Bridge Superstructure Tolerance to Total and Differential Foundation Movements (2018)

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NCHRP Project 12-103 14 whether the current guidance (regardless of its initial intent) could accurately reflect tolerable TD support movements. 3 Definition and Construction of 3D FE Bridge Suites 3.1 Sampling of Parameters to Define a Bridge Suite (Task 2.1.1) To allow the impact of LD and TD support movements to be fully examined, it is necessary to sample the parameters of interest (Table 1-2) to develop a representative “bridge suite” or sample of bridges. A hybrid approach of LHS and Full-Factorial DoE was used to effectively and efficiently cover the parameter space (see Figure 3-1). Figure 3-1 - Representation of the hybrid sampling approach employed in this project. For continuous parameters, LHS, was employed for the sampling. This method divides each parameter range into n number of bins with equal probability density. In the case of this study, each parameter was assumed to have a uniform (rectangular) probability distribution and thus all bins were of equal size. To develop a set of samples, LHS draws a random sample from each bin, and then randomly pairs samples from different parameters. Although LHS belongs to the Monte Carlo (MC) family of sampling Divide Continuous Parameters into n intervals Draw a random sample from each interval for each parameter Randomly combine samples from each parameter to generate n sets of continuous parameters Generate a full-factorial sample (I.e. all possible combinations of discrete parameters) 2 Bridge Types x 3 levels of Continuity 6 Samples of Discrete Parameters LHS Sampling of Continuous DoE Sampling of Discrete Pair each sample of continuous parameters with each sample of discrete parameters Total number of samples within Bridge Suite = 6*n

NCHRP Project 12-103 15 approaches, the use of bins ensures a more uniform coverage of the parameter space (McKay et al. 1979) and thus it tends to be more efficient (i.e. better sampling) than traditional MC methods. Figure 3-2 gives a matrix plot of the continuous parameter samples generated using LHS. This plot shows histograms of each parameter on the diagonal and then correlation plots between parameters in the off-axis cells. As can be observed from this figure, the LHS approach was able to cover the defined sampling space in a uniform manner. Figure 3-2 - Matrix plot of continuous parameters. Since the discrete parameters assume such a small number of values, it is feasible to examine every combination of these parameters through a full factorial approach. Uncoupling the discrete parameters from the continuous parameters allows for finer sampling of the latter. In two cases the sampled parameters were correlated, (i.e. related to one another). These cases were (1) girder spacing and bridge width, and (2) diaphragm spacing and bridge length. Specifically, the “spacing” parameters needed to divide equally into the width/length parameters (since the number of girders and diaphragms must be integers). To enforce this relationship within the sample sets, two assumptions were made as follows: 1. In the case of girder spacing and bridge width, it was decided to modify the randomly sampled bridge width parameter to ensure the number of girders was an integer value. Specifically, the bridge width value obtained using LHS was adjusted by first dividing the value by the sampled girder spacing and rounding to the nearest whole number. This number gives the total number

NCHRP Project 12-103 16 of girder spaces, which is then multiplied by the actual girder spacing to obtain the adjusted bridge width. 2. In the case of diaphragm spacing, the number of interior diaphragm rows was taken as the maximum number of diaphragms that can fit within the span length with a maximum diaphragm spacing of 20ft. To carry out this sampling approach, the Research Team began by generating a sample set of 100 samples using LHS on the continuous parameters. The sample set (or “bridge suite”) was then paired with each of the 6 combinations of variables obtained from the full factorial sampling approach for a combined total of 600 samples. Each sample contains bridge configuration information (length, skew, continuity, etc.) To assess convergence a second, independent set of 600 samples was also drawn and the results were compared to the original sample using standard convergence tests (see Section 4.6). There are many additional parameters outside of the sampled parameters that are required to develop a design. These design constants were not varied since they did not show appreciable influence over tolerable support movement during the Phase I sensitivity analyses. Such parameters included deck thickness, overhang, maximum spacing of diaphragms (for steel girder bridges only), and various material properties. Table 3-1 through Table 3-3 provide a complete listing of these parameters as well as the numerical values used for each simulation model. Table 3-1 - Table of design constants for steel bridges only. Design Constant Value Diaphragm Spacing 20 ft Steel Density 490 pcf Steel Modulus 29000 ksi Steel Yield Strength 50 ksi Table 3-2 - Table of design constants for PS bridges. Design Constant Value Beam Type PS Concrete Bulb-T (PCBT) Pre-Stressing Ultimate Strength 270 ksi Pre-Stressing Yield Strength 243 ksi Pre-Stressing Steel Modulus 28500 ksi

NCHRP Project 12-103 17 Table 3-3 - Design constants for steel and PS concrete bridges. Design Constant Value Concrete Density 150 pcf Deck Concrete Compressive Strength 4 ksi Barrier Height 27 inches Barrier Width 12 inches Deck Thickness 8 inches 3.2 Automated LRFD (Task 2.1.2) The goal of this task was to develop member sizes (e.g. steel girder dimensions, pre-stressing force/eccentricity, etc.) for each bridge configuration produced by the hybrid sampling approach described in Section 3.1. To ensure the relevancy of the overall study, it is imperative that the “designs” developed in this task are representative of those produced by designers following AASHTO LRFD. In addition to meeting the requirements of AASHTO LRFD, it is common for designers to employ heuristic rules or fabrication constraints such as sizing increments for webs, flanges, etc. that act to provide varying levels of reserve capacity beyond those required by the specifications. However, since the next larger size is used in design due to sizing increments, an additional capacity (which results in excess conservatism above and beyond the requirements of the specifications) is realized and therefore such heuristic sizing rules were ignored within this research. Rather, the member sizing methods employed by this research generated bridges that precisely met (but did not exceed) the governing limit state. The following sections provide the details associated with the automated member sizing approach employed within this research. 3.2.1 Approximation of Demands Using Single Line Girder (SLG) Model Member actions due to dead load and live load were obtained using the commonly employed SLG model. For the dead load of steel multi-girder bridges, a unit distributed load was applied to the SLG model for each bridge sample and the resulting member actions (moments and shears) were obtained. The dead load demand was obtained by scaling the actions from the unit distributed load by the actual distributed dead load of the bridge sample (determined using a tributary width assumption). Since common practice employed un-shored construction, the dead load force effects were assumed to be carried only by the girder (i.e. no composite-action was assumed).

NCHRP Project 12-103 18 Dead load demands for PS concrete multi-girder bridges were computed in the same manner as steel girder bridges, with one exception; for multiple-span continuous PS concrete bridges, the initial dead load demands were computed with the spans acting individually (i.e., simple spans) while the superimposed dead load and live load demands were computed with the spans acting continuously. This was done to reflect the common construction practice of making the girders continuous after the deck is constructed with an integral connection. In many cases, both multi-span steel and PS concrete multi-girder bridges are not made continuous. That is, they are composed of multiple simple spans. Due to the lack of continuity, each span of these bridges (from a tolerable support movement standpoint) are essentially equivalent to a simple span bridge. As a result, in this research all multi-span bridges examined were continuous for steel bridges and continuous-for-live-load for PS concrete bridges. For live load, single-lane member actions were obtained by applying point loads across the entire length of the SLG model representing the axle loads of the design truck plus dynamic impact factor as per the AASHTO LRFD Bridge Design Specifications. When applicable, a distributed load representing the design lane load was also applied to the SLG model. Single-lane member actions (moments and shears) were then calculated for each load case combination with the appropriate load distribution factors applied. Live load distribution factors for moment and shear (with skew reduction when applicable) were calculated in accordance with AASHTO LRFD Article 4.6.2.2.2. The resulting demand envelopes for dead load, super-imposed dead load, and live load were used to evaluate the applicable design limit states. 3.2.2 Automated Sizing of Steel Members To size steel plate girder sections, a built-in optimization algorithm in Matlab called fmincon was employed. Matlab documentation notes that the fmincon algorithm “attempts to find a constrained minimum of a scalar function of several variables starting at an initial estimate” (Mathworks). To size steel plate girders, the variables of the function were the flange width, flange thickness, web thickness, and for multiple span continuous bridges, the thickness of a cover plate in the negative moment region. The reason a cover plate was included was to allow the plate girder to have different capacities in the positive and negative moment regions and thus remove any excess conservatism due to enforcing a constant cross-section.

NCHRP Project 12-103 19 The scalar, or “objective” function, within the member-sizing algorithm was taken as the area of the steel section. In the same manner that the typical designer may attempt to find the most economical section that satisfies all constraints set by AASHTO LRFD, the fmincon algorithm attempts to find the combination of variables (i.e. plate girder dimensions) that satisfy all constraints while minimizing the area (i.e. minimizing the objective function). Using fmincon and the proper sizing constraints, a cross-section can be sized such that it is the least conservative section possible that still passes all the requirements of the AASHTO LRFD. That is, the section is sized to minimize any additional capacity greater than what is needed to satisfy all of the applicable limit states. Using a section designed in this manner will provide conservative estimates of tolerable LD and TD support movements since actual bridges typically employ heuristic sizing increments that provide additional capacity. Table 3-4 provides the constraints used for evaluating the section, which were taken from provisions of AASHTO LRFD. Table 3-4 - Constraints used for steel girder sizing, adopted from AASHTO LRFD Bridge Design Specifications. Sizing Constraints AASHTO LRFD Provision Depth Criteria L ∗ 0.033 ≤ D୥୧୰ୢୣ୰ L ∗ 0.040 ≤ Dୱୣୡ୲୧୭୬ 2.5.2.6.3 Ductility D୮ ≤ 0.42 ∗ Dୱୣୡ୲୧୭୬ 6.10.7.3

NCHRP Project 12-103 20 Proportioning Limits D୵ t୵ ≤ 150 b୤ 2t୤ ≤ 12 b୤ ≥ D୵ 6 t୤ ≥ 1.1 ∗ t୵ 0.1 ≤ I୷ୡI୷୲ ≤ 1 6.10.2 Web Thickness Limit t୵ ≥ 0.3125" 6.7.3 Strength I Positive Flexure M୳ ≤ M୬ if ଶୈౙ୲౭ ≤ 3.76ට ୉ ୊౯ fୡ, ୲ ≤ F୬ if ଶୈౙ୲౭ ≥ 3.76ට ୉ ୊౯ 6.10.6 Strength I Negative Flexure fୡ ≤ F୬ 6.10.8 Service II Positive Flexure fୡ, ୲ ≤ 0.95 ∗ F୷ 6.10.4 Service II Negative Flexure fୡ, ୲ ≤ 0.80 ∗ F୷ 6.10.4 Shear V୳ ≤ V୬ 6.10.9 Fatigue I γ(∆f) ≤ (∆F)୘ୌ 6.6.1.2

NCHRP Project 12-103 21 Where, L = span length D୥୧୰ୢୣ୰ = depth of the steel section Dୱୣୡ୲୧୭୬ = depth of the composite section D୮ = depth of compression at the plastic moment for the composite section D୵ = depth of the web Dୡ = depth of the web in compression t୵ = thickness of the web b୤ = flange width t୤ = flange thickness I୷ୡ , I୷୲ = moment of inertia of the compression and tension flanges, respectively E = Young’s modulus of elasticity F୷ = Steel yield strength M୳ = factored ultimate moment demand M୬ = nominal moment capacity fୡ, ୲ = compression/tension stress in the flange F୬ = nominal compressive/tensile stress capacity V୳ = factored ultimate shear demand V୬ = nominal shear capacity (∆F)୘ୌ = fatigue stress limit γ(∆f) = fatigue stress range 3.2.3 Automated Sizing Pre-Stressed Concrete Members The framework for sizing PS concrete multi-girder bridges is different than for steel bridges. Rather than using the fmincon optimization algorithm, a step function algorithm was developed for identifying the smallest section that satisfies the appropriate constraints. This was required since both the available cross-sections and the number of pre-stressing strands are discrete in nature, and thus cannot be optimized in the same manner as steel plate dimensions which are continuous. The smallest standardized section appropriate for the span length is initially chosen as specified in Table 3-5. This table is based on recommendations provided by Pre-Stressed Concrete Committee for Economical Fabrication, synthesized from local DOT specifications (VDOT 2005). Table 3-5 - Standardized PCBT sections for each maximum span length. Section Max Span (ft) PCBT 29 60 PCBT 37 80 PCBT 45 100 PCBT 53 115

NCHRP Project 12-103 22 PCBT 61 125 PCBT 69 135 PCBT 77 145 PCBT 85 150 PCBT 93 155 An iterative approach was used in selecting the number of pre-stressing strands as well as the concrete compressive strength. An initial minimum compressive strength of 4 ksi and a minimum number of eight pre-stressing strands was assigned. The relevant section properties (PS strand area, eccentricity, etc.) and associated pre-stress demands were calculated and the section was evaluated based on the provisions of AASHTO LRFD. If all design constraints were met, the section was accepted along with the compressive strength and number of pre-stressing strands for that iteration. Otherwise, the number of pre-stressing strands was increased and the section was re-evaluated. This process continued until the section satisfied the design constraints or the maximum allowable number of pre-stressing strands was met for that section. If the maximum number of allowable strands was reached before all design constraints were met, the concrete strength was increased and the process was repeated starting from the minimum number of pre-stressing strands. If the maximum allowable compressive strength of 10 ksi was reached before all design constraints were met, the next largest section was selected and the process repeated until an acceptable design section was identified. When evaluating the sections at each iteration, shear and end negative moment demands were not included since the associated capacities are highly dependent on the harping or de-bonding patterns, stirrup placement and other reinforcement design decisions, which are at the discretion of the designer. Instead, the shear and negative moment capacities were set equal to the design demands. That is, the associated capacities of the cross-section were taken as the demand calculated using the SLG model. This essentially assumes that the designer correctly accounted for these force effects, and therefore the girders will have corresponding capacities greater than or equal to the demands. Figure 3-3 presents the framework for the automated sizing of PS concrete members. Table 3-6 gives the constraints used for evaluating the section, taken from provisions of AASHTO LRFD.

NCHRP Project 12-103 23 Figure 3-3 - Framework for sizing of PS concrete girders. Table 3-6 - Constraints used for PS concrete girder sizing, adopted from AASHTO LRFD. Sizing Constraints AASHTO LRFD Provision Compressive stress limit at transfer: ௧݂௥௔௡௦௙௘௥ < 0.6 ∗ ݂′௖௜ 5.9.4.1.1 Compressive stress limit due to the sum of the effective pre-stress, permanent loads: ௉݂ௌା஽௅ < 0.45 ∗ ݂′௖ 5.9.4.2.1 Compressive stress limit due to the sum of the effective pre-stress, permanent loads, and 5.9.4.2.1

NCHRP Project 12-103 24 transient loads (Service I): ௌ݂௩ଵ < 0.6 ∗ ݂′௖ Compressive stress limit for live load plus one-half the sum of the effective pre-stress and permanent loads: ݂ଵ ଶ(௉ௌା஽௅) < 0.40 ∗ ݂′௖ 5.9.4.2.1 Tensile stress limit due to the sum of the effective pre-stress, permanent loads, and transient loads (Service III): ௌ݂௩ଷ < 6ට݂ᇱ௖ 5.4.9.2.2 Deck stress limit under full load: ௗ݂௘௖௞ < 0.6 ∗ ݂′௖ 5.9.4.2.1 Flexure limit for the Strength I load combination: ܯ௨ ≤ ߮ܯ௡ 5.7.3.2 Minimum reinforcement check” ݉݅݊ ൜1.33 ∗ ܯ௨ܯ௖௥ ൠ ≤ ߮ܯ௡ 5.7.3.3.2 Where, ௧݂௥௔௡௦௙௘௥ = Compressive stress at transfer ௉݂ௌା஽௅ = Compressive stress due to sum of effective pre-stress, permanent loads ௌ݂௩ଵ = Compressive stress due to sum of effective pre-stress, permanent loads, and transient loads (Service I) ݂ଵ ଶ(௉ௌା஽௅) = Compressive stress for live load plus one-half the sum of effective pre- stress and permanent loads ௌ݂௩ଷ = Tensile stress due to the sum of effective pre-stress, permanent loads, and transient loads (Service III) ݂′௖ = specified compressive strength of concrete for use in design ݂′௖௜ = specified compressive strength of concrete at time of initial loading or prestressing ௗ݂௘௖௞ = Deck stress under full load ܯ௨ = Ultimate factored moment demand ߮ܯ௡ = Factored nominal moment capacity ܯ௖௥ = Cracking Moment

NCHRP Project 12-103 25 3.2.4 Software Validation Validation of the member-sizing processes described above was a critical task of the project. In order to draw reliable conclusions about tolerable support movements, the process adopted must be capable of producing member sizes that are consistent with all the requirements of AASHTO LRFD without additional and/or arbitrary conservatism. In an effort to validate the member-sizing approach, researchers from the University of Delaware (UD) acted as independent partners to provide a “peer review” of the model design philosophy and assumptions utilized in the development of the software (Berti 2015). A “one-to-one” approach was used within this validation effort. Specifically, several designs were conducted by hand, and key parameters of these designs (see Table 3-7) were compared to the member sizes produced by the automated process. Using Microsoft Excel®, the research personnel from UD developed a design spreadsheet based on the SLG design method. The spreadsheet required cross section dimensions, load cases, and bridge orientation information as inputs, and calculated all relevant section properties as well as the flexural and shear strength of the girder following the AASHTO LRFD. To size the girders using the spreadsheet, calculated moment and shear capacities were checked against the factored load cases to determine the most efficient girder cross section. In this case, efficiency was defined as a minimization of the cross- sectional area. Using all of the provisions listed in Tables 3-1 and 3-4 above, this one-to-one comparative approach validated the automated calculation of each of the design parameters presented in Table 3-5. Table 3-7 - List of "one-to-one" validation checks. “One-to-One” Validation Checks • flange area • web area • girder depth • girder moment of inertia • girder section modulus • girder capacity (lateral torsional, local buckling resistance calculations) • composite moment of inertia • composite section modulus • composite section capacity (plastic neutral axis and plastic moment calculations) • single line dead load computations • single line live load computations

NCHRP Project 12-103 26 In addition to the “one-to-one” validation approach, the UD research personnel performed an independent, line-by-line auditing of the source code for the automated member-sizing software to validate all components of the software. This audit was conducted to ensure that there were no typographical or syntax errors and all appropriate equations were present and being calculated properly. 3.3 Automated 3D FE Modeling of Bridge Suite (Task 2.1.3) To permit the execution of the large parametric study defined in Phase I, the Research Team developed an automated software tool to construct, analyze, and extract results from 3D FE models of multi-girder bridges. This approach makes use of Strand7 (www.strand7.com), a commercially-available FE simulation software, which provides a seamless interface with Matlab through an Application Programming Interface (API). The API allows Matlab to dictate the model construction and results extraction activities that are normally done through manual means. With this approach, all manual interaction with the simulation software is eliminated and replaced by an automated interaction via Matlab. This allows for rapid creation of models with several different design configurations, which will be needed in developing the large bridge suite previously described. To generate a 3D FE model of a bridge, the software requires that a number of parameters and common practices be explicitly defined. Specifically, the required information can be grouped into three categories based on how it will be obtained: 1) Bridge configuration (achieved through sampling the parameters to be carried out under T2.1.1). 2) Member sizes (achieved through automation of the bridge design specifications to be carried out under T2.1.2) 3) Common details of secondary elements (e.g. barrier and sidewalk sizes, diaphragm type and spacing, deck thickness, etc. defined using heuristics) Once defined, the automation of the FE model construction and results extraction process proceeds in step-by-step manner as shown in Figure 3-4.

NCHRP Project 12-103 27 Figure 3-4 - Step-by-step approach to automated FE model construction. 3.3.1 Model Form A wide range of modeling techniques are available to simulate the behavior of common multi-girder bridges. The element-level model is the most common class of 3D FE models employed for constructed systems (Aktan 2013), and is therefore employed for this study based on the results of the sensitivity study conducted in Phase I. This model employs one-dimensional elements (beam/frame elements) to model girders, diaphragms, and barriers, and two-dimensional (shell elements) to model the deck and sidewalks. In an element-level model link elements are used to connect the beam and shell element components (i.e. girders to the deck, diaphragms to the girders, etc.) to remain consistent with the 3D geometry of the structure. The schematic in Figure 3-5 shows how the 3D geometry of the bridge is simulated using the beam, shell, and link elements described. Step 1 – Initialization of the API (opens communication between Strand7 and Matlab) Step 2 – Definition of global geometric grid, mesh size, and nodal locations Step 3 – Definition of member geometry and nodal connectivity Step 5 – Definition of element geometric and material properties Step 4 – Definition of continuity and boundary conditions (use of links, offsets, releases as appropriate) Step 7 – Execution of analysis cases and extraction of key responses (element actions/stresses, global displacements) Step 6 – Definition of loads and load cases (both magnitude and spatial locations)

NCHRP Project 12-103 28 Figure 3-5 - Schematic representation of the element-level model used for simulating a typical multi-girder structure (Masceri 2015). 3.3.2 Modeling the Superstructure For analysis using the element-level model, girders and diaphragms are simulated using one- dimensional (1D) beam elements. The section geometry and material properties of the girders and diaphragms are assigned to the beam element property using the Strand7 API. The beam elements are placed at the centroid of the sections they aim to simulate. To define the spatial location of the top and bottom surfaces of the girder flanges rigid links are provided between the beam elements and these locations. These rigid links are then available to connect the beam elements with cross frames, the deck, or bearings. Depending on the type and configuration of the cross frames/diaphragms, the associated beam elements can connect to the top or bottom flange nodes or to the girder centroid node. If connections exist in other locations on the girder, nodes may be placed at the connection location on the girder and linked to the top/bottom flange nodes or girder centroid node (diaphragm). Figure 3-6 shows a schematic of a typical girder-cross frame connection.

NCHRP Project 12-103 29 Figure 3-6 - Typical girder-cross frame connection (Masceri 2015). The concrete deck is modeled using three-node (triangular) and four-node (quadrilateral) shell elements depending on the configuration needed. These shell elements are assigned a bending and membrane thickness equivalent to the thickness of the concrete deck. Deck nodes are located at the center of the shell element thickness. Composite action of the deck is enforced by connecting the girder nodes and the deck nodes with rigid link elements. The steel reinforcement within the deck is not explicitly considered in the model. 3.3.3 Defining Boundary Conditions at the Supports The boundary conditions in all models were defined to provide the least amount of restraint while keeping the model stable. Assigning the least restraint to the model avoids problems associated with self-equilibrating forces (locally) that can greatly influence results, but which are not representative of actual bridge behavior. This is achieved by restraining all supports in the vertical direction, restraining the exterior girders in the longitudinal direction at one end of the bridge model, and restraining the central girder in the transverse direction at the same end of the bridge model. In this manner, local self- equilibrating forces are avoided as all longitudinal and transverse reactions are necessarily zero. This method of assigning boundary conditions represents an idealization that permits the automation of analysis and results extraction. Figure 3-7 shows a two-span continuous structure as an example of boundary conditions imposed in this manner.