Uncertainty in Geometric and Structural Parameters

The distributions for geometric and structural parameters are mostly derived from the NBI and are based on review of a significant number of plans pertinent to bridges across design eras for the bridge classes considered in this study.

5.2.2.1 NBI based Bridge Geometric Parameters

Empirical distributions for bridge geometric parameters such as maximum span length, deck width, and column height were presented in Section 3.3 of Chapter 3. Although NBI provides information on the number of spans and probability mass functions were derived and presented in Chapter 3, this study uses the mode statistic for the number of spans due to the complexity involved in parameterizing this variable. The median value modification factor prescribed in HAZUS-MH (2011) is recommended to be used to determine fragilities for bridges with spans not equal to the mode statistic adopted here.

5.2.2.2 Abutment Backwall Height

Most of the structural parameters are attributed to uniform distribution due to a lack of significant data or information that can be used to associate a distribution of any other type. Based on the review of bridge plans, the height of the backwall in the case of diaphragm and seat abutments is assumed to be uniformly distributed between 3.5 ft and

151 5.2.2.3 Column Reinforcement Ratios

The longitudinal and transverse reinforcement ratios in the bridge columns are sampled from uniform distributions with limits established based on the review of bridge plans. Table 5.1 details the parameters of the uniform distribution describing the longitudinal and transverse reinforcement ratios. In the pre 1971 design era, the transverse reinforcement consists of #4 stirrups at 12 in on center, which was a common standard irrespective of the column size or reinforcement. Hence this parameter was not varied in the simulations for the bridges in this design era. Further, MSCC slab bridges employ integral pile columns whose cross-section is standard and hence the reinforcement is not varied in this case.

Table 5.1: Distributions for longitudinal and transverse reinforcement ratios in bridge columns

Bridge class Design era Longitudinal reinforcement ratio Transverse reinforcement ratio u1* u2* u1* u2*

Pre 1971 1.4 2.4 N.A. N.A.

MSCC-BG 1971-1990 1.0 3.7 0.30 0.90

Post 1990 1.0 3.5 0.40 1.70

Pre 1971 1.08 3.61 N.A. N.A.

MSCC-IG 1971-1990 1.18 5.31 0.31 1.07

Post 1990 1.49 5.35 0.31 1.61

Pre 1971 1.08 3.61 N.A. N.A.

MSCC-TG 1971-1990 1.18 5.31 0.31 1.07

Post 1990 1.49 5.35 0.31 1.61

*u1, u2 are the parameters describing a uniform distribution representing lower and upper

bounds. 5.2.2.4 Gaps

The gap between the superstructure and abutment backwall is assumed to be uniformly distributed. As mentioned in Section 3.5.6 of Chapter 3, the gap uniformly ranges between 0 and 1.5 in across all bridge classes and design eras. However, in the case of the MSCC-BG bridges, simulations are performed for two ranges of gap sizes: smaller gaps uniformly distributed between 0 and 1.5 in and larger gaps uniformly distributed between 1.5 in and 6.0 in. Further, the transverse gap between the

superstructure and shear keys is assumed to be uniformly distributed between 0 and 1.5 in for the case of MSCC-BG and MSCC-IG bridge classes.

5.2.2.5 Restrainer Attributes

The length and initial slack of the restrainer cables are assumed to be random variables sampled from uniform distributions. The length of the cables is bounded between 8 ft and 20 ft and samples are drawn at 2 ft increments. The initial slack is sampled from a uniform distribution bounded between 0.25 in and 1.0 in.

5.2.2.6 Pile Effective Stiffness

Piles form an integral part of the foundation system beneath the abutments. Translational springs characterizing by the pile stiffness are provided in the longitudinal and transverse directions at the abutments. As stated in previous chapters, piles could be of many different types such as driven steel H section piles, CIDH concrete piles, PC piles or PPC piles. Based on input from the Caltrans design engineers (Caltrans, 2010- 2012), the stiffness of the piles is assumed to follow a lognormal distribution with a logarithmic standard deviation, ζ, of 0.3. The median value is taken as 65 kips/in for steel H sections and 80 kips/in for all of the aforementioned concrete piles. It should be noted that the stiffness adopted here is much higher than the 40 kip/in value used in previous studies (Choi, 2002; Nielson, 2005; Padgett, 2007).

5.2.2.7 Foundation Translational and Rotational Spring Stiffnesses

The soil profile changes vastly over a wide geographic area and the stiffness of the foundation translation and rotational springs depends on the soil profile at a particular location. In order to obtain realistic estimates of bridge performance within a class, it is imperative to capture a wide range of soil profiles. Other factors such as the type of foundation system (see section 3.5.5), end conditions of the columns (pinned vs.

153

foundation springs. Appendix A documents the different soil profiles considered in this study along with details of the common foundation systems. The different foundation systems and the soil profiles are modeled and analyzed in LPILE (2012) using substantial input from Shantz (2011) and Table 5.2 summarizes the parameters for the truncated normal distribution describing the stiffness of the foundation translation and rotational springs.

5.2.2.8 Other Bridge Structural Attributes

Several other attributes are uniformly distributed between the simulations such as type of backfill soil: sand versus clay; pile class: Class 45 versus Class 70; pile type: PC versus PPC piles. The type of backfill soil affects the hyperbolic force deformation response of the abutment in terms of the initial stiffness, ultimate strength and the deformations. The class and type of pile dictates the pile geometry and reinforcement details (amount and layout) and therefore affects the strength and stiffness characteristics. The type of girder (Standard I- versus Bulb-Tee) is also assumed to be uniformly distributed among the simulations due to their existence in the California bridge inventory. The type of girder affects the deck geometric properties such as cross-sectional area, moment of inertia and the mass.

Table 5.2: Probability distributions for foundation translational and rotational spring stiffnesses

Foundation type Bridge class

Distribution type Translational spring stiffness (kip/in)

Rotational spring stiffness (kip-in/rad)

µ σ µL µ σ µL

Pile extension

16 in dia integral pile column

MSCC-SL, MSCC-TG

Truncated normal* 30 20 2 8×104 _3×104 _2×104

Pile shafts

6ft dia – 1% long. steel – Fixed top MSCC-BG Truncated normal 600 350 100 5×106 _3×106 ₀

6ft dia – 1% long. steel – Free top MSCC-IG Truncated normal 250 125 50 7×106 _2×106 _3×106

6ft dia – 3% long. steel – Fixed top MSCC-BG Truncated normal 700 400 200 6×106 _4×106 ₀

6ft dia – 3% long. steel – Free top MSCC-IG Truncated normal 300 150 80 1×107 _3×106 _5×106

8ft dia – 1% long. steel – Fixed top MSCC-BG Truncated normal 900 500 200 6×106 _4×106 ₀

8ft dia – 1% long. steel – Free top MSCC-IG Truncated normal 400 200 80 1.4×107 _4×106 _7×106

8ft dia – 3% long. steel – Fixed top MSCC-BG Truncated normal 1300 600 250 7×106 _5×106 ₀

8ft dia – 3% long. steel – Free top MSCC-IG Truncated normal 500 250 100 2.3×107 7×106 1×107

Pile group – pile cap and piles

6ft dia column – 1% long. steel MSCC-IG, MSCC-BG

Truncated normal 1700 800 400 4.1×107 _1.2×107 _2.2×107

6ft dia column – 3% long. steel MSCC-IG, MSCC-BG

Truncated normal 1400 600 600 6.5×107 _1×107 _5×107

3ft dia column – 1.5% long. steel MSCC-IG, MSCC-BG,

MSCC-TG

Truncated normal 800 600 175 0 0 0

155 5.2.3 Uncertainty in Other Parameters

5.2.3.1 Mass

Mass factor is a parameter used to capture the uncertainty in mass from incidental sources and is applied as a factor to modify the mass of the superstructure. It should be noted that the mass factor does not account for the variations due to changes in bridge geometric parameters such as span length, deck width, column height etc., which are explicitly accounted for in the analytical modeling procedure. Various incidental sources accounting for the mass factor include the presence of parapets and barrier rails, variable deck slab thickness, electric poles and other equipment, re-pavement procedures, variation in the material densities etc. The mass factor is assumed to be uniformly distributed with bounds of 1.1 and 1.4. The bounds are established by estimating the additional mass observed from the review of bridge plans.

5.2.3.2 Damping

The recommendations of Fang et al. (1999) for tall buildings are extended to bridges (Nielson, 2005; Padgett, 2007) and the uncertainty in damping is modeled using a normal distribution. Bavirisetty et al. (2003) estimated the 2nd _{and 98}th _{percentile of}

damping ratio in bridges to be 0.02 and 0.07 respectively and using these recommendations, the damping ratio is sampled from a normal distribution with mean, µ, of 0.045 and standard deviation, σ, of 0.0125.

5.2.3.3 Direction Factor

Previous studies (Nielson, 2005; Padgett, 2007; Ramanathan et al., 2010) considered the angle of incidence of the seismic load as a uniform random variable. However, recent studies by Mackie et al. (2011) demonstrated the negligible effect of the angle of incidence on the mean ensemble response of bridge components. Hence, the

incidence angle is not considered as a major source of uncertainty in this study. However, the fault normal and fault parallel components of the ground motion are randomly applied along the longitudinal and transverse axes of the bridge i.e., 50% of the simulations have the fault normal component applied along the longitudinal bridge axis while 50% of the simulations have the fault parallel component applied along the longitudinal axis.