Spring-Based Soil–Structure Interaction Modeling of Pile–Abutment Joints in Short-Span Integral Abutment Bridges with LR and RSM

Abstract

Integral abutment bridges (IABs) are increasingly adopted in transportation infrastructure due to their durability, reduced maintenance needs, and cost-effectiveness compared to conventional bridges. However, their reliable performance under live loads is strongly influenced by the nonlinear soil–structure interaction (SSI) at the pile–abutment joint, which remains challenging to quantify using conventional analysis methods. This study develops simplified spring-based models to capture the SSI behavior of pile–abutment joints in short-span IABs. Predictive equations for joint rotation, deflection, moment, and shear are formulated using Linear Regression (LR) and Response Surface Methodology (RSM). Unlike prior studies relying solely on FEM or traditional p–y curves, the novelty of this work lies in deriving regression-based spring constants calibrated against FEM analyses, which can be directly implemented in standard structural software. This approach significantly reduces computational demands while maintaining predictive accuracy, enabling efficient assessment of pile contributions and global bridge response. Validation against finite element method (FEM) results confirms the reliability of the simplified models, with RSM outperforming LR in representing nonlinear parameter interactions.

1. Introduction

Integrated Abutment Bridges (IABs) have emerged as an effective alternative to conventional bridges because they offer numerous advantages such as accelerated bridge construction, structural resiliency, and efficiency. One of the main benefits of IABs is the reduction of positive moments and deflections at mid-span by generating negative end moments in the bridge girders. These negative moments are transferred to the ground through the pile-abutment connection, making its deformation behavior critical to overall performance. This response is influenced by the relative flexural stiffness of structural components and the mechanical properties of surrounding soil, which together govern load transfer and deformation.

Traditional bridges rely on expansion joints and bearings to accommodate movements caused by temperature changes, traffic loads, and other effects. While essential for serviceability, these components often lead to high construction and maintenance costs due to water ingress, corrosion, and frequent repairs [1]. IABs eliminate these elements by rigidly connecting girders and piles to abutments, creating a monolithic structure that transfers movements through soil-structure interaction. This jointless system enhances durability, reduces maintenance, and improves load distribution, ride quality, and constructability [2,3].

IAB adoption is widespread in the U.S. and Europe, though design practices vary. U.S. agencies typically use flexible steel H-piles oriented for weak-axis bending, allowing for thermal expansion, while European countries often employ concrete-filled pipe piles and incorporate pile sleeves to minimize thermal stresses. Backfill practices also differ, with U.S. standards favoring loose or granular fill and compressible inclusions, while European designs often require simultaneous backfilling to balance lateral pressures [4].

Integral bridges have been successfully implemented in medium to long spans, particularly in the U.S., where lengths of 290–358 m have been achieved [5,6]. Abutment heights vary widely, and studies have shown that thermal movements are primarily accommodated by abutment rotation rather than translation [7]. While joint-free systems induce cyclic bending and axial loads in the piles, advancements in detailing (such as pile orientation, material improvements, and connection alternatives) have mitigated issues like wingwall cracking and pile yielding [8].

Economically, IABs offer significant savings. For example, Vermont’s VTrans study reported 10–20% lower construction costs for three IABs compared to conventional bridges, not accounting for indirect benefits such as reduced maintenance and faster construction [3].

In parallel, advanced modeling methods such as RSM have gained popularity for both optimizing structural performance and predicting structural responses with greater accuracy. RSM employs multiple regression models to conduct statistical experiments, enabling high correlation between input parameters and output responses, as demonstrated by Kuntoğlu et al. [9]. RSM encompasses various experimental design techniques, including Central Composite Design (CCD) and Box–Behnken Design (BBD), each with unique advantages depending on the nature of the problem. CCD is particularly effective for capturing complex variable interactions and curvature in the response surface, as noted by Adamu et al. [10], whereas BBD is more limited in its ability to explore variable extremes, making it less suitable for studies requiring broad parameter ranges [11].

Despite their advantages, modeling IABs is challenging due to the complex, nonlinear interaction between the superstructure, substructure, and soil. Accurate representation demands detailed simulation of soil-structure behavior, involving a large set of geometric, material, and boundary parameters. Although the present study adopts a simplified linear-elastic soil model, it is acknowledged that soil is a multiphase medium in which void ratio, moisture content, and pore pressure strongly influence stiffness and load transfer behavior. These factors are well established in advanced multiphase soil–structure interaction models [12,13] but are beyond the scope of this study. In IAB design, piles are typically modeled as cantilever beams with a fixed end located below the ground surface. The location of this fixity point is commonly assumed to occur at a depth of approximately ten times the pile diameter (D). Although this assumption is not explicitly prescribed in the AASHTO LRFD Bridge Design Specifications [14], it is widely adopted in design practice and supported by previous studies. Empirical observations, such as those reported by Pando et al. [15], indicate that lateral pile deflections tend to diminish significantly within approximately 8–10 times the pile diameter. Similarly, analyses based on Reese and Van Impe’s methods [16] show that most lateral resistance is mobilized within the upper 5–10 D. While these values are used as practical rules-of-thumb, they are influenced by soil stiffness, pile flexibility, and loading conditions. As Rajapakse [17] discusses, in sandy soils the so-called “critical depth” for pile capacity is often approximated as 10D for loose sands, 15D for medium-dense sands, and 20D for dense sands. Although this critical depth concept relates to axial end-bearing rather than lateral fixity, it reinforces the fact that the “10D rule” is a heuristic approximation whose value depends strongly on soil density and stiffness. Thus, while the 10D assumption is commonly adopted in practice, it should be applied cautiously and verified against site-specific soil conditions and analytical methods. Beyond linear-elastic or p–y-based models, recent studies have advanced coupled hydro-mechanical and thermo-hydro-mechanical formulations capable of capturing pore pressure evolution, capillary effects, and transient saturation [18,19]. While these approaches offer comprehensive representation of multiphase soil behavior, they require significant computational effort and expertise. By contrast, the present study develops simplified regression-based spring models, prioritizing efficiency and practical usability for preliminary analysis of short-span IABs.

This study aims to develop practical and efficient spring models that accurately represent pile–soil interaction in integral abutment bridges. These models are calibrated against finite element method (FEM) simulations to replicate the static behavior of the pile–abutment joint while incorporating the influence of the superstructure. Simplified spring models are commonly used in preliminary structural analyses, as they reduce computational effort and facilitate early-stage design decision-making. The approach adopted in this study transforms highly indeterminate systems into more tractable analytical representations and is particularly suitable for short-span IABs with lengths up to approximately 98 ft (30 m). Extrapolation beyond this limit may introduce uncertainties not captured in the present dataset.

The analysis focuses on short-span, single-span IABs without skew, subjected to vehicular live loads. Both cohesive (clay) and non-cohesive (sand) soils are considered under the assumption of deep groundwater conditions. The bridge configurations used in this study are based on Polat [20], which analyzed single- and multi-span IABs using FEM under varying soil and pile stiffness. That study demonstrated the role of IABs in extending allowable span lengths by mitigating mid-span positive moments. The two single-span bridges (Bridges I and II) from that work are used as references here. Their critical design parameters including substructure and superstructure stiffness, soil type, abutment height, and span length were systematically varied to represent practical design scenarios.

A two-dimensional (2D) linear-elastic modeling approach was adopted due to its efficiency and proven ability to capture fundamental structural responses in integral abutment bridges (IABs). Fennema et al. [21] demonstrated that 2D models show strong agreement with 3D simulations, particularly for thermal and soil-structure interaction effects. Similarly, Kim et al. [22], through long-term monitoring of operational IABs, confirmed that 2D static and time-history analyses yield reliable predictions of substructure behavior. Given their accuracy and computational efficiency, 2D models are well-suited for evaluating IAB performance in preliminary and parametric studies.

The study omits backfill-abutment interaction as the preliminary investigation, which resulted in conservative estimates. Results from supplementary analysesindicate that including backfill leads to reduced span moments and pile head deformations due to the stiffening effect of the soil, while slightly increasing shear forces at the pile head because of the added restraint. Similar behavior is documented in the literature; for example, Huang [7] observed that abutment substructures provide rotational restraint, resulting in end-span behavior intermediate between simple-span and fixed-pinned conditions.

2. Proposed Spring Model

Figure 1 illustrates the developed force components and the corresponding spring model of the pile-abutment joint. Part (a) shows the free-body diagram with applied moment (M), shear (V), and axial force (N), while part (b) presents the idealized spring representation. As axial deformation is not considered in this study, the axial force component (N) is excluded from the spring model. The objective is to predict the pile-abutment joint responses, including rotation (θ), deflection (Δ), moment (M), and shear (V), using statistical modeling techniques. Regression equations are developed to quantify the effects of geometric, material, and structural parameters under lateral deformation. Spring stiffness values are then derived from the predicted responses using Linear Regression (LR) and Response Surface Methodology (RSM).

3. Reference Bridge Models

The bridge models employed in this study are based on typical bridges in North America that were previously examined by Polat [20] to evaluate span moment reduction when configured as IABs. Although the two reference bridges were adopted from Polat [20] the design variables (K_s, EI_p, EI_g, H, and L) were systematically varied using CCD, enabling the regression equations to be applied to a wide range of short-span IAB configurations. The geometric and structural properties of the reference bridges serve as the basis for defining axial levels in the numerical design matrix, developed using CCD in the next section. These configurations guide the selection of realistic parameter ranges for statistical modeling using LR and RSM.

Bridge I is a single-span slab-on-girder structure with a span of 45 ft (13.72 m) and a width of 32 ft (9.75 m), as shown in Figure 2. Its superstructure consists of precast channel-section girders placed adjacently without gaps, supporting an 8 in (203 mm) thick slab deck. The figure also includes composite section details for an interior girder, showing centroid location and transformed properties. The substructure comprises abutments without wingwalls, supported by 8 steel HP piles. The abutments are 5 ft (1.52 m) high and 3 ft (0.91 m) thick, rigidly connected to both girders and piles.

Bridge II is a solid slab bridge constructed entirely of reinforced concrete, with a span length of 50 ft (15.24 m) and width of 46 ft (14.02 m), as illustrated in Figure 3. The slab serves as both superstructure and deck and is 26.4 in (670 mm) thick. The substructure includes abutments supported by 10 steel HP piles. These abutments are 10 ft (3.05 m) high and 2.5 ft (0.76 m) thick and are also rigidly connected to the piles.

For both bridges, a uniform pile length of 30 ft (9.14 m) was used in the analyses. The soil conditions were modeled using four clay consistency levels and three sand types. A detailed summary of the bridges’ structural and material properties is provided in

Table 1. Structural Properties and Material Specifications for Bridges I and II.

Property	Bridge I	Bridge II
Span length	45 ft (13.7 m)	50 ft (15.2 m)
Bridge width	32 ft (9.8 m)	46 ft (14.0 m)
Girder type	Channel	Concrete slab
Girder width	4 ft (1.2 m)	-
Girder height	28 in (711 mm)	-
Slab thickness	8 in (203 mm)	26.4 in (670 mm)
Abutment height	5 ft (1.5 m)	10 ft (3.0 m)
Abutment thickness	3 ft (914 mm)	2.5 ft (762 mm)
Number of piles	8 HP steel	10 HP steel
Pile length	30 ft (9.1 m)	30 ft (9.1 m)
Girder concrete strength (fc′)	5000 psi (34.5 MPa)	-
Girder modulus of elasticity (Ec)	4286 ksi (29.6 GPa)	-
Deck concrete strength (fc′)	4000 psi (27.6 MPa)	4000 psi (27.6 MPa)
Deck modulus of elasticity (Ec)	3834 ksi (26.4 GPa)	3834 ksi (26.4 GPa)
Substructure concrete strength (fc′)	4000 psi (27.6 MPa)	3500 psi (24.1 MPa)
Substructure modulus of elasticity (Ec)	3834 ksi (26.4 GPa)	3587 ksi (24.7 GPa)
Composite Section M.of.Inertia (Ig)	86,988 in2 (561,000 cm2)	-

.

4. Experimental Design Using RSM, LR

The displacement and force responses of the pile-abutment joint in an IAB are influenced by complex, nonlinear interactions among material and geometric design variables, including substructure and superstructure flexural stiffness, soil stiffness, span length, and abutment height. RSM offers an effective framework for modeling such nonlinear behavior, providing high predictive accuracy and the ability to capture intricate parameter interactions. In comparison, LR is a simpler and more computationally efficient approach, particularly suitable when variable interactions are minimal or predominantly linear.

4.1. Application of RSM

In this study, RSM is used to predict pile-abutment joint responses in terms of displacements and forces. The method identifies the most influential parameters and models their interaction effects on structural responses. The governing input variables for IAB response include soil stiffness (K_s), pile flexural stiffness (EI_p), girder flexural stiffness (EI_g), abutment height (H), and bridge span (L). These variables were selected based on their fundamental role in the structural analysis of indeterminate elastic frames, which represent the assumed behavior of IABs. Frame deformations are primarily governed by flexural rigidity (a function of EI and span length, L), while soil response is represented by discrete springs that deform axially and are distributed along the soil depth. Thus, K_s, EI_p, EI_g, H, and L together capture the dominant mechanisms influencing global bridge–soil interaction.

RSM employs regression-based statistical analysis and offers an efficient approach to modeling quadratic response functions, especially when compared to full factorial designs [23]. While both CCD and Box–Behnken Design (BBD) are commonly used in RSM, though BBD is less suitable for variables with more than three levels. Accordingly, CCD is adopted in this study for its ability to accommodate higher-level variable complexity and support robust optimization.

4.2. Central Composite Design

The CCD methodology provides a second-order model for predicting outputs by accounting for both individual and interaction effects of input variables. The output variables, rotation, deflection, moment and shear, are expressed as a quadratic function in Equation (1). n represents the number of input variables, ${\beta }_0$ is the constant coefficient, and ${\beta }_i, {\beta }_{ii}, {\beta }_{ij}$ are regression coefficients for linear, quadratic, and interaction terms, respectively. $X_i$ denotes the coded level of the inputs.

$$Y={\beta }_0+{\displaystyle \sum }_{i=1}^n{\beta }_i+{\displaystyle \sum }_{i=1}^n{\beta }_{ii}{X}_i^2+\sum {\displaystyle \sum }_{i<j}^n{\beta }_{ij}{X}_i{X}_j$$

(1)

The CCD includes $2^n$ factorial points, 2n axial points, and one center point (cp), leading to the total number of experiments estimated using Equation (2):

$$TE=2^n+2n+cp$$

(2)

In this study, with five input variables, 43 numerical experiments are required, consisting of 32 factorial points, 10 axial points, and 1 center point.

Table 2. Input parameter levels in CCD for $K_s, E{I}_p, E{I}_g$, H, and L.

			Levels
No	Variables	Units (US/SI)	Axial	Factorial			Axial
			(−2)	Low (−1)	Center (0)	High (1)	(+2)
1	Soil stiffness (Sand), Ks	kip/in (kN/m)	20 (3504)	60 (10,512)	100 (17,520)	140 (24,528)	180 (31,536)
2	Pile flexural rigidity, EIp	106 k-in2 (109 N-m2)	50 (5.65)	100 (11.3)	150 (16.95)	200 (22.6)	250 (28.25)
3	Girder flexural rigidity, EIg	106 k-in2 (109 N-m2)	1000 (113)	2000 (226)	3000 (339)	4000 (452)	5000 (565)
4	Abutment height, H	in. (mm)	36 (914.4)	48 (1219.2)	60 (1524)	72 (1828.8)	84 (2133.6)
5	Bridge span, L	in. (mm)	500 (12,700)	550 (13,970)	600 (15,240)	650 (16510)	700 (17,780)

presents the input values distributed across five levels: center level (0), low-high levels (±1), and axial levels (±α), where α is defined as 2 and calculated using Equation (8). The studied inputs include $K_s$, $E{I}_p$, $E{I}_g$, H, and L.

$$\mathsf{\alpha }=\sqrt[4]{2^n}$$

(3)

Table 3. List of numerical analyses model parameters obtained from central composite design (1 in = 25.4 mm, 1 kip = 4.448 kN, 1 kip-in = 0.113 N·m).

	2-D IAB Model Properties					Analyses Results (Sand)				Analyses Results (Clay)
Case No	${\mathit{K}}_{\mathit{s}}$	$\mathit{E}{\mathit{I}}_{\mathit{p}}$	$\mathit{E}{\mathit{I}}_{\mathit{g}}$	$\mathit{H}$	$\mathit{L}$	$\mathit{\theta }$	$\mathit{\Delta }$	$\mathit{M}$	$\mathit{V}$	$\mathit{\theta }$	$\mathit{\Delta }$	$\mathit{M}$	$\mathit{V}$
	(Sand)	× 106	× 106			× 10−6	× 10−6		× 10−2	× 10−6	× 10−6		× 10−2
	kip/in	kip-in2	kip-in2	in.	in.	rad	in.	kip-in	kips	rad	in.	kip-in	kips
1	100	150	1000	60	600	853	−27,410	4772	−7411	740	−30,500	4565	−8322
2	60	100	2000	48	550	774	−28,310	3053	−4734	743	−26,830	3030	−5291
3	140	100	2000	48	550	681	−23,440	3196	−5946	608.1	−19,530	3166	−7201
4	60	200	2000	48	550	638	−21,340	3935	−5124	620.8	−20,620	3875	−5552
5	140	200	2000	48	550	561	−17,350	4015	−6234	509.7	−14,710	3924	−7274
6	140	100	2000	48	650	966	−31,200	4436	−8184	860.6	−25,550	4359	−9798
7	60	100	2000	48	650	1100	−38,340	4287	−6604	1060	−36,190	4240	−7344
8	60	200	2000	48	650	900	−27,900	5445	−7027	875	−26,970	5353	−7589
9	140	200	2000	48	650	788	−22,300	5506	−8451	716.8	−18,640	5357	−9774
10	140	200	2000	72	550	437	−21,000	3621	−5968	383.6	−17,150	3431	−6824
11	60	200	2000	72	550	517	−26,900	3660	−5041	494.5	−25,440	3555	−5440
12	60	100	2000	72	550	634	−35,600	2923	−4790	595.4	−32,860	2851	−5306
13	140	100	2000	72	550	534	−28,100	2955	−5823	456.4	−22,330	2820	−6850
14	140	100	2000	72	650	750	−37,100	4038	−7888	641.4	−29,000	3826	−9179
15	60	200	2000	72	650	724	−35,100	5000	−6823	692	−33,160	4848	−7337
16	60	100	2000	72	650	894	−47,700	4040	−6576	838.6	−43,860	3926	−7246
17	140	200	2000	72	650	611	−27,000	4906	−7990	537.9	−21,770	4632	−9063
18	100	150	3000	36	600	696	−16,700	3604	−5408	659.9	−15,110	3641	−6375
19	100	150	3000	60	500	379	−17,700	2399	−3912	348.7	−15,610	2378	−4603
20	20	150	3000	60	600	715	−34,000	3139	−3577	708	−33,790	3087	−3798
21	100	250	3000	60	600	493	−19,880	4133	−5843	458.2	−17,610	4037	−6642
22	100	50	3000	60	600	739	−36,470	2321	−4942	665.7	−31,370	2365	−6170
23	180	150	3000	60	600	515	−21,000	3531	−6457	455	−16,920	3427	−7697
24	100	150	3000	60	600	566	−24,470	3484	−5624	519.3	−21,380	3429	−6552
25	100	150	3000	60	700	775	−31,300	4661	−7456	710.2	−27,100	4563	−8615
26	100	150	3000	84	600	460	−28,390	3264	−5529	410.1	−23,990	3132	−6326
27	140	100	4000	48	550	456	−16,640	2185	−4096	420.3	−14,380	2239	−5142
28	60	200	4000	48	550	436	−15,680	2741	−3600	426.9	−15,220	2719	−3935
29	140	200	4000	48	550	395	−13,310	2903	−4556	367.5	−11,570	2905	−5457
30	60	100	4000	48	550	499	−19,180	2000	−3120	485.3	−18,400	2012	−3538
31	140	200	4000	48	650	569	−17,700	4085	−6348	528	−15,200	4062	−7525
32	140	100	4000	48	650	664	−22,900	3116	−5799	608.8	−19,440	3163	−7189
33	60	100	4000	48	650	731	−26,800	2887	−4478	709.2	−25,640	2893	−5052
34	60	200	4000	48	650	630	−21,300	3898	−5083	617.1	−20,640	3859	−5535
35	140	200	4000	72	550	323	−17,120	2781	−4645	289.7	−14,370	2703	−5470
36	60	100	4000	72	550	432	−25,670	2042	−3370	412.6	−24,140	2027	−3806
37	140	100	4000	72	550	379	−21,420	2167	−4310	334.8	−17,750	2148	−5283
38	60	200	4000	72	550	370	−20,880	2701	−3760	357.5	−19,920	2649	−4105
39	140	100	4000	72	650	545	−29,200	3039	−6001	479.2	−23,790	2981	−7257
40	60	200	4000	72	650	530	−28,300	3786	−5232	511.2	−26,880	3705	−5688
41	60	100	4000	72	650	625	−35,600	2902	−4764	595.2	−33,310	2867	−5347
42	140	200	4000	72	650	461	−22,800	3855	−6378	412.7	−18,850	3723	−7431
43	100	150	5000	60	600	432	−20,000	2728	−4445	432	−20,000	2728	−4445

presents the numerical experiment parameters for 43 IAB cases generated using CCD, with input variables including $K_s, E{I}_p, E{I}_g, H$, and L. FEM analyses were conducted for each configuration to produce datasets suitable for LR and RSM modeling.

The study considers two distinct soil types, sand and clay, to account for differences in soil stiffness distribution along the pile length. In sand, stiffness increases uniformly with depth, whereas in clay, it increases linearly up to a certain depth and then remains constant. To reflect these variations accurately, all numerical analyses were performed separately for each soil type.

5. Finite Element Modeling

5.1. Pile-Soil Interaction

Soil-structure interaction was modeled using spring elements representing soil stiffness, with properties calibrated based on empirical studies and existing literature. The p-y method was employed to capture soil behavior. Multilinear soil springs derived from p-y curves, as validated in previous studies [21], were integrated into the numerical models to accurately represent nonlinear soil-pile interaction. This approach, widely recognized in the literature, provides a robust framework for simulating lateral pile behavior under various soil conditions.

P-Y Method

The p-y method, developed by Reese and Van Impe [16] for computer program use, is widely adopted for modeling the nonlinear behavior of pile-soil interaction. It employs the Winkler approach, treating the soil as discrete, independent resistances rather than a continuum. This method has proven to provide reasonable responses [16].

The p-y method correlates the soil resistance (p, force per unit length of the pile) and pile deflection (y) through a nonlinear curve, representing real soil behavior. The method assumes that soil pressure corresponds to pile deflection, influenced by the stiffness of both the pile and soil. This correlation is illustrated in Figure 4.

In this study, the approximate method of Greimann, Abendroth, Johnson and Ebner [24] was used in the development of p-y curves, which adapts the modified Ramberg-Osgood model. Using this method, the expressions [24] that enable us to get the spring stiffness for soft and stiff clay are as follows:

For soft and stiff clay, the soil stiffness is calculated as:

$$k_h={\displaystyle \frac{p_u}{y_{50}}}$$

(4)

where the ultimate soil resistance is calculated from the following expression:

$$p_u=min\left\{\begin{array}{c}9·c_u·B_p\\ \left(3+{\displaystyle \frac{\gamma }{c_u}}·z+{\displaystyle \frac{0.5}{B_p}}·z\right)·c_u·B_p \end{array}\right.$$

(5)

For very stiff clay the soil stiffness is calculated as:

$$k_h={\displaystyle \frac{p_u}{2·y_{50}}}$$

(6)

where the ultimate soil resistance is calculated from the following expression:

$$p_u=min\left\{\begin{array}{c}9·c_u·B_p\\ \left(3+{\displaystyle \frac{\gamma }{c_u}}·z+{\displaystyle \frac{2}{B_p}}·z\right)·c_u·B_p \end{array}\right.$$

(7)

For sand, the soil stiffness is obtained by the following equation:

$$k_h={\displaystyle \frac{J·\gamma ·k}{1.35}}$$

(8)

Details of the soil spring parameters, representative undrained shear strength values, and their correlations are provided in Appendix A.

To assign the spring elements, the beam elements representing piles were divided into 30 discrete elements, each 1 ft. (0.3 m) long, for a total length of 30 ft (9.14 m) from the ground surface. This allowed for the assignment of linear-elastic springs to the nodes of adjacent discrete elements, representing the soil’s response to the piles. The soil properties considered in the analysis of all the bridge models are given in

Table 4. Soil parameters used in the analysis.

Soil Type	Unit Weight (ɣ)	Cohesion (c)	Angle of Internal Friction (Ø)	Strain at 50% Strength (ε50)
	(pcf)/(N/m3)	(Psi)/(MPa)	(Deg)
Soft Clay	120/18,852	2.6/0.018	-	0.02
Medium Clay	120/18,852	5.2/0.036	-	0.01
Stiff Clay	120/18,852	10.4/0.072	-	0.007
Very Stiff Clay	120/18,852	20.8/0.143	-	0.005
Loose Sand	120/18,852	-	30	-
Medium Sand	120/18,852	-	35	-
Dense Sand	120/18,852	-	40	-

.

5.2. Two-Dimensional Frame Modeling of Bridges

Figure 5 illustrates the 2D frame model of a typical bridge developed in SAP2000. Soil-pile interaction was represented by assigning discrete spring stiffnesses at regular intervals (1 ft or 0.3 m) along the length of the piles. The bridge was analyzed under AASHTO HL-93 live load conditions, which include a design truck with a front axle load of 8 kips (35.6 kN) and rear axles of 32 kips each (142.3 kN), a tandem load of two 25-kip axles (111.2 kN) spaced 4 ft (1.22 m) apart, and a uniform lane load of 0.64 kip/ft (9.34 kN/m) over a 10-ft (3.05 m) lane width. These loads were applied following AASHTO-recommended combinations to capture the most critical structural effects. The figure also depicts the deformed shape of the bridge model under the combined action of truck and lane loads, demonstrating the system’s response to representative live load scenarios.

For each analysis case, influence line analyses were performed to evaluate the rotation at the pile-abutment joint under moving truck load conditions. These analyses identified the critical positions of the AASHTO HL-93 truck load along the bridge span that produced the maximum rotational response. Truck axle loads were then applied as concentrated point loads at these critical positions, while lane loads were uniformly distributed along the girder length. Since the reference bridges were assumed to have two traffic lanes, both truck and lane loads were applied in duplicate to reflect the full loading configuration.

5.3. Typical Pile Response and Spring Model Representation

A representative pile behavior obtained from 2D numerical analysis is presented as an illustrative case under selected soil and pile properties. In this analysis, seven different steel H-pile sections (HP10x42, HP10x57, HP12x53, HP12x63, HP14x73, and HP14x89) were modeled interacting with a single soil type, medium sand, under AASHTO HL-93 truck loading conditions. For Bridge I, the corresponding pile responses in terms of lateral displacements and internal forces are shown in Figure 6 and Figure 7, respectively. Figure 6 illustrates the variation in lateral deflection and rotation along the pile length, while Figure 7 presents the corresponding shear force and bending moment distributions. As plotted, the piles develop a characteristic double curvature under the defined vehicular loads. At a certain depth below the ground surface, pile rotation reduces to zero, indicating the location of the inflection point, beyond which the curvature reverses. A second point of zero rotation typically occurs deeper, with its position influenced by both pile stiffness and soil properties.

The distributions in Figure 7 indicate that the maximum shear force and bending moment both develop at the pile head. Likewise, Figure 6 shows that the largest rotation is concentrated at the pile head. In contrast, Figure 6 demonstrates that the maximum lateral deflection occurs at a depth of about 6.5 ft (2 m) below the ground surface. These patterns are consistent with the expected pile–soil interaction mechanisms in short-span IABs, noting also that the applied vehicle loading was intended to create maximum rotations at the pile–abutment joint.

Figure 8 illustrates the spring model representations for the pile-abutment joint response obtained for each analysis set. The target rotational spring stiffnesses were determined by dividing the pile head moments by their corresponding rotations, while the target translational spring stiffnesses were calculated by dividing the pile head shear forces by the lateral displacements.

6. Design Optimization Using LR and RSM

Both LR and RSM analyses were performed using the same data set, as presented in Table 3, and under similar conditions. Additionally, response optimization was performed to identify the most influential factors affecting the outputs. The Minitab software (version 21.1.0) [25] was employed for the design process.

6.1. Development of Predictive Equations Using LR and RSM

LR and RSM models were developed to predict the analysis outputs (i.e., θ, Δ, M, and V) based on the selected input variables (i.e., Kₛ, EIₚ, EIg, H, and L).

Table 5. LR equations for θ, Δ, M, and V based on IAB properties (K_s, EI_p, EI_g, H, and L) for sand and clay soils. All coefficients are in consistent units: θ (rad × 10⁻⁶), Δ (in × 10⁻⁶), M (kip-in), V (kip × 10⁻²). For SI units: 1 in = 25.4 mm, 1 kip-in = 0.113 N·m, 1 kip = 4.448 kN.

	Sand Soil				Clay Soil
Variable	Θ (rad × 10−6)	Δ (in × 10−6)	M (kip-in)	V (kip × 10−2)	Θ (rad × 10−6)	Δ (in × 10−6)	M (kip-in)	V (kip × 10−2)
R2 (%)	98.84	92.73	97.21	97.13	94.99	91.90	96.87	97.08
Constants	256.3	5748	−2657	4705	293.2	−1020	−2323	5187
Linear Terms
$K_s$ (kip/in)	−1.071	70.01	1.430	−15.157	−1.49	99.87	1.069	−22.75
$E{I}_p$ (kip-in2)	−1.133	72.20	8.598	−3.789	−0.962	59.71	8.048	−2.057
$E{I}_g$ (kip-in2)	−0.10765	3.225	−0.5504	0.8399	−0.09235	2.903	−0.5053	0.9265
$H$ (in)	−5.196	−248.3	−8.23	−0.45	−5.467	−193.4	−11.42	3.46
$L$ (in)	2.107	−73.25	11.436	−17.799	1.95	−63.32	11.055	−19.952

presents the LR equations, while the corresponding RSM equations are provided in

Table 6. RSM regression equations for θ, Δ, M, and V based on IAB properties (K_s, EI_p, EI_g, H, and L) for sand and clay soils. All coefficients are in consistent units: θ (rad × 10⁻⁶), Δ (in × 10⁻⁶), M (kip-in), V (kip × 10⁻²). For SI units: 1 in = 25.4 mm, 1 kip-in = 0.113 N·m, 1 kip = 4.448 kN.

	Sand Soil				Clay Soil
Variable	Θ (rad × 10−6)	Δ (in × 10−6)	M (kip-in)	V (kip × 10−2)	Θ (rad × 10−6)	Δ (in × 10−6)	M (kip-in)	V (kip × 10−2)
R2 (%)	99.81	98.82	99.58	99.78	99.61	99.55	99.79	99.79
Constants	−609	12,769	276	−2372	−268	68,654	−342	−3510
Linear Terms
$K_s$ (kip/in)	−1042	28.4	6.44	−14.74	−1.667	−16.1	9.77	−22.15
$E{I}_p$ (kip-in2)	−1412	66.4	5.38	−4.95	−1.608	31.9	5.13	−1.51
$E{I}_g$ (kip-in2)	−0.2014	3.30	−0.567	0.842	−0.1992	8.96	−0.583	0.458
$H$ (in)	0.75	−384	11.3	−9.9	−1.37	−686	15.4	−12.4
$L$ (in)	4.80	−71	−0.46	7.2	3.98	−247.4	1.29	12.7
Quadratic Terms
${K_s}^2$	0.00659	−0.529	−0.01033	0.0669	0.00967	−0.4537	−0.01524	0.0871
$E{I_p}^2$	0.00432	−0.406	−0.01741	0.0053	0.00423	−0.2039	−0.01536	−0.0101
$E{I_g}^2$	0.000017	0.000103	0.000087	−0.000121	1.70 × 10−5	−0.0007	0.000073	−0.000020
$H^2$	0.0090	2.73	0.0571	−0.041	0.0267	5.04	0.0554	−0.079
$L^2$	0.00042	−0.038	0.01289	−0.0239	0.00098	0.1096	0.01159	−0.0304
Interaction Terms
$K_s·E{I}_p$	0.001781	−0.1309	−0.00769	0.0114	0.00282	−0.2011	−0.01066	0.02081
$K_s·E{I}_g$	0.000306	−0.01670	0.000659	0.000812	0.000415	−0.02223	0.001059	0.000867
$K_s·H$	−0.00378	1.172	−0.0578	0.0779	−0.00285	1.502	−0.0884	0.159
$K_s·L$	−0.003844	0.2447	−0.00050	−0.0377	−0.00542	0.3558	−0.00322	−0.05547
$E{I}_p·E{I}_g$	0.000338	−0.01911	−0.000467	−0.000969	0.000274	−0.01468	−0.000476	−0.001184
$E{I}_p·H$	0.00698	0.592	−0.0606	0.0597	0.00828	0.37	−0.06	0.0522
$E{I}_p·L$	−0.004375	0.2708	0.02375	−0.00374	−0.00371	0.2181	0.0227	0.00137
$E{I}_g·H$	0.001542	0.0022	0.005698	−0.00725	0.001521	−0.00518	0.006161	−0.00935
$E{I}_g·L$	−0.000307	0.00619	−0.001407	0.002036	−0.000277	0.00479	−0.001274	0.002065
$H·L$	−0.02052	−0.673	−0.0477	0.0322	−0.02136	−0.503	−0.0569	0.0496

. These models were used to estimate bridge responses and compared against FEM results. Note that the predicted values for θ and Δ should be multiplied by 10⁻⁶, and those for V by 10⁻², to convert them to actual units consistent with FEM output.

The LR and RSM results exhibit strong predictive capability, with high R² values obtained across all response variables. For sand soil, LR produced R² values of 98.84% for θ, 92.73% for Δ, 97.21% for M, and 97.13% for V, which indicates strong accuracy. For clay soil, LR yielded 94.99% for θ, 91.90% for Δ, 96.87% for M, and 97.08% for V, also demonstrating high reliability. RSM further improved predictive performance across both soil types. For sand soil, RSM achieved 99.81% for θ, 98.82% for Δ, 99.58% for M, and 99.78% for V. For clay soil, RSM achieved 99.61% for θ, 99.55% for Δ, 99.79% for M, and 99.79% for V. These values show that while LR provides robust predictive models, RSM attains near-perfect fits for all response variables and thus offers a more reliable representation of nonlinear relationships among the design parameters.

6.2. RSM and LR Predicted Response

Figure 9 compares the LR-predicted responses for rotation (a), deflection (b), moment (c), and shear (d) with the FEM results for sand soil. While LR demonstrates reasonable accuracy for predicting rotation and deflection, notable discrepancies emerge in the scatter plots for moment and shear, particularly at higher response magnitudes. For instance, deviations in moment and shear predictions exceed 10% in several cases, especially under extreme loading conditions.

Figure 10a–d presents the RSM-predicted responses for the same variables. The scatter plots indicate strong correlations between the predicted and FEM-obtained values, with most data points closely aligning along the equality line. Minor deviations appear at dataset extremes, suggesting model limitations under these conditions. Nonetheless, percentage errors for rotation predictions generally remain below 5%.

The comparison of clay soil results (Figure 11 and Figure 12) reveals similar variations. RSM predictions closely match the FEM responses across all response variables, whereas LR maintains acceptable accuracy for rotation and deflection but shows relatively larger discrepancies for moment and shear predictions. These outcomes demonstrate RSM’s superiority over LR in capturing nonlinear interactions and complex parameter dependencies within IAB systems.

6.3. Interaction of Variables with Responses Obtained by RSM

Figure 13, Figure 14 and Figure 15 present interaction plots illustrating how paired variables influence pile-abutment joint responses (θ, Δ, M, and V) in sand-type soil. Each plot shows the mean response at different levels of one factor as the other is varied. Similar trends were observed for clay-type soil and are therefore not shown.

Figure 13 demonstrates that θ decreases with increasing soil stiffness (Kₛ), pile rigidity (EIₚ), and girder rigidity (EIg). For example, increasing Kₛ from 20 to 180 kip/in (3.5 to 31.5 kN/mm) at EIₚ = 100 × 10⁶ kip-in² (42 × 10⁶ kN·m²) reduces θ from 0.000770 to 0.000585 rad. Similarly, θ is sensitive to geometric variables: increasing bridge span (L) increases θ, while taller abutments (H) reduce it.

In Figure 14, Δ also decreases with higher Kₛ, EIₚ, and EIg. For instance, increasing EIₚ from 50 to 250 × 10⁶ kip-in² (21 to 105 × 10⁶ kN·m²) at Kₛ = 100 kip/in (17.5 kN/mm) reduces Δ from −0.035 in (−0.9 mm) to −0.021 in (−0.54 mm). In contrast, increases in H and L generally lead to higher deflections, especially for longer spans.

Figure 15 shows that M is moderately affected by stiffness variables but strongly influenced by geometry. For example, increasing EIg at Kₛ = 100 kip/in reduces M from 4876 to 2670 kip-in (551 to 302 kN·m), while increasing L from 500 to 700 in (12.7 to 17.8 m) at H = 60 in (1.52 m) increases M from 2410 to 4698 kip-in (272 to 531 kN·m).

Figure 16 indicates that V generally increases with Kₛ and EIₚ due to higher load transfer efficiency. For instance, increasing Kₛ from 20 to 180 kip/in at EIₚ = 100 × 10⁶ kip-in² causes V to rise from −35.96 to −61.13 kip (−160 to −272 kN). Conversely, increased EIg reduces V, especially at lower stiffness ranges. Geometric effects vary: longer spans consistently increase V, while taller abutments tend to reduce it under fixed span conditions.

Overall, the interaction plots confirm that increasing structural and soil stiffness reduces rotational and deflection responses but may elevate moment and shear due to load redistribution. Span length consistently amplifies demands, while abutment height provides moderating effects depending on the response type.

7. FEM vs. RSM Predictions

RSM predictions for pile-abutment rotation, deflection, moment, and shear responses of Bridge I were evaluated under varying pile parameters (seven different HP piles: HP10x42, HP10x57, HP12x53, HP12x63, HP14x73, HP14x89) and different soil conditions (sand types: loose sand, medium sand, and dense sand; and clay types: soft clay, medium clay, stiff clay, and very stiff clay). The results, presented in Figure 17 and Figure 18, illustrate the prediction accuracy and numerical error values, categorized by soil type and pile flexural stiffness. For sand types (Figure 17), the errors between RSM and FEM predictions were generally small, with the rotation response for loose sand exhibiting the lowest error range (2.1% to 5.5%). For the medium sand condition, the predictions from the RSM and FEM models showed strong agreement, with most errors remaining below 5%. In contrast, for the dense sand condition, error rates exhibited an increasing tendency, particularly for the deflection response, where the maximum error reached 16.6%. Moment predictions were highly accurate across all sand types, with error rates predominantly below 8%. Shear predictions displayed a mixed tendency, with minimal errors (0.8% to 6.6%) observed for loose sand, while higher errors occurred for medium and dense sands, reaching up to 12.1%. While 2D–3D discrepancies are inherent, the additional LR and RSM approximations introduced only modest error increases, with cumulative deviations remaining below 15% for most soil conditions and response variables. For very stiff soils, cumulative errors may approach 20–27%, which indicates that the models remain within acceptable engineering limits but should be applied with caution in extreme stiffness ranges or for longer spans.

For clay types (Figure 18), RSM predictions for rotation response demonstrated high accuracy for soft and medium clay conditions, with error rates below 5% in most cases. However, error rates increased to as high as 17.7% for stiff and very stiff clays. Deflection response predictions showed a clear tendency of increasing error rates with clay stiffness, remaining below 8% for soft and medium clay conditions but reaching 26.8% for very stiff clay. For moment response, consistent prediction accuracy was observed across all clay types, with errors remaining below 12%. Predictions for shear force followed a similar tendency to deflection response, with error rates increasing for stiffer clay types. The highest errors were observed for very stiff clay, exceeding 18%, highlighting potential limitations of the RSM model in capturing shear behavior in high-stiffness materials. Overall, the RSM model performed well for both sand and clay types, particularly for loose sand and soft clay, where errors were minimal. However, challenges were noted in predicting deformation and shear responses for dense sand and very stiff clay.

8. Implementation of Spring Model for Pile-Abutment Joint

To demonstrate the implementation of the spring model for a pile-abutment joint, the 2D frame model of Bridge I was modified to include rotational and linear springs. To further clarify the procedure, a flowchart has been developed (Figure 19), illustrating the step-by-step process of deriving the spring stiffness constants from the regression equations (Table 5 and Table 6) and implementing them within structural analysis software.

These springs were attached to the bottom joints of the abutment element to replicate the pile-soil interaction model behavior, as illustrated in Figure 20. The stiffness values of the springs were derived from the pile-abutment joint responses predicted for a given soil, substructure and superstructure bridge parameters by the RSM regression equations (Table 6). Rotational spring stiffness was calculated by dividing the pile head moments by the pile head rotations (K_θ = M/θ), while linear spring stiffness was determined by dividing the pile head shear forces by the lateral deflection at the pile head (K_Δ = V/Δ). The analysis was conducted under loading conditions similar to those used in the experimental design. For demonstration purposes, medium and stiff clay soils were assumed as the foundation soil types, and seven different HP pile sections were considered for comparison.

The comparison between the predictions of the spring model and FEM results, along with the corresponding error values, is presented in Figure 21. The spring model demonstrated good accuracy in predicting FEM results for medium clay. Error rates for pile-abutment joint rotation, deflection, and bending moment are generally below 5%. However, the shear forces were moderately predicted with increased error rates. For stiff clay, the errors in the spring model’s predictions, particularly for deflection and moment, were slightly higher compared to medium clay, ranging between 5% and 13.3%. Rotational response predictions also exhibited slightly greater errors in stiff clay, with values reaching up to 6.3%. On the other hand, shear force predictions in stiff clay were more accurate than those in medium clay, with error rates ranging between 5.3% and 13.7%.

9. Conclusions

This study developed and validated predictive models to analyze and optimize the behavior of pile–abutment joints in short-span IABs. Using LR and RSM, practical and computationally efficient spring-based models were established to represent the nonlinear characteristics of pile–soil interaction. The proposed models provide viable alternatives to conventional FEM analyses, particularly for early-stage design and parametric studies. The principal findings of the study can be summarized as follows:

• The complex and nonlinear behavior of pile–soil interaction can be effectively approximated using the proposed spring models, in which stiffness values are derived from RSM equations.
• RSM provides better prediction accuracy for pile–abutment responses of short-span IABs under live loads and varying soil and structural conditions, significantly outperforming LR in capturing complex parameter interactions.
• The novelty of this study lies in deriving regression-based spring constants calibrated against FEM analyses, which is a practical and resilience-oriented technique for bridge designers. Unlike prior studies relying solely on FEM or traditional p–y curves, the proposed model enables direct use of predictive spring constants in design software.
• Prediction errors for soft clay and very stiff clay exceed 20% in some cases, reaching up to 27% for very stiff clays. Extrapolation to high-stiffness soils should therefore be applied cautiously, with engineering judgment and conservatism as required by design codes.
• For other soil conditions, prediction errors remain below 20% for rotation and deflection, and below 15% for moment and shear. The models are most reliable within the ranges K_s = 20–180 kip/in (3.5–31.5 kN/mm), EI_p = 50–250 × 10⁶ kip·in² (0.14–0.72 × 10⁶ kN·m²), EI_g = 1000–5000 × 10⁶ kip·in² (2.9–14.3 × 10⁶ kN·m²), H = 3–7 ft (0.91–2.13 m), and L = 42–58 ft (12.7–17.8 m). A design safety factor of 1.2–1.3 is recommended when applying regression-derived stiffness values.
• Sensitivity analyses revealed that increasing soil stiffness reduces pile–abutment rotation and deflection, slightly increases moments, but considerably raises shear demand.
• Increasing pile rigidity reduces joint rotation and deflection but increases moments and shear forces.
• Increasing girder flexural rigidity reduces joint rotation, deflection, moments, and shear forces.
• Increasing abutment height slightly reduces moments and joint rotation but increases deflection, with negligible influence on shear.
• Increasing span length amplifies joint rotation, deflection, moments, and shear forces due to larger lever arms and higher load demands.
• Prediction errors increase with span length due to amplified lever-arm effects and greater sensitivity of pile-abutment responses to geometric nonlinearity. Regression models capture these effects within the studied range of 42–58 ft (12.7–17.8 m), but extrapolation to longer spans introduces cumulative deviations.
• The proposed regression-based spring model is restricted to static analysis of short-span IABs. It does not capture dynamic effects such as seismic or thermal cycling, nor does it consider pile-to-pile interaction effects. These aspects should be addressed in future research.
• The findings regarding the influence of soil stiffness on pile–abutment rotation, deflection, moment, and shear are valid only for the dry/static soil conditions modeled herein. Pore pressure evolution, seasonal groundwater fluctuations, and long-term environmental effects are not accounted for in this study, as they are beyond its scope.

Overall, the proposed RSM- and LR-based spring models provide a reliable and efficient framework for estimating pile–abutment joint behavior in short-span IABs.