Effects of thermal cycles and time-dependent behavior of the deck on steel network arch bridges

Abstract

Network arches are a type of arch bridges with inclined hangers, in which some hangers cross each other at least twice. This paper aims to evaluate the effects of thermal fluctuations and volumetric changes of the concrete deck on steel network arches. Fifty-nine variants of a reference network arch bridge were developed, assuming identical arch geometries but varying hanger arrangements. Each bridge was modeled using the finite element method and subjected to heat flux due to solar radiation and changes in ambient temperature in addition to heat transfer via conduction, convection, and radiation. The time-dependent deformations of the concrete deck were implemented using a rate-type formulation. Results obtained during a one-year period after the assumed end of construction showed that stress changes in the arch bridge due to ambient and time-dependent effects were not very sensitive to the hanger layout. Neglecting concrete creep and shrinkage was found to cause a 15-20% underestimation of maximum stresses in the concrete deck, whereas a 40-60% error was observed in both the deck and rib when thermal effects were ignored. The radial arrangement of hangers, already known to perform favorably under live loads, was also found to provide the smallest bending moments and the fewest number of relaxed hangers due to thermal and time-dependent effects.

Keywords

arch bridge network arch creep shrinkage thermal effects

Introduction

Arches are one of the oldest structural forms used to carry relatively large loads in bridges. The load-transfer mechanism in an arch bridge makes use of a mainly compressive path for the vertical loads to the ground and generates small bending moments, which makes arches structurally efficient. However, horizontal reaction forces are generated at the supports of arch bridges, requiring suitable geotechnical conditions (Barker and Puckett, 2021).

To employ arches in a wider variety of settings, tied arches were developed. In tied arch bridges, a horizontal member, referred to as “the tie”, transfers the horizontal force between the two ends of the curved element, referred to as “the rib”. In most tied arch bridges, the deck is located between the arches and is suspended from the rib by tensile elements that are called “hangers” (Yousefpour et al., 2017).

Network arch bridges are a relatively new type of tied arch bridge with inclined hangers, in which some hangers cross each other at least twice. These arches were introduced by Tveit in the 1960s. Other than being aesthetically pleasing and having the structural benefits of tied arch bridges, network arches are remarkably efficient because their hanger arrangement causes less bending moment in the rib and the tie, leading to smaller sections for both elements (Tveit, 1987). A typical network arch bridge is illustrated in Figure 1.

Figure 1.

Configuration of a typical network arch bridge.

Before the introduction of network arches, tied arch bridges were built either with vertical hangers or inclined hangers that crossed each other once, in a layout commonly referred to as the Nielsen arrangement. The use of inclined hangers provides a means for transferring shear between the rib and the tie, which reduces the bending moments in these elements. However, the efficiency of arch bridges with vertical and Nielsen arrangements may diminish when these structures are subjected to non-uniform live loads, e.g., when only half of the span is under live load. This loading condition may lead to compressive forces in many hangers on the non-loaded half of the bridge, causing them to relax. As a result, significantly greater bending moments will be generated in the arch rib and the tie (Tveit, 2014).

Network arch bridges have the clear benefit of possessing hangers inclined in two opposite directions, so when hangers inclined to one side tend to go into compression, hangers inclined to the other side stay in tension. As a result, network arches are considerably more efficient than arches with vertical or Nielsen hanger arrangements. Comparative studies by Tveit (2014) showed that replacing the vertical hanger arrangement with the network arrangement could save up to 70% in the weight of steel. The efficiency of network arches has made them suitable choices not only for steel construction but also for precast concrete (Yousefpour et al., 2015, 2017).

Multiple studies have been carried out to optimize hanger arrangement in network arch bridges. Brunn and Schanack (2003) introduced an optimized hanger layout called the radial arrangement, in which the hanger intersections lie on the arch radii, leading to nearly equal hanger forces in all hangers. The optimal cross angle, i.e., the angle between the hangers and their intersection lines in the radii, was reported as 45°, which resulted in the lowest bending moments in the structure.

De Zotti et al. (2007) confirmed the advantages of the radial arrangement regarding lower bending moments but reported that cross angles between 28° and 37° yield the lowest internal forces in the rib and tie. Another study by Schanack and Brunn (2009) showed that in the most unfavorable cases of loading, the bending moment in network arch bridges is approximately 10% lower than tied arch bridges with vertical hangers, and despite the higher complexity of construction, the former would be more economical.

Network arch bridges are a rather modern line of structures; thus, their long-term behavior is still relatively unknown. Two major factors that impact the long-term behavior of these structures are the time-dependent volumetric changes, i.e., creep and shrinkage of the deck concrete and the effect of temperature changes.

Creep is defined as the long-term deformation of concrete that occurs as a result of sustained stress. The time-dependent strain at constant temperature that occurs without stress is referred to as shrinkage. Both phenomena are known to affect the deformations of structural members and distribution of forces and stresses (Gilbert and Ranzi, 2010). As network arch bridges are highly statically indeterminate, it is essential that the effect of these factors be investigated to ensure their safety and serviceability.

The temperature within structural members, especially those exposed to the environment, is constantly changing due to solar radiation and ambient temperature changes (Zhu et al., 2020b). Differences in exposure to sunlight, as well as material properties and member shape and size cause a non-uniform temperature distribution in bridges, which in turn leads to thermal stresses (Cao et al., 2011; Westgate et al., 2015).

The effects of thermal changes on different types of bridges have been investigated in previous studies. Catbas et al. (2008) assessed environmental monitoring data from long-span truss bridges and showed that strains due to thermal changes may easily exceed those induced by live loads Chen (2008) investigated the effect of daily and seasonal thermal loads on steel bridges, mainly box-girders, by means of field monitoring and numerical analysis. The results showed thermal stresses can cause the concrete deck to crack in a composite steel girder bridge. Design recommendations were developed for the analysis of box girder bridges under thermal changes. Wang et al. (2013) conducted a numerical study on the effects of temperature on the creep behavior of CFST bridges, which showed that temperature effects cannot be neglected in concrete creep calculation as the difference in results is considerable.

Wang et al. (2016) proposed a 3D finite element model to assess thermal stresses and the displacement of concrete box-girder arch bridges subject to solar radiation and convection. The results showed that the thermal stress has a nonlinear distribution along the depth of the cross section and is large enough to affect structural safety. Hedegaard et al. (2016) studied the time-dependent behavior of concrete bridges under cyclic temperature and showed that thermal fluctuations noticeably alter the magnitude of time-dependent stresses and strains, and the combination of thermal gradient in the depth of concrete sections and creep and shrinkage may lead to large residual stresses. Zhu et al. (2020b) investigated temperature effects on steel truss bridges and proposed a formula for the relationship between temperature distributions and temperature-induced strains based on long-term monitoring data.

Previous studies on the effects of ambient conditions on network arches have been very limited. Perhaps the only field studies on this type of bridge were carried out by Yousefpour (2015) and Yousefpour et al. (2015, 2017) on the world’s first precast concrete network arch bridge. The results of these studies showed that creep and shrinkage have a noticeable effect on the stresses in the members of a concrete network arch, especially the tie. However, time-dependent changes mostly occurred during construction and were predicted to have a rather small effect on the structure afterwards. Moreover, the magnitude of stresses caused by daily temperature cycles in the summer was substantially greater than those by live loads and was comparable to self-weight stresses.

Most design calculations are not sophisticated enough to accurately incorporate the creep and shrinkage, or non-uniform heating or cooling of the structure due to solar radiation or thermal changes that occur on a daily or seasonal basis. Therefore, for structures such as network arches, it is essential that the sensitivity of the structure to time-dependent and thermal effects be investigated to guide engineers in identifying potential conditions that warrant advanced analyses.

The present paper makes use of validated finite element (FE) simulations of network arches to identify the effects of environmental and time-dependent changes on steel network arches and determine hanger arrangements that may be more sensitive to such changes.

Methodology

The reference structure used in this research is a steel network arch bridge with a concrete deck, based on the bridge designed by Brunn and Schanack (2003), as shown in Figure 2. The bridge has a span of 100 m, an arch-rise of 17 m, and a width of 10.15 m between the hangers, with 44 hangers in each arch (22 in each direction). Each side of the deck has 7 longitudinal DYWIDAG 6827 tendons (with 27 strands in each tendon, each with a cross-sectional area of 150 mm²) The profile of the arch rib consists of three circular curves, with the two end curves having slightly greater radii. Only the self-weight of the bridge, superimposed dead loads, and prestressing loads were applied to the bridge. The bridge was assumed to be in Minneapolis, USA and longitudinally positioned in the direction of global north.

Figure 2.

Layout of the base structure.

Models of 59 variants of this reference structure with different numbers and arrangements for the hangers were developed in the FE software Abaqus (Dassault Systemes, 2020). These variants were investigated for their sensitivity to time-dependent and ambient effects for the duration of one year, starting from the end of construction. Analyzing the bridge over a one-year period allows for the effects of daily and seasonal temperature changes to be taken into account. First, a thermal analysis was carried out to calculate nodal temperatures considering solar radiation, ambient temperature changes, and wind speed. Next, the resultant nodal temperatures were applied to a mechanical model, and user subroutines were employed to incorporate the time-dependent behavior of concrete. The procedures for simulating thermal and time-dependent effects are discussed in the following.

Modeling thermal effects

Calculations of solar radiation

Incident solar radiation on a horizontal surface includes a beam component and a diffuse component. Beam radiation is the solar radiation that directly strikes a surface without changing direction while the diffuse radiation is that scattered by the atmosphere (Duffie et al., 2020)

To calculate incident solar radiation on the bridge, data recorded by the National Renewable Energy Laboratory Wilcox (2012) were used. The NREL database includes hourly data for total incident solar radiation on a horizontal surface and its diffuse component, from which the beam component can be calculated using equation (1) (Elbadry and Ghali, 1983):

I_{b h} = I_{h} - I_{d h}

(1)

In which $I_{h}$ is incident solar radiation on a horizontal surface, and $I_{d h}$ and $I_{b h}$ are the diffuse component and the beam component, respectively. For an inclined surface, incident solar radiation is calculated using equation (2).

I_{t} = I_{b} + I_{d} + I_{r}

(2)

In this equation, $I_{b}$ is the beam component, $I_{d}$ is the diffuse component, and $I_{r}$ is the reflected component, which depends on the reflective properties of objects surrounding a surface. Equation (3) is used to calculate the beam component of solar radiation. In this equation, $θ$ is the solar incidence angle, i.e., the angle between the surface normal and the sun’s rays, and $β$ is the Solar Zenith Angle, which is the angle between the sun’s rays and an upward vector perpendicular to the earth’s surface.

I_{b} = \frac{I_{b h}}{\cos β} \cos θ

(3)

To calculate $\cos θ$ and $\cos β$ , equations (4) and (5) are used, respectively (Hsieh, 1985).

\begin{array}{l} \cos θ = \cos δ \cos H \sin L \sin ω \cos ψ + \sin δ \cos L \sin ω \cos ψ \\ + \cos δ \sin H \sin ω \sin ψ + \cos δ \cos H \cos L \cos ω \\ + \sin δ \sin L \cos ω \end{array}

(4)

\cos β = \cos δ \cos H \cos L + \sin δ \sin L

(5)

In these equations, $δ$ is the declination angle; $H$ is the solar hour angle; $L$ is the latitude of the point under investigation; $ω$ is the angle between the surface and a horizontal plane; and $ψ$ is the solar Azimuth angle, which for the northern hemisphere, is defined as the clockwise angular distance from the global south direction to the projection of the sun’s rays on a horizontal plane passing the point.

The declination angle is defined as the angle between the equator and a line connecting the centers of the earth and the sun, which can be approximated in degrees by equation (6) (Goswami, 2015):

δ = 23.45 \sin [\frac{360}{365} (d_{n} + 284)]

(6)

where

d_{n}

is the day number in the Gregorian calendar.

The solar hour angle for any point on the earth is the angle between a meridian passing the point and the direction the sun directly strikes that point and expresses the time in the form of an angle. Each hour is approximately 15° and the hour angle at solar noon is zero. The solar hour angle is therefore calculated using equation (7).

H = 15 ° (L S T - 12)

(7)

where H varies between −180° and 180° and is negative before solar noon and positive afterwards.

L S T

is local solar time, which is calculated using equation (8) (Hsieh, 1985):

L S T = L C T + 4 (L_{s t d} - L_{l o c}) + E T - D S T

(8)

where

L C T

is the local civil time,

L_{s t d}

is the longitude of the standard meridian of the local time zone,

L_{l o c}

is the longitude of the point considered,

E_{t}

is the equation of time, approximately calculated by equation (9) in minutes, and

D S T

is 1 during daylight saving time and 0 otherwise (Hsieh, 1985). In equation (9),

Z

is obtained from equation (10) where

n

is the number of the day within the year.

E_{t} = 9.87 \sin 2 Z - 7.53 \cos Z - 1.5 \sin Z

(9)

Z = \frac{360}{364} (n - 81)

(10)

The direct component of the solar radiation is calculated using equation (11).

I_{d} = I_{d h} \frac{1 + \cos ω}{2}

(11)

In this study, equation (12) was used to approximate the reflective component, assuming all surrounding surfaces have the same reflective properties and reflect the sun’s rays in all directions equally, and multiple reflections from different surfaces were neglected.

I_{r} = r_{g} I_{h} \frac{1 - \cos ω}{2}

(12)

where

r_{g}

is the reflectivity of surrounding surfaces which was taken as 0.2 (Chen, 2008).

Once $I_{b}$ , $I_{d}$ , $I_{r}$ were calculated as above, $I_{t}$ was determined and applied as a surface heat flux to each bridge surface in the heat transfer models. In addition to this heat flux, interactions were defined for the surfaces of the model to incorporate heat transfer by radiation according to equation (13) and by convection according to equation (14).

q_{r a d} = ε σ (T^{4} - T_{a}^{4})

(13)

q_{c o n v} = h_{c} (T - T_{a})

(14)

where

q

is the energy transferred per unit area per unit time in

w / m^{2}

;

ε

is the surface emissivity;

σ

is the Stephan Boltzmann constant, equal to

5.67 \times 10^{- 8}

(\frac{w}{m^{2} K^{4}})

;

T

and

T_{a}

are the absolute temperatures of the bridge surface and the ambient air in

K

; and

h_{c}

is the convective heat transfer coefficient, which is inputted as the surface film coefficient. This coefficient depends on factors such as wind speed as well as surface material and configuration (Hulsey and Emanuel, 1978). In this study,

h_{c}

was assumed based on equation (15) in

\frac{w}{m^{2} K}

based on the work of Ibrahim (1995), in which

u

is the wind speed in

\frac{m}{s}

h_{c} = {\begin{cases} 4.67 + 3.83 u & top surfaces of bridge deck \\ 2.17 + 3.83 u & soffit surfaces \\ 3.67 + 3.83 & outer surfaces of steel webs and concrete slabs \\ 3.5 & inside surfaces \end{cases}

(15)

Validation of thermal calculations

For verification of thermal calculations, data provided by Chen (2008) from monitoring the temperatures within a twin box girder bridge constructed in Ferguson Structural Engineering Laboratory (FSEL), Austin, Texas were used. The cross section of the bridge, which was exposed to ambient conditions, is presented at the top of Figure 3.

Figure 3.

Comparison between calculated and measured temperatures for the FSEL bridge.

A model of the bridge from Chen’s research (the FSEL Bridge) was developed, as shown in Figure 3 with material properties shown in Table 1 and hourly ambient temperature, solar radiation, and wind speed data from NREL (Wilcox, 2012). To save on analysis time, only 1 m of the bridge length was modeled. Comparison between results from the numerical model and measured data is also provided in Figure 3, which shows the good performance of the modeling approach in predicting temperatures at different locations of the bridge.

Table 1.

Thermal material properties in the heat transfer analysis.

Model	Material	Thermal conductivity (w/m.K)	Specific heat (J/kg.K)	Absorptivity	Emissivity	Density (kg/m³)
FSEL bridge	Steel	46	486	0.7	0.9	7850
FSEL bridge	Concrete	1.5	960	0.5	0.8	2400
Network arch	Steel	53	443	0.7	0.9	7850
Network arch	Concrete	1.5	900	0.5	0.8	2400

Thermal modeling of the network arch bridge

To calculate nodal temperatures during the period of one year for the network arch bridge with different hanger configurations, recorded daily thermal data for the assumed location of the bridge were extracted from April 2003 through April 2004. The use of this period was governed by the availability of complete solar data to the public. However, it is believed that generally similar conclusions would be drawn if data from different 1-year periods are used.

A model of the reference structure was developed, in which the rib length was divided into 6 parts. For each part, incident solar radiation was calculated and applied to the FE model through creating surface heat flux loads. To simulate convection, surface film condition interactions were defined on the bridge using hourly wind speed to calculate surface film coefficient and considering ambient temperature. Radiation was simulated by defining surface radiation interactions which depend on emissivity and ambient temperature. Material properties used for the thermal modeling of this bridge are presented in Table 1, which are primarily based on Eurocode 3 (CEN European Committee for Standardization, 2005b) and Eurocode 2 (CEN European Committee for Standardization, 2004a).

The general layout of the FE model is shown in Figure 4 To save on calculation time and storage space, only half of the bridge was modeled for parametric investigation. In addition, for simplicity, prestressing tendons and reinforcing bars within the deck were not modelled in the thermal simulation, as their effect was considered negligible due to their small cross sections.

Figure 4.

Layout of meshed FE half model for the base network arch bridge.

To ensure suitable simulation of boundary conditions in the half-bridge model, the entire bridge was first modeled and analyzed for thermal effects for a period of one month. Comparing nodal temperatures showed less than 5% difference between the entire bridge and the half-bridge models.

For steel rib and the concrete deck, DC3D8 (8-node linear heat transfer brick) elements were used for thermal analyses. For hangers, DC1D2 (2-node heat transfer link) elements were employed. The concrete deck was meshed with three elements in its depth that had dimensions of 800 by 800 mm in the horizontal plane. The rib had a maximum mesh size of approximately 650 mm along the span.

Once each model was analyzed for thermal effects, nodal temperatures from the model were extracted and used as predefined fields for the mechanical models to carry out analyses for elastic and time-dependent effects. Typical results from the thermal analysis of the case study bridge are presented in Figure 5, which shows temperature ranges of 82°C, 83°C, and 62°C for the arch, hangers, and the deck, respectively.

Figure 5.

Ambient temperature, temperature of one hanger and at upper mid points of the arch and deck in a typical FE model.

Mechanical modeling

A mechanical model of each variant of the base structure was developed with mesh sizes that were identical to those in the thermal model. In this model, 8-node solid brick (C3D8R) elements were used for the rib and deck. For hangers and prestressing tendons, 2-node linear 3D truss (T3D2) elements were employed. The modulus of elasticity of concrete at 28 days was assumed as 33550 MPa, whereas the modulus of elasticity of steel was taken as 200000 MPa. The Poisson’s ratios of steel and concrete were assumed as 0.3 and 0.15, respectively; and the coefficient of thermal expansion was assumed as 10⁻⁵ (1/°C) for concrete and 1.2 $\times$ 10⁻⁵ (1/°C) for steel in the arch, hangers, tendons, and reinforcing bars.

The two ends of the arch were embedded in the deck to create a rigid connection in all directions. The bridge model was assumed to have a fixed support at one end and a pinned support at the other, through creating displacement boundary conditions at the ends of the arch.

The prestressing force in the tendons was applied via assigning low temperatures to the elements comprising the tendons at the beginning of the mechanical analysis. Because tendons were not a part of the thermal model, changing their temperature in the mechanical model because of the thermal analysis results was not a concern.

The time-dependent behavior of concrete was implemented in the model by means of user subroutines USDFLD and UEXPAN. User subroutine USDFLD and utility routine GETVRM were used to access solution data, most importantly stresses, at each integration point, which were stored in state-variables to be later passed onto UEXPAN.

UEXPAN is a user subroutine that can be implemented to define incremental strains based on temperature, predefined field variables, and state variables. In the mechanical model, the expansion properties of the concrete were defined via this user subroutine. Moreover, in each increment, with a maximum time step of 4 h, UEXPAN was employed to calculate the concrete time-dependent and thermal strains and apply them to the model. Values that were needed for the next iteration were also stored in state variables at the end of each increment.

USDFLD was also used for the definition of field-variables to assign a varying modulus of elasticity to concrete over time. To appropriately simulate creep strains in an incremental solution approach, a rate-type formulation of the Eurocode 2 (CEN European Committee for Standardization, 2004b) equations was employed, using the procedure that follows.

Calculation of creep and shrinkage strains

In general, calculations of concrete creep strains are based on an assumption of linear viscoelastic behavior for stress levels lower than half of compressive strength of concrete (Gilbert and Ranzi, 2010) The creep response is characterized by the compliance function $J (t, τ)$ , which is the strain at time $t$ due to unit stress applied at time $τ$ . If the stress is variable over time, it is common to superimpose the effects of increments of stress over time through integration, as expressed in equation (16), in which $ε (t)$ is total strain at time $t$ and $σ (τ)$ is stress at time $τ$ .

ε (t) = \int_{0}^{t} J (t, τ) d σ (τ)

(16)

Using this formulation in the finite element model would require a large storage space and excessive solution time, as in each increment, the whole stress history would have to be recalled and used for calculations. To solve this problem, a rate-type approach is used, in which the compliance function is approximated by a Dirichlet series (Bažant et al., 2011; Bazant and Wu, 1973; Di Luzio et al., 2020; Zhu et al., 2020a). Using the rate-type formulation, the creep problem is turned into a step-by-step procedure in which strain change in each increment only depends on the stress change in that increment and a number of parameters stored from the previous increment.

Approximating the Eurocode 2 creep strain function with a Dirichlet series would yield (Zhu et al., 2020a):

{Δ ε_{n}}^{c c} = \sum_{j = 1}^{m} [1 - e^{- λ_{j} Δ t_{n}}] η_{j, n} + {Δ σ_{n}}^{c} \sum_{j = 1}^{m} ψ_{j} ({\bar{t}}_{n}) [1 - e^{- λ_{j} (t_{n} - {\bar{t}}_{n})}]

(17)

where

m

is herein taken equal to 3 and other parameters are calculated as follows.

Δ t_{n} = t_{n} - t_{n - 1}

(18)

{\bar{t}}_{n} = \frac{t_{n} + t_{n - 1}}{2}

(19)

λ_{j} = 10^{- (j - 1)} λ_{1} with λ_{1} = 1

(20)

η_{j, n} = {\begin{cases} Δ σ_{0} ψ_{j} (t_{0}) & n = 1 \\ e^{- λ_{j} Δ t_{n - 1}} η_{j, n - 1} + {Δ σ_{n - 1}}^{c} ψ_{j} ({\bar{t}}_{n - 1}) e^{- λ_{j} (t_{n - 1} - {\bar{t}}_{n - 1})} & n \geq 2 \end{cases}

(21)

where

ψ_{j} (t_{0})

is calculated using equation (22).

ψ_{j} (t) = \ln (10) C_{0} C_{1} (t) A (\frac{1}{λ_{j}})

(22)

in which

C_{0}

and

C_{1} (t)

are parameters determined using Eurocode 2 formulation, as defined below, and

A (\frac{1}{λ_{j}})

is continuum retardation spectrum of order 3 and is obtained from the following formula (Di Luzio et al., 2020):

A (α) = - \frac{{(- 3 α)}^{3}}{2} \frac{0.001 β_{H} {(\frac{3 α}{3 α + β_{H}})}^{0.3} [1800 {(3 α)}^{2} + 1260 β_{H} (3 α) + 357 {β_{H}}^{2}]}{{(3 α)}^{3} {(3 α + β_{H})}^{3}}

(23)

where

β_{H}

is calculated using equation (24) in which

α_{3}

is a factor to consider the effect of concrete strength.

C_{0}

is calculated using equation (25).

β_{H} = {\begin{cases} 1.5 [1 + {(0.012 R H)}^{18}] h_{0} + 250 \leq 1500 & for f_{c m} \leq 35 MPa \\ 1.5 [1 + {(0.012 R H)}^{18}] h_{0} + 250 α_{3} \leq 1500 α_{3} & for f_{c m} > 35 MPa \end{cases}

(24)

C_{0} = \frac{φ_{R H} β_{(f_{c m})}}{E_{c i}}

(25)

where

E_{c i}

is the modulus of elasticity of concrete at the age of 28 days and

β_{(f_{c m})} = 16.8 / \sqrt{f_{c m}}

with

f_{c m}

taken as the 28-days compressive strength of concrete. The factor

φ_{R H}

is calculated based on the relative humidity

R H

in percent according to equation (26).

\begin{array}{l} φ_{R H} = 1 + \frac{1 - \frac{R H}{100}}{0.1 \sqrt[3]{h_{0}}} & for f_{c m} \leq 35 MPa \\ φ_{R H} = [1 + \frac{1 - \frac{R H}{100}}{0.1 \sqrt[3]{h_{0}}} α_{1}] α_{2} & for f_{c m} > 35 MPa \end{array}

(26)

In this equation, $α_{1}$ and $α_{2}$ take into account the influence of concrete strength and $h_{0} = 2 A_{c} / u$ in mm is the notional size of the member, where $A_{c}$ is the cross-sectional area and $u$ is the perimeter in contact with the atmosphere. $C_{1} (t)$ is calculated for the current age of concrete, $t$ as follows:

C_{1} (t) = β (t) = \frac{1}{1 + t^{0.2}}

(27)

In Eurocode 2, total concrete shrinkage strain $ε_{c s}$ consists of two components, autogenous shrinkage strain $ε_{c a}$ and drying shrinkage strain $ε_{c d}$ . Autogenous shrinkage strain is obtained from equation (28) through (30) whereas drying shrinkage occurs is calculated using equations (31) through (34).

ε_{c a} (t) = β_{a s} (t) ε_{c a} (\infty)

(28)

β_{a s} (t) = 1 - \exp (- 0.2 t^{0.5})

(29)

ε_{c a} (\infty) = 2.5 (f_{c k} - 10) {\times 10}^{- 6}

(30)

ε_{c d} (t) = β_{d s} (t, t_{s}) k_{h} ε_{c d, 0}

(31)

β_{d s} (t, t_{s}) = \frac{(t - t_{s})}{(t - t_{s}) + 0.04 \sqrt{{h_{0}}^{3}}}

(32)

ε_{c d, 0} = 0.85 [(220 + 110 α_{d s 1}) \exp (- α_{d s 2} \frac{f_{c m}}{f_{c m 0}})] 10^{- 6} β_{R H}

(33)

β_{R H} = 1.55 [1 - {(\frac{R H}{{R H}_{0}})}^{3}]

(34)

In these equations, $t_{s}$ is the age of concrete at the start of drying; $R H$ is ambient relative humidity in percent, and $R H_{0}$ is taken as $100$ ; $α_{d s 1}$ and $α_{d s 2}$ depend on the type of cement; and $k_{h}$ is a coefficient varying between 0.7 and 1 depending on the notional size $h_{0}$ . The magnitude of creep and shrinkage strains within the model were calculated using these equations and implemented in the model using user subroutine UEXPAN for each integration point within the model in each solution increment.

Checking the time-dependent calculation procedure

Given the complexity of formulation for the calculation of creep strain increments, a simple FE model of a 3000 by 500 by 500 mm beam was developed for checking the appropriate functioning of the user subroutines. The beam was assumed to be constructed using materials identical to those in the arch bridge model and was subjected to the 100-days load history shown in Figure 6(a). Results from the FE model were compared to the analytical integration-based creep prediction from Eurocode 2. As presented in Figure 6(b), the strain difference between analytical solution and the FE model results was less than 5%; hence, the creep calculations were deemed satisfactory.

Figure 6.

Checking the proper function of user subroutines for creep: (a) loading history; (b) output strain history.

Hanger arrangement layouts

The case study bridges, i.e., the variants of the reference structure, were assumed to have hanger arrangements shown in Figure 7 and underwent thermal-mechanical analyses to investigate the effects of ambient conditions and time-dependent changes. Each variant was assigned an ID based on its hanger arrangement, as described in Table 2.

Figure 7.

The 59 hanger arrangements assumed for network arches.

Table 2.

Model ID guide.

Layout	ID parameters
Radial	$R a d i a l ‐ \bar{x x} ‐ \bar{y y}$
	$\bar{x x}$ : number of hangers in each direction
	$\bar{y y}$ : cross angle
VNT	$VNT ‐ \bar{x x} ‐ \bar{y y} \bar{z z}$
	$\bar{x x}$ : number of hangers in each direction
	$\bar{y y}$ : $r_{V N T} \times 100$
	$\bar{z z}$ : $λ_{V N T} \times 100$
VAT	$VAT ‐ \bar{x x} ‐ \bar{y y} S \bar{z z}$
	$\bar{x x}$ : number of hangers in each direction
	$\bar{y y}$ : $φ_{1}$
	$S$ (Sign): “P” for $Δ φ \geq$ 0, and “N” for $Δ φ <$ 0 $\bar{z z}$ : $Δ φ \times 100$
Vertical	$Vertical ‐ \bar{x x}$
Vertical	$\bar{x x}$ : number of hangers in each arch

The arrangements in Figure 7 can be categorized into four groups: (1) the vertical hanger arrangement (with 44 hangers in each arch); (2) radial arrangements; (3) arrangements with varying node distance along the tie (VNT); and (4) arrangements with varying angle along the tie (VAT). These arrangements are described below.

Radial arrangements

In this hanger layout, which is recommended by Schanack and Brunn (Schanack and Brunn, 2009), hangers intersect at points that lie on equidistant radii of the circular arch, as presented in Figure 8(a). The angle with which the hangers cross their intersection line is referred to as the cross angle. Fourteen radial arrangements were considered in this study, 11 of which contained 22 hangers in each direction of each arch with cross angles varying from 0° to 50°, and the other 3 were assumed to possess a constant cross angle of 40° with 11, 33, and 44 hangers in each direction of each arch.

Figure 8.

Details of hanger arrangements and stress outputs.

VNT arrangements

In the VNT layout, the distance between hanger nodes increases along the tie when moving from one end of the arch to the other. The increments for the change in distance are assumed to be larger at the ends and smaller for sections that are closer to the middle. The geometry of an ellipse was used to produce different arrangements of this type based on the ratio of the semi-minor axis to the semi-major axis, $λ_{V N T} = b / a$ , and a parameter to consider the implemented range of the ellipse $r_{V N T} = Δ x / x_{m}$ , as shown in Figure 8(b) (Brunn and Schanack, 2003). Twelve hanger arrangements were assumed to follow this layout, with $λ_{V N T}$ and $r_{V N T}$ ranging from 0.2 to 0.8.

VAT arrangements

The VAT layout is produced by varying the angle between hangers and the tie, $φ_{n}$ , with n being the hanger number from 1 to 22 for hangers that are inclined in one direction. The increment for this angle change, $Δ φ = φ_{n} - φ_{n + 1}$ , was assumed as a variable, as shown in Figure 8(c). In this study, a total of 32 hanger arrangements with this layout were assumed, for which the starting angle $φ_{1}$ varied between 50° and 85°, and $Δ φ$ was taken as −1, 0, 1, 2, or 3.

Results and discussion

Once the thermal and mechanical analyses were complete for the case study bridges, stresses at three chosen sections of each arch were extracted. As shown in Figure 8(d), these sections include those at one end of the rib and deck, at quarter length from one end and the sections located at midspan. Due to symmetry, these sections were investigated only in one half of the arch.

A variety of outputs were extracted from the model at each monitored section. The Mises stresses at two opposite corners of the steel rib sections and average stress and strain in the longitudinal direction at the top and bottom of the concrete deck sections were calculated and compared between the 59 models.

One of the parameters used for comparison between the models was the normalized stress range (NSR), which is the absolute value of the difference between the maximum and minimum stress at a specific point, divided by its average stress value. NSR is a measure of stress variability in each model due to time-dependent and thermal effects.

Moreover, the average normal stress (ANS) within each cross section was calculated as a measure of axial force within the section. To obtain a measure of bending moment within the cross sections, the absolute difference between the normal stresses at the top and the bottom of each section was divided by ANS to calculate a normalized stress gradient (NSV). ANS and NSV are measures that show how sensitive axial forces and bending moments in each arch are to time-dependent and thermal effects.

The tensile force within all hangers was also extracted. Tension in hangers in two opposing directions at the middle of the arch and the two ends were monitored for comparison, along with the number and position of hangers that relax throughout the monitoring period. The monitored hangers in a typical case study bridge are identified in Figure 8(d).

The following section compares the performance of arches with different hanger arrangements. Note that the discussions herein focus merely on the effects of thermal and time-dependent effects on the case study bridges, and the effects of live loads are not investigated. Therefore, the superior performance of a particular arrangement with respect to these ambient effects does not necessarily signal a better overall performance for the arch with that arrangement.

Stress changes over time

Changes in deck and rib stresses in four case study bridges are presented in Figure 9. This figure shows that stress values in the deck in all variants with network hanger arrangements were very similar. At the mid-top point of the deck, i.e., the top of the deck at the midspan, normal stresses change more rapidly when vertical hangers are used. Conversely, the opposite occurs at the mid-bottom point. Stress values within the arch show more noticeable differences among the studied variants, and the vertical arrangement appears to be more sensitive to time-dependent and thermal effects at midspan.

Figure 9.

Stress history (normal stress for the deck and mises stress for the arch) at 4 points for 4 representative variants.

Figure 10 shows typical NSR (normalized stress range) in the deck and arch rib with respect to total hanger length for arrangements with 44 hangers. From this figure, it appears that stress changes in the rib or tie are not very sensitive to hanger layout for network arches with mid-range hanger lengths, i.e., those which do not have very large or very small inclination angles. However, for variants with a total hanger length smaller than 300 m, a notable difference can be seen in NSR with radial arrangements (with most having smaller NSRs in the deck) and VAT arrangements (with most having larger NSRs in the rib). This includes radial arrangements with cross angles between 0 and 20°, and VAT arrangements with a small $∆ φ$ while having a large initial angle, such as variants VAT-80P00 and VAT-70N10. This shows a lower sensitivity to time-dependent and thermal effects considering stress variation for these radial arrangements and a higher sensitivity for the mentioned VAT arrangements.

Figure 10.

Changes in NSR as a function of hanger length in variants with 44 hangers.

Changes in NSR with respect to initial hanger inclination for VAT variants are shown in Figure 11. In the Mid-Bottom point of the deck, a significantly greater NSR is observed for VAT variants with a $∆ φ$ of 2° if the initial angle is less than 65°, and those with a $∆ φ$ of 3° if the initial angle is less than 75°. These conditions correspond to a smaller inclination angle for most hangers and a smaller total hanger length. In the Mid-Top and ¼-Bottom points, the same variants had a smaller NSR. In the arch rib, the difference in NSR was notable only for ¼-Bottom point, where variants with $∆ φ$ less than 1 degree show greater NSR.

Figure 11.

NSR for VAT variants with 44 hangers with respect to initial hanger inclination.

Another notable observation from Figure 10 is that compared to vertical tied arch bridges, typical network arches do not have a particular disadvantage when evaluating thermal and time-dependent effects. At the mid points of both the rib and deck, most network variants had a smaller NSR compared to the vertical hanger variant. While the vertical variant showed smaller NSR values at the bottom of the quarter point in the arch rib, the design is often governed by midspan forces and stresses and is therefore unlikely to be affected by greater stresses at the quarter point.

Significance of time-dependent deformations of concrete

To assess the significance of time-dependent deformations of concrete, the stress histories within the four example variants already used in Figure 9 were calculated based on 4 sets of assumptions: (1) considering thermal effects as well as concrete creep and shrinkage (TCS); (2) considering thermal effects and concrete creep (TC); (3) considering thermal effects and concrete shrinkage; (TS), (4) considering only thermal effects (T).

Figure 12(a) presents the typical ratios of maximum stresses based on the TC, TS and T sets of assumptions to those based on the TCS assumption for the deck and the rib, respectively for four example bridge models. From Figure 12(a), it appears that in network arches, neglecting concrete creep and shrinkage will lead to an underestimation of maximum longitudinal stress by about 15 to 20% at the Mid-Bottom and ¼-Top points of the deck. This error seems to be slightly smaller in arches with vertical hangers. However, neglecting creep and shrinkage lead to an overestimation of stresses in the arch, of approximately 5 to 10%.

Figure 12.

Maximum stress (longitudinal in the deck and mises in the arch) based on different sets of assumptions divided by maximum stress based on TCS for the representative arch bridge models.

Moreover, considering concrete creep but not shrinkage yields maximum stresses close to the case in which neither is considered. Neglecting creep while considering shrinkage deformations will cause an overestimation in the range of 2 to 10% at some points in network arches. This is likely due to the positive effect that long-term creep has on relieving stresses caused by concrete shrinkage. Overall, it appears that the effect of concrete shrinkage is more significant than concrete creep. In addition, maximum stress due to creep and shrinkage is not strongly dependent on hanger arrangement in network arches with inclined hangers.

Significance of thermal effects

To evaluate the significance of thermal effects, the stresses were also calculated once without thermal effects being considered, i.e., assuming shrinkage and creep are occurring at a constant temperature. Results based on this assumption are denoted as “CS” in this paper. The maximum stress range within the 8 cross sections studied were determined based on the CS assumption and compared with those under the TCS assumption. The results are presented in Figure 12(b).

Figure 12(b) shows that neglecting thermal effects causes an underestimation of typically 40 to 60% of the stress range in the deck and arch rib. Overall, all hanger arrangements investigated appear to show similar trends with respect to sensitivity to thermal effects. In other words, the sensitivity to thermal effects does not appear to be a function of the hanger layout.

Comparing the arrangements based on ANS and stresses in midspan hangers

Figure 13(a) shows the maximum ANS, which is an indicator of axial forces within the arch elements throughout the 1-year monitoring period as a function of total hanger length in different arches. Note that all arches represented in the figure had 44 hangers. As the figure shows, maximum ANS in the tie (deck) was relatively insensitive to the arrangement used for the hangers except for cases with highly inclined hangers, which corresponds to a large total hanger length. This observation is likely because the average normal stresses within the deck are primarily governed by the prestressing force.

Figure 13.

Changes in stress parameters as a function of hanger length in models with 44 hangers in each arch.

The maximum ANS in the arch rib sections shows a slightly ascending trend with the increase in the length of hangers, but for the most part, the maximum ANS shows less than 20% variability among different arches. Similar to the deck, large hanger lengths or more inclined hangers were associated with a more significant increase in ANS compared to others.

The maximum stress in hangers positioned at the middle of the arch is shown in Figure 13(c). As visible in the figure, arches with similar hanger lengths but different arrangements showed relatively similar maximum stresses in the hangers. Nevertheless, an increase in the hanger lengths, which corresponds to an increase in the inclination angle of hangers, resulted in an increase in the maximum hanger stresses with very similar ascending trend among different layouts. In other words, the maximum stress within hangers was more sensitive to the hanger inclination than to the hanger layout used. Based on Figure 13(a) and (c), it appears that using greater amounts of materials for hangers does not necessarily result in a more efficient network arch.

Comparing the arrangements based on NSV

The values of NSV for different arches with each of the four layouts are illustrated in Figure 13(b). The figure shows that for VNT and VAT arrangements, except for very large or very small hanger lengths, the values of NSV were not very sensitive to the hanger lengths or inclination. In arches with VAT hanger arrangements, to minimize NSV, it appears better to maintain a balance between initial hanger angle and $∆ φ$ so that the gradual angle variation does not lead to very small or very large inclination at the ends. This was also recommended by Tveit for a most desirable network arch performance (Tveit, 2014). Employing $∆ φ = 1$ seems to result in relatively good performance in all sections regarding NSV.

Figure 13(b) also shows that models with the radial hanger arrangement had considerably lower stress gradient, i.e., bending moments, in the arch compared to all other arrangements except the vertical model at the middle of the arch. The sensitivity of bending moments to hanger layouts and hanger lengths was relatively small in the deck, where stress variations in the middle and ¼ of the deck for radial models were similar to those for other hanger arrangements. This shows that designers need to be less concerned with bending moment variation when designing network arches with radial hangers.

A closer look at NSV as a function of cross angles for radial arrangements is provided in Figure 14(a), which shows that radial models with cross angles between 0 and 10° and those between 20 and 35° had the lowest values of deck NSV at the midspan and quarter points, respectively. For the arch rib at midspan, the lowest NSVs occurred for cross angles between 40 and 50° and at quarter points for cross angles between 10 and 15°. Overall, radial arrangement with cross angles between 20 and 35° seem to provide an optimal performance in terms of lower NSV under thermal and time-dependent effects.

Figure 14.

Normalized stress variation (NSV) as a function of hanger parameters in arches with radial arrangement.

A comparison between NSV values for radial models with the same cross angle (40°) but a varying number of hangers is provided in Figure 14(b), which shows insignificant differences between models with different number of hangers in all four sections. Therefore, the value of bending moment in the rib and the tie appears to be more dependent on the type of arrangement than on the number of hangers.

Comparing the arrangements based on hanger relaxation

The number of hangers that relaxed due to time-dependent or thermal effects through the monitoring period in different arrangements is shown in Figure 13(d). In some models, some hangers positioned at the ends of the arch and inclined inward repeatedly shift between tension and compression due to daily temperature variation, especially in the summer when the difference between maximum and minimum daily temperature was greater. The position of these relaxed hangers can be seen in Figure 15.

Figure 15.

Position of relaxed hangers in all arrangements.

From Figure 15 it can be observed that only some VNT and VAT hanger arrangements exhibit hanger relaxation due to time-dependent and thermal effects, and no relaxed hangers were detected in the radial and vertical arrangements. In many of these arrangements, some inward-facing hangers relax because of possessing excessive inclination angle. For the VAT models with larger $∆ φ$ , which had a similar configuration to the radial arrangement, no relaxed hangers were detected. In other models of this category, an average of 10 hangers relaxed, with the greatest number of relaxed hangers being 16. In VNT models, almost all models had about 6 to 8 relaxed hangers during the monitoring period.

As Figure 15 shows, in some arrangements, the relaxed hangers are not symmetric. In this figure, all hangers that relaxed even once are marked. Therefore, some hangers might have relaxed very few times while symmetric hangers on the other side have not, due to the slight difference in nodal temperatures along the bridge, which may cause a small difference in deformations.

Note that the hangers identified in Figure 15 relax only due to ambient and time-dependent effects. Cyclic relaxation of hangers, especially when occurring on a daily basis, is expected to be exacerbated by non-uniform loading on the bridge. This condition may severely impact the efficiency of the arch and cause fatigue concerns within hangers. Therefore, incorporating thermal and time-dependent effects reveal another advantage for the radial arrangement over other arrangements.

Conclusions

This study investigated the effect of ambient effects such as thermal changes and volumetric changes of concrete on the long-term behavior of steel network arch bridges with concrete decks. Different variations of a reference structure were developed and modeled using the Finite Element method, with varying arrangements for hangers. The effects of solar radiation, ambient temperature changes, heat transfer phenomena, and shrinkage and creep of the concrete deck were applied to the model based on validated procedures. The most significant conclusions of this study are as follows:

• Stress changes in the rib or tie are not very sensitive to hanger layout for network arches with mid-range hanger lengths, i.e., those which do not have very large or very small inclination angles.

• Compared to tied arch bridges with vertical hangers, typical network arches do not have a particular disadvantage in terms of sensitivity to thermal and time-dependent effects despite their high degree of static indeterminacy.

• The network arch bridges are much more sensitive to thermal effects than to time-dependent deformations of concrete. Neglecting concrete creep and shrinkage results in an approximately 15 to 20% underestimation of maximum stresses, whereas a 40 to 60% error in stresses may occur if thermal effects are neglected.

• In all VNT and some VAT arrangements, some hangers repeatedly relax under cyclic thermal changes. However, no hanger relaxation was observed in the radial or vertical arrangements. Cyclic hanger relaxation is considered detrimental to the efficiency of network arches, thus making the VNT and VAT arrangements less favorable.

• The lowest bending moments due to thermal and time-dependent effects were observed in the radial and vertical arrangements. However, the vertical arrangement is not considered efficient under nonuniform live loads because some hangers may relax and become ineffective.

Overall, based on the response to ambient changes and time-dependent effects, radial arrangements showed the best performance in terms of bending moments and number of relaxed hangers. Cross angles between 20 and 35° are recommended, as these angles tend to result in lower bending moments in most sections.

The above conclusions are based on the numerical analysis of one reference bridge. Further studies need to be conducted on the long-term behavior of network arch bridges with different geometries under thermal cycles through monitoring and testing. The impact of different material types should also be investigated to assess ambient effects, particularly regarding hanger relaxation and fatigue.

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 Babol Noshirvani University of Technology; BNUT/396026/00.

ORCID iDs

Yasmin Bagheriehnajjar

Hossein Yousefpour

Data availability statement

The data, models, and code that support the findings of this study are available from the corresponding author upon reasonable request.*

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