1 Introduction

Integral abutment bridges (IABs) are a form of construction whose superstructures are directly linked to the substructure. They have been used for more than 90 years in many countries and under different seismic conditions (Naji et al. 2020). IABs were introduced in the 1930s in the United States and expanded during the 1960s in many parts of the world (Mahjoubi and Maleki 2020; Dicleli 2016; Khosravikia et al. 2018). As reported by Barghian et al. (2020), the North American Study Tour Report presented several types of abutments: embedded-wall abutments, the full-height frame, spread footings on reinforced earth walls, stub abutments and vertical walls with semi-integral. A bridge receives lateral force due to seismic load and difference in temperature, which affects contraction-related and bridge expansion to soil–structure interaction (SSI) and a variety of materials. In a typical single-span bridge, the substructure and superstructure are integrally constructed, without expansion joints as a rigid connection. This connection transfers most loads from the superstructure to abutments and pile foundations. IABs use relatively flexible pile foundations such as a single pile foundation row to handle superstructure expansion and contraction due to laterally induced loads, backfill strain, creep, and shrinkage. However, as soil piles, abutments, and superstructure act as a combined framework to withstand the loading on the bridge, and the SSI properties such as soil stiffness has a significant influence on load distribution (Naji et al. 2020).

Expansion joints and moving bearings at the ends of the deck are replaced by control joints situated at the end of the approach slab, where the structure is not adversely affected by joint leakage. When the foundation has greater flexibility and less resistance to longitudinal displacements, stress from longitudinal forces can be minimized.

IABs have many advantages over conventional bridges. One important benefit of this type of bridge is the removal of expansion joints and bearings (Arsoy et al. 1999; Frosch et al. 2005; Abdel-Fattah and Abdel-Fattah 2019). The most troublesome aspect of bridges is expansion joints, creating numerous serviceability and structural issues. Furthermore, deck length will increase or decrease due to seasonal temperature changes. Since integral bridges are of a continuous unit system, stresses are transferred from the bridge to the soil under the expansion and contraction caused by temperature changes. The amount of force generated in the backfill depends on the expansion and contraction of the bridge. The increase/decrease amount of the deck length is directly related with backfill specifications. Therefore, the reduction or expansion of the bridge and the force created in the backfill are in interact with each other (Perić et al. 2016). SSI at railway bridges with integral abutments studied by Bigelow et al. (2017). They proposed that there are two advantageous aspects of selecting bridges with integral abutments in general. Firstly, there is an economic factor, because, during their life span, these bridges often tend to be less costly to build, simpler to maintain, and more economical to own. The second key factor concerns the efficiency of the framework. IABs with rigidity properties, which produce load-carrying behavior such as structural frames, have a reduced field moment when subjected to vertical loads, smaller deflections in the mid-span, and smaller abutment rotation angles than traditional bridges with bearings and joints (Bigelow et al. 2017; Tam and Le 2019).

The seismic risk assessment of the bridge systems within the road network is considered to be an important tool for the proper maintenance of the entire network, as well as for the effective mitigation of the direct and indirect socioeconomic losses that may arise as a result of a major earthquake. The most important components of such a highway system are basically bridges since the related uncertainty in the prediction of their fragility directly spreads to the level of the network. Therefore, any simplification of the interaction between earthquake excitation and the dynamic properties of the soil–bridge system in the reliability chain of major the interaction can affect the assessment of the seismic risk of the network (Lesgidis et al. 2017). The numerical simulation of SSI at each pier foundation and abutment-embankment support of the bridge is one of the main elements in the aforementioned reliability chain. It is well known that the nature of SSI is inherently frequency-dependent and, as such, the frequency content of the incoming seismic motion at each bridge support is influenced by the complex dynamic impedance matrix, especially as soil responds inelastically (Saitoh 2007; Pal and Baidya 2018; Makris et al. 1994). Within the context of seismic risk assessment of bridges for both long structures and bridges, various SSI simulation approaches have been implemented. The simulation of the entire soil domain-structure system by a finite element model (FEM) will provide a consistent representation and justification for the semi-infinite soil's dynamic properties and failure mechanisms, allowing the entire system to be evaluated (Sextos et al. 2003).

Field evidence from past earthquakes suggests that the dynamic response can be changed by SSI effects and thus affect the seismic output of bridges. While the effects of SSI have been a subject of interest for researchers, and several detailed solutions are currently available, there is still uncertainty as to the impact of the SSI on the seismic response of bridges, as presented by the conflicting results of numerous research studies (Stefanidou et al. 2017). Collaboration of soil–bridge systems is essentially a case-dependent, multi-parametric problem, and is dependent on various parameters such as type of foundation, structural characteristics, structure-to-soil stiffness, and soil stiffness, as well as frequency, materials, and length and intensity of earthquake ground motion (Nakhaei and Ghannad 2008; Kwon and Elnashai 2006). Consideration of SSI effects requires detailed analytical models, including all major parameters defining the physical problem and all essential structural components of the studied system, given the substantial epistemic and aleatory uncertainty associated with the parameters. During a seismic analysis of bridges, various types of interactions need to be considered, including deck–abutment, soil–foundation–pier, and abutment–embankment, while close coupling between soil conditions and spatially variable soil movements strongly affects longer bridges (Mylonakis et al. 2006; Kotsoglou and Pantazopoulou 2007; Shamsabadi et al. 2005; Taskari and Sextos 2015).

Soil–foundation–pier interaction can consist of soil–pile or pile–soil–pile (i.e., piles-group) interaction for bridges with deep foundations, depending on the method under consideration, while a more simplistic approach based on wave propagation formulations can be adapted for shallow foundations. The influence of SSI on the fragility analysis of bridges has been discussed in several studies (Stefanidou et al. 2017; Lesgidis et al. 2017; Xie and DesRoches 2019). In the case of rigid structures situated on soft soils, these effects are more noticeable, as in the case of bridges with a comparatively light superstructure and a heavy substructure, irrespective of soil stiffness. For seismically isolated bridges and bridge foundations with limited rotational rigidity around their transverse axes, consideration of SSI effects was also found to be significant. Moreover, the significance of the SSI consideration is related to the ratio between the time of the structure and the predominant period at the site and duration of earthquake ground motion, as well as to its frequency content (Lesgidis et al. 2017, 2018; Rahmani et al. 2018; Lim and Jeong 2018).

In this study, due the small width–length ratio of the bridge, a two-dimensional nonlinear FEM was employed to achieve a parametric study to examine the effect of different parameters on the behavior of the IAB. FEM considered a complete bridge including substructure, superstructure, and soil around it. Seismic analysis of the IAB was conducted on the displacement performance philosophy using nonlinear time history analysis. Such analysis requires the geotechnical engineer’s prediction of abutment soils, which is essentially nonlinear with regard to the movement between the bridge structure and backfill. The main objectives of this study were to study the sensitivities and influences of key parameters that significantly affect IAB response and the effects of bridge parameters on IAB performance under seismic loads.

2 Methodology

A complete IAB structure is the superstructure of the bridge that is directly related to its substructure. When subjected to lateral loading, the superstructure and substructure shift towards or away from the backfill, and interaction analysis of the soil base structure is advantageous for the actions of IAB subjected to seismic forces. In this study, 2D FEMs are planned for a single period to investigate the seismic load responses of IABs. As shown in Fig. 1, soil–abutment interaction, soil–pile interaction (SPI), connections of abutment back wall, and connections of pile abutment are crucial factors leading to FE model.

Fig. 1
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Component of critical zone in modeling

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The important matter is clarifying the critical sections in a model of an IAB which performs as an integrated structure. A model of a pile is implemented in the bridge to verify FEM. Then, p–y curves of the pile at various heights below the ground are measured. The pile model is then formulated in FE software (i.e., ANSYS), and the same lateral forces are installed on the pile. The next step is to verify the model formulation procedure under dynamic load. Since the seismic behavior of an IAB is the focus of the present study and loading is performed under a dynamic load, it is necessary to implement a method to formulate a soil model which could be effective for the model under a dynamic load whose precision can be measured with high probability. The model formulation of the back wall follows the same procedure. The connection between the abutment and the back wall is a very critical part of modeling, which is the next critical stage of model formulation. Therefore, the \(\left( {M - \phi } \right)\) diagram is applied to draw the \(\left( {M - \theta } \right)\) diagram which is the necessary for model formulation of nonlinear behavior of connections.

Soil–pile (SP) and soil–abutment interactions, known as SSIs, are related to geotechnical behavior, while abutment back wall and pile abutment connections are related to the yielding of structural connections. SSI is generally a more influential parameter than the yielding of structural connections because soil exhibits nonlinear and hysteretic behavior even over a small range of deformations. In addition, the yielding of structural connections may never occur in a lifetime of bridge structures, if properly designed. The classical p–y curve technique is assumed to deliver a load-deformation curve of laterally loaded SPI behaviors. p–y curves are generated using the COM624P computer program developed by Wang and Reese (1993). The p–y curve method is an SSI analysis method based on the modulus of subgrade reaction. The p–y curve method was originally developed using finite difference techniques. The substitution of nonlinear p–y curve springs in the governing equation was done instead of using a traditional linear Winkler spring. An iterative solver was implemented to achieve a numerical solution in this transition. Although a nonlinear spring generated from the p–y curve method is considered uncoupled, validation of this method was performed against full-scale experiments explicitly to include a continuum effect (Wang and Reese 1993). A series of p–y curves along the pile length are generated by pile analysis software (i.e., AllPile (AP)). AllPile is a Windows-based analysis program that can handle most types of piles, from steel pipes, HP piles and precast concrete piles to drilled shafts and shallow foundations. Lateral analysis in AP software utilizes the finite difference method for the modeling of SSI. In lateral loading, the pile shaft deflects and exerts pressure on the adjacent soils, resulting in the generating of lateral resistance (pressure) between soils and the pile. By integrating lateral pressure, shear, moment, rotation, and deflection, an analysis of the pile and soil interaction and behavior is obtained. For validation of FE software (i.e., ANSYS) and AP software, one pile of implemented IAB is chosen and modeled with FE software by p–y curves taken from AP software; then the output of FE and AP is compared. Bridge 222, the IAB selected for modeling which is located in Pennsylvania in the United States and the implemented plan is shown in Fig. 2. Soil parameters were obtained from the soil profile of Bridge 222 for clay above the water table and sand, respectively, as illustrated in Fig. 3. Samples of p–y curves generated by AP software are presented in Fig. 4.

Fig. 2
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A section of Bridge 222

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Fig. 3
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Soil properties for Bridge 222 at abutment 1 (Pugasap 2006)

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Fig. 4
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p–y curve for clay and sand

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The multilinear curves represent a nonlinear soil spring in FE software. The COMBIN39 element type was adopted as an appropriate nonlinear one-dimensional element in FE. A lateral force of 44.5 kN, which produces a working range of actual pile movements of Bridge 222 and a free-end boundary condition was applied at the pile head, as displayed in Fig. 5. The maximum pile moment occurs at the top of the pile. The value is dependent on pile deformations and abutment rotation. Pile behavior can be considered similar to a cantilevered member with a free end at the first point of inflection in the pile and partial release of end moment restraint at the pile top depending on abutment rotation, with soil-springs attached along this length (Shedge and Kumar 2022; Salman and Issa 2019; Abdelrahman et al. 2018; Xie et al. 2017; Zhang et al. 2008; Civjan et al. 2007; Teguh et al. 2006). Both AP and FE software display highly similar response to the lateral forces on the pile head (Fig. 6). Maximum displacement was 2.44 mm by FE software and 2.65 mm by AP software, while the maximum moment was 24.5 kN·m and 25.6 kN·m by FE and AP software, respectively. The FEM predictions of pile behavior were similar to AP, with percentage differences of 8% and 8.5%, for a maximum displacement and maximum moment, respectively. Element length in the FE pile model was relatively coarse (33 mm) compared to the length used in AP (30 mm). Therefore, some small differences in moments at a depth of approximately 3 m are expected to appear, where a short distance between two adjacent inflection points occurs. It is not possible to select elements with the same length because element length in AP is automatically determined by software, but in FE, several extra nodes with minor element length are generated because the user limits the coordinates of nodes. This increases error probability; hence, it is recommended to avoid such element length difference.

Fig. 5
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Single lateral force applied to the head of the pile

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Fig. 6
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Lateral displacement and pile bending moment due to 44.5 kN loads at the pile head

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The dynamic response of piles in bridges with pile foundation is a function of the characteristics of the loading, dynamic pile–soil interaction behavior and dynamic characteristics of the piles structural system. The seismic soil–pile structure interaction analysis is the primary step in evaluating the seismic behavior of pile-supported structures. The pile–soil interaction problem when an earthquake occurs with its accompanying stresses is one of the most significant sources of nonlinear dynamic response analysis of bridges, which has gained attention in recent years and several studies have investigated the characteristics of ground motion input and the mechanism of pile–soil interaction to determine seismic load design for structures that are pile-supported (Shamsi et al 2021; Souri et al. 2022; Li et al. 2022). p–y analysis is used to analyze the dynamic response of piles and should allow for a variety of soil properties with depth, nonlinear soil behavior, nonlinear behavior of pile–soil interfaces and energy dissipation through hysteretic damping. The method of validating the reliability of p–y analysis followed in this study is evaluated with an experimental test that involves a series of dynamic centrifuge model tests of pile-supported structures in the soft ground followed by an evaluation of the capability of the p–y analysis method, which is done to obtain a reliable picture of the SPI effects. Experiments were conducted by Boulanger et al. (1999) at the University of California, Davis, using a large servo-hydraulic shaking table. The profile of the soil consisted of two horizontal layers of soil with a lower layer of fine, evenly graded Nevada sand, 11.10 m thick with Cu equal to 1.5 and D50 equal to 15 mm, where Cu is the ultimate shear stress of soil and D50 is the relative density in average particle size. The sand dryness density was 1.66 mg/m3 at a relative density (Dr) of 75–80%. The upper layer was of reconstituted bay mud 6.39 m thick with a plasticity index of 48%.

The single pile-supported system employed in this test was a 670 mm steel pipe with 19 mm wall thickness and a total length of 20.57 m and a superstructure with a mass of 49,100 kg connected to an extension of the pile of 3.81 m above the ground surface. Figure 7 presents the single pile–soil profile and superstructure systems. The pile structure was separated into 12 elastic–plastic beam elements, with 11 elements below the soil surface and one element above ground. Each sub-ground pile node was linked to a set of parallel nonlinear p–y elements on each side of the pile modeled with COMBIN39 for nonlinear springs. Interface elements were employed to join the pile and soil nodes at each level and on each side of the pile. Acceleration time histories at various soil layers derived from the free field site response analyses using the EERA program.

Fig. 7
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Profile of soil (Boulanger et al. 1999)

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EERA is an implementation of the equivalent linear concept of earthquake site response analysis that is an add-on in Excel. EERA input was used to define the earthquake acceleration time history as well as the geometry and properties of the soil profile. With this application, it is possible to obtain the time history acceleration and displacement in different layers for that the user chose. At this stage of study, this application software is used to determine the desirable time history analysis.

The soil profile is defined in AP software and then a p–y curve is created for the pile dimensions in various heights of soil. As is shown in Fig. 8, the soil profile at the 0.2 m height is composed of clay, and at the 11.2 m height it is made up of sand with certain specifications which were used in the experiment. Furthermore, the ground movement of the soil layers was measured due to earthquake excitations performed on the bedrock in site response analysis. The results of this free-field analysis (acceleration or displacement time histories at different soil layers) are shown in Fig. 9 and become the input excitation at the FEM support nodes.

Fig. 8
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p–y curve for sand and clay

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Fig. 9
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General view of modeling for nonlinear dynamic loads on the pile

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3 FE Modeling

The bridge discussed in this study is a single-span bridge on the I-99 extension in Pennsylvania Bridge 222 (Fig. 2) and includes element types, mesh sizes, section properties, and material properties for component elements. A composite slab, abutment, girder section, back wall, and piles for the bridge were combined and modeled using the ANSYS BEAM3 element, which is a uniaxial element capable of tension, compression and bending. Each node of the element possesses three degrees of freedom: translations in the nodal x and y directions and rotation about the nodal z-axis. Table 1 shows a short description of the structure of the instrumented integral abutment bridges of four prescribed concrete girders with a cast-in-place deck and without angle at the end on both abutments. The abutments are supported by a single row of weak axis-oriented steel piles by HP12 × 74. An ANSYS BEAM3 element was used to combine a composite slab and four girder sections. This beam member was subdivided into 10 pieces, and all were located on a composite elastic neutral axis. The elastic modulus of girders was used as a reference modulus so that the slab and parapet widths were transformed using corresponding modulus ratios. AASHTO LRFD was used to determine a concrete modulus of elasticity based on a concrete girder strength of 55.2 MPa. Table 2 presents the material properties of the superstructure.

Table 1 Descriptions of the IAB
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Table 2 Material properties for superstructure models
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As shown in Fig. 10, three parts including: A1 (SPI), A2 (soil–abutment interaction) and A3 (abutment back wall connections), respectively, were applied to the bridge components. A1 hysteresis element shows that a nonlinear spring was connected to two adjacent pile elements through a single node to represent SPI. A single degree of freedom UX was used for these elements, as we were only interested in the longitudinal movement. Similar to the A1 case, the A2 element was connected to two adjacent abutment elements inside the line segment C–D and two back wall elements inside segment A–B, through a single node to represent soil–abutment interaction. One degree of freedom UX was enabled for each A2 element with an element spacing of approximately 0.3 m. Unlike the A2 case, a single A3 element was used to connect the bottom of the back wall component (node B) to the top of the abutment component (node C). One degree of freedom in the z direction was applied for this case to represent the hysteretic behavior of the abutment back wall connection. For the other degrees of freedom, UX and, UY a coupling of nodes B and C were employed. It is noted that nodes B and C overlap. Properties of the A1 element were obtained primarily from p–y curves, generated by using AP. These properties depended on depth, embankment slope, soil overburden, pile stiffness, and soil properties. Figure 11 presents diagrams of soil profiles and properties for two abutments of the bridge on each side of the bridge. As it is shown, the soil is composed of three layers: stiff clay, sand, and bedrock. p–y curve specifications are related to characteristics of soil such as stiffness and overburden. As it was discussed earlier, the length of the pile element is defined as 0.3 m in FE software. Figure 12 shows the p–y curve at different levels in increments of 0.3 m.

Fig. 10
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Schematic representation of substructure modeling

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Fig. 11
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Soil properties for Bridge 222 (Pugasap 2006)

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Fig. 12
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p–y curves for the pile at abutment 1

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3.1 Material Properties of Substructure

All material properties of the instrumented bridge are shown in Table 3 and Fig. 13 shows soil backfill interaction modeling.

Table 3 Material properties for substructure models
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Fig. 13
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Soil–structure (pile, abutment, back wall) interaction view

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3.2 Loading

The bridge is located in Pennsylvania, United States, which is a moderate seismic zone and has a PGA of 0.15 g. As it is clearly seen in Fig. 14, the loading in FE time history of the lateral displacement is applied on each node. Nonlinear time history analyses are conducted utilizing the scaled component of the Kobe earthquake to apply to the longitudinal direction of the bridge. Figure 15 indicates the time history of accelerations at a height of 2 m below the surface of the soil profile and Fig. 16 shows its equivalent displacement. The largest lateral displacements above and below the abutment walls during the time history analysis are presented in Table 4 for abutments, showing how generally the maximum wall lateral displacements are greater at the bottom and well matched with the pile deflection. Consequently, all lateral and longitudinal loads applied to the superstructure of the bridges are transmitted directly to abutment embankments. Furthermore, the highest positive and negative moments in the head of piles for the duration of the time history analysis are presented in Table 5. It appears that there is a normal reduction in the girder moments along the bridge within the span due to the stiffness of the soil at the back of the abutment and in the area surrounding the piles, like the pile deflection and abutment displacement. Table 6 presents the high and low soil stiffness properties of clay.

Fig. 14
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Applied loads in FE software

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Fig. 15
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Time history accelerations in the soil profile during the event of 0.15 g at 2 m depth

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Fig. 16
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Time history displacement in the soil profile during the event of 0.15 g at 2 m depth

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Table 4 Maximum abutment displacements
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Table 5 Moment of pile head and girder axial forces
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Table 6 Soil properties
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4 Results and Discussion

This study performed 81 sets of parametric investigation of IABs using a two-dimensional numerical FEM. The considered parameters include bridge span, backfill height, backfill stiffness, and clay stiffness around the pile listed in Table 7. The specific locations of critical responses are shown in Fig. 17.

Table 7 Parametric study cases
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Fig. 17
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Critical locations

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4.1 Effects of Bridge Span on Critical Response

The bridge span shows significant effects on the pile lateral force (FP); for 18.3, 34.5, and 65.4 m, IABs with abutment height of 3.0 m were found to be 12, 30, and 45 kN with high stiffness backfill, respectively. As the bridge span was increased, the pile bending moment increased substantially as well. The maximum average positive moment (MP) for the 18.3 m span was 32.13 kN·m, whereas for the 34.5 m span it was 42.38 kN·m, and lastly the 65.4 m span results showed 71.13 kN·m. The maximum average negative moment (MP) for the 18.3 m span was −30.5 kN·m, and for the 34.5 m span it was −42.38 kN·m, while for the 65.4 m span it was −79.13 kN·m. With increases in bridge span, both compressive and tensile girder axial forces (Pg) increased considerably. The maximum average compressive girder axial force found was 389.7 kN and 795.1 kN for spans of 18.3 m and 65.4 m, respectively. The maximum average tensile girder axial force was 250.0 kN and 522.1 kN with a span of 18.3 m and 65.4 m, respectively. As the bridge span increases, both positive and negative girder bending moment at the abutment (Mge) increase significantly. Table 8 summarizes the analyses and Fig. 18 displays the absolute value of outputs relative to the seismic load applied in the longitudinal direction as parametric modeling cases considered by bridge span including the stiffness of the soil around the pile. For instance, when a span of 18.3 m is subjected to ground motion, the maximum lateral force in the longitudinal direction is obtained as 12.5 kN for the case where the stiffness of the soil around the pile is high. However, for spans of 34.5 and 65.4 m using the same ground motion and scales and including the SPI performance in the structural models, the maximum lateral forces were 35.5 and 45.1 kN, respectively. The differences between the maximum displacements found from the spans of 18.3, 34.5, and 65.4 m were 30% and 24%, respectively, as shown in Fig. 18d.

Table 8 Time history results in the critical locations
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Fig. 18
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Effects of bridge span on critical response with consideration soil stiffness around the pile

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4.2 Effects of Backfill Materials on Critical Response

In a seismic case, the nonlinear force–displacement capability of the bridge abutment is built primarily from mobilized passive pressure behind the back wall of the abutment. Therefore, proper abutment backfill system modeling is important and allowance for nonlinear stiffness must be made. To determine the effect of backfill height on the critical response of the bridge after time history analysis, we selected three levels of height as shown in Table 9. For the domain of backfill height, all backfill stiffness behind the abutment selected high density as soil parameters described in Table 7. This is because of the lateral displacement revealed by the change in the backfill height of the bridge. Time history analysis was applied in all cases and pile displacements (ΔP) for three spans (i.e., 18.3 m, 34.5 m, and 65.4 m) and results presented in Fig. 19. The variables selected in the above tables were bridge span and backfill height changes in the three levels. In Fig. 19a, three bridges with the same span of 18.3 m were analyzed. Considering the conditions of equality to the stiffness of backfill with high-stiffness sand behind the abutment, abutment height has no significant influence on the lateral displacements (ΔP). This process is repeated in Fig. 19b and c. By changing the height of the abutment with the accompanying span of the bridge, the structural stiffness increased, but this factor is not related to the lateral displacements.

Table 9 ID Cases for back wall height effect
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Fig. 19
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Lateral displacements for pile 1

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The backfill height influences pile lateral forces (FP). The average pile lateral force for 18.3 m span and 3.0 m height of backfill was 9.54 kN, for 4.6 m it was 12.16 kN, and for 6.1 m it was 13 kN. In addition, backfill height does not influence pile bending moments (MP). The average change in pile moment between 3.0 and 6.1 m was −6.37 kN·m. However, the increase in backfill height led to larger displacement when the soil stiffness around the pile was low. The results of the time history analysis are presented in Table 10. Backfill height significantly influences girder axial forces (Pg). The average increase in tensile and compressive girder axial forces (Pg) between the considered cases with 18.3 m and 65.4 m was 410 kN and −165 kN, respectively. An increase of backfill height significantly decreases tensile axial force (Pg), while backfill height has little influence on compressive axial forces. The maximum average tensile girder axial force reduction between 3.0 and 6.1 m was −325 kN. Also, backfill height affected the girder bending moments at the abutment (Mge). An increase of backfill height significantly decreases the positive girder bending moment at the abutment. The maximum average positive moment decreased between 3.0 and 6.1 m and was 482 kN·m for the span of 18.3 m, while the average negative girder moment was 233 kN·m for the span bridge. Figure 20 shows the absolute values of outputs relative to the seismic load applied in the longitudinal direction as parametric modeling cases considered by backfill height including the soil stiffness around the pile. Figure 20 shows that simplification of the structural model leads to a substantial clarification of the full value of the IAB. For example, when the back wall height is 3.0 m subjected to ground motion, the maximum lateral force in the longitudinal direction is 30.8 kN for the case where the soil stiffness around the pile is high. However, for back wall heights of 4.6 and 6.1 m analyzed using the same ground motion and scales and including the SPI behavior in structural models, the maximum lateral forces obtained were 40.0 and 41.7 kN, respectively. The changes between maximum displacements found from 3.0, 4.6, and 6.1 m heights of backfill were 37.8% and 85.6%, respectively (Fig. 20b).

Table 10 Time history results in critical location
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Fig. 20
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Effects of soil stiffness on different parameters (lateral forces, displacements, pile head moments, girder axial forces, and abutment head moment)

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5 Summary of Backfill Height

The results show that backfill height does not have a significant impact on pile lateral forces and pile bending moments (MP). Backfill height significantly affects girder axial forces. An increase in backfill height decreases tensile axial forces (Pg), while it shows little effect on the compressive axial forces. Backfill height significantly influences the girder bending moments. The results of the time history analysis are shown in Table 11. The maximum average compressive girder axial force increased between low-stiffness and high-stiffness clay, and it was 73.95 kN. The average increase in tensile girder axial forces was 123.41 kN. The superstructure forms a convex curve during bridge expansion while it forms a concave curve during bridge contraction. This shape formation is more obvious in single-span bridges.

Table 11 Time history results considering backfill stiffness effect
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Figure 21 shows the absolute value of outputs relative to the seismic load applied in the longitudinal direction as a parametric modeling case on 65.4 m length and 3.0 m height of backfill considered by backfill stiffness including soil stiffness around the pile. When backfill stiffness is highly subjected to ground motion, the maximum lateral force in the longitudinal direction obtained is 45.01 kN for the case where soil around the pile is hard. For intermediate- and low-stiffness backfill, the same ground motion was applied and the scale included SPI behavior in structural models. The maximum lateral force found for those cases was 49.78 and 46.23 kN, respectively. The difference between maximum displacements obtained from high, intermediate, and low stiffness backfill were 82.26% and 96.87%, respectively (Fig. 21b).

Fig. 21
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Effects of soil stiffness around the pile

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6 Conclusion

Integral abutment bridge construction is an increasingly common alternative to conventional bridge construction, as it removes expansion joints, bearings and their related maintenance problems. However, in the absence of expansion joints and bearings, the abutments and piles must be capable of accommodating lateral movements of the bridge due to thermal expansion and contraction and seismic movements. The present study developed numerical modeling to investigate seismic analysis and probabilistic simulations of IABs. To validate the numerical modeling, we examined lateral force and dynamic load response to the finite element model process in both static and dynamic conditions. A constructed bridge was selected as an example. A model formulated for a pile was implemented in the bridge to verify software used in this study. Afterward, we examined the p–y curves of piles at various heights below the ground. As indicated in the NCHRP curve (Barker et al. 1991), there is no distinction in the type of rigid-body movement that generates the displacement measured at the top of the abutment: abutments can experience pure translational, pure rotational, or combined translational-rotational response to generate a top displacement. In this study, we did not consider rotational and translational degrees of freedom. The parametric study consisted of four variables, namely the bridge span (short, medium, and long spans of 18.3, 35.4, and 64.5 m, respectively), value of the backfill pressure (3.1, 4.6, and 6.1 m, respectively), stiffness of soil mixture backfills (high, intermediate, and low stiffness), and soil density around the piles (high, intermediate, and low). Because of the small width–length ratio of the bridge, a 2D model of IAB with soil springs around the piles and abutments was developed. By applying seismic loading, maximum pile deflection and abutment displacement occurred at the top of the pile where it was connected to the abutment. This deflection is affected by the banded stiffness of sand behind the abutment and the stiffness of clay around piles. IAB was modeled with all the critical details in FEM software. Then, for the defined SSI, p–y curves were used to verify to the software. The Beam3 element was used for pile, abutment, back wall, and superstructure. Also, the COMBIN39 element was used for nonlinear springs of soil. In subsequent sections, we used AllPile software to obtain the p–y curve of piles, and then the diagram of force–displacement for abutments was drawn. Finally, the model was formulated, and time history dynamic loads were applied to the bridge. The parametric study considered several parameters (i.e., bridge span, backfill height, backfill stiffness, soil type around piles, and soil stiffness around piles, using three variables of clay stiffness around piles [i.e., high, intermediate, and low] from each parameter, resulting in 81 parametric study cases).

Based on the analytical results, several conclusions were drawn:

  • Under seismic loads, the maximum moments in the pile were at the pile–abutment interfaces. The pile moments decreased when abutment backfill was high-stiffness sand and increased when piles were placed in dense clay. Therefore, this moment is greatest for the case with piles in high-stiffness clay and low-density backfill, and smallest for the case with piles in intermediate-stiffness clay around the pile and high-stiffness backfill.

  • Bridge span played a significant role in the performance of IAB (i.e., pile lateral forces, pile bending moments at pile head, tensile and compressive girder axial forces, and bending moments at the abutment).

  • Backfill height, which is important for the distribution of induced forces transmitted into the soil, should be considered together with other parameters.

  • The maximum axial girder moments at the superstructure generally decrease when the combined stiffness of soil behind the abutment and around piles increases, similar to pile deflection and abutment displacement. However, the maximum abutment head moments decreased when abutment backfill was dense and increased when piles were located in hard clay, similar to pile moments.

  • Dense sand backfilling behind abutments is recommended, since it decreased the pile deflections, pile lateral forces, abutment displacements, abutment head moments, and in particular pile bending moments.

  • Upon evaluating the p–y backbone curves in relation to depth, it became evident that as depth increased, there was a corresponding increase in soil resistance and a decrease in pile displacement. This observation seems to be due to the fact that the largest shear force is generated at the head of the pile, as a result of the load imposed by the pile cap and the upper structure. Furthermore, it is important to note that soil-structure interaction (SSI) studies are currently limited to issues with similar p–y curves. Therefore, it is essential to generate individual p–y curves for each structure.