Integral abutment bridge response under thermal loading

Engineering Structures 32 (2010) 1495–1508

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Integral abutment bridge response under thermal loading WooSeok Kim a,∗ , Jeffrey A. Laman b,1 a

Parsons Brinckerhoff, One Penn Plaza, New York, NY, United States

b

Department of Civil and Environment Engineering, Penn State University, University Park, PA, United States

article

info

Article history: Received 7 May 2009 Received in revised form 6 January 2010 Accepted 6 January 2010 Available online 16 March 2010 Keywords: Integral abutment Bridge Nonlinear Numerical model Time-dependent

abstract This paper presents a parametric study and approximate integral abutment bridge (IAB) response prediction models. IABs are complex structures due to the nonlinearity and uncertainties in bridge materials and soil boundaries. Current design specifications and guides do not provide clearly defined analysis methods; therefore, there is a need for easily implemented preliminary analysis and accurate two-dimensional (2D) analysis methods. Based on a calibrated, nonlinear, 2D numerical modeling methodology including backfill–abutment interaction, soil–pile interaction and critical construction joints, a parametric study of 243 analysis cases was performed, consistent with the AASHTO 75-year bridge life. Analyses were performed under thermal load and gradient, and backfill pressure with time-dependent effects of concrete creep and shrinkage and prestressing steel relaxation included. The parametric study considered five parameters: (1) thermal expansion coefficient, (2) bridge length, (3) backfill height, (4) backfill stiffness, and (5) pile soil stiffness. Each of the five parameters was evaluated at three distinct magnitudes to cover the normal range of bridge construction. The parametric study revealed that the thermal expansion coefficient, bridge length and pile soil stiffness significantly influence the IAB response as measured by: (1) girder axial force, (2) girder bending moment, (3) pile lateral force, (4) pile bending moment, and (5) pile head/abutment displacement. The influences of backfill height and backfill stiffness are not relatively significant. The study results provide practical, preliminary estimates of the bridge response and ranges for preliminary IAB design and analysis. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Integral abutment bridges (IABs) undergo thermal movements and develop stresses due to daily and seasonal temperature changes. Bridge behavior and response is not easily predicted due to the complexity of nonlinear soil–structure interaction, timedependent effects, and uncertainties in temperature fluctuation. The parameters most influencing the IAB responses to thermal load must be established and investigated over commonly encountered ranges. This will then lead to reliable approximate IAB analysis methods that permit improved prediction of the superstructure axial force and moment, pile moment, and abutment rotation and displacement. A parametric study was performed to establish the prestressed concrete girder IAB response due to (1) superstructure temperature loads, (2) superstructure temperature gradient loads, and (3) concrete time-dependent loads. Simulating the AASHTO prescribed 75-year bridge life, the present study investigated the

∗

Corresponding author. Tel.: +1 212 631 3869. E-mail addresses: [email protected] (W. Kim), [email protected] (J.A. Laman). 1 Tel.: +1 814 863 0523; fax: +1 814 863 7304. 0141-0296/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2010.01.004

maximum and minimum long-term IAB responses of (1) the girder axial force, (2) the girder bending moment, (3) the pile lateral force, (4) the pile bending moment, and (5) the pile head/abutment displacement. The numerical modeling methods used in this study were validated and calibrated against IAB measurements at four Pennsylvania bridge sites [1] (see Fig. 1). Detailed numerical modeling methodologies and comparisons to field measurements are presented in [2,1]. The parametric study establishes the importance of each parameter and investigates the influence of key parameters on the bridge response. Parametric studies [3–6] with a limited number of parameters have been performed; however, the present study evaluates bridges with longer lengths and pile supports in clay or sand. Based on a preliminary analysis, five key parameters were selected to cover the range of common IAB geometries. The selected parameters include (1) the thermal expansion coefficient, (2) the span length, (3) the backfill height, (4) the backfill stiffness, and (5) the pile soil stiffness. As presented in Table 1, three values of each parameter have been selected, resulting in 243 study cases. As presented in Fig. 2, the present study adopted a common bridge superstructure cross-section that consists of four precast, prestressed concrete girders with concrete compressive strength fc0 = 55.2 MPa (8 ksi) and a cast-in-place concrete deck with fc0 = 27.6 MPa (4 ksi). A common IAB foundation configuration

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Table 1 Parameters and values definition for study. Thermal expansion coefficient, (α ) ×10−6 /°C (×10−6 /F°)

Total length, L (m (ft))

Backfill height, H (m (ft))

Backfill stiffness, B

Pile soil stiffness, P

5.4 (3.0) 9.9 (5.5) 14.4 (8.0)

18.3 (60) 61.0 (200) 121.9 (400)

3.0 (10) 4.6 (15) 6.1 (20)

High in Table 2 Intermediate in Table 2 Low in Table 2

High in Table 3 Intermediate in Table 3 Low in Table 3

3 × 3 × 3 × 3 × 3 = 243 cases.

Fig. 1. 2D nominal numerical model.

Fig. 2. Typical cross-section and bridge dimensions for parametric study.

[7] was adopted consisting of a single row of 11 steel HP piles supporting a cast-in-place wall abutment with fc0 = 20.7 MPa (3 ksi) supporting girders and a backwall with fc0 = 27.6 MPa (3.5 ksi). Each girder section was designed for the respective bridge spans in accordance with [8]. The pile shear forces and moments reported in this study are due entirely to thermal loading. Responses due to traffic or pile driving are not evaluated here. In addition, in the evaluation of the maximum axial force due to backfill pressure, fill in front of the abutments has not been considered. 2. Numerical modeling 2D numerical models were employed for the present study to simulate the AASHTO 75-year design period. Because IAB response such as concrete creep and shrinkage and soil yielding has been observed to be unrecoverable over time, the numerical models perform a time-history simulation of the AASHTO 75-year

design period. The analysis includes the time-dependent effects of concrete creep and shrinkage and prestressing steel relaxation over the 75-year time period. Maximum and minimum responses during the 75 years were identified to determine prediction models for maximum and minimum response. The 2D IAB model based on Kim [2,1] was validated through an extensive field monitoring program and calibration effort. Bridge members are modeled using beam elements based on the cross-sectional properties as shown in Fig. 1. The 2D numerical model includes soil–structure interaction and the abutment-to-backwall construction joint properties. Soil–pile interaction incorporates force–displacement (p–y) curves based on [9]. For backfill–abutment interaction, passive and active Rankine’s theory was adopted to represent the passive and active pressure. Thomson and Lutenegger [10] reported that the wing-wall orientation to the main abutment wall has an influence on the passive earth pressure. However, 2D numerical models without wing-walls have been shown to simulate IAB behavior as compared to measured response [1,3–6]. Soil–pile interaction is modeled using a condensed pile model with

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Fig. 3. Typical bridge construction timeline.

a lateral, nonlinear spring and a rotational spring. The definition of the force–displacement and moment–rotation relationship for the pile was established using LPILE, [26] a commercially available and widely used software package. The typically low reinforcement construction joint between the backwall and the abutment (see Fig. 1) was modeled using joint elements based on moment–curvature properties [11,2,12]. 3. Parameter selection and discussion The selection of critical IAB parameters and parameter magnitude for the present study has been based on previously conducted studies and observed IAB behavior [13,14,12]. The magnitude of the thermal expansion coefficient and the bridge length are the parameters most affecting IAB performance according to [8] and the AASHTO Guide Specifications [15]; therefore, both of these parameters are included in the parametric study. Pile soil stiffness is identified as an important factor in determining IAB performance according to Dicleli and Albhaisi [7] and Fennema et al. [16]. Also, studies by Civjan et al. [3], Dicleli and Albhaisi [7] and Faraji et al. [5] observed that the backfill stiffness and backfill height have a significant effect on IAB performance; therefore, these parameters are also included. Backfill stiffness and pile behavior are directly related to the soil density (γ ), friction angle (φf ), undrained shear strength (cu ), and other engineering soil properties. Therefore, backfill soil properties and soil layers surrounding HP piles are also selected as key parameters in the parametric study. 3.1. Bridge construction timeline The bridge construction timeline influences the IAB behavior due to construction temperature and time-dependent effects inherent in the concrete deck, backwall, and superstructure. Prior to completion of construction, all structural elements are substantially free to undergo creep and shrinkage with no significant effect on the substructure. After the concrete backwall, girders and deck become structurally integrated, the bridge superstructure and substructure become a unit, developing a complex frame action resisting gravity loads as well as horizontal thermal load and backfill pressure. The bridge construction timeline adopted for this study is presented in Fig. 3. Deck concrete placement typically lags girder manufacture by 60 to 180 days. Backwall placement typically lags deck placement by 10 to 100 days. Deck and backwall placement are generally the last components completed; therefore, the parametric study assumes that the deck concrete is placed 100 days after girder manufacture and that the backwall is placed 20 days after the deck placement. Construction schedules other than Fig. 3, i.e. different construction temperature influences, are discussed in the following section.

Fig. 4. Backfill height parameter.

available, [8] recommends α equal to 10.8 × 10−6 /°C (6.0 × 10−6 /°F) with a range of 5.4 to 14.4 × 10−6 /°C (3 to 8 × 10−6 /°F). Nilson [17] reported that α generally ranges from 7.2 to 12.6 × 10−6 /°C (4 to 7 × 10−6 /°F). Based on over 200 sources, Oesterle et al. [18] reported a mean α of 8.77 × 10−6 /°C (4.87 × 10−6 /°F) and a range of 4.1 to 14.6 × 10−6 /°C (2.3 to 8.1 × 10−6 /°F). As presented in Table 1, the present study considered the minimum, mean and maximum α as 5.4, 9.9 and 14.4 × 10−6 /°C (3.0, 5.5 and 8.0 × 10−6 /°F), as representative of expected concrete thermal expansion coefficient values. 3.3. Bridge length and backfill height The two bridge dimensional parameters considered in the parametric study are the bridge length, L, and abutment backfill height, H. Bridge lengths were selected to represent short to medium-long bridges. Backfill heights were selected to represent stub type to medium-high abutments. The bridge lengths considered are 18.3, 61.0 and 121.9 m (60, 200 and 400 ft). As presented in Fig. 4, H represents the total backfill height from the bottom of the abutment/pile cap to the bottom of the approach slab. The roadway elevation was maintained for each IAB while the abutment below the girder seat height (h2 in Fig. 4) was varied due to different girder heights (h1 in Fig. 4) for different span length bridges. As a result of this convention, the construction joint between backwall and abutment was located at different elevations as the span length changed. 3.4. Soil boundaries

3.2. Thermal expansion coefficient

The properties of the abutment backfill and soil surrounding piles were also evaluated as parameters in the parametric study. Soil–structure interaction depends on the soil properties and is a factor that determines the IAB response. Soil–structure interaction modeling is, however, problematic for a parametric study because the soil response is dependent on many parameters, including soil density, friction angle and undrained shear strength. The abutment backfill parameter magnitudes were established based on the soil density and friction angle. For soil surrounding piles, high and low soil stiffness based on the soil density, friction angle and undrained shear strength were established irrespective of cohesive and/or cohesionless materials.

The thermal expansion coefficient, α , determines the superstructure strain in response to concrete temperature changes. Estimates of the concrete thermal expansion coefficient vary significantly due to variations in concrete mix properties, aggregate properties and proportions, water to cement ratio, relative humidity, age of concrete, and other factors. When test data is not

3.4.1. Abutment backfill Typical abutment backfill soil properties (B) were investigated for low, intermediate, and high level stiffness [19,25,20,14,21] as presented in Table 2. Established magnitudes of backfill soil density and friction angle matched well with field-derived properties [13, 14,21]. To determine the subgrade modulus, field measurements

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(a) Bridge 109.

(b) Bridge 203.

(c) Bridge 211. Fig. 5. Backfill pressure versus displacement of abutment. Table 2 Backfill properties and range determination. Property

Intermediate

Density, γ (kN/m (pcf)) Angle of friction, φf (degree) Subgrade modulus, Kh (MN/m3 (pci)) 3

18.7 (119) 34 12 (43.8)

Table 3 Soil layer properties and range determination. High 19.3 (123) 37.4 18 (65)

Low

Property

18.2 (116) 30.6 6 (22.5)

Sand density (kN/m (pcf)) Clay density (kN/m3 (pcf)) Angle of friction (sand) (degree) Undrained shear strength (clay) (kN/m2 (psi)) Elastic modulus, K (MN/m3 (pci)) ε50 (mm (in))

[14,21] were analyzed using a backfill pressure to abutment displacement ratio at a given depth. As presented in Fig. 5, each ratio, or slope, of backfill pressure to abutment displacement maintains a constant slope for all three field-evaluated bridges. The approximated slope of 12.0 MN/m3 (43.8 pci) was obtained as a representative value, which is the same as the subgrade modulus from the literature in Table 2. Based on the field measurement investigation and the published literature, backfill properties were adopted as presented in Table 2 low, intermediate, and high stiffness for the parametric study. 3.4.2. Soils surrounding piles The present study established typical soil properties for the determination of soil–pile interaction stiffness magnitudes in the parametric study. Eleven HP 310 × 110 (HP 12 × 74) weak axis oriented piles with a yield strength of 345 MPa (50 ksi) were taken as a standard configuration for the parametric study. Piles were taken as driven more than 7.6 m (25 ft) below the bottom of the abutment as a consequence of previous analytical and experimental studies [22,5,6] reporting that the depth of pile fixity points is generally between 3.0 and 4.6 m (10 to 15 ft) from the pile head. The soil layer at the top of the pile is taken as a 6.1 m (20 ft) deep clay or sand layer. The second soil layer material depth and properties do not significantly influence the pile behavior and the depth is taken as 1.5 m (5 ft) to reach bedrock. Three soil stiffnesses have been considered for each of three overburden conditions (backfill heights: 3.1, 4.6 and 6.1 m). Soil properties representing clay and sand surrounding piles were

3

Intermediate

High

Low

19 (121) 19 (121) 35

22 (142) 22 (142) 42

16 (100) 16 (100) 28

121 (17.5)

193 (28)

48 (7)

271 (1000)

353 (1300)

190 (700)

0.20 (0.008)

0.13 (0.005)

0.25 (0.01)

selected at high, intermediate and low stiffness, as presented in Table 3. The soil–pile lateral and rotational stiffness for translational and rotational nonlinear spring definition was derived from a moment–rotation curve with fixed pile head lateral displacement and force–displacement curve with fixed pile head rotation from LPILE results, as presented in Fig. 6. The high and low magnitudes, regardless of cohesive or cohesionless soil materials, were adopted for the high and low stiffness numerical model definition. 4. Loads The superstructure thermal load, thermal gradient, concrete time-dependent loads, and backfill pressure were considered in the 2D numerical analysis for the parametric study. The bridge thermal load is affected by the ambient temperature, solar radiation, wind speed and direction, precipitation, and humidity. Thermal conductivities and structure mass also affect the bridge temperature. The thermal gradient across the superstructure also significantly influences the IAB response [6]. The present study utilized a linear temperature profile equivalent to the AASHTO temperature profiles to facilitate numerical modeling. To include time-dependent effects, concrete creep and shrinkage effects and prestressing steel relaxation were considered in the numerical

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(a) Moment–rotation (H = 3.0 m).

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(b) Force–displacement (H = 3.0 m).

(c) Moment–rotation (H = 4.6 m).

(d) Force–displacement (H = 4.6 m).

(e) Moment–rotation (H = 6.1 m).

(f) Force-displacement (H = 6.1 m). Fig. 6. Rotational and lateral pile capacity as derived from LPILE.

analysis based on [23] and the age-adjusted elastic modulus method (AAEM). 4.1. Temperature fluctuation In the present study, the thermal loading was determined by a sinusoidal temperature fluctuation range based on actual field data in Pennsylvania collected from 2002 through 2008, which is less than the AASHTO design temperature range. Eq. (1) presents the superstructure temperature, T (t ), at year t, over a 75-year bridge life as T (t ) = Tm + A sin (ωt + φ) ,

t = 1 to 75 years

(1)

where Tm = annual mean temperature (7.5 °C, 45.5 °F), A = annual temperature amplitude (16.7 °C, 30 °F), ω = frequency (2π ), t = analysis time (year) and φ = phase lag (−π ). The simulation included the simultaneous effects of temperature fluctuation and time-dependent effects at 7-day time steps up to the AASHTO 75year design life.

To investigate the influence of initial temperature at construction completion when the backwall is placed and integral behavior initiates, three cases were compared for the 75-year simulation presented in Fig. 7: (1) case 1, during spring = 7.5 °C (45.5 °F); (2) case 2, during summer = 24.2 °C (75.5 °F); and (3) case 3, during fall = 7.5 °C (45.5 °F). An initial construction completion temperature during winter was not considered because bridge specifications require concrete cure at a minimum concrete temperature of 10 °C (50 °F) and completion of bridge construction in January or February is unusual. The initial temperature at completion of construction significantly influences both the initial and long-term responses. In addition, the initial abutment displacement difference between cases 1 and 2 or cases 3 and 2 due to the difference in construction temperature is maintained over the bridge life. The difference in abutment displacement between cases 1 and 2 or cases 3 and 2 at 75 years is similar to the initial abutment displacement difference. Based on the numerical simulation, the higher the construction temperature, the larger the

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Fig. 7. Displacement shifting due to construction temperature. Table 4 Temperature gradient. Girder section

Temperature gradient

Temperature °C (°F) Top fiber

Bottom fiber

AASHTO III (18.3 m (60 ft))

Positive (ATG_P ) Negative (ATG_N )

13.1 (23.6) −5.9 (−10.6)

−1.2 (−2.1)

BT-72 (61.0 m (200 ft) and 121.9 m (400 ft))

Positive (ATG_P ) Negative (ATG_N )

−4.0 (−7.3)

8.3 (14.9)

1.8 (3.3)

−1.1 (−1.9) −1.1 (−2.0)

TTG (t ) = 0.5ATG sin(ωt + φ), if 0.25 ≤ fraction of t ≤ 0.75, then ATG = ATG_P , else ATG = ATG_N .

abutment displacement the bridge experiences during bridge life because a higher construction temperature means a larger temperature decrease. 4.2. Thermal gradient A superstructure temperature gradient was also included in the numerical model, recognizing that the temperature in the superstructure is not uniform. The deck and girder are irregularly shaped sections; therefore, the thermal transfer and the resulting temperature profile are both non-uniform and nonlinear. Based on the [8] multi-linearly varying temperature profile, an equivalent linearly varying temperature gradient was created to develop equivalent axial and bending strains in the superstructure. The temperature gradient values, ATG , at the top and bottom fiber were calculated for different girder sections and are presented in Table 4. The temperature gradient at t years, TTG (t ), was superimposed on the constant superstructure temperature and was applied throughout for the 75-year bridge life simulation. 4.3. Time-dependent load Time-dependent loads include concrete creep, concrete shrinkage, and prestressing steel relaxation. The AAEM [24] was utilized to compute the time-dependent effects as a time-varying concrete modulus based on the [23] creep coefficient and aging coefficient. The girder and deck section properties, pre-tensioning forces, elastic and time-dependent loss through time, dead load of girder and deck, and age of girders and deck at placement were considered in the concrete creep stress analysis. Stress loss due to

prestressing steel relaxation was computed using the [8] intrinsic relaxation with a reduced relaxation coefficient suggested by Ghali et al. [24]. Total strains from time-dependent effects are represented in the 2D numerical model as equivalent temperature loads based on the superstructure thermal expansion coefficient. Equivalent temperature for time-dependent strains, at the top and bottom fibers at time t, are presented in Fig. 8. Equivalent temperatures are computed at multiple sections to closely simulate the behavior along the span, including any strand drape. Each time-dependent strain is converted to an equivalent temperature change by dividing each by the thermal expansion coefficient. The total strains at the top and bottom fibers of each girder section and thermal expansion coefficient were computed based on an assumed deck concrete placement date of 100 days from girder manufacture. Fig. 8 presents the total strain at the top and bottom fibers of girders for span lengths of 18.3 and 61.0 m (60 and 200 ft). 5. Discussion of results A total of 243 parametric study cases of the 2D numerical model have been investigated considering the key parameters of (1) thermal expansion coefficient, (2) span length, (3) backfill height, (4) backfill stiffness, and (5) pile soil stiffness. The critical responses of (1) the girder axial force, (2) the girder bending moment, (3) the pile lateral force, (4) the pile moment and (5) the pile head/abutment displacement were determined and compiled. The girder average axial force and average maximum (or positive) and minimum (or negative) moment with respect to each of the five key parameters were computed to investigate the thermally

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(a) Top fiber (L = 18.3 m).

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(b) Bottom fiber (L = 18.3 m).

(c) Top fiber (L = 30.5 m).

(d) Bottom fiber (L = 30.5 m). Fig. 8. Time-dependent strain along bridge girder.

induced axial tension and compression and the maximum and minimum bending response. The envelope of both the maximum and the minimum response indicates the response limits for the girder axial force, girder bending moment, pile moment and pile lateral force. Both the average maximum responses and the response envelopes for the pile lateral force, pile moment and pile head/abutment displacement were computed and presented. 5.1. Girder axial force and moment Both the girder axial force and the girder central moment between an abutment and an adjacent pier or abutment are significantly influenced by the thermal expansion coefficient and the bridge length, but are insignificantly influenced by the backfill height and pile soil stiffness. As the bridge length increases, both the girder axial force and the girder central moment increase (Fig. 9(a) and (b)). The influence is most apparent for the girder axial force. The study case with α = 14.4 × 10−6 /°C (8.0 × 10−6 /°F), L = 121.9 m (400 ft), H = 3.0 m (10 ft), B = high value, and P = high value produced both the maximum tension and compression girder axial forces of 883 kN/girder (199 kips/girder) and 2189 kN/girder (492 kips/girder), respectively. These forces translate to a tensile and compressive stress of 0.862 MPa (0.125 ksi) and 2.136 MPa (0.31 ksi) or 23.3% and 8.6% of the [8] service stress limits, respectively. The bridge length also significantly influences the negative girder central moment, which increases as the bridge length increases. However, the positive bending moment was not significantly influenced by changes in the bridge length throughout the parametric study cases. Short bridges and stiff pile soils tend to produce higher positive bending moments. The maximum negative moment is 2213 kN m/girder (1632 kip ft/girder) (α = 14.4 × 10−6 /°C (8.0 × 10−6 /°F), L = 121.9 m (400 ft), H = 3.0 m (10 ft), B = high value, and P = high value), which translates to 0.347 MPa (0.050 ksi) compressive stress (1.4% of the AASHTO LRFD service stress limit) at the bottom fiber and 0.242 MPa (0.035 ksi) tensile stress (6.5% of the AASHTO limit) at the top fiber. The maximum positive moment is 184 kN m/girder

(136 kip ft/girder) (α = 5.4 × 10−6 /°C (3.0 × 10−6 /°F), L = 18.3 m (60 ft), H = 6.1 m (20 ft), B = high value, and P = low value), which translates to 0.045 MPa (0.007 ksi) (1.2% of the AASHTO limit) tensile stress at the bottom fiber and 0.036 MPa (0.005ksi) (0.14% of the AASHTO limit) compressive stress at the top fiber. The girder axial force and girder central moment are inversely related to the backfill height (see Fig. 9(c) and (d)). A higher backfill results in a lower positive girder axial force. However, the girder central moment is not sensitive to the backfill height variation. The overall influence of backfill height was not as significant as that of the bridge length, and the stress ranged from 0.0 to 0.321 MPa (0.0 to 0.047 ksi). However, as α , L, and P decrease and A increases, the girder moment due to the backfill height increases. The girder axial force and girder central moment increase as the pile soil stiffness increases (see Fig. 9(e) and (f)). Most notably, the positive axial force and both positive and negative moments increased as the pile soil stiffness increased. However, the average negative axial forces remained constant throughout the parametric study cases. The girder end moments at an abutment are higher than the girder central moment between an abutment and an adjacent pier or opposite abutment (see Fig. 9(g), (h), (i)). The study case with α = 14.4 × 10−6 /°C (α = 8.0 × 10−6 /°F), L = 121.9 m (400 ft), H = 3.0 m (10 ft), B = high value, and P = high value produced a maximum positive moment of 2189 kN m/girder (1615 kip ft/girder), which translates to 0.343 MPa (0.050 ksi) tensile stress (9.3% of the AASHTO limit) at the bottom fiber and 0.239 MPa (0.035 ksi) compressive stress (1.0% of the AASHTO limit) at the top fiber. The study case with α = 14.4×10−6 /°C (α = 8.0×10−6 /°F ), L = 121.9 m (400 ft), H = 6.1 m (20 ft), B = high value, and P = low value produced the maximum negative moment of 4387 kN m/girder (3235 kip ft/girder), which translates to 0.687 MPa (0.100 ksi) compressive stress (2.8% of the AASHTO limit) at the bottom fiber and 0.479 MPa (0.069 ksi) tensile stress (12.9% of the AASHTO limit). The maximum combined stresses of girder axial forces and girder moments are presented in Table 5. The study case with α = 14.4 × 10−6 /°C (α = 8.0 × 10−6 /°F ), L = 121.9 m (400 ft),

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Fig. 9. Bridge responses.

H = 3.0 m (10 ft), B = high value, and P = high value produced the maximum tensile stress of 0.861 MPa (0.125 ksi) (23.2% of the AASHTO limit) at the mid-span top fiber and 1.205 MPa (0.175 ksi) (32.5% of the AASHTO limit) at the end-span bottom fiber. The study case with α = 14.4 × 10−6 /°C (α = 8.0 × 10−6 /°F), L = 121.9 m (400 ft), H = 6.1 m (20 ft), B = high value, and P = low value produced a maximum compressive stress of 2.226 MPa (0.323 ksi) (9.0% of the AASHTO limit) at the top fiber of the

mid-span and 2.823 MPa (0.409 ksi) (11.4% of the AASHTO limit) at the bottom fiber of the end-span. 5.2. Pile lateral force and moment Both the pile lateral force and moment are significantly influenced by the thermal expansion coefficient, bridge length, and pile soil stiffness. The bridge length significantly influences the pile

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Fig. 9. (continued)

Table 5 Maximum combined stresses of the axial force and bending moment. Location

Fiber

Compressive stress kPa (ksi) Stress

Tensile stress kPa (ksi) Stress

AASHTO limit (%)

Girder center

Top Bottom

2.23 (0.323) 2.43 (0.353)

9.0 9.8

0.86 (0.125) 0.86 (0.124)

23.2 23.1

Girder end

Top Bottom

1.95 (0.283) 2.82 (0.409)

7.9 11.4

0.62 (0.090) 1.21 (0.175)

16.7 32.5

lateral force and pile moment at the pile head (see Fig. 10(a) and (b)). The study case with α = 14.4 × 10−6 /°C (8.0 × 10−6 /°F), L = 121.9 m (400 ft), H = 3.0 m (10 ft), B = low value, and P = high value produced the largest pile shear force of 425 kN (96 kips). The study cases with α = 14.4 × 10−6 /°C (8.0 × 10−6 /°F), L = 121.9 m (400 ft), and P = high value reached the yield moment of the pile, which is equal to 172 kN m (127 kip ft). The HP 310×110 (HP 12 × 74) with Fy = 0.345 MPa (50 ksi), approximately 8.8% shear capacity and 100% of the pile section bending capacity was consumed. The backfill height and backfill stiffness do not significantly influence the abutment displacement or backfill pressure, as shown in Fig. 10(c) and (d). However, a short abutment results in a higher pile lateral force than a tall abutment when the pile soil stiffness is high. The pile soil stiffness significantly influences the pile lateral force and pile moment, as presented in Fig. 10(e) and (f). When the

AASHTO limit (%)

pile soil stiffness is high, the maximum pile lateral force and pile moment are 15 and 4 times larger than the minimum, respectively. 5.3. Pile head/abutment displacement The pile head/abutment displacement is primarily influenced by the magnitude of the thermal expansion coefficient, bridge length and pile soil stiffness, as observed from Fig. 11(a) to (c). While higher pile soil stiffness reduces the displacement, a higher backfill height results in a larger displacement when the pile soil stiffness is lower, as observed from Fig. 11(b) and (c). The case with α = 14.4 × 10−6 /°C (8.0 × 10−6 /°F), L = 121.9 m (400 ft), H = 6.1 m (20 ft), B = low value, and P = low value produced the extreme displacement of 37 mm (1.45 in). The study results were compared to the free expansion of the superstructure at the pile head and centroid of the superstructure (Fig. 12). The influences of the bridge length and thermal expansion coefficient effects were clearly observed.

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Fig. 10. Pile responses.

The pile head displacement in Fig. 12(a) is highly dependent on the bridge length, thermal expansion coefficient and pile soil stiffness. However, the displacement at the centroid of the superstructure in Fig. 12(b) is primarily dependent on the span length and thermal expansion coefficient. Also, the time-dependent effects that are not considered for free expansion significantly influenced the displacements at the centroid so that the displacements in all parametric study cases are larger than the free expansion displacements.

6. IAB response prediction model Based on the parametric study results, approximate IAB response prediction models for (1) the bridge total axial force, (2) the bridge total bending moment, (3) the total pile lateral force, (4) the total pile moment, and (5) the pile head/abutment displacement were developed using regression analyses and are presented in Tables 6 and 7. The coefficient of determination, R2 , was used to evaluate the model accuracy. Several equation forms, linear and

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Fig. 11. Pile head displacement.

(a) At the pile head.

(b) At the centroid of the superstructure. Fig. 12. Abutment displacement.

nonlinear, were considered. In the development process a p-value test based on 95% acceptance was performed to attempt to reduce the number of parameters by identifying insignificant parameters. To improve the accuracy of the approximate prediction models, certain responses were separated into three equations based on the backfill height or bridge length. As presented in Tables 6 and 7, each of the prediction models exhibits a high R2 as compared to the numerical model results. Fig. 13 presents the numerical simulation data and provides a comparison between the prediction models

and the parametric study results. Based on the IAB response prediction models, each member response can be computed using the total response divided by the expected number of members (number of girders or number of piles). 7. Summary and conclusions The present study developed a numerical modeling methodology for long-term simulation of IABs and demonstrated characterization methods for the key parameters. The numerical

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Table 6 IAB response prediction models. Prediction model

Range

R2

31α + 16L − 491H + 453P + 165 ≥ 0 −29α 0.65 L0.75 B0.15 P 0.25 −39α 0.75 L0.60 B0.30 P −0.05 −149α 0.60 L0.45 B0.35 P −0.20

for H ≤ 3.8 m for 3.8 ≤ H ≤ 5.4 m for H ≥ 5.4 m

0.90 0.94 0.92 0.95

93α − 850H + 940P − 19 ≥ 0 180α + 160H + 330P − 4700 ≥ 0 45α + 1000P − 3500 ≥ 0 −162α 0.35 L0.60 P 0.20 −460α 0.40 L0.35 −4950α 0.1 P −0.40

for L ≤ 39.7 m for 39.7 ≤ L ≤ 91.5 m for L ≥ 91.5 m for H ≤ 3.8 m for 3.8 ≤ H ≤ 5.4 m for H ≥ 5.4 m

0.90 0.74 0.75 0.87 0.80 0.56

100α + 50L + 1400P − 4500 ≥ 0 89α + 58L + 1500P − 5400 ≥ 0 160α + 61L + 2100P − 9200 ≥ 0 −38α 0.70 L0.80 B0.10 P 0.20 −43α 1.00 L0.65 B0.10 P −0.08 −900α 0.55 L0.25 B0.04 P −0.70

for H ≤ 3.8 m for 3.8 ≤ H ≤ 5.4 m for H ≥ 5.4 m for H ≤ 3.8 m for 3.8 ≤ H ≤ 5.4 m for H ≥ 5.4 m

0.84 0.73 0.64

Total pile lateral force (kN)

15α 0.5 L0.7 B−0.1 P 0.9 ≤ Np Fy_pile 216α 0.2 L0.3 B−0.1 P 0.7 ≤ Np Fy_pile 1720L−0.1 B−0.1 P 0.6 ≤ Np Fy_pile

for H ≤ 3.8 m for 3.8 ≤ H ≤ 5.4 m for H ≥ 5.4 m

0.96 0.89 0.85

Total pile head moment (kN m)

33α 0.5 L0.5 H −0.1 P 0.6 ≤ Np My_pile

Pile head displacement (mm)

0.04α 0.3 L1.2 P −0.05 0.03α 0.7 L1.0 P −0.15 5.3α 0.3 L0.1 P −1.0

Abutment displacement at centroid

0.12α 0.3 L1.0 H 0.04 P −0.03

Response (+) Bridge axial force (kN)

(−)

(+) Bridge bending moment at girder center (kN m) (−)

(+) Bridge bending moment at girder end (kN m) (−)

0.91

0.83 for H ≤ 3.8 m for 3.8 ≤ H ≤ 5.4 m for H ≥ 5.4 m

0.99 0.89 0.56 0.99

Note 1: (+): maximum axial force or positive moment; (−): minimum axial force or negative moment. Note 2: α : (thermal expansion coefficient) ×106 /°C; L and H: meter; B and P: low = 1, intermediate = 2, high = 3 in Tables 2 and 3. Fy_pile : pile shear capacity; My_pile : pile moment capacity; Np : number of piles. Note 3: Single member response = (total response)/(number of members).

Table 7 IAB stress response prediction models. Response Maximum stress (kPa)

Location Girder center Girder end

Minimum stress (kPa)

Girder center Girder end

R2

Fiber

Prediction model

Top Bottomb Bottoma

−183 + 13α − 0.2α + 9L − 0.03L − 152H + 2H + 22B − 4B + 222P − 24P −332 + 16α − 0.2α 2 + 10L − 0.04L2 − 149H + 0.7H 2 + 23B − 4B2 + 275P − 31P 2 −623 + 21α − 0.4α 2 + 14L − 0.05L2 − 89H − 8H 2 + 33B − 7B2 + 352P − 42P 2

0.96 0.96 0.96

Top Bottomb Bottoma

−23α 0.6 L0.5 B0.2 −42α 0.6 L0.45 H 0.04 B0.2 P −0.02 −31α 0.7 L0.5 B0.2

0.84 0.78 0.79

2

2

2

2

2

Note 1: (+): maximum or tensile stress; (−): minimum or compressive stress. Note 2: α : (thermal expansion coefficient) ×106 /°C; L and H: m; B and P: low = 1, intermediate = 2, high = 3 in Tables 2 and 3. a Maximum or minimum stress throughout the girder. b Maximum or minimum stress at the mid-span of the exterior span.

model included key features of IABs: backfill–abutment interaction, pile–soil interaction, construction joint, temperature variation, temperature gradient, and time-dependent effects. Based on this numerical method, a 243-case parametric study with five parameters was performed. The key parameters evaluated are (1) the thermal expansion coefficient, (2) the bridge length, (3) the backfill height, (4) the backfill stiffness, and (5) the pile soil stiffness. Based on the results, a regression analysis was performed for predictions of (1) the bridge total axial force, (2) the bridge total bending moment, (3) the total pile lateral force, (4) the total pile moment, and (5) the pile head/abutment displacement. The parametric study using a 75-year bridge life revealed the following conclusions and recommendations. 1. The magnitude of the thermal expansion coefficient significantly influences the girder axial force, girder moment, pile lateral force, pile moment and pile head/abutment displacement. 2. The bridge length significantly influences the girder axial force, pile lateral force, pile moment and pile head displacement. The influence of the bridge length on the girder positive moment at mid-span is relatively weak.

3. The backfill height and backfill stiffness parameters are relatively insignificant influences on the bridge responses studied. 4. An increase in pile soil stiffness increases the girder moment, pile lateral force and pile moment and reduces the pile head displacement. The pile lateral force, even in the extreme case, consumes only 8.8% of the pile shear capacity; however, the restraint of stiffer pile soil stiffness causes the piles to reach the yield moment due to a large abutment rotation. 5. During a 75-year bridge life, the combined stress of girder axial force and girder moment at the mid-span can reach 23.1% of the AASHTO limit for tensile stress at the bottom fiber and 9.0% of the AASHTO limit for compressive stress at the top fiber. At the end-span, the combined stress can reach 11.4% of the AASHTO limit for compressive stress at the bottom fiber and 16.7% of the AASHTO limit for tensile stress at the top fiber. The inclusion of the effect of thermal stresses in the design of prestressed girders is justified and recommended. 6. The developed IAB response prediction models are accurate and rapidly provide critical bridge responses. Therefore, the prediction models can be utilized in a preliminary IAB design within the scope of this study.

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(a) Maximum girder axial force.

(b) Minimum girder axial force.

(c) Maximum girder moment at the girder end adjacent to the abutment.

(d) Minimum girder moment at the girder end adjacent to the abutment.

(e) Maximum girder moment at the girder mid-span in the end span.

(f) Minimum girder moment at the girder mid-span in the end span.

(g) Pile lateral force.

(h) Pile moment.

Fig. 13. Relationship between the approximate analysis and the 2D model prediction.

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(i) Pile head displacement. Fig. 13. (continued)

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