ANALYTICAL MODEL DEFINITION

In order to evaluate the vulnerability of piers to instability during seismic attack analytical models that capture all key aspects of hysteretic behavior, including modes of strength and stiffness degradation must be developed. In this study the finite element analysis package OpenSees was employed.

Each pile element within the pier bent is modeled according to the schematic shown in Figure 3. The model consists of centerline beam-column elements (see Figure 3b) with fiber-discretized cross-sections (see Figure 3c). The use of fiber sections adds some computational cost, but allows full P-M interaction throughout the analysis and the ability to monitor material strains at any point within the element and cross-section. At the cross-sectional level the concrete was divided into confined and unconfined fibers, which were both modeled using a Kent-Scott-Park (Kent and Park 1971) backbone (the Concrete01 model in OpenSees) with the enhanced strength and ductility of the confined concrete determined according to the Mander model (Mander et al 1988). The mild steel (dowel bar) and

Component Strain in

Plastic Hinge _{OLE CLE DE}

Pile-to-deck connection

concrete strain εc ≤0.005 εcc= 0.005+1.1ρs≤ 0.025 Νο limit

In-ground concrete strain εc ≤0.005 εcc= 0.005+1.1ρs≤ 0.008 εcc= 0.005+1.1ρs≤ 0.025

Pile-to-deck connection

reinforcing steel strain εs≤0.015 εs ≤0.6εsmd ≤0.06 εs≤0.8εsmd≤0.08 In-ground prestressing

tendon strain εp≤0.015 εp≤0.025 εp≤0.035 εc = Maximum unconfined (cover) concrete compression strain.

εcc = Maximum confined (core) concrete compression strain. εs = Maximum steel tensile strain.

εsmd = Maximum uniaxial tensile strain capacity of mild steel. εp = Maximum total prestressing steel tensile strain.

prestressing tendons were modeled using the Giuffré-Menegotto-Pinto model (Menegotto and Pinto 1973) as implemented in OpenSees by the Steel02 material model. Prestressing was modeled by utilizing the initial strain feature of the Steel02 model. Dowel bar fracture was accounted for using the Fatigue material implemented in OpenSees, which accounts for low-cycle fatigue using a linear strain accumulation model based off of the Coffin- Manson log-log relationship (Coffin 1954), (Manson 1953). A modified rainflow cycle counter was used to track strain amplitude.

The pile-to-deck connection was modeled using a fiber based concentrated plasticity hinge located at the soffit of the deck. A concentrated plasticity model is attractive here for two reasons. First the spread of plasticity in the pile-to- deck connection is limited because a lap splice region exists directly adjacent to deck soffit, where the pile prestressing strand overlaps with the mild steel in the connection. Secondly because (as will be seen in the following section) the pile-to-deck connection plastic hinge exhibits a softening response, a concentrated plasticity hinge prevents numerical concentration issues from arising through the use of a predetermined fixed plastic hinge length. It is well documented (Coleman and Spacone 2001), (Almeida et al. 2010) and (FIB 2008) that distributed plasticity elements become non-objective when a softening local (moment-curvature) response exists. This creates a situation where the curvature demand becomes dependent on the discretization of the element (i.e. the maximum predicted curvature increases without bound with increased member discretization).

Figure 3 – Schematic Analytical Model

While several regularization schemes have been developed (Coleman and Spacone 2001), (Scott and Fenves 2006) a simple method of mitigating numerical concentration issues is to use an element with a predefined plastic hinge length wherever a plastic hinge is expected to form. The connection plastic hinge length was calculated using [4, which has been developed through experimental research on pile-to-deck connections at the University of California, San Diego (Krier 2008) and has been adopted by the POLB.

b ye p f d L =0.25 (

f

f

The remainder of the pile length was modeled using distributed-plasticity force-based beam-column elements (Neuenhofer and Filippou 1997). This formulation captured the spread of plasticity along the embedded length of the pile allowing a variable-length in-ground plastic hinge to form. Each element was 0.5D (12 in or 300 mm) long

and had three Gauss-Lobatto integration points1_{. A sensitivity analysis was conducted to ensure that element length} provided accurate results with minimal computational cost, whereby the element length was incrementally reduced from 1.0D to 0.1D. It was found that for a given displacement demand the maximum curvature within the in-ground plastic hinge converged with element lengths shorter than 0.5D.

Local soil structure interaction was captured using p-y springs (see Figure 3d) spread over the embedded length of the pile at 0.5D (12in or 300mm spacing). The p-y relationship is determined from a macroelement, which combines elastic, plastic, and gap springs in series. The gap component of the spring consists of a nonlinear closure component and a nonlinear drag component. Viscous damping is included in the elastic far field component of the spring to approximate radiation damping (Boulanger et al. 1999). This p-y formulation allows the spring to approximate the API sand model (American Petroleum Institute 2007) under dynamic loading conditions. Parameters for the p-y curve generation were taken to represent submerged sand with a friction angle of 37 degrees under cyclic loading. Sample p-y curves at various depths are shown in Figure 4.

Figure 4 -- Sample p-y Curves

The deck was modeled with a fiber based centerline beam-column element defined by the tributary width of the deck between pile bents. This resulted in a 20ft wide (6.1 m) by 30-inch (750 mm) deep cross section, which effectively acts as a rigid link between the piles. The seismic mass was applied as a discrete point mass at the intersection of the deck and pile centerlines. The mass was taken to only act in the horizontal direction. Axial loads on the piles were applied at the same location by converting the mass to a weight applied as a point load. The ground motion was applied uniformly to all fixed boundary conditions on the pile; this included the pin support at the pile tip and the restrained end of the p-y macroelements. While this does not account for the variation of the seismic waves with depth within the soil column adjacent to the pile, it does provide a reasonable estimate for the homogeneous stiff soil assumed within this study.

Geometric nonlinearity was accounted for in the model so that structural (P-Δ) and element (P-δ) geometric nonlinearity effects could be captured. Structural (P-Δ) effects were captured using the OpenSees PDelta linear geometric transformation which accounts for P-Δ effects due to axial load on an element. Member (P-δ) effects were captured through the use of multiple beam-column elements along the unsupported length of the pile.

One of the advantages to the fiber-discretized cross section is the automatic consideration of elastic damping due to energy dissipation from the elastic hysteretic response. This circumvents the need to apply a fictitious viscous

1

Three point Gauss-Lobatto quadrature places integration points at each end and the middle of each element. The integration weights are 4/3 at the end points and 1/3 at the middle point.

damping to the linear-elastic portion of the hysteresis by treating inherent damping as a hysteretic response (Priestley et al. 2007), (Charney 2008). During a free-vibration of the system it was found that piers in this study had elastic damping ratios of approximately 4% of critical. This corresponds well to expected values for prestressed concrete systems without nonstructural components. Additional damping due hydrodynamic drag on the piles was neglected.