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

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NCHRP Project 12-103 30 Figure 3-7 - Typical boundary conditions employed in each model. 4 Estimation of Maximum Tolerable Support Movement (Task 2.2) The methodology for estimating the maximum tolerable support movement is similar to the typical load rating methodology in which the capacity of the structure is evaluated against the calculated demands for each limit state. This approach begins by calculating the excess capacity associated with each limit state. This is done by subtracting the factored demand force effects from the factored resistance. The tolerable support movement is then defined as the excess capacity divided by the force effects associated with a unit support movement. This approach is summarized by Equation 4-1. Equation 4-1 – Formula for calculation tolerable support movement. ∆௧௢௟ = ߶ܴ − ߛܦ ܵଵ Where, ∆tol = tolerable support movement [in.] ϕR = factored resistance for the limit state being evaluated (units dependent on limit state, e.g. [k-in.]) γD = factored demand force effects for the limit state being evaluated (units dependent on limit state, e.g. [k-in.]) S1 = force effects per unit support movement (units dependent on limit state, per inch of support movement, e.g. [k-in./in.])

NCHRP Project 12-103 31 This expression is based on superposition and thus assumes linear response. The tolerable support movement computed from Equation 4-1 will be in units of length (e.g. in.). As an example, in the case of a flexure limit state, the numerator would be in units of moment (e.g. kip-in) and the units of the denominator would be in units of moment per unit support movement (e.g. kip-in/in). This expression is formulated for each Strength and Service limit state shown in Table 1-3 using the force effects associated with dead load, super-imposed dead load, live load, and support movements computed using the 3D FE models described in Section 3. 4.1 Definition of Longitudinal- and Transverse-Differential Support Movements The two support movements investigated for this study were vertical translation at a single support (i.e. LD support movement) and a linearly varying translation in the TD of a single support (see Table 4-1). A LD support movement occurring at an abutment or at an interior pier is when all bearing locations at a single support undergo the same translation. A TD support movement is when one side of a single support undergoes a larger vertical translation than the opposite side. These two movements govern the limit states that are being considered in this study (listed in Section 1). Other support movements, such as horizontal movements in the longitudinal direction, are certainly important; however, these types of movements are accommodated by the movement systems provided within bridges. As a result, their tolerable limits would be defined based on the excess deformation capacity of the movements systems (inclusive of joints and bearings) as opposed to strength and service limit states. It is anticipated that the recommendations to the AASHTO LRFD to be developed under Phase III will address these types of movements.

NCHRP Project 12-103 32 Table 4-1 - Support movements investigated in this project. Support Movement Representation Longitudinal-Differential Support Movement Transverse-Differential Support Movement 4.2 Simulation of Longitudinal- and Transverse-Differential Support Movements For two and three-span continuous bridges, responses from LD and TD movements at the abutment and at the interior pier(s) were investigated. LD movement of a support was simulated in the model by applying a unit translation in the vertical downward direction to each bearing node sequentially at each support location. For TD movement, one of the exterior bearing locations was held stationary and the opposite bearing location was translated vertically downward a unit. The interior bearing movements were then linearly interpolated between these exterior bearing locations. For TD support movements, two cases were considered. The first held the bearing location furthest from the origin stationary, which resulted in a rotation of the support about the longitudinal axis. The second case held the bearing closest to the origin stationary. This was done to ensure that the worst case for TD support movement was considered as these two cases produce different results for skewed bridges. Figure 4-1 depicts the two cases of TD support movement for an example structure with five girders (looking in the longitudinal direction).

NCHRP Project 12-103 33 Figure 4-1 - Schematic representation of a rotational translation for an example structure with five girders. 4.3 Simulation of Dead Load Two dead load cases were simulated in order to be able to consider the limit states shown in Table 1-3: initial dead load and superimposed dead load. Each load case was defined by manipulating the mass and stiffness through material properties (density and modulus of elasticity) of specific components of the structure. The following subsections provide descriptions of each dead load case as well as the application of each case. 4.3.1 Initial Dead Load (DC1) Included in the initial dead load case is the self-weight of the steel components (girders, diaphragms, and connections) as well as the self-weight of the deck. For analysis of this load case, the stiffness of the steel components are included but the stiffness of the deck is not. This is achieved by setting the modulus of elasticity for the deck to an arbitrarily small value before running the linear static solver in Strand7, ensuring that the dead load of the un-cured concrete is accounted for while providing no stiffness. Table 4-2 summarizes the state of each component for this load case. 0.75 0.5 0.25 1.0 0.75 0.5 0.25 1.0 x y

NCHRP Project 12-103 34 Table 4-2 - Bridge component states for analysis of initial dead load. Component(s) Mass Stiffness Density Modulus Girders, Diaphragms, Connections Yes Yes Specified density of component(s) Specified modulus of component(s) Deck Yes No Specified density of component(s) 1.0 psi 4.3.2 Superimposed Dead Load (DC2) Superimposed dead load considers only the self-weight of the components that are constructed after the deck has cured (e.g. sidewalks and barriers). The stiffness of these components is ignored by setting their modulus of elasticity to an arbitrarily small value. The stiffness of the cured concrete deck is included but the mass of the deck and all steel components (girders, diaphragms, and connections) is ignored for this load case by setting the density of each component to 0. This ensures that the dead load for each component is only accounted for only once. Table 4-3 - Bridge component states for analysis of superimposed dead load. Component(s) Mass Stiffness Density Modulus Girders, Diaphragms, Connections No Yes 0 lb/in3 Specified modulus of component(s) Deck No Yes 0 lb/in3 Specified modulus of component(s) Sidewalks, Barriers Yes No Specified density of component(s) 1.0 psi 4.4 Simulation of Live Load Live load is applied to the model via a series of point loads representing the HL-93 live load model specified in AASHTO LRFD. This live load model includes both design truck loads (including the tandem load) and lane loads, which are required to be positioned within lanes. The number of lanes is determined by the roadway width, such that the maximum number of 12 ft. wide lanes is assigned. Truck loads consist of six point loads that total 72 kips and are amplified by a dynamic impact factor of

NCHRP Project 12-103 35 1.33. Lane loads are defined as a 0.64 kip/ft (longitudinally) and are applied as a series of point loads to simulate the lane load being spread uniformly across the width of the lane. The application of live loads was done individually for each lane and for each type of load (design truck and lane load). That is, analyses were conducted separately for each lane loaded by both the design truck and lane load. To simulate the potential crowding of trucks, the design truck was placed along both edges and along the center of the lane. To combine these individual analyses, the structural responses for the design truck placed within each lane are enveloped to ensure the largest demands are captured. To combine these responses with the lane loads or to combine the results from multiple lanes being loaded super position is employed. When multiple lanes are loaded, the structural responses are modified by the multiple presence factors as per AASHTO LRFD. Figure 4-2 illustrates how this process simulates lane crowding for a three-lane bridge. Figure 4-2 – Transverse truck load positioning for an example 3 lane bridge. The longitudinal placement of trucks (within each lane) was carried out to maximize the moment and shear demands on the structure. Figure 4-3 illustrates these placement locations which result in (a) maximum positive moment of Span 2, (b) maximum negative moment and shear over the center support (for shorter bridges), (c) maximum positive moment of Span 1, (d) maximum negative moment and shear over center support for longer bridges, (e) maximum shear over first abutment, and (f)

NCHRP Project 12-103 36 maximum shear over second abutment. Loads were applied in this fashion for each axle spacing of the design truck as well as the specified tandem truck. Figure 4-3 – Longitudinal truck load positioning for an example two-span continuous bridge. 4.5 Results Extraction The following sections describe the details associated with the extraction of force effects from the FE models. Specifically, these sections discuss (a) the regions of interest, (b) the extraction of moments from composite sections, (c) the extraction of moments from non-composite sections, and (d) the extraction of shear forces.

NCHRP Project 12-103 37 4.5.1 Response Regions of Interest Results were extracted from the entire length of each girder; however, only specific regions experience adverse force effects from support movements. These regions depend on the continuity of the bridge as well as the type/location of the support movement. For example, a LD support movement at an abutment of a continuous bridge induces a negative moment in the girders within the adjacent span, which serves to decrease positive moment demand in the mid-span region as well as the total shear demand at the location of the support movement. At the same time, this support movement increases the negative moment and shear demand over the pier. Limitations on the support movements of an abutment are generally governed by flexure or shear limits over the pier. In some cases, highly-skewed bridges may exhibit positive bending in some locations due to a movement occurring at the abutment. This is especially true for TD support movements. Moreover, TD support movements occurring at an abutment of highly skewed bridges can increase the shear force within the exterior girder at the abutment that does not undergo any translation. Figure 4-4 depicts the regions that experience adverse force effects due to LD and TD support movements occurring at the abutment. Figure 4-4 – Response regions of interest for movements occurring at the abutment. A LD support movement at a pier within a continuous bridge induces positive moment in the girders of the adjacent spans, which serves to decrease the negative moment and shear demands over the pier. Conversely, this type of support movement increases both the positive moment demand in the mid- span region and the shear demand at the supports adjacent to the one undergoing a displacement. As a result, for this type of support movement, the positive moment region or shear at the adjacent supports will generally govern. Similar to support movements at an abutment, in highly-skewed bridges, TD support movements occurring at a pier can increase the shear force over the pier within the girder that

NCHRP Project 12-103 38 does not undergo any translation. Figure 4-5 depicts the regions that experience adverse force effects due to LD and TD support movements occurring at the pier. Figure 4-5 - Response regions of interest for movements occurring at the pier. 4.5.2 Moment on Composite Sections For composite moments, components of the moment were extracted from the beam and adjacent shell elements along the entire length of the girder. To compute the full composite moment at a cross- section, three response components were extracted from each location. These components are major axis bending moment in the beam element (M1), axial force in the beam element (P), and bending moment in the adjacent shell elements (M22). The adjacent shell elements are those in a transverse row within the effective width of each longitudinal member (as defined by AASHTO LRFD). The bending moment in the adjacent shell elements is only considered for load cases where the stiffness of the deck is considered (i.e. all except initial dead load). Equation 4-2 gives the calculation used to determine the total moment that acts on the cross-section where y represents the distance between the centroid of the deck and the centroid of the beam. Equation 4-2 - Formula for calculating the total moment action on a cross-section. ܶ݋ݐ݈ܽ ܯ݋݉݁݊ݐ,ܯ்௢௧௔௟ = ܯଵ + ܲݕ +෍ܯଶଶ

NCHRP Project 12-103 39 4.5.3 Moment on Non-Composite Sections Moments needs to be extracted from two types of non-composite sections. The first is the dead load moments within the girders. Since the deck is assigned an arbitrarily low stiffness for these analyses, the total moment in the girder is simply the primary bending moment of the associated beam element. The second type is the negative moment within the girders under super-imposed dead load, live load, and support movements. In this case the model (as is common practice) simulates these sections as composite; however, the capacity of these sections is computed assuming they behave in a non- composite manner. To meet these requirements, the demand moments are extracted from the model in the same manner as moments on composite sections (see the previous section). For limit states that are dependent on flexural stresses, the demand moments extracted from the model are transformed into stresses using the non-composite section properties. Specifically, the total moment on the cross-section and the section modulus (S) for the top or bottom of the non-composite cross-section are used to compute the extreme fiber stresses as presented in Equation 4-3. Equation 4-3 - Formula for calculating the extreme fiber stress for the top or bottom flange of a cross-section. ܧݔݐݎ݁݉݁ ܨܾ݅݁ݎ ܵݐݎ݁ݏݏ, ௧݂,௖ = ܯ்௢௧௔௟ ܵ As per AASHTO LRFD, these extreme fiber stresses are used in the calculation of the Service II and Strength I (negative moment regions and positive moment regions for non-compact sections) limit states for steel bridges. For PS concrete bridges these extreme fiber stresses are used in the calculation of the Service I and III limit states. 4.5.4 Shear in Girders The method of extracting shear responses depends on the location of where the response is being extracted. Shear responses over abutments (exterior) supports are taken as the reaction at the support. This conservatively assumes that the total reaction is completely resisted by the shear capacity of the girder. Shear responses over interior supports are taken as the absolute maximum shear response in the beam element assuming there is no contribution from the concrete deck. The decision to extract shear responses using these two methods was a result of findings from Phase I of this project where it was found that shear responses over abutment (exterior) supports were dependent upon the discretization of the model. This was not the case, however, for responses over interior supports.

NCHRP Project 12-103 40 4.6 Results Convergence Results convergence was assessed by comparing the empirical cumulative density function (ECDF) for the results of superstructure tolerance for each support movement scenario (type/location) and limit state. The ECDFs are compared using a two-sample Kolmogorov-Smirnov (KS) test with the Matlab function kstest2. The KS test returns the p-value for the null hypothesis that the data in two given vectors are from the same continuous distribution at a specified significance level (Chakravarti 1967). The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed statistic under the null hypothesis (Mathworks). Thus, a larger p-value indicates greater statistical likelihood that the data in the two vectors come from the same continuous population. The two-sample KS test is a nonparametric hypothesis test that evaluates the difference between the ECDFs of two distributions. The criteria for passing convergence are defined in Table 4-4. Table 4-4 - Passing criteria for assessing results convergence. Test Passing Criteria Kolmogorov-Smirnov (KS) Test p-value > 0.8 A large α value was chosen in order to make it easier to reject the null hypothesis that the two sample distributions come from the same continuous distribution. This places the burden of truth on the null hypothesis (that the two sample distributions come from the same population). For cases where the results do not pass the criteria above, additional bridge suites were created and analyzed until convergence was achieved. Once all results were found to converge, it was concluded that more samples would not add any meaningful information to the data. Figure 4-6 provides an example ECDF for Strength I Flexure tolerance to a LD support movement occurring at the abutment of a two-span continuous steel multi-girder bridge. The plot shows very good agreement for Suites 1 and combined Suites 1 & 2. With a KS p-value of 0.99, convergence was achieved and no additional samples were needed. Convergence was assessed for both movement types, occurring at either support location, and for each limit state. Once convergence was achieved for each case, no additional samples were needed. Table 4-5 and Table 4-6 provide the convergence results for each bridge suite. The accompanying ECDF plots are included in Appendix C and Appendix D of this report for steel and PS concrete, respectively.

NCHRP Project 12-103 41 Table 4-5 - Convergence data for each data set of steel multi-girder bridges. Sample Set Movement Limit State Converged at N Samples KS-Value Simple Span Steel TD at Abutment Strength I Flexure 200 0.9990 Service II 200 0.9990 Strength I Shear 200 0.9853 Two-Span Steel LD at Abutment Strength I Flexure 200 0.9999 Service II 200 0.9990 Strength I Shear 200 0.9653 TD at Abutment Strength I Flexure 200 1.0000 Service II 200 0.9953 Strength I Shear 200 0.9853 LD at Pier Strength I Flexure 200 1.0000 Service II 200 0.9953 Strength I Shear 200 0.9653 TD at Pier Strength I Flexure 200 0.9999 Service II 200 0.9853 Strength I Shear 200 0.9853 Three-Span Steel LD at Abutment Strength I Flexure 200 0.9999 Service II 200 0.9999 Strength I Shear 200 0.9999 TD at Abutment Strength I Flexure 200 0.9999 Service II 200 0.9999 Strength I Shear 200 0.9953 LD at Pier Strength I Flexure 200 0.9953 Service II 200 0.9953 Strength I Shear 200 0.9990 TD at Pier Strength I Flexure 200 0.9990 Service II 200 0.9990 Strength I Shear 200 0.9331

NCHRP Project 12-103 42 Table 4-6 - Convergence data for each data set of PS concrete multi-girder bridges. Sample Set Movement Limit State Converged at N Samples KS-Value Simple Span PS TD at Abutment Strength I Flexure 200 0.8885 Service I/III 200 0.9653 Strength I Shear 200 1.0000 Two-Span PS LD at Abutment Strength I Flexure 200 1.0000 Service I/III 200 1.0000 Strength I Shear 200 0.9853 TD at Abutment Strength I Flexure 200 1.0000 Service I/III 200 0.9999 Strength I Shear 200 0.9903 LD at Pier Strength I Flexure 200 1.0000 Service I/III 200 0.9999 Strength I Shear 200 0.9953 TD at Pier Strength I Flexure 200 0.9953 Service I/III 200 0.9903 Strength I Shear 200 0.9999 Three-Span PS LD at Abutment Strength I Flexure 200 1.0000 Service I/III 200 1.0000 Strength I Shear 200 1.0000 TD at Abutment Strength I Flexure 200 1.0000 Service I/III 200 1.0000 Strength I Shear 200 0.9888 LD at Pier Strength I Flexure 200 1.0000 Service I/III 200 0.9999 Strength I Shear 200 0.9999 TD at Pier Strength I Flexure 200 1.0000 Service I/III 200 0.9990 Strength I Shear 200 0.9871