Towards a seismic capacity design of caisson foundations supporting bridge piers

Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

Contents lists available at ScienceDirect

Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn

Towards a seismic capacity design of caisson foundations supporting bridge piers Athanasios Zafeirakos, Nikos Gerolymos n National Technical University of Athens, Greece

art ic l e i nf o

a b s t r a c t

Article history: Received 14 July 2014 Received in revised form 3 September 2014 Accepted 4 September 2014

The present work investigates, by means of finite element analysis in 3D, the nonlinear response under lateral monotonic and slow-cyclic loading of caisson foundations supporting bridge piers in cohesive soils. The study is performed with respect to the combined moment (M)–horizontal load (Q) on the foundation, and involves similar rigid cubic caissons carrying a column-mass superstructure of varying height (H) and pier-to-deck joint rigidity. The latter is simulated by a rotational spring at the deck level with parametrically varying stiffness, relating to the rigidity of the connection. The lateral load is imposed at the deck level, and the resulting M–Q load path on the caisson is mapped within the bounds of the respective failure envelope. The analysis revealed that due to pier-to-deck joint stiffness and the nonlinear coupling between the rotational and translational degrees of freedom at the foundation, the loading at the caisson head follows a nonlinear path characterized by the mobilization of a specific failure mechanism (identified as the “inverted” pendulum failure mode), triggered by the “negative effective height” effect. Interestingly, all load-paths display certain characteristics, such as a particular “overstrength” in bearing capacity that is mobilized by the foundation, irrespective of the stiffness properties and constraints at the superstructure system. Regarding the response under cyclic loading, the M–Q loops are shown to be well enveloped by the monotonic “backbone” curve for all examined cases. Finally, from the numerical results, a closed-form expression for the load-path is formulated, and a methodology to be used towards the seismic design of caisson foundations is proposed. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Caisson foundations 3D finite element analysis Failure envelope Pier-to-deck joint stiffness Nonlinear load path Preliminary seismic design approach

1. Introduction The seismic design of critical civil engineering structures (e.g. bridges, offshore platforms) requires the thorough understanding of the mechanics, from the material level to the complex interaction between structural, foundation and soil elements. Admittedly, the contemporary advancements in material science and technology have facilitated the feasibility of major projects. However, the costeffective earthquake design of structures, requires the role of the nonlinear soil–foundation–structure interaction (SFSI) on the final performance to be explicitly considered. Thus, for the realistic estimation of maximum and residual displacements of foundation– superstructure systems, which in essence constitutes the core of performance-based design philosophy, all sources of nonlinearities should be taken into account; namely nonlinearities developed above and below ground level. In fact, there is a growing awareness among the geotechnical and structural engineering communities towards incorporating the effects of soil compliance into advanced static and dynamic response analysis techniques (e.g. [30–36]). Perhaps

n

Corresponding author. E-mail address: [email protected] (N. Gerolymos).

http://dx.doi.org/10.1016/j.soildyn.2014.09.002 0267-7261/& 2014 Elsevier Ltd. All rights reserved.

surprisingly, the field of performance based design of deep embedded foundations has received little attention (e.g. [1–5]). For caisson foundations, which have been widely used as supports and anchors for major structures (especially bridges) in soft soils, the necessity for considering inelastic foundation response stemmed from recent strong earthquake events (e.g. the Great Hanshin earthquake), where it was evidenced that their massive dimensions do not guarantee elastic response (e.g. [16,20,28]). According to the conventionally employed analysis methodologies, the foundation is substituted with a set of linear, or even nonlinear, springs attached at the base of the superstructure. Yet, these springs are usually uncoupled, resulting in most cases in an erroneous foundation response, as will be shown in the next section. Nowadays, the improvement in computational efficiency would suggest the three-dimensional (3-D) finite elements (FE) as the obvious method for solving complex nonlinear SFSI problems. Still, the success of the FE method lies in a complete geotechnical site investigation and a rigorous constitutive modeling, restricting thus its use in common practice. To date, the analysis methods that attract increasing attention from geotechnical and structural engineers are based on the strain-hardening plasticity theory. These propose a single macroelement, a term first used in geomechanics by Nova and

180

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

prior to a comprehensive performance based design in which the response of the foundation–superstructure system will be evaluated in terms of displacement or other damage measures.

Montrasio [6], to describe the macroscopic soil–foundation response to applied forces and moments. Placed at the base of the superstructure, the macroelement reproduces the nonlinear interaction phenomena of material (soil yielding) and geometric (slippage and gapping at soil–foundation interface) nature. It, essentially, consists of a generalized inelastic stiffness matrix, in which the translational and the rotational degrees of freedom are fully coupled. The behavior of the foundation is thus described in an elegant and comprehensive manner (e.g. [6,7,26,37–40]). One of the main components of a macroelement is the representation of the ultimate limit states under combined vertical–horizontal and moment (N–Q–M) loading are by a failure envelope (or bearing strength surface). Analytical expressions for the failure criterion of deeply buried foundations have been proposed by Cassidy et al. [8] (skirted caissons in clay), Zhang et al. [9] (deeply buried footings in clay), [41,42] (spudcans in clay), and by the authors ([3]) for rigid caissons in cohesive soil with embedment ratio D/BZ 1. This study utilizes the concept of macroelement modeling towards developing a seismic design procedure for caisson foundations supporting bridge piers. The proposed methodology considers the full incorporation of the foundation compliance effects, with due consideration to the nonlinear coupling between all degrees of freedom at the foundation level, into a force-based design. The procedure aims at the preliminary design of caisson foundations (in terms of capacity)

1.1. Motivation The motivation of the present study is unveiled by means of an illustrative example. Fig. 1 depicts a bridge pier founded on a caisson foundation. The superstructure (pier, deck) is modeled as a single degree of freedom (SDOF) oscillator. For the design of the foundation, two of the aforementioned methodologies will be (qualitatively) employed and compared: (i) a set of uncoupled nonlinear springs, and (ii) a rigorous FE model, schematically represented by the coupled elasto-plastic stiffness matrix at the base of the pier. The comparison of the response under lateral loading, which in this case is transferred at the foundation level as a pair of shear force (Q) – overturning moment (M¼QH), is shown on an indicative bearing strength surface of the caisson, plotted in normalized form with respect to the pure horizontal (Qu) and moment (Mu) capacities of the foundation (e.g. [10,11]). The oblique shape of the strength surface owes to the coupling between the translational and the rotational degrees of freedom. The graph reveals that modeling the foundation with uncoupled springs results in a non-admissible state, outside of the strength surface. Note that in this example the path corresponds to a rotational spring failure (Mu) prior to the respective horizontal one

Q

Q

Q H

Q

u M M

Un-coupled non-linear springs

/M

Finite element method (coupled non-linear springs)

1.5

1

FE analysis un-coupled nonlinear springs analysis

0.5

0 0 -0.5

0.5

1

1.5

2 Q/Qu

-1

-1.5

-2 Fig. 1. Motivation of the study: a caisson foundation supporting a bridge pier-deck subjected to lateral loading. The pier is simulated as a SDOF oscillator. Indicative results are plotted for the load path at the caisson, when the foundation is modeled by: (i) un-coupled nonlinear springs, and (ii) the rigorous finite element method. The coupled (actual) failure envelope is shown with the solid black line, while the dashed grey line represents the un-coupled failure envelope.

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

(QoQu). In contrast, the “actual” load path tracks the failure envelope with accuracy. At this point it is noted that when the superstructure is modeled as a SDOF, the load path will always be tracked in the first quadrant (where MQ40). The shape of the failure envelope, however, reveals that admissible load states lie also in the fourth quadrant (MQo0), in which, more importantly, the loading combinations result in a bearing capacity of the foundation exceeding Qu or Mu (“overstrength”). We therefore extended the investigation into the nonlinear SFSI of systems that mobilize the increased capacity of the foundation. A caisson foundation supporting a bridge pier is again considered, as shown in Fig. 2. This time, however, a more realistic assumption is adopted. More specifically, it is anticipated that the deck applies certain kinematic constraints on the soil–foundation– pier system, owing to its rotational stiffness. An appropriate way to model these constraints is by attaching a rotational spring at the deck level. When the system is subjected to lateral loading (Q) applied at the deck, the deck, and more specifically the deck–pier joint, will develop a moment reaction to the loading, proportional to its stiffness KR. Consequently, the moment loading on the foundation may differ significantly from the case where a SDOF is considered (M¼ QH). The graph shows the indicative load paths derived from the prior used methodologies. The discrepancy in the response is now obvious: although both methodologies are

181

capable of capturing a nonlinear response of the foundation, the assumption of uncoupled springs leads again to a non-admissible state within the first quadrant. In marked contrast, the “real” load path, as calculated from the rigorous FE solution, manifests that the nonlinear coupling between the translational and the rotational degrees of freedom at the foundation mobilize the “overstrength” of the caisson; the load path now terminates near the apex of the failure envelope. The observed response is attributed to the “negative effective height” effect, the meaning of which will be discussed in the following section.

1.2. The meaning of “negative effective height” The response of a bridge-pier subjected to a pseudo-statically imposed seismic horizontal force Q at the deck level is analyzed herein. The bridge–pier system comprises a single-column bent of height H, supported by a caisson. Five idealized versions of the actual system are considered with respect to the top (pier–bridge– deck support connection) and bottom (foundation compliance) boundary conditions: 1. A top free to rotate column (appropriate for modeling the lateral response of relatively long-spanned bridges, or when

m

Q

Q

KR

EI

N, w ,

LRP

Q, u

Q N

/M

2

1

FE analysis un-coupled nonlinear springs analysis

0

0

0

0.5

0.5

1

1

1.5

2

Q/Qu

-1

-2 Fig. 2. Motivation of the study: a caisson foundation supporting a bridge pier-deck subjected to lateral loading. The deck now applies kinematic constraints at the pier-deck joint, simulated by a rotational spring. Indicative results are plotted for the load path at the caisson, when the foundation is modeled by: (i) un-coupled nonlinear springs, and (ii) the rigorous finite element method. The coupled (actual) failure envelope is shown with the solid black line, while the dashed grey line represents the un-coupled failure envelope.

182

2.

3.

4.

5.

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

where [Kss], [Ksb], [Kbs] and [Kbb] are the four stiffness sub-matrices corresponding to the superstructure (s) and its base (b). [Kf] and [Kd] are the foundation stiffness matrix (e.g. [12,13,28,29]) and the stiffness matrix representing the pier-to-deck connection rigidity:

the bridge columns and beams are connected through a hinge) fixed at its base (e.g. foundation in rock). A top free to rotate column on a compliant foundation, represented by a set of coupled linear springs associated with the swaying KH, rocking KM and cross-swaying-rocking KHM, KMH motion of the caisson. A beam column fixed at its base, the motion of which is hampered at its top against rotation via a rotational spring KR representing the pier-to-deck connection rigidity. Similar to model 3 with the difference of a compliant base represented by a set of uncoupled linear swaying KH and rocking KM springs. Similar to model 4, but with consideration of a fully-coupled linear stiffness matrix for the foundation compliance.

2 12 EI c 4 H2 ½K bb  ¼ H H6

6 H

4

2 12 EI c 4  H2 ½K bs  ¼ H  H6

3

2 12 EI c 4 H2 ½K ss  ¼ H  H6

5;

6 H

2

 H6 4

3 5

3 5;

½K sb  ¼ ½K bs T

ð2Þ

The goal of the analysis is to calculate the effective height of the bridge–pier–foundation system, determined as the ratio of the overturning moment Mb to the horizontal force Q b transmitted to the head of the caisson. The response of the bridge–pier, namely the (absolute) displacement and rotation of the pier top (deck) and bottom (caisson head), can be computed from the following equation: 8 9 0> > >   > " > > < ½K bb  þ K f 0= ¼ > ½K sb  > >Q > > ; : > 0

9 8 Ub > > > #> > > < θb =

½K bs  Us > ½K ss  þ ½K d  > > > > > ; : θs

ð1Þ Fig. 4. Overview of the 3D finite element model with the parameters used in the analysis.

KR

KR

Q

KR

H

KH KHM

Model 1

KH

KH KM

Model 2

KM

Model 3

KM

KHM

Model 4

Model 5

1.25

Heff / H

1

0.75

0.5

0.25

0 0.0001 -0.25

0.001

0.01

0.1

1

GS /G0 Model 1 Model 2 Model 3 Model 4 Model 5

Fig. 3. Effective height Heff, normalized by the geometric height H, of various idealized bridge pier–foundation systems subjected to lateral loading, as a function of soil compliance. Observe, that for large deformations and near failure conditions (Gs/G0-0), the effective height obtains negative values (“negative” effective height effect) when coupling between the degrees of freedom at the caisson head are fully taken into consideration (model 5).

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

" ½K d  ¼

0

0

0

KR

# ;





"

Kf ¼

KH

K HM

K MH

KM

#

After some algebra, the following expressions for the effective height Heff ( ¼Mb/Qb) are obtained: Models 1 and 2

ð3Þ

H ef f ¼ H

The boundary conditions are as follows:

ð4Þ

Model 3

 Model 1: Ub ¼θb ¼0, [Kd] ¼[0],[Kf] ¼ [Inf]  Model 2: [Kd] ¼[0] (Qb ¼ –KHUb  KHMθb,   

183

H ef f ¼

Mb ¼–KMHUb  KMθb) Model 3: Ub ¼θb ¼0, [Kf] ¼[Inf] Model 4: KHM ¼ KHM ¼ 0 (Qb ¼ –KHUb, Mb ¼  KMθb) Model 5: (Qb ¼ –KHUb  KHMθb, Mb ¼–KMHUb  KMθb)

H 2EI c þ K R H 2 EI c þ K R H

ð5Þ

Model 4 H ef f ¼

H K M ð2 EI c þ K R H Þ 2 K R EI c þ K M EI c þ K R K M H

ð6Þ

Model 5 Hef f ¼ Table 1 Geometric properties of the pier columns. 2

ð7Þ In an attempt to demonstrate the physical meaning of the negative effective height, Eqs. (4)–(7) are applied to an actual case study involving a major highway bridge on caisson foundations in soft rock. The bridge deck was constructed by the cantilever method and is rigidly connected to the piers. The bridge has a total length of 200 m

4

H (m)

A (m )

I (m )

6 17 55

13.85 22.90 40.72

3.98 15.27 41.74

2 EIc K R K HM  2 EI c K 2HM H  K R K 2HM H 2 þ K R K H K M H 2 þ 2 EIc K H K M H   2 EI c K R K H  K R K 2HM H  EI c K 2HM þ EI c K H K M þ K R K H K M H

Q, udeck Q, ucaisson

H = 6m

/Mu 2

1.5

Kratio = 0.0 Kratio = 0.10

Response at failure

1

Kratio = 0.25 Kratio = 0.40

0.5

Kratio = 1.0 0 0

0.5

1

1.5

2

Q/Qu

-0.5

failure point -1

-1.5

-2

Q/Q u , ucaisson

Q/Qu , udeck -u caisson

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

1.8 1.5 1.2 0.9 0.6 0.3 0 0

0.2

0.4

0.6

u (m)

0

0.2

0.4

0.6

u (m)

Fig. 5. FE analysis results for a caisson foundation supporting a bridge pier with H¼ 6m, subjected to parametrically varying constraints in rotation (Kratio) imposed at the deck level. A lateral force Q is imposed on the pier (at the deck level) until complete failure of the system. Interestingly, for all studied cases the “inverted” pendulum failure mechanism is manifested.

184

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

approximation of the actual non-linear response of the caisson. The following observations are worthy of note:

and a central span of 90 m. The tallest of the two supporting piers is about 30 m high with a rectangular hollow cross-section of 5 m  7.3 m  0.75 m. The bending rigidity of the pier is EIc E1.8  109 kN m2 and the stiffness of the rotational spring (accounting for the rotational rigidity of the deck at the connection with the pier) is estimated to KR E0.2  109 kN m/rad, resulting in a Kratio ¼ 0.9 (the definition of Kratio is given in the next section). The foundations are squared reinforced concrete caissons of B (width)¼D (depth of embedment)¼ 10 m and the shear modulus of rock at very small deformations is estimated to G0 ¼1 GPa. Assuming homogeneous elastic soil conditions, the stiffness matrix of the foundation with respect to the top of the caisson can be computed by the following expressions [12,13,27]: K H  9:4G0 B; K M  10:8G0 B3 ; K HM ¼ K MH   6:3G0 B2

 The effective height for models 1 and 2 is equal to the geometric height H, irrespective of the caisson compliance.

 The normalized effective height for model 3 is calculated to



ð8Þ

Numerical substitution of EIc, H, KR and G0 into Eqs. (4)–(8) results to the plots of Fig. 3, which shows the normalized (with respect to the geometric height H) effective height ratio as a function of the shear modulus ratio Gs/G0. Gs is an equivalent secant shear modulus accounting for the decrease in foundation stiffnesses due to soil and interface nonlinearities, in an equivalent–linear

Heff E0.62H implying a nearly inflexible rotational stiffness of the deck. An infinitely rigid deck would result in an effective height ratio of 0.5 (fixed-end beam boundary conditions), while in the case of a very flexible joint (e.g. support on elastomeric bearings) the effective height ratio would be equal to 1 (cantilever beam boundary conditions). When the rotational rigidity of the pier-to-deck connection and the foundation compliance are both considered (models 4 and 5), the response at very small deformations (Gs /G0 E1 quasi elastic response) is similar to that for fixed-base conditions (model 3) irrespective of the coupling of the foundation degrees of freedom (coupled or uncoupled springs). There is a gradual deviation of the predictions by the two models, however, with increasing soil nonlinearity. Indeed, as the Gs/G0 ratio decreases, the effective height tends to zero for model 4 while exhibits negative values for model 5 (fully-coupled foundation stiffness matrix) when Gs /G0 becomes smaller than 0.3%. The negative

Q, udeck H = 17m

Q, ucaisson

/Mu 2

Kratio = 0.0

1.5

Kratio = 0.10 1

Kratio = 0.25

Response at failure

Kratio = 0.40 0.5

Kratio = 1.0

0 0

0.5

1

1.5

2

Q/Qu

-0.5

failure point -1

-1.5

-2

Q/Q u , ucaisson

Q/Q u , udeck - u caisson

1.8

1.8

1.5

1.5

1.2

1.2

0.9

0.9

0.6

0.6

0.3

0.3 0

0 0

0.2

0.4

0.6

0.8

1

u (m)

0

0.2

0.4

0.6

0.8

1

u (m)

Fig. 6. FE analysis results for a caisson foundation supporting a bridge pier with H¼ 17 m, subjected to parametrically varying constraints in rotation (Kratio) imposed at the deck level. A lateral force Q is imposed on the pier (at the deck level) until complete failure of the system. Interestingly, for all studied cases the “inverted” pendulum failure mechanism is manifested.

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

values for Heff imply that the overturning moment and shear force at the head of the caisson are counteractive (of the opposite sign). The conditions in the response that should be satisfied for a bridge pier to exhibit the negative effective height effect, are: (1) negative cross-swaying-rocking stiffnesses (an inherent property of the embedded and deep foundations), and (2) extensive soil yielding and bearing capacity mobilization. The latter observation will serve as the basis in developing a methodology for the preliminary seismic design of caisson foundations supporting bridge piers. It will be shown that a “negative effective height” response could be manifested even for very small values of KR (e.g. response of a long-spanned bridge in the transverse direction) at the cost, however, of larger displacements.

185

paper is directed. By means of a series of numerical analyses in 3-D, the response of caisson foundations supporting a pier-to-deck system is investigated, with due consideration to nonlinearities developed below (soil, caisson–soil interface) and above ground (pier, deck–pier joint). From the analysis, a closed-form expression for the “actual” monotonic load path, in M–Q space, is developed. It is furthermore shown that the monotonic curve envelopes the load paths derived from slow-cyclic loading with striking precision. Finally, a methodology for the preliminary seismic response evaluation, in capacity terms, of caisson foundations supporting bridge piers is proposed.

2. Problem definition and method of analysis 2.1. Problem definition

1.3. Scope of work The above analysis demonstrated that the simplistic approaches, usually employed in practice, and refer to (a) the separation between structural and geotechnical design, and (b) the modeling of foundation elements, cannot reproduce many essential features of the actual foundation behavior. It is to this end that the present

The studied problem is portrayed in Fig. 4. It involves a bridge pierto-deck system supported by a cubic caisson of side D¼ 10 m in a 20 m thick 2-layer cohesive soil stratum. The caisson is modeled as a rigid body, with loads and displacements relating to a single load reference point (LRP), as shown in Fig. 2. The sign convention obeys a right-handed axes and clockwise positive convention, as proposed by

Q, udeck

Q, ucaisson

/Mu

H = 55m

2

1.5

Kratio = 0.0 Kratio = 0.10

Response at failure

1

Kratio = 0.25 Kratio = 0.40

0.5

Kratio = 1.0

0 0

0.5

1

1.5

2

Q/Qu

-0.5

failure point -1

-1.5

-2

Q/Q u , udeck - ucaisson

Q/Q u , ucaisson 1.8

1.8

1.5

1.5

1.2

1.2

0.9

0.9

0.6

0.6

0.3

0.3

0

0 0

0.2

0.4

0.6

0.8

1

u (m)

0

0.2

0.4

0.6

0.8

1

u (m)

Fig. 7. FE analysis results for a caisson foundation supporting a bridge pier with H ¼55 m, subjected to parametrically varying constraints in rotation (Kratio) imposed at the deck level. A lateral force Q is imposed on the pier (at the deck level) until complete failure of the system. Interestingly, for all studied cases the “inverted” pendulum failure mechanism is manifested.

186

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

Butterfield et al. [14]. Three different pier heights, to represent three different loading conditions with respect to moment loading on the foundation, are used; namely H¼6 m, 17 m and 55 m. The mass of the deck, m, is given the value of 2700 Mg, corresponding to a static factor of safety of FSV ¼ 5 [4]. The kinematic constraints imposed by the pier-to-deck joint are simulated by a rotational spring attached at the mass level. The respective axial stiffness was neglected which, although admittedly an idealized case, is considered adequate for representing the deck connection (in the longitudinal direction) to the abutments with elastomeric joints.

2.2. Finite element modeling The problem is analyzed with use of the commercially available finite element code ABAQUS. Both caisson and soil are modeled with 3D 8-noded brick elements, assuming elastic behavior for the former and elasto-plastic for the latter. Fig. 4 shows a half caisson cut through one of the orthogonal planes of symmetry. The size of the finite element mesh is 5B  5B  5B carefully weighing the effect of boundaries on the caisson's response and the computational time [4]. Zero-displacement boundary conditions prevent the out of-plane deformation at the vertical sides of the model, while the base is fixed in all three coordinate directions. The caisson is connected to the soil with special contact surfaces, allowing for realistic simulation of the possible detachment and sliding at the soil–caisson interfaces. The mesh for the soil– caisson consists of 12,500 elements. To improve the computational cost without jeopardizing the accuracy of the analysis, the surfaceto-surface contact interaction was modeled by exponential (“softened”) pressure–overclosure relationship through the direct constraint enforcement method that makes use of Lagrange

multipliers. For more details on the contact interaction algorithm, the reader may refer to ABAQUS manual [15]. The caisson material is modeled as isotropic linear elastic, with a unit weight of γ¼25 kN/m3, a Young's modulus of Ec ¼ 3  108 kPa and a Poisson's ratio of νc ¼ 0.15. However, to ensure rigid body motion appropriate kinematic constraints have been applied between the top central node and all the peripheral nodes of the caisson. The pier columns are modeled with 3-D Timoshenko beam elements, with elasticity modulus Ec ¼3  108 kPa and νc ¼0.15. The geometric properties of the piers, namely the cross-sectional areas (A: m2) and moments of inertia (I: m4), are given in Table 1. To investigate the effect of per-to deck stiffness on the response of the foundation, the stiffness of the rotational spring, KR, was appropriately modified so that the stiffness ratio K ratio ¼ K R = ð4Ec I=HÞ, where (4EIc/H) is the flexural stiffness of the column, yields the following values: Kratio ¼0.1, 0.25, 0.5, 1.0, for every case examined. Moreover, the case of Kratio ¼0 (inactive rotational spring) is also provided, for reference. At the first stage of the analysis, the pier column and the deck (rotational spring) are considered to respond in a linear elastic fashion. The insight into the effect of structural nonlinearity will be given in the ensuing sections.

2.3. Constitutive model Soil behavior is described by an elasto-plastic constitutive model [16], which is a reformulation of that originally developed by Armstrong and Frederick [17]. The model is available in the material library of ABAQUS. It uses the Von Mises failure criterion along with a nonlinear combined kinematic–isotropic hardening law and an associated plastic flow rule.

/Mu 2

KR

1.5

R

R

1

0.5

0 0

0.5

1.5 Q/Qu

1

2

-0.5

-1

-1.5

-2

Non-linear pier

Non-linear spring

P, i

R, i

KR, i 1, i

2, i

1, i

2, i

Fig. 8. Schematic representations of: (a) the moment sustained by the pier-deck joint, ΜR. (b) the bi-linear moment–rotation and moment–curvature laws used to model the spring and the pier, respectively.

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

The clayey soils are saturated responding in undrained fashion with shear strength Su ¼ 65 kPa at the upper 6 m, Su ¼ 130 kPa at the lower 14 m and a constant stiffness to strength ratio Eu =Su ¼ 1500. The unit weight of the soil materials is taken γ¼20 kN/m3. To approximate the constant volume response under undrained

Table 2 Predefined bilinear moment-rotation values for the rotational spring. H (m)

MR (MN m)

Kratio

θ1 (rad)

θ2 (rad)

6 17 55

198 276 1070

0.25 0.40 1.0

0.00264 0.002307 0.003568

1 1 1

187

conditions, a Poisson's ratio of νE 0.5 (¼ 0.475 to avoid numerical instabilities) was assumed. The authors in [4] validated the constitutive model employed herein for the nonlinear response of embedded and caisson foundations against field tests of suction caissons in clay [18]. From the validation process, it was deduced that the model is capable of capturing the response observed in the field trials with satisfactory agreement, in both small and large amplitude lateral loading, including the development of caisson–soil interface nonlinearities (gapping).

2.4. Method of analysis At first, a series of finite element (FE) static pushover-type of analyses are carried out to derive the bearing strength surface of

/Mu

/Mu

1

0.5

=6m

Kratio = 0.25

deck

0.4

0.5

0.3 0.2

0

caisson

0.1

0

0.5

1

1.5

2

0

Q/Qu

-0.5

-0.1 0

0.02

0.04

: rad -1

-1.5

-2

/Mu

/Mu

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

= 17 m

Kratio = 0.40

0.5

0 0

0.5

1

1.5

2

caisson

0

Q/Qu

-0.5

deck

0.02

0.04

: rad

-1

-1.5

-2

/Mu

/Mu

3

1

deck

2.5

= 55 m

2

Kratio = 1.0

0.5

1.5 1

caisson

0.5

0

0

0 -0.5

0.5

1

Kratio = 0.0

1.5

2

Q/Qu

0

0.02

0.04

: rad

Linear spring -1

Non-linear spring

-1.5

-2 Fig. 9. FE results for the load path at the caisson foundation, assuming a linear elastic pier and a nonlinear spring. The corresponding evolution of the moment sustained by the caisson (solid black line) and at the deck (solid grey line) with respect to the caisson's rotation are plotted alongside.

188

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

the caisson–soil system in M–Q space [3,19,20]. Firstly, the soil undergoes geostatic loading and then a part of the soil is replaced by the foundation, on which a vertical load is applied till the specified value of the factor of safety (FSV ¼5) is reached. Afterwards, the vertical load is kept constant and a combination of horizontal force Q and overturning moment M is applied at the head of the caisson until failure. By a series of force-controlled analyses the strength surface is ultimately determined. The surface is presented normalized with respect to the pure moment capacity Mu (with no horizontal loading) and the pure horizontal capacity Qu (no moment loading) of the foundation. The analysis yielded the following values for the pure capacities: Qu ¼44 MN, and Mu ¼ 430 MN m. The whole soil–foundation–superstructure system is then subjected to lateral loading at the deck level, monotonically increasing until system failure. The analysis terminates either due to bearing capacity failure of the foundation or due to

Table 3 Predefined bilinear moment–curvature and moment–rotation values for the pier column and rotational spring, respectively. H (m)

MP (MNm)

κ1 (rad/m)

κ2 (rad/m)

55 55

215 195

0.0000544 0.0000550 107 0.0000493 0.0000510 85

MR (MNm)

Kratio θ1 (rad) 0.40 1.0

θ2 (rad)

0.00089 1 0.000283 1

failure of the superstructure. Throughout the analysis, the load path on the foundation is being constantly tracked in M–Q space.

3. Results 3.1. Monotonic loading: effect of column height and deck stiffness Figs. 5–7 portray the computed load paths at the foundation level with respect to the rotational stiffness ratio (Kratio), for each of the pier heights examined. This section deals with elastic response of the superstructure (pier column and deck). To facilitate the interpretation of the results, the response is also presented in terms of the evolution of the horizontal loading Q (normalized with Qu) with respect to the horizontal displacement at the foundation level (ucaisson) and at the deck (udeck). Noticeably, the presence of the rotational spring, in conjunction with the nonlinear response of the foundation, has a distinct effect on the shape of the load paths. Considering moment equilibrium at the pier base (caisson top), the following condition is obtained: M ¼ Q H  MR

ð9Þ

By dividing each term of the equation with Q, Eq. (1) becomes M MR ¼ H Q Q

ð10Þ

/Mu 1

= 55 m

Kratio = 1.0

/Mu pier

0.5 0.4 0.3

deck

0.2 MR

MP

0.5

0.1

MR

0 0

0.001

0.002

: rad

0 0

0.5

Q/Qu

1

/Mu

1

= 55 m

Kratio = 0.40

/Mu

pier

0.5 0.4 0.3

MP MR

0.5

0.2

MR

deck

0.1 0 0

0.001

0.002

: rad 0 0

0.5

Q/Qu 1

Kratio = 0.0 Linear spring, Linear pier Non-linear spring, Non-linear pier Fig. 10. FE results for the load path at the caisson foundation, assuming nonlinear response of both the pier and the pier-deck joint. The corresponding evolution of the moment sustained by the caisson (solid black line) and at the deck (solid grey line) with respect to the caisson's rotation are plotted alongside.

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

189

Yet, as revealed from the evolution of the horizontal displacement ucaisson with respect to the lateral loading Q, each load path tracks the failure envelope at a different point. The shorter piers carrying a rotationally stiff deck (Kratio-1) exhibit a quasi elastoplastic Q  u response. This is congruent with the respective load paths, which can be tracked within the failure envelope (elastic response) until the ultimate bearing capacity load is reached. Note, however, that (a) as the deck becomes more rotationally flexible (Kratio-0.10), and (b) as the pier height increases (large initial moment loading on the foundation), the loading paths reach the failure envelope at an earlier point, and then follow it asymptotically until the termination point. In the respective Q  u graphs, this behavior is manifested from the quasi tri-linear shape of the plots; the first yielding point corresponds to the point where the load path first reaches the failure envelope. The discrepancy in the loading paths with respect to the pier height is further demonstrated by the snapshots of plastic strain magnitude at failure. Interestingly, all load paths terminate at about the same point (which will be designated as point C in the sequel) on the failure locus, eventually mobilizing the “inverted pendulum” failure mechanism. Regarding the horizontal displacements at the deck level, they follow closely the respective ones computed at the top of the caisson, when short piers carrying rotationally stiffer decks are considered. However, it is evident that with an increase in the pier height and a decrease in the deck's rotational stiffness, the deck displacements exceed the respective ones of the caisson even by a factor of 2.

Eq. (10) thus reveals that the load path in the M–Q space is a function of the pier height (H) and a term (MR/Q) that depends on both the stiffness of the rotational spring (KR) as well as the nonlinear stiffness of the foundation. In the case of KR ¼ 0, Eq. (10) yields the height of a cantilever beam, bearing no dependence on the stiffness of the foundation. It thus becomes evident that the mapping of the nonlinear SFSI effects (nonlinear load path) in the M–Q space is “triggered” by the rotational rigidity at the deck. Given that the moment sustained by the deck increases constantly (in proportion to its stiffness, KR) during the lateral loading, the following can be inferred from the graphs: The nonlinear load path on the foundation exhibits an initial increase until a maximum value in the MQ 40 region, after which the path is directed towards the “overstrength” (the moment loading decreases and re-develops in opposite sign, MQ o0). As anticipated, the maximum “positive” moment increases with increasing pier height (higher lever arm), but decreases with increasing Kratio (the stiffer the deck, the more moment loading it sustains). The slope of the tangent to the M/Mu–Q/Qu curve at the origin (“elastic” region) does not correspond to the pier height, H, but to the initial effective height of the pier column (Heff,0). This, in turn, can be calculated, for each examined case, by the methodology presented in the previous section using Eq. (7). Quite strikingly, the load paths appear to terminate at a specific point on the failure envelope, irrespective of the stiffness ratio and the pier height. In terms of kinematic mechanisms that accompany failure, this point corresponds to mobilization of the “inverted pendulum” mechanism [20], as can be visualized by the contours of plastic strain magnitude (herein corresponding to Kratio ¼ 0.25), shown at the top of each figure. For the studied case of caisson foundations with embedment ratios D/B¼ 1, the coordinates of the   “inverted” pendulum point are approximated by: ðπ=2Þ  ð2=3Þ in the normalized M–Q space. The identification property of the “inverted” pendulum failure mode is that the rotation pole of the caisson, zp, is located at its base (zp ¼D).

3.2. Accounting for structural nonlinearity At this section, the numerical investigation is directed towards the effect of the nonlinear response of the superstructure on the load path at the foundation level. The presentation of the methodology and the results precedes a graphical illustration (Fig. 8a) that serves to elucidate the contribution of each of the load-carrying

I: Pier failure II: Pier-deck joint failure

III: Bearing capacity failure /Mu 1

0.5 R

I

II

0 0

0.5

III 1

1.5

2

Q/Qu -0.5

III -1

-1.5

-2

Fig. 11. Schematic illustration of the possible failure modes of a caisson foundation–pier–deck system, accounting for nonlinearities above and below ground level.

190

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

mechanisms within the foundation–superstructure system. Yet, the emphasis is again given to the response of the foundation. The caisson and the pier base are subjected to moment loading that is represented by the load path. The moment that develops at the pier–deck joint (MR), however, is given by the vertical distance between the load path and the (extended) line corresponding to the respective (for that specific pier height) cantilever beam (KR ¼0). It should be noted that Fig. 8a portrays an ideal case, in which the pier and the deck respond elastically, and they are therefore capable of carrying substantial loading. A more realistic, though simplified, nonlinear response is now introduced for the superstructure, by means of a bi-linear moment– rotation law for the deck and a bi-linear moment–curvature law for the pier (the ultimate bending moment capacity of which is MP), as shown in Fig. 8b. The analysis aims at gaining insight into the effects of the interplay between the nonlinearities, developed above and below ground level, on the load path at the foundation. Hence, the values of the bi-linear laws are deliberately pre-defined, so that structural failure precedes bearing capacity failure of the foundation.

To begin with, we consider a nonlinear rotational spring attached to the top of an elastic pier column. Three characteristic cases are examined, the properties of which, along with the predefined values of the bilinear moment–rotation model for the spring, are given in Table 2. The results are shown in Fig. 9. For reference, each characteristic response is accompanied with the respective ones considering (a) cantilever beam (KR ¼0), and (b) a linear rotational spring. Moreover, the evolution of the normalized moment loading on the foundation (solid black line) and on the deck (solid grey line) with respect to the caisson rotation, are given alongside. For clarity reasons, the negative (in sign) moment developed at the rotational spring is plotted in absolute values. The computed response can be interpreted as follows: as long as the spring is loaded within its elastic regime, the loading on the foundation follows the respective nonlinear path (dashed line). However, when the moment capacity of the spring (MR) is reached, denoting the formation of a plastic hinge, the spring cannot sustain any further increase in moment loading. The additional moment is

/Mu 2 1.5 1 0.5 0 0

0.5

1

1.5

2

Q/Qu

-0.5 -1 -1.5 -2

/Mu 2

0.5 0.4 0.3

1.5 Q Q

1

,

M Q M , Q M

B

0.5

Heff, 0 Qu Mu

0.2

M

A

0.1 0 0

0.1

0.2

0.3

0 0 -0.5

0.5

1

1.5

2

C

2 , 2 3

Q/Qu

-1

-1.5

-2 Fig. 12. (a) Collection of the numerically calculated load paths examined in this study. (b) Graphical representation of the three conditions applied for the extraction of a closed-form expression for the load path. The particular coordinates at point C belong to the failure envelope of caisson foundations with D/B¼ 1.

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

now imposed onto the foundation as positive (in sign) loading, and the system exhibits the response of a caisson carrying a cantilever beam; the load path is tracked parallel to the respective one of a cantilever with height H, and continues until bearing capacity failure of the foundation occurs. However, caution should be exercised during the design of such systems, since spring failure corresponds to a pier-to-deck joint failure. Hence, appropriate provisions should be adopted, mainly to accommodate for large displacements at the deck level. The response is now examined accounting for nonlinearity in both structural, elements, namely the pier and the deck. Two characteristic cases are considered: a pier height of H¼ 55 m, kinematically restrained at the deck level by a rotational spring that yields a stiffness ratio of (a) Kratio ¼0.40, and (b) Kratio ¼1.0. The values for the bi-linear models (as schematically shown in Fig. 8b) are given in Table 3. Given that the determination of shear strength, and especially of shear deformation characteristics of (e.g. reinforced concrete) structures are still controversial issues [21], the interplay between shear and flexure in the inelastic regime is not taken into account and the controlled mode of failure of the piers is assumed to be flexure-dominated. The results, provided in Fig. 10, highlight that contrary to the “ductile” progressive failure of the superstructure–foundation system observed following the yielding of the deck, column yielding leads in an abrupt system failure. This “brittle-type” system response is further illustrated from the graphs of the normalized moment loading at the pier base (solid black line) and on the deck (solid grey line) with respect to the caisson rotation, which are plotted alongside the load paths. In summary of the above, Fig. 11 schematically represents the nonlinear SFSI effects, and identifies the possible failure modes of the system; point I signifies pier column failure; point II corresponds to yielding at the pier-to-deck joint, followed by a ductile /Mu

191

foundation response; point III indicates bearing capacity failure of the foundation. 3.3. Analytical description of the load path Based on the analysis presented in the preceding sections, an analytical representation of the load path in M–Q space is attempted. Hence, a generalized expression of the following form is considered:    M Q ¼ γ ð1 þ αÞ 1  e  βðQ =Q u Þ  α ð11Þ Mu Qu where the coefficients α, β and γ control the shape of the load path, and depend on: the properties of the pier (H, I), the rotational stiffness of the spring (KR), and the pure uniaxial capacities of the foundation (Q u, Mu), which incorporate the effects of soil conditions and foundation's geometry. The determination of the three unknown coefficients requires the consideration of appropriate conditions. These conditions are identified from Fig. 12a, where the collection of the load paths for all examined cases is plotted, and detailed in Fig. 12b. More specifically: (1) Point C, is the point on the failure envelope associated with the “inverted” pendulum failure mechanism, on which the load paths terminate. Point C can be considered as an attractor of the caisson response at failure, irrespective of the stiffness properties and constraints at the superstructure system. Its coordinates can be derived from the following conditions: Point C lies on the failure envelope. Therefore, satisfaction of the respective equation (f¼0) for caisson foundations [3,20,24,25] requires that:

n1 n2

  Q M Q M f¼ þ þ n3 1 ¼ 0 ð12Þ Qu Mu Qu Mu /Mu

1

1

Kratio = 1.0

0.5

Kratio = 0.40

0.5

0

0 0

0.5

1

1.5

2

Q/Qu

-0.5

0

0.5

1

1.5

-1

-1

-1.5

-1.5

-2

-2

2

Q/Qu

-0.5

H 55m 17m

/Mu

/Mu

1

0.5

Kratio = 0.10

0.5

0

0 0

-0.5

6m

1

Kratio = 0.25

0.5

1

1.5

2

Q/Qu

0 -0.5

-1

-1

-1.5

-1.5

-2

-2

0.5

1

1.5

2

Q/Qu

Fig. 13. Comparison between numerical (solid lines) and analytical (dashed lines) results for the nonlinear path at the foundation in M–Q space.

192

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

coordinates of point C:

prescribing the bearing strength surface as an ellipsoid in the normalized M–Q space at a particular vertical load level (N). The coefficients n1, n2 affect the skewness with respect to the axis of symmetry, while n3 controls the size (expansion–contraction) of the surface. They are functions of the embedment ratio D/B, vertical load and interface conditions ([24,25]). It has been shown ([3,20]) that for fully bonded interface conditions n1 ¼n2 ¼2 (zero skewness), while n3, for any practical reason,

 1:98 depends only on the embedment ratio: n3 ¼ 1:84  0:21 D=B . Apparently, for nonlinear interface conditions (gapping and slippage) the coefficients are described by more complex functions, as suggested in [24]. Invoking normality conditions for the incremental plastic deformation of the caisson (with respect to its head) at failure, as has been conjectured by several researchers (e.g. [3] for deeply embedded foundations, D/B41 and [8,26] for shallow embedded foundations, D/Bo1), the ratio of the plastic displacement increments can be obtained by differentiating Eq. (12) with respect to Q and M, respectively. However, the incremental plastic displacement to rotation ratio for rigid caisson foundations at failure is by definition equal to the depth of the rotation pole, zp, which in turn is equal to the embedment depth of the caisson (zp ¼D) when the “inverted” pendulum failure mechanism is manifested. This results in an additional equation that must be satisfied by the

δup ð∂f =∂Q Þ ðMn3 =M u Q u Þ þ ðn1 signQ ðjQ j=Q u Þn1  1 =Q u Þ ¼ ¼ ¼D δϑp ð∂f =∂MÞ ðQ n3 =M u Q u Þ þðn2 signMðjMj=M u Þn2  1 =M u Þ ð13Þ in which δup and δϑp are the incremental plastic displacement and rotation at failure, respectively, with respect to the caisson head. Eqs. (12) and (13) form a nonlinear algebraic system, from which the coordinates of point C can be derived. Substituting the solution of the aforementioned system (ðQ =Q u ÞjC , ðM=M u ÞjC ) into Eq. (11), gives the equation for the first condition:      Q  M   βðπ=2Þ  α ¼ γ ð 1 þ α Þ 1 e ð14Þ Q  M  u C

u C

(2) Point A, that corresponds to the slope of the tangent to the load path at the initial stages of loading. The tangent represents the load path when considering a cantilever beam of initial effective height Heff,0. Given that M ¼ H ef f ;0 Q , the equation of the tangent in the normalized M/Mu–Q/Qu space can be rewritten as follows:

 M Q Q ¼ H ef f ;0 u ð15Þ Mu Mu Q u

Q

Kratio H

Caisson: D/B = 3

/Mu 1.5

H = 70m, Kratio = 0.40 1

H = 100m, Kratio = 0.25

0.5 0 0

0.5

1

1.5

2

2.5

-0.5

3

3.5

Q/Qu

-1 -1.5

C 2.36, 1.65

-2 -2.5 -3 Fig. 14. FE analysis results for a caisson foundation (with D/B ¼3), embedded in a homogeneous cohesive soil (Su ¼ 50 kPa) and supporting alternatively two bridge piers: (1) with height H¼ 70 m, and rotational rigidity of the deck-to-pier connection Kratio ¼ 0.40, and (2) with height H¼100 m, and rotational rigidity of Kratio ¼ 0.25. The bridge piers are subjected to a horizontal force Q (applied at the deck level) until complete failure of the caisson. Interestingly, for both studied cases the “inverted” pendulum failure mechanism is manifested.

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

where H ef f ;0 ðQ u =M u Þ is the respective slope. By differentiating Eq. (11) with respect to Q/Qu, one obtains: h   i ∂ðM=M u Þ ¼ γ ð1 þ αÞ 1 þ βe  βðQ =Q u Þ  α ∂ðQ =Q u Þ

Eqs. (14), (17), (18) constitute a system of nonlinear algebraic equations, the solution of which yields the parameters α, β and γ, and thus the equation of the load path from the beginning of loading until complete failure of the caisson. The numerical (solid lines) and the analytically derived (dashed lines) results regarding the load path are plotted in Fig. 13, for all cases examined in this paper. As evident from the graphs, the results from the proposed methodology fit the finite element results rather satisfactorily. So far, only the response of a caisson with an embedment ratio of D/B ¼1 has been investigated. To examine the generalization of the developed design algorithm, the response of two caisson (with embedment ratio of D/B ¼3) supported bridge piers was also analyzed, attempting to provide evidences on the uniqueness of the “inverted” pendulum failure mode (at point C) as the only possible termination load path point at complete failure of the caisson. The square section caisson has a width of B¼ 10 m and a height of D ¼30 m embedded in a homogeneous cohesive soil with undrained shear strength of Su ¼50 kPa and a constant stiffness to strength ratio Eu =Su ¼ 1000. The response for two bridge piers was examined with respect to the height of the pier, H, and rotational rigidity of the deck-to-pier joint connection (in terms of the Kratio): (1) H¼70 m and Kratio ¼0.40, and (2) H¼ 100 m and Kratio ¼ 0.25. The piers are assumed to behave elastically. The results are portrayed in Fig. 14. Remarkably, the termination of the load path for both structural systems once again is associated with the

ð16Þ

At the region of initial lateral loading, where Q/Qu-0, the slope of the tangent can be substituted into Eq. (16), resulting in the following condition: H ef f ;0

Qu ¼ γ ½ð1 þ αÞð1 þ βÞ  α Mu

ð17Þ

(3) Point B, which marks the maximum positive (in sign) value   of M/Mu, with coordinates ðQ B =Q u Þ; ðM B =M u Þ . Furthermore, point B is also the point of intersection between (a) the nonlinear load path, and (b) the (dashed) line that connects the point where the linear load path of the respective cantilever (KR ¼ 0) meets the   failure envelope ðQ f =Q u Þ; ðM f =M u Þ , and the abscissa at f0:5; 0g. Hence, and after some algebra that is detailed in the Appendix, the following expression for the third condition is derived:



  1 βð1 þ αÞ  2HQ u α HQ u ln þ1 ¼  αγ  ðγ þ αγ Þ β α βð1 þ αÞ HQ u M u HQ u M u ð18Þ

: rad

=6m Kratio = 0.40

193

/Mu 2

-0.03

-0.02

0 -0.01

0

0.01

0.02

0.03

0.02

0.03

1.5 -0.05

1 0.5

w: m -0.1

0 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Q/Qu

-0.5 -1 -1.5 -2

: rad

=6m Kratio = 0.10

/Mu 2

-0.03

-0.02

0 -0.01

0

0.01

1.5 -0.05

1 0.5

w: m -0.1

0 -2

-1.5

-1

-0.5

0 -0.5

0.5

1

1.5

2

Q/Qu

-1 -1.5 -2 Fig. 15. Slow cyclic loops in M–Q space and corresponding w θ loops, for caissons supporting a pier with H¼ 6 m and pier–deck joint stiffness ratios Kratio ¼ 0.10 and 0.40.

194

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

mobilization of the “inverted” pendulum failure mechanism. The coordinates of point C can now be approximated by: {2.36,  1.65}. It is important to note that the generalized form of the load path (Eq. (11)), as well as the resulting equations system (Eqs. (14), (17) and (18)), can be applied as a general design scheme for caisson foundations supporting bridge piers in cohesive soils. Its generalized form allows for its applicability to all soil types (cohesive or frictional) and layering conditions, but that would require the detailed identification of the failure envelope (Eq. (12)) which is out of the scope of the present study.

space, together with the corresponding settlement-rotation (w  θ) loops, are shown in Figs. 15 and 16. It is rather interesting to observe the remarkable agreement with which the monotonic path envelopes the cyclic loops, in all examined cases. The following are worthy of note:

 When shorter piers and stiffer decks imposing increased rota-

3.4. Response under cyclic loading: characteristic results The investigation of the response of the caisson–superstructure systems under cyclic loading serves as a preliminary evaluation of the possible effects of seismic loading. Slow cyclic results are thus shown for four representative cases: pier height H ¼6 m and 17 m, with pier-deck rotational stiffness ratio Kratio ¼ 0.10 and 0.40. Since the emphasis is given on the response of the foundation, the pierto-deck systems were modeled as linear elastic. The lateral load imposed at the deck level increased gradually until the bearing capacity of the foundation–superstructure systems was almost reached; however, collapse was prevented. The loops in M–Q



tional rigidity are considered, the M–Q loops at the foundation follow a smooth path. However, for taller piers carrying more rotationally flexible decks, thus resulting in greater moment loading on the foundation, the loops are wider, more irregular, and the residual settlements are substantial. Yet, small residual rotations were observed at the end of the loading for all examined systems; a clear indication of the restoring role of the significant dimensions and weight of the caisson. From the M–Q loops, the development of phenomena associated with strong inelastic response of the foundation (e.g. gapping) can also be identified. More specifically, their pinched shape denotes that the plastic deformations, developed in the resisting part of the soil during loading at one direction, form an irrecoverable gap as the direction of loading changes. Consequently, when re-loading occurs, the passive-type soil resistance can only be mobilized after the closing of the gap. In : rad

= 17 m Kratio = 0.40

/Mu 2

0

-0.08

-0.04

0

0.04

0.08

-0.05

1.5 -0.1

1 -0.15

0.5

w: m

-0.2

0 -2

-1.5

-1

-0.5

0

0.5

1

1.5

-0.5

2

Q/Qu

-1 -1.5 -2

: rad = 17 m Kratio = 0.10

-0.08

/Mu 2

0 -0.04

0

0.04

0.08

-0.05

1.5

-0.1

1

-0.15

0.5

-0.2

w: m

0 -2

-1.5

-1

-0.5

0 -0.5

0.5

1

1.5

2

Q/Qu

-1 -1.5 -2 Fig. 16. Slow cyclic loops in M–Q space and corresponding w θ loops, for caissons supporting a pier with H¼ 17 m and pier-deck joint stiffness ratios Kratio ¼ 0.10 and 0.40.

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

the case of caissons supporting SDOF cantilever beams, the result is an overall reduction in the capacity of the foundation, in-between the large loading amplitudes. Yet, the presence of the rotational spring at the deck level offers “overstrength” to the system, by redistributing the moment loading between the deck and the foundation. Perhaps expectedly, clear evidence for the detachment of the caisson from the surrounding soil is manifested only during the strongly nonlinear response of the foundation supporting the tall H ¼17 m pier with Kratio ¼0.10, in which the loops exhibit a flattened shape at certain parts.

4. Practical aspects and implications for design The results of the study highlighted important aspects regarding the lateral response of caisson foundations supporting bridge pier-to-deck systems, when the complex interplay between nonlinearities developed below and above ground level are accounted for. It was shown that in the case of a deck imposing constraints in rotation at the top of a pier, the combined base shear–moment loading that is transferred at the foundation follows a nonlinear path in M–Q space. Most importantly, it was found that when the superstructure is given sufficient strength, the caisson foundation exhibits a particular “overstrength” in both moment and horizontal load capacities, irrespective of the stiffness properties of the pier-to-deck system. Given that actual design involves finite strength of the structural elements, this finding has serious

195

implications for the cost-effective seismic design of caisson foundations. The objective of this section is therefore to present a methodology which incorporates the findings of this study, so that it can be utilized as a preliminary seismic capacity design of caisson foundations. The methodology, which is presented by means of a flowchart in Fig. 17, is developed for the typical scenario met in practice, where the task is to design a caisson foundation for a given pierto-deck superstructure and a specific seismic design load. To facilitate the understanding, Fig. 8a should be studied alongside. Taking into account that according to the modern bridge code requirements [22,23]: (a) the moment capacity of the pier base is greater or equal (e.g. in steel columns) than the respective capacity of the top of the pier, and (b) pier-to-deck failure should be avoided, and any development of plastic hinge is directed to the top of the pier, it becomes obvious that the critical component in the superstructure is the top of the pier. However, as clearly shown in Fig. 8a, the pier top is subjected to substantial moment loading, MR, as compared to the pier base, M. Regarding the caisson foundation, an optimum design in terms of cost-effectiveness would take advantage of the “overstrength” that can be mobilized, in order to reduce the caisson's dimensions, and thus its cost. The foundation is therefore designed so that its “overstrength” corresponds to the bearing capacity under: (i) A horizontal load Q¼Samg, where Sa is the design spectral acceleration.

Fig. 17. Flow chart of the proposed methodology for the preliminary seismic design of caisson foundations.

196

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

(ii) A moment load M ¼  ð2=3ÞM u (for caissons with D/B ¼1) corresponding to a bending moment at pier top MR, which is equal to the respective moment capacity (from structural properties). Simple geometry based on Fig. 8a yields:

 π Qu 2 H MR ¼ þ Mu ð19Þ 2 Mu 3 From (i) and (ii) the pure capacities (Qu, Mu) are calculated, and hence the dimensions of the caisson can be determined. At this point, the particularly significant condition regarding the static safety factor (FSV Z2) should be checked. If the condition is satisfied, the preliminary design ends. If, on the other hand, the condition is not met, the dimensions of the foundation should increase accordingly. Yet, an increase in the dimensions results in an increase in Qu and Mu, respectively. In terms of the normalized failure envelope, this can be visualized by the ordinate, MR, moving towards the left (Q o ðπ=2ÞQ u ). In this case, however, the loading state could even reach the first quadrant (QoQu), where it is possible that the moment loading at the pier base is greater than that at its top (M4MR). Thus, the respective capacity should also be checked against M. In order to estimate the exact moment load that is transferred at the pier base, the failure envelope and the nonlinear load path must be drawn. The former can be based on already published works (e.g. [3,24]), while the latter on Eq. (11), given the properties of the foundation–superstructure system. In summary of the above, a capacity-based approach is proposed, involving a trial-and-error optimization procedure. Consideration of the expected performance of the foundation (e.g. developed displacements) belongs to a later design stage (performance-based design).

5. Conclusions The aim of the present study was to qualitatively and quantitatively express the nonlinear SFSI effects in the case of rigid caisson foundations supporting bridge piers subjected to lateral loading. A series of finite element analysis in 3D was therefore employed to assess the performance, with respect to the combined M–Q loading at the foundation, of rigid caissons carrying a columnmass superstructure. The pier height (H), the embedment ratio of the caisson (D/B) and the pier-to-deck joint rigidity (KR) varied parametrically, with the latter being modeled by a rotational spring at the deck level. The foundation–superstructure systems were then subjected to lateral monotonic and slow-cyclic loading at the deck level. Throughout the loading, the resulting M–Q load paths at the caisson were constantly monitored and plotted on the normalized of M/Mu–Q/Qu space, along with the failure envelope of the foundation. To evaluate the effects of the interplay between foundation and structural nonlinearities on the load path, the latter was also introduced by means of bi-linear moment–rotation and moment– curvature laws for the spring and the pier, respectively. The main conclusions of the study can be summarized as follows:



the foundation, that trigger the “negative effective height” effect, the loading at the caisson follows a nonlinear path in M–Q space. This load-path was found to display certain characteristics, the most remarkable of which is the particular “overstrength” in bearing capacity that can be mobilized by the foundation at failure conditions. It was shown that all load paths terminate at a specific point on the failure locus associated with the “inverted” pendulum failure mechanism irrespective of the stiffness properties and constraints at the superstructure system. From the analysis results, a closedform expression for the load path was formulated and validated in all examined cases. The load paths under slow-cyclic pushover loading track the respective monotonic path with striking agreement, justifying thus the use of the latter as a “backbone” curve for seismic analysis.

Based on the results, the paper developed and proposed a methodology to be used towards the seismic capacity design of caisson foundations, accounting for nonlinearities both at the foundation level as well as at the superstructure level.

Aknowledgements A. Zafeirakos acknowledges the financial support from the European Union (European Social Fund – ESF) and Greek National Funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) – Research Funding Program: Heracleitus II. Investing in knowledge society through the European Social Fund. Nikos Gerolymos acknowledges the financial support from the PEVE 2008 Research Project funded through the SEVE/NTUA, under Contract number 65/1694. Nikos Gerolymos is also thankful to his former undergraduate student Mr. Charilaos Paraskevoulakos, acknowledging the contribution of his diploma theses to this research study.

Appendix A Determination of the condition regarding point B in Fig. 12b At point B, the load path exhibits a maximum value and the slope of the tangent equals zero. Therefore, Eq. (14) yields: h   i 0 ¼ γ ð1 þ αÞ 1 þ βe  βðQ r =Q u Þ  α

ðA1Þ

from which the following QB/Qu coordinates are obtained: Q B ln β þ lnð1 þ αÞ  ln α ¼ β Qu

ðA2Þ

 It was highlighted that contrary to conventional design, in which



uncoupled springs are used to model the foundation elements, accurate prediction of the foundation response can be achieved only if the nonlinear coupling between the horizontal and the rotational degrees of freedom is accounted for. Due to the pier-to-deck joint stiffness and the nonlinear coupling between the rotational and translational degrees of freedom at

To extract the equation of the straight line that intersectswith the nonlinear  load path at its maximum, ðQ B =Q u Þ; ðM B =M u Þ , the coordinates ðQ f =Q u Þ; ðM f =M u Þ should first be determined. These are the coordinates of the intersection between the failure envelope and the load path of the respective cantilever (KR ¼0), which is given by Eq. (13). Regarding the expression for the failure

A. Zafeirakos, N. Gerolymos / Soil Dynamics and Earthquake Engineering 67 (2014) 179–197

envelope at the first quadrant (MQ40), it can be approximated by a straight line with the form: M Q ¼ 1 Mu Qu

ðA3Þ

From the system of Eqs. (13) and (A3), the following coordinates for their intersection are derived: Qf Mu ¼ Q u M u þ HQ u

ðA4Þ

Mf HQ u ¼ M u M u þHQ u

ðA5Þ

Hence, the expression for the (dashed) line, that intersects the nonlinear load path at point B, is obtained, as follows:

 M  2HQ u Q HQ u ¼ ðA6Þ þ M u HQ u  M u Q u M u þ HQ u   Since point B, with coordinates ðQ B =Q u Þ; ðM B =M u Þ , marks the intersection of the lines expressed by Eqs. (11) and (A6), the solution of the respective system yields:

 Q B  2HQ u HQ u  αγ þ e  βðQ r =Q u Þ ð  γ αγ Þ ¼ γ þ αγ þ ðA7Þ Q u HQ u  M u HQ u  M u Substituting Eq. (A2) into Eq. (A7), yields Eq. (18). References [1] Hutchinson TC, Chai YH, Boulanger RW, Idriss IM. Inelastic seismic response of extended pile-shaft-supported bridge structures. EERI Earthq Spectra 2004; 20(4):1057–80. [2] Silva PF, Manzari MT. Nonlinear pushover analysis of bridge columns supported on full-moment connection CISS piles on clays. EERI Earthq Spectra 2008;24(3):751–74. [3] Gerolymos, N, Zafeirakos, A, Souliotis, C. Insight to failure mechanisms of caisson foundations under combined loading: a macro-element approach. In: proceedings of the 2nd international conference on performance-based design in earthquake geotechnical engineering, Taormina, Italy; 28–30 May 2012. Paper no. 11.10. [4] Zafeirakos A, Gerolymos N. On the seismic response of under-designed caisson foundations. Bull Earthq Eng 2013;11(5):1337–72. [5] Zafeirakos A, Gerolymos N, Drosos V. Incremental dynamic analysis of caisson–pier interaction. Soil Dyn Earthq Eng 2013;48:71–88. [6] Nova R, Montrasio L. Settlements of shallow foundations on sand. Géotechnique 1991;41(2):243–56. [7] Randolph MF, Gourvenec S. Offshore geotechnical engineering. 270 Madison Avenue, New York, NY 10016, USA: Spon Press/Taylor & Francis; 2011 (ISBN: 978-0-415-47744-4). [8] Cassidy MJ, Randolph MF, Byrne BW. A plasticity model describing caisson behavior in clay. Appl Ocean Res 2006;28:345–58. [9] Zhang Y, Bienen B, Cassidy MJ, Gourvenec S. Undrained bearing capacity of deeply buried flat circular footings under general loading. J Geotech Geoenviron Eng, ASCE 2012;138(3):385–97. [10] Gourvenec S, Randolph MF. Effect of strength non homogeneity on the shape of failure envelopes for combined loading of strip and circular foundations on clay. Géotechnique 2003;53(6):575–86. [11] Bransby MF, Yun GJ. The undrained capacity of skirted strip foundations under combined loading. Géotechnique 2009;59(2):115–25. [12] Gazetas G. Formulas and charts for impedances of surface and embedded foundations. J Geotech Geoenviron Eng, ASCE 1991;117(9):1363–81. [13] Gerolymos N, Gazetas G. Winkler model for lateral response of rigid caisson foundations in linear soil. Soil Dyn Earthq Eng 2006;26(5):347–61.

197

[14] Butterfield R, Houlsby GT, Gottardi G. Standardized sign conventions and notation for generally loaded foundations. Géotechnique 1997;47(5):1051–4. [15] ABAQUS, Inc. ABAQUS Analysis User's Manual, Version 6.10, Providence, RI, USA; 2010. [16] Gerolymos N, Gazetas G. Static and dynamic response of massive caisson foundations with soil and interface nonlinearities–validation and results. Soil Dyn Earthq Eng 2006;26(5):377–94. [17] Armstrong PJ, Frederick CO. A mathematical representation of the multiaxial Bauschinger effect. Technical Report RD/B/N 731. Berkeley, UK: Central Electricity Generating Board; 1966 (Reprint in: Armstrong PJ, FrederickCO. A mathematical representation of the multiaxial Bauschinger effect. Mater High Temp 2007;24(1):1–26). [18] Houlsby GT, Kelly RB, Huxtable J, Byrne BW. Field trials of suction caissons in clay for offshore wind turbine foundations. Géotechnique 2005;55(4):287–96. [19] Gourvenec, S. Bearing capacity under combined loading – a study of the effect of shear strength heterogeneity. In: proceedings of the 9th Australian and New Zealand conference on geomechanics, Auckland, New Zealand; 2004. p. 527–33. [20] Karapiperis K, Gerolymos N. Combined loading of caisson foundations in cohesive soil: finite element versus Winkler modeling. Comput Geotech 2014;56:100–20. [21] Mergos PE, Kappos AJ. Seismic damage analysis including inelastic shear– flexure interaction. Bull Earthq Eng 2010;8:27–46. [22] California Transportation Agency. Seismic design criteria. Ver.3.1, Caltrans, California; 2004. [23] European Committee for Standardization. Eurocode 8: design of structures for earthquake resistance – Part 2: bridges. CEN, Brussels; 2005. [24] Karapiperis K. Insight to the numerical modeling of the lateral response of caisson foundation to static and cyclic loading [Diploma thesis]. NTUA; 2012 [in English]. [25] Souliotis Chr. Development of macroelement for lateral response of caisson foundation to static and cyclic loading [Diploma thesis]. NTUA; 2012 [in English]. [26] Martin CM, Houlsby GT. Combined loading of spudcan foundations on clay: numerical modelling. Géotechnique 2001;51(8):687–99. [27] Gerolymos N, Gazetas G. Development of Winkler model for static and dynamic response of caisson foundations with soil and interface nonlinearities. Soil Dyn Earthq Eng 2006;26(5):363–76. [28] Varun Assimaki D, Gazetas G. A simplified model for lateral response of large diameter caisson foundations-Linear elastic formulation. Soil Dyn Earthq Eng 2009;29(2):268–91. [29] Tsigginos C, Gerolymos N, Assimaki D, Gazetas G. Seismic response of bridge pier on rigid caisson foundation in soil stratum. Earthq Eng Eng Vib 2008; 7(1):33–44. [30] Bienen B, Cassidy MJ. Advances in the three-dimensional fluid-structure-soil interaction analysis of offshore jack-up structures. Mar Struct 2006;19(2–3):110–40. [31] Cassidy MJ, Eatock TR, Houlsby GT. Analysis of jack-up units using a constrained new wave methodology. Appl Ocean Res 2001;23:221–34. [32] Cassidy MJ, Taylor PH, Eatock TR, Houlsby GT. Evaluation of long-term extreme response statistics of jack-up platforms. Ocean Eng 2002;29(13):1603–31. [33] Cassidy MJ, Byrne BW, Randolph MF. A comparison of the combined load behavior of spudcan and caisson foundations on soft normally consolidated clay. Géotechnique 2004;54(2):91–106. [34] Martin, CM, Houlsby, GT. Jackup units on clay: structural analysis with realistic modelling of spudcan behaviour. In: proceedings of the 31st offshore technology conference, Houston, OTC 10996; 1999. [35] Vlahos G, Cassidy MJ, Martin CM. Numerical simulation of pushover tests on a model jack-up platform on clay. Géotechnique 2011;61(11):947–60. [36] Williams MS, Thompson RSG, Houlsby GT. Non-linear dynamic analysis of offshore jack-up units. Comput Struct 1998;69(2):171–80. [37] Gottardi G, Houlsby GT, Butterfield R. The plastic response of circular footings on sand under general planar loading. Géotechnique 1999;49(4):453–70. [38] Houlsby GT, Cassidy MJ. A plasticity model for the behavior of footings on sand under combined loading. Géotechnique 2002;52(2):117–29. [39] Cassidy MJ, Byrne BW, Houlsby GT. Modelling the behavior of circular footings under combined loading on loose carbonate sand. Géotechnique 2002;52(10):705–12. [40] Bienen B, Byrne BW, Houlsby GT, Cassidy MJ. Investigating six degree of freedom loading of shallow foundations on sand. Géotechnique 2006;56(6):367–79. [41] Zhang Y, Cassidy MJ, Bienen B. Jack-up push-over analyses featuring a new force resultant model for spudcans in soft clay. Ocean Eng 2014;81:139–49. [42] Zhang Y, Cassidy MJ, Bienen B. A plasticity model for spudcans in soft clay. Can Geotech J 2014b; 51: 629-646; dx.10.1139/cgj-2013-0269.