply-supported beam and cantilever beam in the following analysis. The cross-sectional size and material properties of the example beams are assumed to be consistent with the corresponding values of the mock-up. Because a low level of instantaneous stress of concrete will amplify the shrinkage effect of composite beams in the time-dependent analysis (Huang et al. 2018), the maximum concrete stress in the control section of example beams at the initial loading is designed to be 0.3 times the ultimate compressive strength of the precast slab concrete. The control section of the simply-supported beam and cantilever beam corresponds to the mid-span section and root section, respectively. Moreover, the assembly age of the concrete deck is set to 180 days after the precast slab concrete had been cast, which is more representative.

The coefficient $λ$ related to the effective flange width is introduced to study the time-dependent regularity of the effective flange width in the assembled deck

λ = \frac{\int_{0}^{y} σ_{c} (y, t) d y}{σ_{\max} (y, t) b}

(23)

where

σ_{\max} (y, t)

is the maximum long-term stress in the neutral layer of the concrete deck.

Situation of beam bending

Taking the simply supported composite beam under a uniformly distributed load as an example in Figure 8, the coefficient $λ$ of the simply-supported beam within the range of 0 < b/L < 0.5 is analysed, where b/L denotes the width-span ratio. The cross-sectional size and material properties of the simply-supported beam are assumed to be consistent with the corresponding values of the mock-up.

Figure 8.

Simply-supported beam example.

Under the conditions of b/L = 0.5 and 0.1, the normal stress distributions of the concrete along the concrete flange in the mid-span section corresponding to the initial loading and ten years of sustained loading are drawn in Figure 9. Moreover, the coefficients $λ$ obtained by the proposed model and traditional model with 0.1 < b/L < 0.5 are given in Table 2. It is worth noting that, compared with the proposed model, the traditional model adopting the traditional analytic strategy ignores the age difference between the precast slab concrete and longitudinal joint concrete. In the traditional model, the age of any concrete cross-section is recognized as the age of precast slab concrete at the corresponding cross-section.

Figure 9.

Long-term stress distribution of the concrete along the concrete flange.

Table 2.

Comparison of the effective flange width coefficients in the simply-supported beam example.

b/L	Instantaneous ①	Ten years of sustained loading
b/L	Instantaneous ①	Proposed model ②	Different rate (②-①)/①	Traditional model ③	Different rate (③-①)/①
0.05	0.99	0.92	−7.9%	0.99	−0.2%
0.1	0.98	0.90	−8.2%	0.97	−0.6%
0.2	0.92	0.84	−9.0%	0.90	−1.9%
0.3	0.84	0.76	−9.3%	0.82	−2.7%
0.4	0.76	0.69	−9.3%	0.74	−2.8%
0.5	0.68	0.62	−9.1%	0.67	−2.6%

The results obtained by the proposed model in Figure 9 show a sudden drop in the long-term stress distribution along the concrete flange at the joint position. The relatively more significant shrinkage and creep of joint concrete lead to greater relaxation of the joint concrete stress with time in the time-dependent analysis. Therefore, the long-term stress of the joint concrete decreases sharply compared with that of the precast slab concrete after ten years of sustained loading. Further, the tensile stress could appear in the construction joint under a low level of instantaneous compressive stress. Moreover, the long-term stress of the precast slab concrete obtained by the proposed model is larger than that of the traditional model. With the width-span ratio of the simply-supported composite beam, the long-term stress distribution of the concrete along the concrete flange became more non-uniform in the mid-span section.

As shown in Table 2, the coefficient $λ$ obtained by the proposed model decreases with time. Due to the time-dependent effect of the new-old concrete, the long-term stress of the joint concrete obtained by the proposed model is accelerated to decay, which aggravates the non-uniform distribution degree of the long-term stress distribution along the concrete flange. After ten years of sustained loading, the coefficients $λ$ obtained by the proposed model are approximately 9% less than the corresponding instantaneous values in the mid-span section. Nevertheless, the coefficient $λ$ of the traditional model has little difference from the corresponding instantaneous value.

Situation of axial compression

Taking the cantilever composite beam under concentrated axial loading as an example in Figure 10, the $λ$ distribution along the length direction is analysed. The cross-sectional size and material properties of the cantilever beam are assumed to be consistent with the corresponding values of the mock-up. At the initial loading and after ten years of sustained loading, the coefficients $λ$ obtained by the proposed model and traditional model, which ignores the new-old concrete age difference, are given in Table 3.

Figure 10.

Cantilever composite beam example.

Table 3.

Comparison of the effective flange width coefficients in the cantilever beam example.

l_x/b	Instantaneous ①	10 years of sustained loading
l_x/b	Instantaneous ①	Proposed model②	Different rate (②-①)/①	Traditional model ③	Different rate (③-①)/①
0.32	0.56	0.53	−7.6%	0.57	−1.1%
0.63	0.74	0.69	−2.5%	0.74	5.4%
0.95	0.86	0.79	−5.7%	0.85	2.1%
1.26	0.93	0.85	−9.3%	0.92	−2.0%
1.89	0.98	0.91	−8.0%	0.98	−0.4%
2.21	1.00	0.92	−7.8%	0.99	−0.3%

Note: l_x denotes the axial distance from the loaded section to any specified cross-section in the cantilever beam example.

Table 3 shows that the coefficient $λ$ of the cantilever beam increases gradually along the length direction from the loaded cross-section to the root section. The coefficient $λ$ of the cantilever beam at the initial loading is close to 1, where l_x = 2.2b. Due to the time-dependent effect of the new-old concrete, the non-uniformity of the normal stress distribution obtained by the proposed model is increased compared with that of the traditional model after ten years of sustained loading. The coefficients $λ$ obtained by the proposed model in all the observed cross-sections are approximately 7.1% less than the corresponding instantaneous value. In contrast, the coefficient $λ$ of the traditional model is almost equal to the corresponding instantaneous value.

The long-term stress distribution of the concrete along the concrete flange exhibits a mutation at the joint position in all the observed cross-sections. The coefficients $λ$ of the cantilever beam along the length direction are always less than 1.0 and asymptotically approaches a value. The AASHTO LRFD bridge design specifications (2018) defined a diffused angle to analyse the effective flange width related to concentrated axial loads. Suppose the diffused angle continues to be applied. In that case, the range end of the diffused angle in the cantilever beam example should be confirmed as the asymptotic value of the coefficients $λ$ along the length direction at ten years of sustained loading, instead of $λ = 1$ at instantaneous loading.

Conclusion

Based on the virtual work principle, a time-dependent shear lag model that considers the different ages of concrete across the deck section is established to analyse the long-term performance of steel-concrete composite beams with assembled decks. The approximate solution of the proposed model is derived by using the finite difference method. Comparing the results obtained by the proposed model and a long-term performance test, the accuracy of the proposed model is verified. Further analysis suggested the following conclusions:

(1) When ignoring the concrete age difference in an assembled deck section of composite beams, the long-term stress of the joint concrete could not be analysed accurately. Due to the time-dependent effect of the new-old concrete, the long-term stress of the joint concrete decreased faster than that of the precast slab concrete with time. The non-uniformity of the distribution of the long-term stress along the concrete flange increased, and tensile stress could appear in the construction joint under a low level of instantaneous compressive stress.

(2) When the composite beams sustained a bending load, the long-term effective flange width of the assembled deck decreased with the width-span ratio of the composite beam. The variation regularity of the long-term effective flange width is the same as that of the corresponding instantaneous value. However, the long-term effective flange width is approximately 8.8% smaller than the corresponding instantaneous value.

(3) When the composite beams sustained an axial load, the long-term effective flange width of the assembled deck was approximately 7.1% smaller than the corresponding instantaneous value. The long-term stress distribution of the concrete along the concrete flange exhibited a mutation at the joint position in any cross-section of the assembled deck. Therefore, the long-term effective flange width of the assembled deck asymptotically approached a value, which was less than 1.0, along the length direction from the loading section.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant number 52008036), the China Postdoctoral Science Foundation (grant number 2020M672462), the Hunan Provincial Natural Science Foundation of China (grant number 2021JJ40581), and the Science and Technology Innovation Program of Hunan Province (grant number 2020RC2052). This financial support is greatly appreciated.

ORCID iD

Dunwen Huang

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