The influence of soil plasticity on the seismic performance of bridge piers on caisson foundations

Soil Dynamics and Earthquake Engineering 118 (2019) 120–133

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The influence of soil plasticity on the seismic performance of bridge piers on caisson foundations

T

D. Gaudio, S. Rampello

⁎

Dipartimento di Ingegneria Strutturale e Geotecnica, Via Eudossiana 18, 00184 Rome, Italy

ARTICLE INFO

ABSTRACT

Keywords: Caisson foundations Bridge piers 3D coupled dynamic analysis Effective stress analysis Soil plasticity

This paper investigates the role of soil plasticity on the seismic performance of bridge piers founded on cylindrical caissons through a numerical study where the geometrical and mechanical properties of the caisson and the pier, as well as the weight of the deck, are varied. Numerical models of the soil-caisson-pier-deck systems are subjected to six real acceleration time histories differing in frequency content, strong-motion duration and intensity. Three-dimensional coupled dynamic analyses are carried out in the time domain in terms of effective stresses, assuming an undrained response for the foundation soils. The studied systems are designed to be characterised by the same safety factors under static vertical loads, to evaluate the seismic performance of different systems endowed with similar mobilised shear strength, and by a low pseudo-static factor of safety to promote the activation of plastic mechanisms during seismic shaking. To reproduce the initial state of stress around the caissons, the effects induced by the construction stages are also simulated in the analyses through a simplified procedure. Seismic performance of the systems is evaluated comparing the maximum and the permanent deck drift ratios with corresponding threshold values related to ultimate limit states. It is shown that seismic performance is strongly related to the ratio between the fundamental periods of the flexible-base system and the soil, as well as to the strong motion duration. The range of systems and input properties for which soil plasticity is significant is identified, thus avoiding excessive overestimates of earthquake-induced displacements and inertial forces on the superstructure.

1. Introduction Over the last 40 years dynamic soil-structure interaction of bridge piers founded on caissons has been increasingly focused on by the scientific community in light of their observed performance during strong earthquakes. Caisson foundations contributed to the survival of some bridges during the Kobe (1995) and Chi-Chi (1999) earthquakes [1,2], while others suffered severe damage, in terms of excessive permanent displacements and rotations, cracks, as well as shear failure due to fault movements [3,4]. Post-earthquake observations evidence that development of plastic deformations induced by strong earthquakes in the soil-foundation system is sometimes unavoidable. Indeed, contrary to current practice in seismic design, triggering of plastic strains in the foundation soils could even be promoted, thus limiting the ductility demand in the pier to protect the superstructure [5,6]. However, this approach requires an accurate control and assessment of the seismic performance of the system, in terms of maximum and permanent values of deck

⁎

displacement or drift and of caisson rigid rotation and settlement, attained during and at the end of the seismic event. Seismic performance of soil-caisson-pier-deck systems is often evaluated assuming a fixed-base pier for bridges of minor importance, thus neglecting the role of foundation-soil compliance, under the assumption that soil-structure interaction effects are beneficial [7]. Furthermore, when considering soil-structure interaction, seismic performance is usually assessed assuming that the soil will behave as a viscous-elastic material, referring to the decoupled substructure approach [8,9]. With this technique, the foundation input motion corresponding to a massless foundation and the impedance functions, describing the stiffness and damping of the foundation elements, are initially evaluated; a dynamic analysis is then carried out where the structure is supported by a compliant base and is subjected to the foundation input motion [e.g.: 10–13]. The substructure method is rigorously valid only for linear systems and may not be fully adequate for complex problem outlines (inclined ground surface or soil layering, varying water levels, etc.) or when a substantial mobilisation of the soil

Corresponding author. E-mail addresses: [email protected] (D. Gaudio), [email protected] (S. Rampello).

https://doi.org/10.1016/j.soildyn.2018.12.007 Received 28 June 2018; Received in revised form 16 October 2018; Accepted 6 December 2018 0267-7261/ © 2018 Elsevier Ltd. All rights reserved.

Soil Dynamics and Earthquake Engineering 118 (2019) 120–133

D. Gaudio, S. Rampello

shear strength can be anticipated. The main drawback stems from the hypothesis that soil non-linearity is not involved during seismic shaking. Consequently, maximum displacements and rotations are typically overestimated by these analyses, while permanent values cannot be evaluated. Non-linear dynamic analyses assessing the seismic performance of bridge piers founded on caissons have already been performed. However, they often refer to specific cases [14,15] or to parametric studies where the soil behaviour is described in terms of total stresses, using a linear elastic-perfectly plastic constitutive model [16–18]. In this paper, a parametric study is presented where the seismic performance of several soil-caisson-structure systems subjected to different natural records is evaluated through three-dimensional nonlinear dynamic analyses. Soil behaviour is described in terms of effective stresses, accounting for non-linear and irreversible soil behaviour from very small strain levels. Seismic performance of the systems is expressed in terms of maximum and permanent values of the deck drift ratio and is related to the properties of both the systems and seismic inputs. Influence of soil plasticity is also inferred comparing the results of the analyses performed in the hypothesis of elastic-plastic soil behaviour with those provided from companion analyses performed assuming that the soil behaves as a viscous-elastic material. Thus, it was possible to identify the properties of a soil-caisson-pier-deck system, subjected to a given seismic input, for which viscous-elastic modelling provides a reliable assessment of its seismic performance, and those for which influence of soil plasticity is important and should be accounted for in the analyses.

Table 1 Viscous-elastic parameters of the structural elements. element

γc (kN/m3)

E (GPa)

ν (-)

ξ (%)

Pier Caisson

25 25

27.3 30.0

0.15 0.15

5.0 1.0

pier may be deemed acceptable when the caisson foundation is underdesigned [e.g.: 16, 17], to promote temporary activation of plastic mechanisms in the foundation soils during seismic shaking. The pier is assimilated to a single-degree-of-freedom system (S.D.O.F.) characterised by flexural stiffness ks and damping ratio ξs = 5%. The mass lumped at the deck height, ms = mdeck + 0.5 mpier, represents those of the deck and the upper half of the pier. The mass of the lower half of the pier is applied at the top of the caisson via a uniform vertical pressure σz (0.5pier), in order to reproduce the initial stress state in the soil. Table 1 reports the parameters assumed for the caisson and the pier: the unit weight of reinforced concrete γc, the Young's modulus E, the Poisson's ratio ν and the damping ratio ξ. A reduced Young's modulus was adopted for the pier to account for the expected flexural cracking induced by seismic shaking. 2.2. Soil model and input parameters Non-linear soil behaviour is described using the Hardening Soil model with small-strain stiffness (HS small) [20], an elastic-plastic constitutive model, with isotropic hardening and Mohr-Coulomb failure criterion, implemented in the Plaxis 3D suite [21]. An isotropic pre-yield para-elastic behaviour is assumed for the soil, where the small-strain shear modulus G0 is expressed as a function of the effective stress state and of soil strength parameters:

2. Problem definition 2.1. Problem layout Fig. 1 depicts a schematic layout of the example problem, where the transverse section of an ideal bridge or viaduct is considered. A bridge pier of height hs, with hollow rectangular cross section, is supported by a cylindrical caisson foundation of diameter D and height H, embedded in a 5-meter-thick layer of gravelly sand (H1 = 5 m) and a 55 meterthick layer of silty clay (H2 = 55 m). Foundation soil is representative of typical alluvial deposits for which caisson foundations are usually designed in the presence of heavy horizontal actions, such as those induced by severe earthquake loading. The pore water pressure regime is assumed to be hydrostatic, with the water table located at the interface between the two layers (zw = H1 = 5 m). Seismic action is described through a horizontal acceleration time history applied at the bedrock depth (Z = 60 m). In the adopted scheme, equally-spaced piers and long-span bridges or viaducts are assumed, so that the assumption of no interaction between two adjacent piers is valid. In such conditions, torsional stiffness of the deck can be ignored [19]. In the analyses, caisson foundations and piers are regarded as a linear viscous-elastic material. Assumption of elastic behaviour for the

G0 = G0ref

c cot c cot

+ 3 + pref

m

(1)

ref

where p = 100 kPa is a reference pressure, σ′3 is the minimum principal effective stress and G0ref and m are model parameters. Hyperbolic laws are adopted in the model to describe the shear modulus decay and the damping ratio increase with shear strain, as shown in Fig. 2. Definition of the elastic behaviour is completed in the model by the shear strain γ0.7, corresponding to a secant shear modulus Gs = 0.722 G0 (Fig. 2), and by the unloading-reloading Young's modulus E′urref, also expressed as a function of the effective stress similarly to the small-strain shear modulus G0. Under cyclic loading a generalisation to 3D conditions of the Masing's rules [22] are followed in the elastic response of the model.

Fig. 1. Schematic layout of the problem.

Fig. 2. HS small model: stiffness decay and damping increase with shear strain. 121

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Table 2 Index and mechanical properties of foundation soils and parameters of the HS small model. soil

γ (kN/m3)

c′ (kPa)

φ′ (deg.)

OCR (dimensionless)

k0 (dimensionless)

G0ref (MPa)

m (dimensionless)

γ0.7 (%)

E′urref (MPa)

νur (dimensionless)

E′50ref (MPa)

E′oedref (MPa)

Gravelly sand Silty clay

20 20

0 20

30 23

10 4.4÷1.5

0.5 1.1÷0.7

145.7 65.7

0.61 0.75

0.024 0.045

174.9 58.2

0.2 0.2

58.3 19.4

58.3 19.4

The plastic response is described in the model by a deviatoric and a volumetric yield surface, fs and fv, with independent isotropic hardening depending on deviatoric plastic strains γp and volumetric plastic strains εvp, respectively. The deviatoric hardening rule is related to parameter E′50, while the volumetric one is controlled by parameter E′oed. Both are given by expressions similar to Eq. (1) but, in contrast to G0, they are not used within a concept of elasticity. Reference values of E′50ref = E′oedref = E′urref/3 were assumed at pref = 100 kPa (Table 2). Table 2 summarises the parameters assumed for the foundation soils including the unit weight γ, the effective cohesion c′ and the angle of shearing resistance φ′. The silty-clay deposit is slightly overconsolidated, with an overconsolidation ratio OCR decreasing from about 4.4 at the top to about 1.5 at the bottom of the layer. For the gravelly sand, the overconsolidation ratio is fictitiously set equal to 10 to exclude the activation of the volumetric yield surface for this layer. The coefficient of earth pressure at rest [23]:

k 0 = (1

sin ) OCRsin

p G0 = S ref pr p

2.17 e 2 (p ) n 1+e

R 0m

(4)

where pref = 100 kPa is a reference pressure, R0 = p′y/p′ is the ratio between the yielding and the current mean effective stresses and S = 407.5, n′ = 0.77 and m′ = 0.32 are dimensionless stiffness parameters related to the plasticity index IP [27]. The model parameters G0ref and m listed in Table 2 were calibrated using Eq. (1) to reproduce the G0 profile of Fig. 3. Parameters γ0.7 and E′urref were instead calibrated to best approximate the assumed curves for the decay of the secant shear modulus Gs and the increase of the damping ratio ξ with the shear strain γ: specifically, the curves proposed by Seed and Idriss [28] (upper bound for Gs and lower bound for ξ) and Vucetic and Dobry [29] (Ip = 25%) were adopted for the gravelly sand and the silty clay layers, respectively. Fig. 4a-b show the good agreement between the experimental curves and those computed using the values of γ0.7 and E′urref listed in Table 2.

(2)

decreases with depth from about 1.1. to 0.7. Undrained shear strength, evaluated accounting for initial stress state and past stress history, yields values of su = 50÷100 kPa, typical of a medium stiff clay, at the depths of the caisson bases (5÷24 m). Fig. 3 shows the assumed profiles of overconsolidation ratio OCR and small-strain shear modulus G0. This latter is obtained using the empirical relationships proposed by Hardin and Richart [24] for gravelly sand, and that by Rampello et al. [25] for the silty clay. In the first case, it is

G0 = A

n

2.3. Selected input motions Two sets of three real seismic records are adopted for the dynamic analyses, representative of Italian, Turkish and Iranian high-intensity seismic events. Each set consists of three horizontal acceleration time histories scaled to the same Arias intensity IA [30]: the first set is

(a)

(3)

in which p′ is the mean effective stress, e is the void ratio and A and n are dimensionless stiffness parameters; values of A = 7000, e = 0.56 and n = 0.5 are selected as representative of round grained sands [26]. For the silty clay, the small-strain shear modulus is computed as:

(b)

Fig. 3. Profiles of overconsolidation ratio and small-strain shear modulus obtained from empirical relationships and the HS small model.

Fig. 4. Calibration of secant shear modulus decay and hysteretic damping increase against empirical relationships: (a) gravelly sand; (b) silty clay. 122

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Table 3 Main properties of scaled seismic input motions selected for dynamic analyses. Recording

F (dimensionless)

amax,inp (g)

vmax,inp (m/s)

IA (m/s)

Tm (s)

Tp (s)

TD (s)

Tolmezzo E-W (1978) Assisi E-W (1997) Adana E-W (1998) Colfiorito N-S (1997) Nocera Umbra N-S (1997) Dayhook N-S (1978)

1.00 2.00 1.05 2.00 1.00 1.45

0.316 0.332 0.292 0.676 0.502 0.573

0.33 0.18 0.24 0.42 0.33 0.37

1.17 1.12 1.17 2.79 2.87 2.84

0.50 0.24 0.62 0.51 0.21 0.46

0.67 0.18 0.67 0.95 0.16 0.38

5.220 4.295 12.990 5.115 4.640 12.870

Fig. 5. Seismic input motions adopted in the analyses: (a)-(c), (e)-(g) time histories of scaled horizontal acceleration; (d) and (h) Fourier amplitude spectra.

characterised by values of IA in between 1.12 and 1.17 m/s, while the second by an Arias intensity of 2.79–2.87 m/s, about 2.4 times higher. These records were selected to cover a wide range of frequency content, strong-motion duration and intensity, though limiting the scaling factor in the range F = 0.5÷2, and to match a site-specific elastic spectrum. The main properties of the scaled seismic inputs are reported in Table 3 where amax,inp and vmax,inp are the peak horizontal acceleration and velocity, respectively, Tm is the mean period [31], Tp is the predominant period and TD is the significant duration [32]. Fig. 5a-h show

the acceleration time histories and the Fourier amplitude spectra of the six input motions adopted in the analyses. The first set of seismic inputs consists of the East-West components of the recordings of Tolmezzo (TLM1 E-W, 06/05/1978) [33], Assisi (AAAL098, 26/09/1997) [34,35] and Adana (TK-1998-0063, 27/06/ 1998) [36] (Fig. 5a-b-c). Acceleration time histories of Assisi and Adana are amplified by a factor F = 2 and 1.05 to attain the same Arias intensity as the Tolmezzo record. Assisi time history is characterised by a mean period Tm twice lower than that of Tolmezzo, as richer in high 123

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D. Gaudio, S. Rampello

frequencies (Fig. 5d), while the Adana record presents a longer strongmotion phase, with a duration TD ≈ 13 s, 2.5 times greater than that of Tolmezzo (Fig. 5c). The second set of accelerograms includes the North-South components of the records of Colfiorito (IT-1997-0004, 26/09/1997) [37], Nocera Umbra (IT-1997-0006, 26/09/1997) [37] and Dayhook (IR1978-0002, 16/09/1978) [36] (Fig. 5e-f-g). The records of Colfiorito and Dayhook are scaled by a factor F = 2 and 1.45 to have the same Arias intensity as Nocera Umbra. Similarly to the first set, the Nocera Umbra record is characterised by a mean period Tm twice lower than that of Colfiorito (Fig. 5h), while the significant duration of the Dayhook record (TD ≈ 13 s) is about 2.5 times longer than that of Colfiorito (Fig. 5g). Hence, the two sets of records span similar ranges of frequency content and duration but with substantially different values of Arias intensity. All the acceleration time histories were corrected to return zero velocities and displacements at the end of the records, using a parabolic baseline correction [38]. The site-specific elastic response spectrum was selected assuming a design working life VN = 100 years, a class of importance III, a subsoil class C (VS,30 = 204 m/s) and a structural damping ratio ξ = ξs = 5%, this corresponding to a return period TR = 1424 years. Fig. 6 compares the assumed and the average response spectra, the latter computed at ground surface from 1D free-field ground response analyses carried out with the linear-equivalent method using each input motion; spectral compatibility is satisfied for periods in the range of 0.2–2.0 s, as prescribed by the Italian Building Code [39].

pier flexural stiffness. The desired values of safety factor were obtained using classical solutions, usually adopted for embedded foundations with H/D ≤ 4, to evaluate the caissons bearing capacity under static and pseudo-static conditions [40,41]. To account for the increase of the natural period of the studied systems due to soil compliance, spectral accelerations were computed using the equivalent period Teq as proposed by Tsigginos et al. [11]:

Teq Ts

1+

2 hs Ts VS, eq

1.18

ms m caisson

0.613

2

hs D

0.5

(5)

where Ts = 2 ms /ks is the period of a fixed-base pier and VS,eq is the equivalent shear wave velocity of the foundation soil defined, for each caisson, down to a depth z = H+ 2D. In Eq. (5) the effect of caisson depth H is included in the caisson mass, the ratio Teq/Ts decreasing as mcaisson and H/D increase. Computed values of periods Ts and Teq are listed in Table 4, the fixed-base pier periods ranging between Ts = 0.72 and 2.76 s, the highest value of Ts being representative of bridges characterised by heavy decks and flexible piers (e.g. the truss bridge on the Makru river, India [42]). The lowest equivalent period Teq = 1.26 s is evaluated for the most rigid pier, of height hs = 30 m, founded on a caisson of diameter D = 12 m with slenderness ratio H/D = 2, while the highest period Teq = 3.62 s is obtained for one of the most flexible piers, with hs = 60 m, D = 12 m and H/D = 0.5. The equivalent periods of the studied systems are from 1.3 to 1.75 times higher than those computed for the corresponding fixed-base piers, the former being computed accounting for soil compliance (Eq. (5)) and non-linear soil behaviour related to the equivalent shear wave velocity VS,eq provided by the 1D free-field ground response analyses; as a consequence, they should be intended as upper-bound values.

2.4. Studied systems Fourteen soil-caisson-pier-deck systems are considered in the parametric study, characterised by two caisson diameters, D = 8 and 12 m, three caisson slenderness ratios, H/D = 0.5, 1 and 2, and three pier heights, hs = 15, 30 and 60 m. The main geometrical and mechanical properties of the systems are listed in Table 4. The pier flexural stiffness ks as well as the deck and pier masses mdeck and mpier are representative of simple and continuous-spanned highway and railway bridges, with span length ranging between 40 and 110 m. Values of ks, mdeck and mpier were back-calculated to obtain given values of the static and pseudo-static global safety factors against bearing capacity, namely FSv = 5.5 and FSe = 0.7. This latter was computed using the design response spectrum of Fig. 6 to evaluate the horizontal load Qs and the moment Ms = Qs·hs of each soil-caisson-pierdeck system. Reducing the response spectrum by a behaviour factor q = 1.5, a value of Fse ≅ 2.4 is obtained for all the systems. Values of FSv in the range of 3–5.5 are commonly used, while a value of FSe = 0.7 was chosen, also adopted by Zafeirakos and Gerolymos [16], to further plastic strains during the seismic event. Some possible combinations of caisson and pier dimensions are not considered in the analyses as they are not realistic or associated to non-realistic values of the deck mass or

2.5. Finite element model The coupled 3D dynamic analyses are performed using the numerical model shown in Fig. 7, implemented in the FE code Plaxis 3D suite [21]. The horizontal acceleration time histories are assumed to act in the plane transversal to the bridge axis (x-direction in the model), making it possible to describe only half of the domain, due to the problem symmetry with respect to the x-axis. Model dimensions X = 12.5·Dmax = 150 m, Y = 6.25·Dmax = 75 m and Z = 2.5·Hmax = 60 m were chosen through a series of preliminary pseudo-static analyses where the contours of soil displacements and stresses were checked so as not to be altered by the mesh boundaries. The FE mesh consists of 10-node tetrahedral elements with linear interpolation for strains and pore water pressure. A mesh refinement is introduced close to the caisson, in a soil volume of dimensions Δx = 33.0 m, Δy = 16.5 m and Δz = 35 m, to improve the accuracy of the analyses. As shown in Fig. 7, the foundation soil is discretised with horizontal layers of variable-thickness, each characterised by a constant overconsolidation ratio (Fig. 3): 2–3 m thick layers are used in the first 20 m, while 5 m thick layers are adopted in the following 40 m. Soil behaviour is described by the elastic-plastic, strain hardening, HS small model [20], while the caisson elements are regarded as a linear elastic material (Table 1). The soil-caisson interface is purely frictional with an angle δ = tan−1 (2/3∙tanφ′) to simulate possible relative sliding. The pier and its lumped mass ms are modelled by one-dimensional beam elements behaving as linear elastic-viscous material (Table 1). A rigid plate element is placed at the top of the caisson, to redistribute the loads directly applied by the lower half of the pier to the foundation. In the static and pseudo-static calculation stages, vertical boundaries are restrained horizontally in the normal direction, while the model base is restrained both horizontally and vertically. Model capability to predict foundation bearing capacity was first checked comparing the results provided by monotonic pushover analyses with published data. Hence, the interaction domain computed by Zafeirakos

Fig. 6. Comparison between the assumed, site-specific, elastic response spectrum provided by the Italian Building Code and the response spectra computed at the ground surface through 1D free-field ground response analyses. 124

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Table 4 Geometric, mechanical and dynamic properties of the studied systems. D (m)

H/D (dimensionless)

hs (m)

ks (MN/m)

mpier (Mg)

mdeck (Mg)

ms (Mg)

Ts (s)

Teq (s)

Teq/Ts (dimensionless)

8

0.5 1

30 30 60 15 30 60 15 30 60 15 30 60 30 60

10.1 11.8 6.2 102.4 46.9 20.8 106.4 37.7 19.8 169.3 78.7 29.9 411.2 192.3

196.7 217.2 1018.9 112.0 422.6 1065.0 113.2 384.8 1399.3 134.6 489.2 1454.0 904.2 3156.8

1278.0 1500.3 698.6 2115.4 1804.8 1162.4 3445.1 3173.5 2159.0 4160.5 3806.0 2841.1 4986.9 2374.3

1376.4 1608.9 1208.1 2171.4 2016.1 1694.9 3501.7 3365.9 2858.6 4227.8 4050.6 3568.1 5439.0 4132.7

2.32 2.32 2.76 0.91 1.30 1.79 1.14 1.88 2.39 0.99 1.43 2.17 0.72 0.92

3.22 2.95 3.59 1.26 1.80 2.47 1.76 2.76 3.62 1.43 2.06 3.05 1.26 1.61

1.39 1.27 1.30 1.38 1.38 1.38 1.54 1.47 1.51 1.44 1.44 1.40 1.75 1.75

2 12

0.5 1 2

lateral boundaries in the direction of application of the seismic input [44]; at the same time, the initial horizontal effective stress is applied at the vertical boundaries normal to the seismic input (y-z plane in Fig. 7), removing the static boundary conditions [45]. On these planes, vertical displacements are also restrained, thus reproducing free-field pureshear conditions at the lateral boundaries. The seismic stages of the analyses are finally performed applying the selected acceleration time histories to the bottom boundary of the mesh. To avoid numerical distortion of the propagating waves during the dynamic analyses, the maximum element height (Δlmax = 4 m) is smaller than 1/7 of the wavelength λmin associated with the highest frequency component of the input wave fmax [46]. A value of fmax = 12 Hz was fixed since negligible energy content is associated to high frequencies (Fig. 5d, h); the input acceleration time histories were then low-pass filtered by removing frequency components higher than 12 Hz. The time step used in dynamic calculations is equal to the sampling interval of the acceleration time histories, Δt = 0.01 s for the input of Tolmezzo and Δt = 0.005 s for the remaining records; the Newmark's time integration scheme is adopted in calculations, with standard values of constants α = 0.25 and β = 0.50 to obtain an unconditionally stable solution [47]. At small-strain levels the HS small model does not provide any hysteretic damping for the foundation soils (Fig. 2): then, an additional viscous damping ratio ξ = 1% is introduced through the Rayleigh formulation following the procedure suggested by Amorosi et al. [48] to calibrate the Rayleigh coefficients αR and βR. The calibration of the FE model was also checked for the dynamic conditions, in terms of boundary conditions, mesh coarseness and viscous damping ratio, through preliminary FE ground response analyses performed with the code Plaxis 3D, simulating a viscous-elastic soil column in free-field conditions. The computed profiles of maximum horizontal accelerations and shear strains were observed to be in a very good agreement with those obtained from ground response analyses carried out with the linear-equivalent method, this confirming the appropriate calibration of the numerical model [49–51].

Fig. 7. Three-dimensional finite-element model implemented in Plaxis 3D.

Fig. 8. Comparison of normalised bearing capacity envelopes from monotonic pushover analyses (D = H = 12 m).

3. Initial conditions The state of initial stress around the caisson is computed accounting, through a simplified procedure, for the reduction of horizontal effective stress induced by caisson construction during the excavation stages (conventional under-excavation or shaft sinking) before installing the steel reinforcement and pouring the concrete. Thus, a volumetric contraction εv is applied to the volume of soil embodied by the caisson perimeter before activating the concrete elements simulating the foundation. The applied volumetric contraction εv is evaluated

and Gerolymos [43] was compared with that obtained for a caisson of diameter D = H = 12 m, unit weight γc = 20 kN/m3, subjected to a vertical load Nhead = 0.2∙Nlim, assuming a friction angle δ = φ′ at the soil-caisson interface and the same soil conditions. Fig. 8 shows the good agreement obtained between the normalised domains of bearing capacity using the equivalent diameter for the caisson cross section area. During the dynamic stages, tied-nodes conditions are assumed at the 125

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D. Gaudio, S. Rampello

initial state of stress close to active limit conditions, soon after shaft excavation, results in larger plastic strains developed during the dynamic stages of analysis; the effects are stronger for deep caissons, with permanent values of deck drift ratios up to 20% higher than those computed neglecting the construction stages. The contours of mobilised to available shear stress, τm/τf, computed at the end of bridge construction, are plotted in Fig. 10 for three caissons of diameter D = 12 m with different slenderness ratios H/D = 0.5, 1, 2; all show high values of mobilised shear strength (τm ≥ 0.8 τf) closely around the caisson shaft, as a result of the stress reduction induced by the excavation stages, and beneath the edges, as expected for a rigid foundation. The close similarity of the different stress contours plotted in Fig. 10 results from the same safety factor FSv = 5.5 adopted to preliminary design the caissons using the bearing capacity formula from the literature [40,41]. Safety of the systems against bearing capacity was also evaluated via the numerical analyses by progressively reducing the effective cohesion c′ and the angle of shearing resistance φ′ by a factor γM until a plastic mechanism was activated; this factor can be seen as the inverse of a degree of mobilisation of the available soil strength (γM = 1/MF = τf/τm), giving a clear and immediate indication of the distance of the structure from a plastic mechanism. For all the systems the analyses provided values of γM av = 1/MF = 3.6 that correspond to an average degree of strength mobilisation MF = τm/τf ≅ 28%, while a lower value of γM av ≅ 3 is evaluated in undrained conditions, as expected. Hence, the conditions of the caissons are similar to each other prior to applying the seismic input, the foundations having the same initial safety margin against bearing capacity, thus allowing a proper comparison of their seismic performance.

Table 5 Volumetric contractions applied to simulate shaft excavation and resulting radial displacements. D (m)

H/D (dimensionless)

H (m)

εv (%)

ur (mm)

8

0.5 1 2 0.5 1 2

4 8 16 6 12 24

0.45 0.90 1.80 0.45 0.90 1.80

6 12 24 9 18 36

12

imposing a value of the ratio between the radial displacement of shaft perimeter and the shaft depth ur/H = 1.5‰ to promote the attainment of active limit conditions at the soil-caisson boundary. Cylindrical conditions of strain are assumed with radial strain εr = 2∙ur/D and volumetric strain v

= 3

r

= 6

ur u H H = 6 r = 0.009 D H D D

(6)

The adopted values of the volumetric contraction are listed in Table 5 together with the resulting average radial displacements ur = εr∙D/2. After the volumetric contraction εv is applied, the caisson concrete and the superstructure are activated in successive stages, assuming bridge construction to occur under drained conditions. Fig. 9 shows the profiles of the radial effective stress σ′r computed at the soil-caisson interface after activating the superstructure, for three caissons of diameter D = 12 m and slenderness ratio H/D = 0.5, 1 and 2 either neglecting (εv = 0) or simulating (εv ≠ 0) the construction stages. Simulation of shaft excavation provides a substantial reduction of effective stress that it is seen to approach the active stress computed using a lower bound solution [52], as the caisson depth increases (H/D = 2); the effective stress is lower than the active limit conditions for depths in the range of z = 18–24 m, because of arching effects [53,54]. In a set of preliminary analyses, it has been seen that computing an

4. Three-dimensional coupled dynamic analyses The 3D coupled dynamic analyses are carried out assuming undrained conditions for the foundation soils, but describing the elasticplastic soil behaviour in terms of effective stresses. According to Zienkiewicz et al. [55], the hypothesis of undrained soil response is seen to be appropriate for the assumed soil permeability (k = 10−5 and 10−9 m/s for the sand and clay layers) and the frequency content of the selected seismic inputs (fp = 1.05÷6.25 Hz). In the dynamic calculation stages, the selected acceleration time histories are applied at the base of the model in the x direction. Computations are prolonged by about 10 s after the end of the seismic motion to allow the residual inertial forces to vanish; a consolidation analysis is then performed to dissipate the excess pore water pressure induced by seismic shaking, thus evaluating permanent displacement and rotation of both the caissons and the piers. Dissipation of excess pore water pressure after the earthquake induced increments of caisson settlement and rotation of 5.3–14.0% and of 0.1–1.0%, respectively. A total of 51 analyses is carried out, where the 14 soil-caisson-pierdeck systems listed in Table 4 are subjected to the 6 acceleration time histories selected for this study (Table 3); the seismic inputs characterised by the highest Arias intensity IA (Colfiorito, Nocera Umbra and Dayhook records) are only applied to the systems with caisson diameter D = 12 m and pier height hs = 30 m. Fig. 11 a-b show the profiles of the maximum acceleration ratio amax/amax,inp and shear strain γmax attained during the seismic action, while Fig. 11c-d show the profiles of the isochrones of permanent shear strain γperm and horizontal displacement uperm computed at the end of the dynamic stage. They are computed for the alignment located on the symmetry plane (y = 0), at a distance of 1 m from the caisson shaft and refer to caissons of diameter D = 12 m, with slenderness ratios H/ D = 0.5, 1 and 2, supporting a pier of height hs = 30 m subjected to the input of Tolmezzo. The profiles of maximum acceleration ratio show that amplification mainly occurs for the squat caissons (H/D ≤ 1) at depths z ≥ 15 m, with maximum ratios amax/amax,inp = 1.5 and 1.25 for H/D = 0.5 and

Fig. 9. Profiles of radial effective stress at the soil-caisson interface after activating the superstructure, simulating (εv ≠ 0) or neglecting (εv = 0) shaft excavation (D = 12 m; H/D = 0.5, 1, 2). 126

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Fig. 10. Contours of relative shear stress after simulating shaft excavation and activating the caisson foundation and the superstructure.

1, the higher value being computed at a greater depth for the shallower caisson. De-amplification of acceleration ratio mainly occurs, instead, from the depth of the caissons base to the ground surface; this may be associated to transient mobilisation of shear strength during seismic shaking, with the development of plastic strains below the base of the caissons, consistent with observations from dynamic centrifuge tests on small-scale models of embedded foundations [56]. The maximum shear strains are seen to attain their maximum just below the caissons’ base, with values increasing with the ratio H/D. The irreversible component of shear strain prevails in that ratios of γperm/ γmax = 85 ÷ 95% are computed, thus indicating that the maximum shear strength is temporarily mobilised at the base of the caisson during seismic shaking. The nearly coincident shapes of the profiles of γmax and γperm indicate that the maximum shear strain is substantially reached at

the end of the seismic input. Higher values of both maximum and permanent shear strain are obtained at the ground surface, with peak values γmax = γperm ≅ 5% for a slenderness ratio H/D = 2, thus providing the largest permanent horizontal displacement and rotation of the caisson, uperm = 1.3 cm and θperm = 1.4‰. The computed results can be ascribed to soil-system dynamic coupling that results in a major development of plastic strains induced by the higher amplification of seismic actions. The equivalent period Teq of a system supported by a compliant base can be identified as the period corresponding to the maximum ratio between the spectral accelerations evaluated at the top and bottom of the piers, R = Sadeck/Sacaisson head [57]; values of Teq = 2.26, 1.97 and 0.99 s are evaluated from Fig. 12 for slenderness ratios H/D = 0.5, 1 and 2. The lowest equivalent period Teq = 0.99 s, obtained for the largest slenderness ratio (H/D = 2), is

Fig. 11. Profiles of maximum acceleration ratio, maximum and permanent shear strain and permanent horizontal displacement (D = 12 m, hs = 30 m – Tolmezzo input). 127

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Fig. 12. Spectral acceleration ratio for caissons of different slenderness ratio (D = 12 m, hs = 30 m – Tolmezzo input).

Fig. 14. Maximum and permanent values of the deck drift ratio versus the period ratio.

4.1. Evaluation of seismic performance The seismic performance of the systems is evaluated through the maximum and the permanent values of the deck-drift ratio, urel/hs, defined as

Fig. 13. Time histories of deck horizontal displacement and caisson rigid rotation, for systems with low and high period ratios Teq/T0 (D = 12 m, hs = 30 m – Tolmezzo input).

urel u = deck hs hs

u caisson head = tan hs

+

uflex hs

(7)

where uflex is the flexural displacement of the pier. Fig. 14 shows the maximum (a) and the permanent (b) deck-drift ratio urel/hs, plotted as a function of the period ratio Teq/T0 for all the 51 studied systems. Full black symbols refer to the inputs of longer duration, Adana and Dayhook (TD ≈ 13 s), while light grey symbols to the inputs of Tolmezzo and Colfiorito, richer in low frequencies (Tm = 0.5 s) than the inputs of Assisi and Nocera Umbra, characterised by a high frequency content (Tm ≈ 0.22 s) and represented by empty symbols. Higher values of both the maximum and the permanent drift ratios are computed for period ratios Teq/T0 of 0.8–1.5 s, for which soil-system dynamic coupling is substantial, while they decrease outside this range for period ratios Teq/T0 ≥ 1.5. Then, the maximum and permanent deck-drift ratios induced by earthquake loading can be reduced endowing the systems with a high period ratio Teq/T0 ≥ 1.5÷2 that, for a given foundation soil and pier height, can be obtained by decreasing the slenderness ratio (H/D) of the caisson foundation, while keeping constant the value of the static safety factor FSv. This can be attributed to the influence of caisson slenderness ratio on the entire system's stiffness. For high values of H/D, soil-foundation compliance tends to be

close to the natural period of the soil column, T0 = 1.15 s, that in turn is in the range of periods provided by the 1D free-field response analyses performed with the linear-equivalent method. Soil-system dynamic coupling then occurs for this system (Teq/T0 ≈ 0.8) with an increase of the inertial forces transmitted to the caisson and of the plastic strains induced in the foundation soils. This is demonstrated in Fig. 13, where the time histories of the horizontal deck displacement udeck and the angle of caisson rigid rotation θ are plotted for two caissons with H/D = 0.5 and 2. According to the mentioned influence of soil-system dynamic coupling, higher values of udeck and θ are computed for the largest slenderness ratio, H/D = 2, for which the ratio Teq/T0 approaches unity: specifically, the maximum and permanent displacements udeck, max = 12.7 cm and udeck, perm = 5.4 cm obtained for H/D = 2, are about 2.3 and 4.2 times higher than the corresponding values computed for H/D = 0.5. Thus caisson foundations could conveniently be designed, with the same safety factors, to decouple the soil and the foundation-structure system, thus reducing the maximum and the permanent displacements and rotations, as discussed in the next section.

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null and the pier's behaviour is similar to that of a fixed-base superstructure, while for low values of the slenderness ratio soil-foundation compliance becomes greater and non-negligible, leading to a flexible system characterised by high period ratios Teq/T0. Comparable values of the maximum drift ratios are obtained using the seismic inputs of Tolmezzo, Assisi, Colfiorito and Nocera Umbra, although the latter two are characterised by a roughly twice greater Arias Intensity, while substantial higher drift ratios, that is the worst seismic performance, are computed for the Adana and Dayhook inputs, characterised by the longest duration TD of the seismic action. A much stronger earthquake, with twice the original Arias intensity, results in a similar seismic performance because the yielding of the foundation soil limits the seismic actions transmitted to the superstructure. Seismic performance is also evaluated in terms of permanent values of the deck drift ratio, urel, perm/hs (Fig. 14b), attained at the end of the dynamic calculation stage. Again, the worst seismic performance is obtained using a seismic input characterised by a long duration, i.e., Dayhook (TD = 12.87 s). Influence of input duration on the pier performance, assessed both in terms of maximum and permanent deck drift ratios, can be associated to the progressive development of plastic strain that occurs in the foundation soil as a result of subsequent and transient mobilisation of shear strength during seismic shaking, as discussed below (see Fig. 15). Data points in Fig. 14a-b are best-fitted assuming a log-normal distribution of the deck drift ratio around its mean value and using an upper-bound exponential relationship, corresponding to the 84th percentile, between the deck drift ratio and the period ratio:

urel = B84 e hs

A Teq / T0

Table 6 Coefficients for upper-bound evaluation of the deck drift ratio. Recording

urel/hs

Α (dimensionless)

Β84 (‰)

Tolm. + Ass. + Colf. + Noc.

max. perm. max. perm.

0.51 0.52 0.41 0.23

9.25 1.58 18.48 4.25

Adana + Dayhook

average value of 8.39‰, while values of A are between 0.23 and 0.52, with an average of 0.42. It is worth noting that the proposed curves should be used for Teq/T0 ≥ 0.75. Computed values of the deck drift ratio urel/hs are compared to some threshold values, as proposed by SEAOC [58]; these are given as a function of the considered limit state, that is, they depend on the earthquake return period and on the structure design working life. Referring to the Ultimate Limit State (ULS), threshold values of 15‰ and 5‰, that also include the caisson rigid rotation, are suggested for the maximum and permanent drift ratios. Seismic performance of all systems, subjected to all adopted seismic inputs, can be deemed satisfactory, in that computed values of the urel/hs are lower than the threshold values mentioned above, despite the high-intensity of ground motions used in the analyses. Permanent deck drift ratios result from irreversible soil behaviour during seismic shaking, that contributes to increase energy dissipation and accumulate permanent deformations. The strain field induced by earthquake loading may be described by the deviatoric strain

(8)

s

where B84 is the deck drift ratio for Teq/T0 = 0 and A represents the slope of the curve in a semi-logarithmic plane; coefficient A is assumed equal to that computed for the average curve (A84 = A50). Computed values of B84 and A are listed in Table 6; the coefficients relative to the maximum and the permanent deck drift ratios are given for the seismic inputs characterised by the lowest (Tolmezzo, Assisi, Colfiorito and Nocera Umbra records) and the highest (Adana and Dayhook records) duration TD of the strong motion phase, as this turned out to be the main parameter affecting the seismic performance of the system, provided the seismic amplitude is large enough to promote soil yielding. Upper bound values of B84 range between 1.58‰ and 18.48‰, with an

=

2 3

xx

v

3

2

+

yy

v

3

2

+

zz

v

3

2

+

1 2

2 xy

+

2 yz

+

2 zx

(9)

where εii and γij, for i ≠ j and i,j = x, y, z are the components of the strain tensor along the three cartesian directions and εv = εxx + εyy + εzz is the volumetric strain. Fig. 15 shows the contours of the plastic deviatoric strain εs developed at the end of the seismic event. They refer to the dynamic analysis carried out for a caisson with diameter D = 12 m and a slenderness ratio H/D = 1, supporting a pier of height hs = 30 m subjected to the inputs of Tolmezzo, Assisi and Adana, all of similar peak ground acceleration and Arias intensity. Small plastic strains (εsp ≤ 0.5%) are computed closely around the edges of the caisson base when using the Assisi input, that is rich in high frequencies

Fig. 15. Contours of permanent deviatoric strain for D = 12 m, H/D = 1, hs = 30 m. Inputs of: (a) Tolmezzo; (b) Assisi; (c) Adana. 129

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(Tm = 0.24 s) compared to that of Tolmezzo (Tm = 0.50 s) for which deviatoric strains εsp ≥ 0.2% develop below the caisson base down to a depth of about 1.2D, with peak values of 1.6% at the caisson edges. Lower plastic strains induced by the Assisi input result in lower values of permanent deck drift ratio in that, although period ratios computed for the two inputs are similar (Teq/T0 = 1.59 and 1.71), it is urel, perm/hs = 0.21‰ for the Assisi input while, using the Tolmezzo input, it is urel, perm/hs = 0.33‰ (Fig. 14b). A substantially different field of deviatoric strain is obtained using the Adana input, characterised by the highest duration of the strong motion phase, with significant plastic strains below the base and around the shaft of the caisson, and peak deviatoric strains εsp = 2.1% (Fig. 15). The worst seismic performance is calculated accordingly in this case resulting in Fig. 14b urel, perm/hs = 1.14‰ for Teq/T0 = 1.98. This confirms the strong influence of the duration TD of the seismic action on the permanent strains developed in the foundation soils and thus on the system performance, provided that plastic strains are attained during earthquake shaking.

Different descriptions of mechanical soil behaviour implicate, as expected, different evaluations of the seismic performance of the systems with lower inertial forces and displacements computed when accounting for the development of plastic strains in the foundation soil. To isolate and assess the influence of irreversible soil behaviour, additional 3D dynamic analyses were performed using the same numerical models of Fig. 7, where the soil layers are assimilated to viscous-elastic material with the operative values of shear modulus G and damping ratio ξ provided by the above mentioned 1D linear-equivalent response analyses. Thus it is possible to evaluate the importance of soil plasticity in defining the seismic performance of different soil-caisson-pier-deck systems, all endowed with the same initial safety factor, and subjected to seismic inputs differing in intensity, frequency content and duration. As an example, Fig. 17 shows the time histories of deck drift ratio urel/hs and normalised bending moment Ms/Ms,lim acting at the top of a caisson with diameter D = 12 m, computed using both the HS small and the linear elastic-viscous model. The limit moments for which caisson bearing capacity is attained under the weight of the superstructure and the horizontal load Qs = Ms/hs are evaluated through pushover undrained FE analyses being Ms,lim = 710.5 MN∙m and 134.5 MN∙m for H/ D = 2 and 0.5, respectively. Plotted time histories refer to systems characterised by the minimum and the maximum period ratios, Teq/T0 = 0.77 and 2.24, obtained for H/D = 2 and hs = 30 m, and for H/ D = 0.5 and hs = 60 m, respectively, both subjected to the Adana input. For both systems, the linear viscous-elastic model predicts values of Ms/ Ms,lim and urel/hs much higher than those computed using the HS small model; specifically, the ratios between the peak values of the normalised bending moment are equal to about 3.1 and 2.2 for the stiffer and the more flexible systems, while those between the deck-drift ratio are equal to 1.3 and 1.4, respectively. By accounting for plastic soil behaviour, as anticipated, lower stresses on the structure and lower displacements are obtained compared with those computed assuming the soil to behave as a viscous-elastic material. However, the difference provided by different constitutive assumptions decreases as the period ratio Teq/T0 increases (Table 7). The normalised bending moment approaches zero at the end of seismic shaking for both systems as the pier is always modelled as a linear viscous-elastic S.D.O.F. system. Temporary attainment of shear strength during seismic shaking, with consequent development of plastic strains, is also evidenced by the undrained stress paths experienced by soil elements close to the foundation base. In Fig. 18 they are plotted in the deviatoric stress plane, in which distances from the origin are equal to τn = √3∙τoct, with τoct being the octahedral shear stress. Stresses ¯ a, b, c = 2/3 a, b, c are the principal stresses projected onto the deviatoric plane, the suffixes a, b and c indicating the principal stress directions: direction a is about the vertical direction (z-axis), while directions b and c are about the horizontal directions (x and y axes) at the beginning of earthquake loading. The initial stress condition (empty triangle) of soil element O (Fig. 18a-b), close to the base of the deeper caisson, is characterised by axisymmetric conditions of extension, the minimum stress being associated to a direction close to the vertical ( ¯ a ). The stress path computed during seismic shaking by the elastic-plastic analysis is irregular, covering a wide range of stress states, and reaches the Mohr-Coulomb strength envelope relevant to element O (p′min = 216 kPa) attaining, at some time instants, a passive limit state; at the end of the dynamic stage, stresses are close to pure shear conditions (light grey square), thus showing a significant change in the state of stress induced by earthquake loading close to the caisson edges. On the contrary, when soil behaviour is described using the linear viscous-elastic model, the stress path trespasses the failure envelopes with much higher values of the principal stresses, though covering a narrow band; at the end of the seismic action, the final stress state plots even outside the strength envelope (light grey circle). For point P (Fig. 18c-d), close to the base of the shallower caisson, stress path moves very closely around the initial state and does not ever approach the Mohr-Coulomb failure surface relevant to element P (p′min

5. Influence of inelastic soil behaviour on the seismic performance Influence of soil plasticity on the seismic performance of the soilcaisson-pier-deck system can be assessed comparing the maximum deck acceleration amax, deck computed by the non-linear dynamic FE analyses with the elastic spectrum calculated at the ground surface (z = 0) via the 1D free-field response analyses performed with the linear-equivalent (LE) method [49,51]. Fig. 16 shows this comparison for the inputs of Tolmezzo, Assisi and Adana. Should the non-linear and irreversible soil behaviour not be involved during the seismic shaking, values of amax,deck would plot on the corresponding free-field spectra, at a period T = Teq relevant to a flexible-base system. This occurrence is only observed for high values of period Teq (> 2 s), that is for very flexible systems (e.g.: tall piers and/or small caisson foundations). On the contrary, for stiffer systems, characterised by lower periods Teq, the computed values of amax,deck lie below the relevant elastic spectra, with substantial differences for periods in the range Teq = 1÷1.5 s, for which soil-system dynamic coupling mainly occurs with a major development of plastic strains and dissipation of seismic energy. The largest differences are observed for the Adana and Tolmezzo inputs, characterised by a mean period Tm close to the second period of the soil column, T1 ≈ 0.5 s. Conversely, values of amax, deck computed with the Assisi input mainly plot around the corresponding elastic spectrum, thus indicating a lower influence of inelastic soil behaviour on the inertial forces transmitted to the deck.

Fig. 16. Comparison between maximum horizontal accelerations at the deck level and spectral acceleration obtained at the ground surface, in free-field conditions, with the linear equivalent method. 130

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Fig. 17. Time histories of deck drift ratio and normalised bending moment at the top of the caisson (D = 12 m) computed using the HS small and the linear viscouselastic model: (a) and (b) stiff system with Teq/T0 = 0.77; (c) and (d) flexible system with Teq/T0 = 2.24 (Adana input).

resulting in lower inertial forces and displacements than that computed assuming viscous-elastic soil behaviour. This is why a rational design should account for the dissipative capability of the foundation soil. By performing 3D non-linear dynamic analyses on 51 soil-caissonpier-deck systems, all endowed with the same initial safety factor against bearing capacity, the seismic performance, that is, the maximum and permanent deck drift ratios urel/hs attained during and at the end of seismic shaking, could be related to the system and the earthquake properties. The deck drift ratio is seen to be a decreasing function of the ratio Teq/T0 between the natural periods of the system and the foundation soil; it also increases with duration of the strong motion phase that mostly affects the computed seismic performance, provided that seismic intensity is large enough to promote plastic strain in the foundation soils. Upper-bound relationships, corresponding to the 84th percentile, are evaluated from the computed results that can be used for preliminary screening analyses since the period ratio Teq/T0 can be easily estimated using analytical and empirical relationships. However, it is understood that the seismic database considered in the analyses needs to be extended to guarantee a more reliable tool. The comparison between the results provided by 3D dynamic analyses performed using an elastic-plastic strain hardening soil model and a viscous-elastic model pin-pointed the range of system and seismic properties for which soil plasticity is relevant and should be accounted for to avoid an excessive overestimate of earthquake-induced displacements and inertial forces acting on the superstructure. Contribution of soil plasticity becomes substantial for values of the period ratio Teq/T0 ≈ 0.8–1.5, for which soil-system dynamic coupling is appreciable, and for seismic inputs characterised by a long strongmotion duration. On the contrary, for more flexible systems (Teq/ T0 > 2), subjected to a seismic input of short duration and with a frequency content far from the fundamental period of the soil, linear viscous-elastic modelling of foundation soils can be deemed accurate enough if carried out using appropriate operative values of the shear modulus and damping ratio, calibrated on free-field ground response analyses.

Table 7 Peak values of normalised bending moment and deck drift ratio computed by different soil models (Adana input).

Teq/T0 = 0.77 (stiff system) Teq/T0 = 2.24 (flexible system)

Ms/Ms, lim urel/hs (‰) Ms/Ms, lim urel/hs (‰)

(1) viscous-elastic

(2) HS small

(1)/(2)

2.75 9.68 1.28 3.87

0.87 7.24 0.59 2.69

3.1 1.3 2.2 1.4

= 122 kPa), thus accumulating less plastic deformations. Conversely, the linear viscous-elastic stress path exceeds the failure surface though the final stress state plots inside the Mohr-Coulomb envelope at the end of the dynamic stage, near the initial stress condition; this confirms the closer agreement between the linear elastic and the plastic soil modelling for the flexible system. Greater plastic strains then develop for stiffer systems due to the transient mobilisation of shear strength during earthquake loading, thus resulting in lower values of the actions transmitted in both the pier and the caisson than those evaluated assuming a linear-elastic soil behaviour. A minor influence of plastic soil behaviour can, however, be inferred for more flexible systems. 6. Conclusions Seismic performance of long-span bridge piers founded on caisson foundations is usually assessed describing mechanical soil behaviour with a linear viscous-elastic model. However, accounting for the inelastic soil behaviour has the twofold advantage of permitting the evaluation of permanent displacements, as well as that of providing a more reliable estimate of the state of stress acting in the pier and the caisson during seismic shaking. Indeed, transient mobilisation of the soil shear strength produces a progressive development of plastic strains that results in permanent displacements, but also limits the maximum accelerations propagating in the foundation soil, thus

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Fig. 18. Undrained stress paths for soil elements located at the base of two caissons computed using the HS small and the linear viscous-elastic model: (a) and (b) stiff system with Teq/T0 = 0.77; (c) and (d) flexible system with Teq/T0 = 2.24 (Adana input).

Acknowledgements

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