A new approach to soil-pile-structure modeling of long-span bridges subjected to spatially varying earthquake ground motion

Abstract

This paper presents a study about the spatial variability effects of ground motion and Soil-Pile-Structure Interaction (SPSI) on the dynamic response of a long bridge. The Spatially Varying Earthquake Ground Motion (SVEGM) is simulated by SIMQKE-II record generator. Target response spectrum and power spectral density function used in the simulation are determined depending on the January 17, 1994, Northridge earthquake. To evaluate the effect of SPSI, the soil surrounding the pile foundation is modelled by frequency- independent springs and dashpots in the horizontal and rotational directions. The effect of soil-pile mass is considered by lumped-mass soil-pile model. A new analytical model is proposed to study the effect of both SVEGM and SPSI on dynamic response of long bridges. The study reveals that the effect of both SVEGM and SPSI in the dynamic analysis and design should be considered simultaneously. A comparison between the results, based on the new suggested model and those given for other similar studies shows that the model could be applicable in calculation of dynamic response of long bridges. Considering the effect of soil-pile contributed mass along with the stiffness and damping is result in applying a model similar to a real soil-pile-structure system. It can simulate the real dynamic behavior of substructure more exactly.

Keywords

Spatially varying earthquake ground motions Soil-Pile-Structure interaction long bridges dynamic response SIMQKE

1 Introduction

Large-dimensional structures, such as long-span bridges, receive different ground motions at different supports in earthquake events. Seismic wave propagation and local site conditions cause spatial variation of ground motion. It may result in pounding or even collapse of adjacent bridge decks owing to the out of phase response. In addition, dynamic Soil-Structure Interaction (SSI) resulting from the interaction of the bridge with the surrounding soil also affects the dynamic bridge response. In most bridges, shallow foundations are not appropriate. Because they do not provide the required capacity or may experience excessive settlements or deformations. In such structures, pile groups are used as foundation system. Pile foundations have to be designed to support not only vertical loads, but also lateral loads due to earthquake, wind and vehicle impact loads. Therefore, soil-pile interaction is added to above factors in dynamic behavior of long-span bridges. From the above reasons, it is very important to consider both Soil-Pile-Structure interaction (SPSI) and Spatially Varying Earthquake Ground Motion (SVEGM) effects in evaluation of the seismic response of long-span bridges.

Several past studies investigated the effect of SVEGM and SSI, separately. Some of thememphasized the importance of the SVEGM effect on long structures [1 –7]. On the other hand, some investigations underlined the importance of SSI effect [8 –13]. Also some studies indicated the effect of SSI on bridge systems when subjected to SVEGM. [14, 15] studied the influence of spatially near-source ground motions and SSI on the relative response of two bridge frames. It was shown that the assumption of uniform ground excitation and fixed base in the analysis and design might not provide a realistic estimation of the response of bridge frames. [16] investigated the effects of SSI and SVEGM on the dynamic characteristics of cable-stayed bridges. The substructure method was used in the analysis. It was concluded that both SSI and SVEGM effects should be considered in the dynamic analysis of cable-stayed bridges. [17] also studied the effects of both SVEGM and SSI on bridge responses. The soil surrounding the pile foundation was modeled by frequency-dependent springs and dashpots. The standard random vibration method was used to estimate peak structural response. It was concluded that SSI is an important factor in structural response analysis and could not be neglected.

The focus of this paper is to present a new analytical model for long-span bridge systems subjected to SVEGM and under the effect of SPSI. For this purpose, ground motion time histories are simulated with SIMQKE II record generator, provided by university of California, Berkeley, based on the Northridge earthquake of January 17, 1994, and applied to the system.

2 Proposed model for the bridge system

Most previous works neglected the effect of pile groups in numerically evaluation of SVEGM and SSI. Figure 1 illustrates the schematic view of two largest decks in the middle of a long bridge standing on pile groups.

Fig.1

(a) Schematic view of the bridge system and (b) Structural model.

Two decks with length d₁ = 100 m and d₂ = 150 m are supported by four isolation bearings which are connected to three elastic piers standing on the pile foundations. The structure of the bridge continues from both sidesand neighboring spans are separated of the considered structure by appropriate expansion joints. The decks are considered as lumped mass model with the total mass of m₁ = 1.2×10⁶ kg and m₂ = 1.8×10⁶ kg. All of the bearings have the same dynamic properties with an effective stiffness K_b1 and equivalent viscous damping C_b1 for the left span, and K_b2 and C_b2 for the right span. The concrete piers with heights of h₁ = 14 m, h₂ = 16 m and h₃ = 15 m are modelled as elastic columns with lumped mass m₃ = m₄ = m₅ = 2×10⁵ kg at the top of each pier. The lateral stiffness of the piers are Kp₃ = 2×10⁸ N/m, Kp₄ = 10⁸ N/m and Kp₅ = 3×10⁸ N/m.As pointed out by the [18], traditionally, the damping ratio of the bridge system is assumed to be 5% without considering the energy dissipation at the bridge supports. In this paper, because the efficiency of the linear elastic model is interested to be studied, just damping ratio of the 5% is considered [17]. This value refers to the internal damping ratio and total damping ratio consists of internal and radiation damping. The most widely used model to perform the analysis of piles under lateral loads, consists in modeling the pile as a series of beam elements and representing the soil as a group of unconnected-concentrated springs perpendicular to the pile which is known as Discrete Winkler Model. [19] developed a rational model that includes the soil contribution to the system inertial properties through a series of lumped mass, consistent with the Discrete Winkler Model, which is used in this study. This simplified lumped model was obtained by performing an approximation of the plane strain model developed by [20]. In the proposed model, the pile-soil interaction is taken into account through three frequency independent elements: a spring with stiffness K_a, a mass with value m_a and a dashpot with coefficient C_a. Figure 2 shows the elements of the simplified model.

Fig.2

Pile element with distributed soil parameters (Pacheco, 2007).

In geotechnical practice, when the response of a pile group is of interest, such pile-soil interaction effects are often assessed through the use of nonlinear methods but in this paper instead of the frequency dependent coefficients for simulating the springs and dashpots, the frequency independent coefficients have been used. This linear model is simple but capable because it can be used in primary design and studies amongst the uncertainties of nonlinear modeling. Also this linear modeling is chosen to detect the error rate between the linear and nonlinear models in different types of soil condition under the structure.

Based on Novak’s analytical expression, the dynamic stiffness of piles can be written as: $K_{u} = G π a_{0}^{2} T (a_{0}, ν, D)$ (1)

Where: $T = - \frac{4 K_{1} (b_{0}^{*}) K_{1} (a_{0}^{*}) + a_{0}^{*} K_{1} (b_{0}^{*}) K_{0} (a_{0}^{*}) + b_{0}^{*} K_{0} (b_{0}^{*}) K_{1} (a_{0}^{*})}{b_{0}^{*} K_{0} (b_{0}^{*}) K_{1} (a_{0}^{*}) + a_{0}^{*} K_{1} (b_{0}^{*}) K_{0} (a_{0}^{*}) + b_{0}^{*} a_{0}^{*} K_{0} (b_{0}^{*}) K_{0} (a_{0}^{*})}$ (2)

Where K_n is modified Bessel function of the second kind of order n, $a_{0} = \frac{ω r_{0}}{V_{s}}$ is dimensionless frequency, ω is vibration frequency in (rad/sec), r₀ is cylinder radius, V_s is shear wave velocity of the soil, ρ is mass density of the soil, $a_{0}^{*} = \frac{a_{0}}{\sqrt{1 + iD}} i$ is complex dimensionless frequency, $b_{0}^{*} = \frac{a_{0}^{*}}{η}$ , $η = \sqrt{\frac{2 (1 - ν)}{1 - 2 υ}}$ , v is Poisson’s ratio of the soil, G is Shear modulus of soil and D is Damping coefficient of soil. By grouping the real and imaginary parts, Equation 1 can be rewritten as: $K_{u} = G [S_{u 1} (a_{0}, ν, D) + i S_{u 2} (a_{0}, ν, D)]$ (3)

Where, S_u1 and S_u2 are real functions. The variation of these functions with a₀, v and D are defined in Novak’s work. The dynamic stiffness of the SDOF system is: $K_{d} (ω) = K - ω^{2} M + i ω C$ (4)

By comparing Equation 1, which describes the soil dynamic stiffness according to the Novak’s model, with Equation 4, which describes the dynamic stiffness of a single degree of freedom system, and introducing a function f, a new dynamic stiffness proposed as below:

$\begin{matrix} K_{u} = G π f (a_{0}, ν, D) = G π {Real [f (a_{0}, ν, D)] \\ + iImag [f (a_{0}, ν, D)]} \end{matrix}$ (5)

In which:

$f (a_{0}, ν, D) = a_{0}^{2} T (a_{0}, ν, D)$

Introducing the following approximations in the real and imaginary parts of the complex function does this: $\begin{matrix} Real [f (a_{0}, ν, D)] \approx α_{k} - α_{m} a_{0}^{2}; \\ Imag [f (a_{0}, ν, D)] \approx α_{c} a_{0} \end{matrix}$

The dynamic stiffness K_u becomes: $K_{u} \approx G π (α_{k} - α_{m} a_{0}^{2} + i α_{c} a_{0})$ (6)

The coefficients α_k, α_m and α_c can be determined by the least square approximation of the function f (a₀, v, D). The subscripts k, m and c corresponded to stiffness, mass and damping coefficients, respectively. The basic equation of the method can be found in most text books on applied numerical analysis [21, 22]. Comparing Equation 6 with Equation 5, one can obtain $K_{u} \approx G π (α_{k} - α_{m} a_{0}^{2} + i α_{c} a_{0}) = K_{a} - m_{a} ω^{2} + i C_{a} ω$ (7)

The equivalent lumped stiffness coefficient K_a, lumped mass coefficient m_a, and the lumped viscous damper coefficient (radiation damping) C_a are: $k_{a} = π G α_{k}$ $m_{a} ω^{2} = G π α_{m} a_{0}^{2} = G π α_{m} \frac{r_{0}^{2} ω^{2}}{(G / ρ)} = ρ π r_{0}^{2} α_{m} ω^{2}$ (8) $\begin{matrix} c_{a} ω = G π α_{c} a_{0} = G π α_{c} \frac{r_{0} ω}{V_{s}} \\ = π r_{0} ρ V_{s} α_{c} = π r_{0} \sqrt{G ρ} α_{c} \end{matrix}$

These coefficients are defined per unit pile length. To apply lumped coefficients at each node of a pile discretized with beam elements, these values have to be multiplied by involvement length for each node. The values for each coefficient are shown in Fig. 3.

Fig.3

Variation of regression coefficients with Poisson’s ratio (Pacheco, 2007).

To determine the effect of SPSI, the soil supporting the piers is modeled as a system of distributed springs and dashpots acting in horizontal direction. To present a simplified, realistic and practical model, frequency independent coefficients indicated in Fig. 3 were used instead of Novak’s frequency dependent coefficients. Also pile caps are modelled by springs and dashpots in horizontal and rotational direction. Frequency independent coefficient for each one is provided by [23]. With the assumption of acting as a retaining wall and occurring the Δ_max, Gazetas coefficients can simulate the soil-structure interaction in front of the pile caps. Δ_max varies from 0.002 h to 0.04 h [24] in which, h is the thickness of the pile cap. With the assumption of the lower band and pile cap thickness of the 1 m in this paper, Δ_max will be 0.002 m or 2 mm which probably could be occurred during the earthquake.

This paper focuses on relative response in horizontal direction because the vertical stiffness of bridge structure is usually substantially larger than that in horizontal direction. Also owing to the fact that the first mode of vibration with large displacements is more important for designing and that is mostly under the effect of horizontal movements, the system assumed to be rigid in vertical direction.

Under these assumptions, the bridge system can be modelled as a seventeen degrees of freedom in order to investigate the effect of SPSI and SVEGM on seismic response of the system as shown in Fig. 1(b): the dynamic displacements u₁ and u₂ of the decks movement, the dynamic response u₃, u₄ and u₅ at the pier top, the horizontal displacement u₆, u₁₀ and u₁₄ of the pile foundation relative to the free field motion ug₁, ug₂ and ug₃, the rotational response φ ₇ , φ ₁₁ and φ ₁₅ of the pier at the foundation level, the dynamic response u₈, φ ₉ , u₁₂, φ ₁₃ , u₁₆, φ ₁₇ at the pile bottom.

3 Spatially varying earthquake ground motion

The programs SIMQKE I and SIMQKE II were used in this study to simulate SVEGM along the bridge supports. One of the most important factors in earthquake records generation is to produce power spectral density function, compatible with ground accelerations for desired soil condition. The program SIMQKE I is used to generate target spectral density function from a response spectra. In this study the target response spectra isNorthridge - 1994 earthquake response spectra. This power spectral density function is given to SIMQKE II as an input data. SIMQKE II as a conditioned earthquake ground motion simulator is designed to generate an array of different spatially correlated earthquake ground motion at an arbitrary set of points, optionally statistically compatible with known or prescribed motions at other locations.

This program is also used in unconditional mode. Unlike conditional mode in which generated ground motions are statistically compatible with, or conditioned by, recorded ground motions at nearby point, in unconditional mode the ground motions are simulated using only the user prescribed space-time statistics. Supplied with a target ground motion spectral density function, which may be evolutionary in nature, the program employs covariance matrix decomposition in the frequency domain followed by best linear unbiased estimation and an inverse fast Fourier transform to efficiently produce the nonstationary, spatially correlated, conditioned or unconditioned ground motions. Complete details of the non-uniform ground motion generation can be found in authors’ previous paper [25].

4 Equations of motion for the proposed model

In the present study, the dynamic response of the long span bridge shown in Fig. 1 is calculated. Firstly, equation of motion is solved in frequency domain, then by use of inverse Fourier transformation responses are derived in time domain. According to the pile group length and the considered site topography, it is assumed that the support motions are approximately equal to the free field motions [26, 27]. Therefore free field ground motions are used as input in the analysis. The equivalent spring-dashpot method is used to analysis SPSI effect. In this method, the super structure and the foundation medium, are treated as two independent models. The connection between the two models is established by the interaction forces acting on the interface. The dynamic equilibrium equations are finally written in terms of interface degrees of freedom. With this background, the dynamic equilibrium equations can be expressed in the matrix form as follows:

$\begin{matrix} [\begin{matrix} M_{ss} & M_{sf} \\ M_{fs} & M_{ff} \end{matrix}] {\begin{matrix} {\ddot{u}}_{s}^{t} \\ {\ddot{u}}_{f}^{t} \end{matrix}} + [\begin{matrix} C_{ss} & C_{sf} \\ C_{fs} & C_{ff} \end{matrix}] {\begin{matrix} {\dot{u}}_{s}^{t} \\ {\dot{u}}_{f}^{t} \end{matrix}} \\ + [\begin{matrix} K_{ss} & K_{sf} \\ K_{fs} & K_{ff} \end{matrix}] {\begin{matrix} u_{s}^{t} \\ u_{f}^{t} \end{matrix}} = {\begin{matrix} 0 \\ P_{f}^{t} \end{matrix}} \end{matrix}$ (9)

In which [M], [C] and [K] are the mass, damping and stiffness matrices, respectively, ${\ddot{u}}$ , ${\dot{u}}$ and {u} are the acceleration, velocity and displacement vectors, respectively and ${P_{f}^{t}}$ is the total nodal forces vector at the base degree of freedom. The subscripts s and f refer to the super structure and the foundation, respectively [34].

The total displacement is written as the sum of two displacement components of quasi-static and dynamic displacement vectors: ${\begin{matrix} u_{s}^{t} \\ u_{f}^{t} \end{matrix}} = {\begin{matrix} u_{s}^{d} \\ u_{f}^{d} \end{matrix}} + {\begin{matrix} u_{s}^{qs} \\ u_{g} \end{matrix}}$ (10)

Where $u_{f}^{d}$ is the interaction displacement vector at the structure-foundation contact points and u_g is the corresponding free field ground motion vector. To define the quasi-static displacement Equation 10 substitutes into Equation 9 and all the dynamic terms are put zero, then:

$[u_{s}^{qs}] = - [K_{ss}^{-} 1] [K_{sf}] {u_{g}} = \frac{1}{ω^{2}} [K_{ss}^{-} 1] [K_{sf}] {{\ddot{u}}_{g}}$ (11)

Equation 9 can be expressed in the frequency domain as: $[Z (i ω)] {u^{d} (i ω)} = [Z_{g} (i ω)] {u_{g} (i ω)}$ (12)

Where:

{u (iω) } = {u₁ (iω) u₂ (iω) u₃ (iω) u₄ (iω) u₅ (iω) u₆ (iω) φ₇ (iω) u₈ (iω) φ₉ (iω) u₁₀ (iω) φ₁₁ (iω) u₁₂ (iω) φ₁₃ (iω) u₁₄ (iω) φ₁₅ (iω) u₁₆ (iω) φ₁₇ (iω)} ^T

and

{u_g (iω) } = { u_g1 (iω) u_g2 (iω) u_g3 (iω) } ^T

Are defined as the dynamic response vector and the input ground motion vector (applied at the support points), respectively. [Z (iω)]and [Z_g (iω)] are the impedance matrices of the dynamic system. The ground motions are applied on the pile groups individually and each pile groups have their own ground motion vary along the bridge decks in the SVEGM model but the ground motions are the same along the piles because of the linear elastic model considered in the analytical model. $[Z (i ω)] = [\begin{matrix} z_{11} (i ω) & \dots & z_{117} (i ω) \\ ⋮ & ⋱ & ⋮ \\ z_{171} (i ω) & \dots & z_{1717} (i ω) \end{matrix}]$

and $[Z_{g} (i ω)] = [\begin{matrix} z_{g 11} (i ω) & \dots & z_{g 13} (i ω) \\ ⋮ & ⋱ & ⋮ \\ z_{g 171} (i ω) & \dots & z_{g 173} (i ω) \end{matrix}]$ (13)

In which: $z_{ij} (i ω) = - ω^{2} m_{ij} + i ω C_{ij} + K_{ij}$ (14)

Where, m_ij, C_ij and K_ij are the mass, damping coefficient and stiffness corresponded to each element, respectively. The above coefficients for the structural degrees of freedom are indicated in Table 1.

Table 1

Material properties for the structural members

Number of mass	Mass m(kg)	Stiffness K_b(N/m)	Damping Ratio C_b(%)	Stiffness K_p(N/m)	Damping Ratio C_p(%)
1	1.2×10⁶	12×10⁶	5	–	–
2	1.8×10⁶	12×10⁶	5	–	–
3	2×10⁵	12×10⁶	5	3×10⁸	5
4	2×10⁵	12×10⁶	5	1.9×10⁸	5
5	2×10⁵	12×10⁶	5	2.4×10⁸	5

To obtain the stiffness, damping and mass matrices, virtual work for a dynamic structural system, also known as the generalized D’Alambert principle is used. Each element of those then will be placed in the primary impedance matrix. $δ W_{int} - δ W_{ext} - δ W_{iner} = 0$ (15)

Where, δW_int is the virtual work of the internal forces, δW_ext is the virtual work associated to the external forces, and δW_iner is the virtual work of the internal forces. For a beam with a distributed force f (x, t), the three virtual work expressions are: $δ W_{int} = \int_{0}^{L} {EIv}^{″} (x, t) \frac{δ θ (x, t)}{dx} dx = \int_{0}^{L} {EIv}^{″} (x, t) \frac{d (δ v (x, t))}{d x^{2}} dx$

$δ W_{ext} = \int_{0}^{L} f (x, t) δ v (x, t) dx$ (16)

$δ W_{iner} = - \int_{0}^{L} \bar{m} v^{″} (x, t) δ v (x, t) dx$

Where, $\bar{m}$ is the distributed mass of the beam per unit length, E is the elastic modulus, I is the moment of inertia of the cross section, v (x, t) is the transverse displacement, and δv (x, t) is the virtual transverse displacement introduced below:

$\begin{matrix} v (x, t) = [N (x)] {q (t)}; δ v (x, t) \\ = [N (x)] {δ q (t)} = {δ q (t)}^{T} {[N (x)]}^{T} \end{matrix}$ (17)

$\begin{matrix} v^{″} (x, t) = [N^{″} (x, t)] {q (t)}; δ v^{″} (x, t) \\ = [N^{″} (x)] {δ q (t)} = {δ q (t)}^{T} {[N^{″} (x)]}^{T} \end{matrix}$ (18)

Where, [N (x)] is the shape function vector and {q (t)} is the end displacements and rotations vector. The internal virtual work becomes:

$\begin{matrix} δ W_{int} = \int_{0}^{L} {EIv}^{″} (x, t) δ v^{″} (x, t) dx \\ = \int_{0}^{L} EI δ v^{″} (x, t) v^{″} (x, t) dx \\ = {δ q (t)}^{T} [\int_{0}^{L} EI {[N^{″} (x)]}^{T} [N^{″} (x)] dx] {q (t)} \end{matrix}$ (20)

By defining the beam element stiffness matrix [K_b]as: $[K_{b}] = \int_{0}^{L} EI {[N^{″} (x)]}^{T} [N^{″} (x)] dx$ (21)

If the force f (x, t) is due to the distributed spring and dashpots, i.e. $f (x, t) = - K_{a} v (x, t) - C_{a} \dot{v} (x, t)$ , the virtual work of the external forces is: $\begin{matrix} δ W_{ext} = - \int_{0}^{L} K_{a} v (x, t) δ v (x, t) dx \\ - \int_{0}^{L} C_{a} \dot{v} (x, t) δ v (x, t) dx \end{matrix}$ (22)

Introducing the finite element approximation shown previously, that for the velocity field results: $\dot{v} (x, t) = [N (x)] {\dot{q} (t)}$ (23) $δ \dot{v} (x, t) = [N (x)] {δ \dot{q} (t)} = {δ \dot{q} (t)}^{T} {[N (x)]}^{T}$ (24)

The external virtual work becomes: $\begin{matrix} δ W_{ext} = - \int_{0}^{L} K_{a} δ v (x, t) v (x, t) dx - \int_{0}^{L} C_{a} δ v (x, t) \dot{v} (x, t) dx \\ = - {δ q (t)}^{T} [\int_{0}^{L} K_{a} {[N (x)]}^{T} [N (x)] dx] {q (t)} \\ - δ q {(t)}^{T} [\int_{0}^{L} C_{a} {[N (x)]}^{T} [N (x)] dx] {\dot{q} (t)} \end{matrix}$ (25)

By defining the soil stiffness matrix [K_s] and the soil damping matrix [C_s] as: $\begin{matrix} [K_{s}] = \int_{0}^{L} K_{a} {[N (x)]}^{T} [N (x)] dx; [C_{s}] \\ = \int_{0}^{L} C_{a} {[N (x)]}^{T} [N (x)] dx \end{matrix}$ (26)

By repeating the above procedure for the acceleration field and defining the virtual work of the inertial forces, the equation of motion for the free vibrations of element expresses as: $[K] {q (t)} + [C] {\dot{q} (t)} + [M] {\ddot{q} (t)} = 0$

Where, $= \int_{0}^{L} {[N^{''}]}^{T} EI [N^{''}] dx + \int_{0}^{L} {[N]}^{T} \times; K_{a} \times; [N] dx$ (27) $[C] = \int_{0}^{L} {[N^{'}]}^{T} C [N^{'}] dx + \int_{0}^{L} {[N]}^{T} \times C_{a} \times [N] dx$ (28)

As the piles in a pile group act in parallel, the lateral and rotational stiffness of the piles in a pile group can be summed up. To simulate the soil filling the space within the piles in every row, springs are used. It means the first part of the stiffness equation should multiply by the number of piles and the second part should multiply by the number of pile rows in pile group.

Therefore, the stiffness and damping coefficients of the rows of piles and the soil within each row can be written as:

$\begin{matrix} [K] = n \times n \int_{0}^{L} {[N^{''}]}^{T} EI [N^{''}] dx \\ + \int_{0}^{L} {[N]}^{T} \times n K_{a} \times [N] dx \end{matrix}$ (29)

$\begin{matrix} [C] = n \times n \int_{0}^{L} {[N^{'}]}^{T} C [N^{'}] dx \\ + \int_{0}^{L} {[N]}^{T} \times n C_{a} \times [N] dx \end{matrix}$ (30)

Where n is the number of piles in each row, [N] is the shape function vector, [N′] and [N′] are the first and second derivative of [N], E is the Young’s modulus of piles, I is the cross sectional moment of inertia, K_a is the spring coefficient and C is the damping ratio of the piles.

Similarly, the mass matrix of substructure system is written as:

$\begin{matrix} [M] = \int_{0}^{L} m_{pile} {[N]}^{T} [N] dx + \int_{0}^{L} m_{a} {[N]}^{T} [N] dx \\ = \int_{0}^{L} \bar{m} {[N]}^{T} [N] dx \end{matrix}$ (31) $\bar{m} = n^{2} ρ_{pile} \times; A_{pile} + n^{2} A_{pile} \times; ρ_{soil} \times; α_{m}$ (32)

In order to compute the spring and dashpot coefficients for the pile caps, [23] coefficients, as recommended in [28], for circular rigid foundation are used as below:

$\begin{matrix} K_{h} = \frac{8 G_{soil} r}{2 - ν_{soi}}; K_{θ} = \frac{8 G_{soil} r^{3}}{3 (1 - ν_{soil})}; K_{h θ} = \frac{0.56 G_{soil} r^{2}}{2 - ν_{soil}} \\ C_{h} = \frac{4.6}{2 - ν_{soi}} ρ_{soil} V_{s} r^{2}; C_{θ} = \frac{0.4}{1 - ν_{soi}} ρ_{soil} V_{s} r^{4}; \\ C_{h θ} = \frac{0.4}{2 - ν_{soi}} ρ_{soil} V_{s} r^{3} \end{matrix}$ (33)

Where r denotes the radius of the equivalent circle and V_s is the shear wave velocity.

The above coefficients are obtained for a circular rigid foundation, but foundations (pile caps) in the proposed model assumed to be square in shape. It is conventional to use an equivalent radius which can be calculated as: $r_{K h} = \frac{a}{\sqrt{π}}; r_{K θ} = \sqrt[4]{\frac{a^{4}}{3 π}}; r_{K h θ} = \frac{a}{\sqrt{π}}$ (34)

Where a is the square dimension.

Therefore, damping and stiffness matrices of the pile cap are: $[K_{cap}] = [\begin{matrix} K_{h} & K_{θ h} \\ K_{h θ} & K_{θ} \end{matrix}]; [C_{cap}] = [\begin{matrix} C_{h} & C_{θ h} \\ C_{h θ} & C_{θ} \end{matrix}]$ (35)

As already mentioned, the cap is massless. The mass, damping and stiffness matrices of the system are assembled from joining the super structure and sub structure coefficient matrices. Finally, the dynamic response of the bridge structure can then be calculated by: ${u (i ω)} = {[Z (i ω)]}^{- 1} [Z_{g} (i ω)] {u_{g} (i ω)}$ (36)

In order to obtain dynamic response in time domain, fast Fourier transformation is applied.

5 Finite element model of the bridge

Numerical model of the considered bridge is developed using ABAQUS 6.11 finite element analysis program. All components of the considered bridge including the 2×2 pile group underneath the piers, the piers, the decks, the capitals, and the supporting soil domain are simulated in a unified model. Solid elements are used to model the soil domain and structural elements. Figure 4 presents 3D model of the considered bridge. The model represents a soil domain of 450 m long (in direction z), 50 m wide (in direction x), and 20 m deep (in direction y). To connect pile elements to the surrounding soil elements, embedded region type of interaction is used. To simulate the real behavior of the soil, advanced nonlinear model is used for constitutive modeling of the foundation soil. The model includes a Drucker-Prager yield surface with a non-associative flow rule and a deviatoric kinematic hardening rule. The above model was successfully used in previous studies to simulate soil-foundation structure interaction problems [29, 30]. Four types of the soil as considered in analytical model is simulated here by this model. Elastic behavior is assigned to the piles, the pile caps, and the decks because in seismic design of bridge systems these elements are capacity protected so that damage is not allowed. In the design process, the bridge piers are usually allowed to yield since any possible damage can easily be detected and repaired. Therefore, the use of an appropriate nonlinear elasto-plastic model for the pier material is necessary. In this study the uniaxial Kent-Scott-Park model [31] is used for modelling of concrete material. Input parameters for all structural and soil constitutive model are not presented here for brevity. They all are compatible with the properties considered in the previous linear analytical model but nonlinear properties are added.

Fig.4

The developed finite element model of the bridge.

5.1 Dimensions of the soil domain

The lateral boundaries must be placed at a location where the effects due to the presence of bridge could be negligible. In other words, free-field conditions at the lateral boundaries of the finite element model should be established. To this end three models are considered for the soil. Trial dynamic analysis are performed for them with soil domain dimensions of 350 m×30 m×20 m, 450 m×50 m×20 m, and 550 m×70 m×20 m (in directions z, x, y shown in Fig. 5). Seconds of tenth to fifteenth of the 1992 Landers earthquake with PGA of 0.78 g including the major part of the motion, i.e. the PGA, is applied to the base of the model. This strong ground motion is selected due to mobilization of the highly nonlinear behavior of the employed materials. Figure 6 shows the time histories of shear force at the middle pier base. It can be observed that almost no difference between the results of the second and third models. Therefore, appropriate and sufficient dimensions of the model are 450 m×50 m×20 m. Then to make sure that free-field conditions at the lateral boundaries are captured, the soil response at the boundaries is investigated. The resulting acceleration time history on top of the soil profile without the presence of the bridge structures is compared to that at a point next to the lateral boundary of the soil-bridge model on the ground surface. Figure 7 shows that the model properly captures the free-field conditions.

Fig.5

3D model of the soil domain.

Fig.6

Sensitivity of the shear forces induced at the pier base to the model dimensions.

Fig.7

Time histories of acceleration on top of the soil profile model compared to that at points next to the lateral boundaries of the soil-bridge model.

5.2 Validity of the model

To confirm the results of this numerical model, time history of the recorded earthquake at the ground surface should be retrieved at a depth of 20 m before applying at the base of the model. Retrieval process was performed by the equivalent linear program PROSHAKE [32]. Therefore, a 20 m deep soil column is modeled in PROSHAKE with comparable properties to those in the free-field of the model. It means the same density of profile and shear wave velocity, representative modulus reduction and damping curve were considered. In retrieval procedure, the recorded earthquake ground motion in the free-field is applied as outcrop motion to the surface of soil column in the PROSHAKE model. The resulting motion at the base of the PROSHAKE model is applied in the form of displacement time history to the base of the ABAQUS model. To avoid resulting different computed motion at the ground surface in the free-field of the ABAQUS model from the corresponding recorded one, iterative process is used for estimation of the modulus reduction and damping curve of the soil profile in the PROSHAKE model. To this end, the choice of these curves from the available curves in the library of PROSHAKE [33, 34] is changed iteratively.

Time history of the recorded and computed acceleration from the last iteration at the ground surface for the 1994 Northridge earthquake is presented in Fig. 8. The comparison shows that this retrieved ground motion can be used as appropriate input motion for the analysis. Figure 9 shows the displacement response spectra at the lateral boundaries in the free-field of the ABAQUS model. It implies that a quite good agreement between the displacement response spectra of the recorded and computed motion is available. It means that the numerical model is generally capable of simulating the seismic responses of the considered system.

Fig.8

Time histories of the recorded and computed accelerations at the ground surface.

Fig.9

Displacement response spectre of motion for the damping ratio of 5% at the lateral boundaries.

6 Numerical results and calculations

The aim of this paper is to investigate the effect of SVEGM and SPSI on the dynamic response of a long-span bridge model. For this reason, ground motion time histories are simulated by the program SIMQKE II. All the non-uniform excitations are conditioned by the first 20.48 seconds of Northridge -1994 earthquake applied at the first support point. For each ground excitation case twenty sets of spatially correlated ground motions are simulated. Figure 10 shows one set of ground motions for the soil type of C. Also Fig. 11 shows the 5% damped acceleration response spectra of the simulated ground motions match well with the target response spectrum.

Fig.10

Simulated ground motions along the bridge supports for soil type of C.

Fig.11

Acceleration response spectrum of the simulated ground motions for soil types of C.

In order to study the effect of SPSI, four different soil types are considered for the bridge supports. It is assumed that all of the supports are founded on homogeneous soil type. Soil properties are shown in Table 2.

Table 2

Parameters of local soil properties

Soil type	Density ρ(kg/m³)	Shear wave velocity Vs (m/s)	Damping ratio ξ
Base rock (A)	2500	2000	0.05
Stiff soil (B)	2000	600	0.05
Medium soil (C)	1950	300	0.05
Soft soil (D)	1900	100	0.05

The longitudinal earthquake excitations are assumed to travel across the bridge model. In the present study investigation of the SVEGM and SPSI are mainly desired. Therefore the following conditions of considering the effect of SVEGM and SPSI are included here:

Uniform excitation and SPSI condition for the soil types of A, B, C and D.

SVEGM excitations and SPSI condition for the soil type of A, B, C and D.

Dynamic response of considered bridge model is obtained for all above conditions and compared with each other.

6.1 Effect of SPSI on dynamic response of the bridge

To study the effect of SPSI, the response of the bridge subjected to uniform ground motion is calculated with or without considering SPSI. Figures 12–15 show the influence of SPSI on displacement of the first deck (m₁) for both analytical and numerical models. It can be seen that the results of the proposed analytical model are close to those of the numerical model for soil types of A and B but they are far from each other for soil types of C and D. It means that this proposed model can be used for dynamic analysis of the structures constructed on base rock or hard soil. Non-zero displacements at the beginning of all diagrams can be observed. It happens because of the relative dynamic displacements produced at all non-support degrees of freedom due to the inertial action and displacements produced at the supports for maintaining elastic compatibility between the foundation and the soil. Also residual relative displacements at the end of time can be observed which refers to the non-zero displacements at the beginning of the time and non-zero input motion values at the end of time.

Fig.12

Displacement of the right deck (m1) for soil type of A.

Fig.13

Displacement of the right deck (m1) for soil type of B.

Fig.14

Displacement of the right deck (m1) for soil type of C.

Fig.15

Displacement of the right deck (m1) for soil type of D.

Displacement results for soil type of A are different from those for other types of soil. It occurs because properties of soil type of A are similar to those for base-rock therefore soil and structure system act as a unified system. Change in free vibration frequency of the soil-structure system may cause it to escape from the resonance.

The effect of SPSI on maximum shear force of the pier 4 for every soil types are also compared in Fig. 16.

Fig.16

Maximum shear forces of the pier 4 for all types of soil in Uniform condition.

The values detetermined from the analyses cases with SPSI effect are significantly larger than those obtained from the fixed-base model, without considering the SPSI effect, except in soil type A where decreasing in maximum shear force can be observed in evaluation of SPSI effect. These values indicate that neglecting SPSI will result in about15%, 49% and 67% underestimating the maximum shear force of the brige pier in B, C and D soil type for uniform excitation, respectively.Also results for the numerical model indicate that proposed analytical model can truly simulate dynamic response of the structures constructed on the hard and stiff soil such as soil type of A and B but it is not very capable for soft soil such as soil type of C and D. Convergence between the numerical and analytical results for maximum shear force are about 98%, 81%, 49% and 6% for soil types of A, B, C and D, respectively.

Generally, in considering the effect of SSI, where displacement increases, base shear force decreases because more displacement of the soil as structure support causes more energy damping and results in lower base shear force applied on the structure base. However, this effect depends on the soil type under the structure. In the analytical model, because of the linear modeling of the soil, results show to be overestimated in the soft soil types where the nonlinear behavior of the soil is significant.

6.2 Relative effects of SPSI and SVEGM on dynamic response of the bridge

To investigate the simultaneousely effects of SPSI and SVEGM on the dynamic behavior of the considerd bridge model, displacements between the decks (m₂-m₁) are analyzed including and excluding the effects of SPSI and SVEGM for A, B, C and D soil types. Figures 17–20 show the influence of non-uniform ground excitations and SPSI on the displacements of the bridge model decks.

Fig.17

Relative displacement of two decks (m₂- m₁), soil type A.

Fig.18

Relative displacement of two decks (m₂- m₁), soil type B.

Fig.19

Relative displacement of two decks (m₂- m₁), soil type C.

Fig.20

Relative displacement of two decks (m₂- m₁), soil type D.

As expected, in harder soil types, displacements of the deck m₁ due to uniform and non-uniform ground motions becomemore similar than those of softer soil types. Also relative displacements between two decks are changed while considering SPSI and SVEGM. Considering the effects of SPSI and SVEGM results in smaller relative displacements for the base rock but larger relative displacemets for other soil types. This effect is more significant in softer soil types. It can be seen that considering the effects of both SPSI and SVEGM changes both amplitudes and phase of the response, specially in soft soil condition.

Figuers 17–20 show that Considering the effect of SVEGM can significantly increase the relative displacement between the bridge decks and if it is not considered in the dynamic analysis and design of the long bridges, it could cause pounding and even structure failure. This effect is more obvious in softer soil types.

The influence of non-uniform ground excitations and SPSI on the maximum shear force of the piers can be observed from the results of fixed-base case (neglecting SPSI) and uniform ground excitation (neglecting SVEGM). Figure 21 shows this comparison for the pier 4.

Fig.21

Maximum shear forces of the pier 4 for all types of soil.

The results show that the effect of SVEGM can increase the base shear force in both fixed base and SSI model. in the model including the SSI effect, considering the SVEGM effect can increase th base shear force about 14% in average and in the model excluding the SSI effect, SVEGM effect can cause 11% increase in the maximum base shear force in average.

As illustrated in the Fig. 21, calculated forces willincrease if the effect of SPSI and SVEGM is considered. Diference between the results are not significant in base rock condition but softer soil conditions result in more different results between SPSI-SVEGM cases and Fixed base-Uniform excitation cases. With respect to the importance of maximum shear force values in structural design, it should be noticed that the results obtained from the dynamic analysis must be accurate and close to reality. Therefore, if the effects of SPSI and SVEGM on the behavior of the considered long-span bridge model is identified, it can be outlined that the influence of SPSI on the dynamic bridge response is apparent and therefore should be considered in the dynamic analysis of long-span bridges. Moreover, SVEGM also have important effects on long-span bridge response.

7 Conclusions

In this paper, dynamic analysis of a long-span bridge subjected to spatially varying earthquake ground motion and under the effect of soil-pile-structure interaction is performed by solving the equation of motion for the proposed model. For the first time, a new simple but capabale model has been proposed to simulate soil-pile system cosidering the pile foundation with distributed mass in a longitudinal bridge model.

A proportionately accurate analytical equation was proposed to determine the dynamic response of long-span bridges including SPSI and SVEGM. The soil supporting the piers is modeled as a system of distributed springs and dashpots acting in horizontal direction. Spatially varying earthquake ground motion were generated by the well-known and powerful record generator SIMQKE II considering the effect of incoherency and wave passage.

The main conclusions drawn from this study can be written as:

The importance of the SPSI effect on the dynamic response of the bridge is investigated in comparison with the fixed-base case. It is observed that the results obtained from the SPSI case are usually amplified in comparison with the fixed-base case. This effect is more significant in softer soil types. It means that the variation of the soil conditions where the bridge supports are located on, has important effect on the bridge dynamic response.

Comparison between the results of the numerical and analytical models shows that the proposed analytical model can be used in rock-base and hard soil but it is not capable in medium and soft soil condition.

If the effects of SPSI and SVEGM are considered simultaneously, it should be noticed that the results will not be as same as which obtained from the addition of the response determined from these effects separately. The effects of SPSI and SVEGM amplify each other specially in softer soil conditions. It is also observed that considering the effect of SPSI with respect to the SVEGM can change the dynamic response of long-span bridges in comparison with the cases in which only SPSI effect or non of them is considered.

In general, the recommendation of fixed-base case with uniform ground excitation in dynamic design regulations of bridges is valid only if SPSI and SVEGM effects are negligible. These assumption can be used in seismic design of bridges on very stiff soil conditions and not very long bridges. Otherwise, this recommendation leads to usually underestimating the dynamic response or even bridge structure damage.

Due to the fact that the effect of SVEGM may cause pounding between the decks, it is necessary to consider this effect when designing the expansion joints between the decks; otherwise, the structure may be damaged or even collapsed.

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