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

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NCHRP Project 12-103 141 6 Pre-Stressed Concrete Multi-Girder Bridges This section presents the tolerable support movement results obtained from the analysis of PS bridges. To structure the results, data sets were segmented by (a) continuity of the structure, (b) limit state, (c) type of support movement (LD or TD), and (d) location of the support movement (at the abutment or at the pier). Controlling limit states were identified for each type and location of support movement, as were the locations within the superstructure where the tolerance controls. In addition to exploratory analysis, linear regression was employed to determine the predictor variables (influential parameters) and their interactions that affect tolerance to LD and TD support movements. Only the controlling limit states/movements are discussed in detail. When applicable, the results were evaluated against the empirical guidance for support movements of simple-span and multiple-span continuous bridges given in the commentary of the AASHTO LRFD Specifications (see Section 2). 6.1 Tolerable Support Movement Influences Throughout the analysis of results discussed in this section, the trends associated with levels of tolerable support movement will be identified and traced to their underlying mechanisms. The three overarching mechanisms that influence the tolerance of PS concrete bridges to support movements are (1) the inherent conservatism implicit within the AASHTO LRFD live load distribution factors, (2) the distribution of dead loads and live loads in highly skewed bridges that are not explicitly accounted for the by the SLG design model, and (3) the uncoupling of stiffness and strength within PS concrete girders. These issues are described in the following sections together with the influence (or lack thereof) of fatigue limit states on the design of PS concrete bridges. 6.1.1 Live Load Distribution Factors In the case of live load distribution factors, the issue related to PS concrete bridges is very similar to the one described for steel bridges in Section 5.1.1. For comparison purposes, Figure 6-1 presents the ratio of SLG and FE model ratings for a suite of PS concrete bridges which essentially shows the level and variability of the conservatism implicit in the live load distribution factors (Masceri 2015).

NCHRP Project 12-103 142 Figure 6-1 - Probability distribution for the ratio of FE-to-SLG Rating factor. 6.1.2 Load Distribution in Highly Skewed Bridges For the dead and live load distributions within highly skewed bridges, this mechanism is similar between PS concrete and steel bridges (see Section 5.1.2) with two notable exceptions. First, PS concrete bridges typically do not have intermediate diaphragms whereas steel bridges are commonly constructed with uniformly spaced cross frames or diaphragms throughout the span. This results in two potential differences between steel and PS concrete bridges in this regard. First, the tendency to distribute additional dead and live load to exterior girders of highly skewed bridges will be reduced for PS concrete bridges, as the transverse stiffness of the bridge is only provided by the concrete deck (rather than the deck plus the diaphragms). Second, since there are no intermediate diaphragms, the discontinuity at a 20o skew angle that separates when intermediate cross frames are placed parallel to the skew or perpendicular to the girder does not exist. The second primary difference is due to the fact that PS concrete bridges are not constructed continuously for dead loads. Rather, these bridges are constructed as simply-supported segments that are rendered continuous following the curing of the concrete deck (i.e. continuous for live load). As a result, while continuous steel bridges experience additional negative dead load moments within the external girders over internal piers, this does not occur in the case of PS concrete bridges since those girders are not continuous when the deck is cast.

NCHRP Project 12-103 143 6.1.3 Uncoupling of Stiffness and Strength in Pre-Stressed Concrete Bridges As illustrated by the results in the following sections, in many cases higher levels of tolerable support movement (governed by flexural limit states) were associated with larger girder spacing for PS concrete bridges. This trend is opposite of what was generally observed for steel bridges. The root cause of this change in behavior is believed to be due to the decoupling between stiffness and strength for PS concrete girders. In the case of steel girders, larger girder spacing (i.e. fewer girders for a given width) requires a larger section to carry the increased load per girder. As a result, both the stiffness of the girders and the strength of the girders increase together. This is not the case for PS concrete bridges. For these bridges, in many cases the cross-section, and thus the elastic stiffness (i.e. EI), does not increase with increasing in capacity. To increase the strength of a PS concrete girder one can simply add additional pre-stressing strands, effectively increasing the capacity without influencing the stiffness of the cross-section. Although in some cases the iterative design process will increase the concrete strength (and thus the elastic modulus), the resulting increase in stiffness is rather small compared to the increase in stiffness associated with additional capacity of steel girders. Thus, for PS concrete bridges, larger girder spacing does not necessarily lead to stiffer PS concrete girders. As a result, PS concrete bridges with larger girder spacing will generally have a reduced stiffness and thus the force effects associated with support movements are smaller. 6.1.4 Elements Governed by the Fatigue Limit State The third mechanism discussed in relation to steel bridges (elements governed by the Fatigue limit state, see Section 5.1.3) does not apply to PS concrete bridges. Although there are fatigue limitations for PS concrete bridges, they apply only to the cracked section. The Fatigue limit state does not control in the design of PS concrete girders (refer to the commentary in AASHTO LRFD 5.5.3.1). Instead, Service III controls the design in the positive moment region. Thus, while steel bridges have additional capacity (for the Strength I and Service II limit states) built-in to their positive moment regions (due to the Fatigue limit state), PS concrete bridges do not. 6.2 Methods for Identifying Tolerable Support Movement Influences The mechanisms discussed in Section 6.1 were identified though an exploratory examination of individual samples driven by the parameters that were found to influence superstructure tolerance to

NCHRP Project 12-103 144 LD and TD movements. As with the analysis of the results for steel bridges, stepwise linear regression was employed as an exploratory tool to determine a subset of predictor variables (bridge configuration parameters) and the combinations of those predictor variables that influence PS concrete bridge superstructure tolerance to support movement. For each data set (movement type/location, limit state, etc.) a regression model was obtained and the influential parameters (and their interactions) were identified based on the p-value given for the t-statistic of each predictor in the regression model. Again, a limiting p-value of 0.05 was chosen. The following sections discuss the results of the stepwise regression. Additional information for each regression model obtained is included in Appendix E. To verify the quality of fit for each regression model, values for the coefficient of determination (R2) are given in Table 6-1 and Table 6-2 for two- and three-span continuous PS concrete multi-girder bridges, respectively. Note: while models were obtained for movements occurring at the pier, these models are disregard given the low levels of tolerable support movements observed for these movements, as discussed in the following sections. Table 6-1 - R2 value of each regression model used to determine the influential parameters and their interactions for two- span continuous PS concrete multi-girder bridges. Movement Type/Location Strength I FlexureLinear Model R2 Strength I Shear Linear Model R2 Service I/III Linear Model R2 LD / Abutment 0.9518 0.9652 0.9443 TD / Abutment 0.9427 0.9487 0.8816 LD / Pier 0.8282 0.8427 0.8816 TD / Pier 0.8550 0.8539 0.4192 Table 6-2 - R2 value of each regression model used to determine the influential parameters and their interactions for three- span continuous PS concrete multi-girder bridges. Movement Type/Location Strength I FlexureLinear Model R2 Strength I Shear Linear Model R2 Service I/III Linear Model R2 LD / Abutment 0.9532 0.9623 0.7593 TD / Abutment 0.9409 0.9424 0.7580 LD / Pier 0.8562 0.8619 0.7580 TD / Pier 0.8702 0.8662 0.5946 6.3 Simple-Span Pre-Stressed Concrete Bridges For simple-span bridges, LD support movement does not induce any adverse force effects and thus is not limited by the either Strength or Service limit states. In the case of TD support movement, the Strength I limit state for shear and the Service I and III limit states controlled (compression and tension

NCHRP Project 12-103 145 stress limits) the superstructure tolerance to TD support movement. The Strength I Flexure tolerance to TD support movements were found to be orders of magnitude greater than the other limit states and thus did not control. Figure 6-2 shows the percentage of the population controlled by each limit state for TD movements occurring at either abutment. Figure 6-2 - Controlling limit state for a TD support movement occurring at the abutment of a simply supported PS concrete multi-girder bridge. Girder spacing was identified as the most influential parameter for bridges that were governed by the Strength I Shear limit state. Skew, span length, and girder spacing were identified as influential parameters for the bridges that were governed by the Service I or III limit states. Sections 6.3.1 and 6.3.2 provide a detailed discussion of the parameters that influence support movement tolerance for the Strength I Shear and Service limit states, respectively. 6.3.1 Strength I Shear Tolerance to TD Movements When the Strength I limit state for shear controls, it controls over the abutment at the exterior girder on the side opposite of where the maximum TD movement occurred (i.e., the stationary girder). This behavior was displayed for nearly the entire population of simply supported bridges (as indicated by Figure 6-3). A considerable number of bridges (approximately 10%) were found to be limited by less than one inch of support movement. Figure 6-4 provides a plot of span length versus the tolerable TD support movement associated with the Strength I limit state for shear. The current AASHTO LRFD

NCHRP Project 12-103 146 criterion for simple-span bridges (0.008L) is shown in this figure as a point of comparison. Evident in figure, approximately 68% of the sample population falls below the current AASHTO LRFD criterion. Moreover, it appears that the use of length as an explanatory variable for tolerance may be inappropriate for the Strength I limit state for shear as no clear trend is observable. Figure 6-3 – Controlling location of Strength I Shear tolerance to a TD support movement occurring at the abutment of a simply supported PS concrete multi-girder bridge. Figure 6-4 – Strength I Shear tolerance to a TD support movement as a function of span length occurring at the abutment of a simply supported PS concrete multi-girder bridge.

NCHRP Project 12-103 147 Figure 6-5 gives the plot of girder spacing versus tolerable TD support movement. The plot shows a positive trend for tolerances as girder spacing increases, indicating that girder spacing may be a more appropriate explanatory variable for estimating the level of tolerable support movement for simple-span PS concrete bridges. This is consistent with the phenomenon associated with the dead load distribution in highly skewed bridges described in Section 6.1.2. That is, the adverse dead load distribution to adjacent girders in highly-skewed bridges is less significant with larger girder spacing. Figure 6-5 – Strength I Shear tolerance to a TD support movement as a function of girder spacing occurring at the abutment of a simply supported PS concrete multi-girder bridge. Linear regression analysis indicated that girder spacing (S) was in fact the most influential parameter. Note: the regression model obtained had an R2 value of 0.732. Figure 6-6 gives the effects of each parameter on tolerance. An effects plot shows the estimated effect on tolerance from increasing each parameter (from its lowest to its highest value), while keeping the remaining parameters at an average value. The horizontal axis gives the main effect on tolerance. The scale ranging from negative to positive, represents a decrease or increase in tolerable support movement, respectively. Each point represents a predictor variable (influential parameter). The location of each point along the horizontal axis indicates whether tolerance decreases or increases when the value of the variable is increased. The error bars indicate a 95% confidence bound on the effects of each variable. Span length (L), width (W), and skew have little effect on tolerance, although their interactions do explain the variability in tolerance as girder spacing increases.

NCHRP Project 12-103 148 Figure 6-6 - Effects plot for tolerance to TD movements occurring at the abutment of a simply supported PS concrete multi- girder bridge. 6.3.2 Service Tolerance to TD Movements Figure 6-7 shows that that Service limit state (Service I or III) controlled at an interior girder for approximately 82% of the population. Investigation into this behavior indicated that interior girders commonly controlled for bridges with higher skew and larger girder spacing when the maximum TD movement occurs at the obtuse corner of a highly-skewed bridge. When the maximum TD movement occurs at the obtuse corner of a highly-skewed bridge, it serves to increase positive bending in the interior girders, thus increasing the tension in the bottom of the girder which causes Service III to control at an interior girder. Figure 6-8 illustrates this behavior. However, in some cases, the TD movement can induce negative bending in the girders effectively increasing the compression in the bottom of the girder causing Service I to control at an interior girder.

NCHRP Project 12-103 149 Figure 6-7 - Controlling location of Service tolerance to a TD support movement occurring at the abutment of a simply supported PS concrete multi-girder bridge. Figure 6-8 - TD movement moment diagram for a highly skewed simply supported bridge. Figure 6-9 gives the plot of span length versus tolerable TD support movement based on Service limit states I and III. Again, the current AASHTO LRFD criterion (0.008L) is included in the plot as a point of comparison. Evident in the plot, a considerable number of observations (approximately 27%) fall below the current AASHTO LRFD criterion. Also, similar to the Strength I limit state for shear, there is no clear trend with span length. Figure 6-10 gives the plot of skew angle versus tolerable TD support movement based on the Service limit states. Tolerable support movement decreases quite rapidly as skew increases. Highly-skewed bridges experience greater bending forces when exposed to TD support movements and thus are less tolerant to this type of demand. Location of maximum TD movement

NCHRP Project 12-103 150 Figure 6-9 – Service tolerance to a TD support movement as a function of span length occurring at the abutment of a simply supported PS concrete multi-girder bridge. Figure 6-10 – Service tolerance to a TD support movement as a function of skew occurring at the abutment of a simply supported PS concrete multi-girder bridge.

NCHRP Project 12-103 151 Figure 6-11 - Effects plot for tolerance to TD movements occurring at the abutment of a simply supported PS concrete multi- girder bridge. The effects plot given by Figure 6-11 above indicates that skew is the most influential parameter. Span length (L), girder spacing (S), and width (W) were found to influence tolerance but to a much lesser extent. Span length and girder spacing become more influential for bridges with lower skew. The interaction plots for the interactions of span length /skew (Figure 6-12) and girder spacing/skew (Figure 6-13) illustrate this behavior. The accompanying scatter plots are given by Figure 6-9 and Figure 6-10. Note: the regression model had an R2 value of 0.714. An interaction plot illustrates the interaction effects between two predictor variables in the fitted model. This plot provides way of showing the effect on tolerance due to the interaction of two influential parameters by assigning all other predictor variables their average value and evaluating the fitted model across the bounds of the two predictor variables considered to obtain the “Adjusted Tolerance”. Each line represents an upper, middle, and lower bound of the interacting parameter. The slope of this line indicates the rate at which tolerance increases or decreases (on average) as the parameter on the horizontal axis increases. Thus, for lower-skewed bridges, tolerance to TD support movement increases with span length. For bridges with lower skew, the superstructure exhibits more uniform load sharing between the girders. The opposite is true for highly-skewed bridges. Note: these interaction lines represent the mean of the data and therefore are only being used here as an exploratory tool for parameter interactions. The lower bound of tolerance is of actual concern.

NCHRP Project 12-103 152 Figure 6-12 - Interaction plot of skew and span length. Evident in Figure 6-13, as skew increases, tolerance to TD movement decreases. The magnitude of the girder spacing affects the extent to which skew influences tolerance. At lower values of skew, larger girder spacing is associated with higher tolerance. The effect of girder spacing becomes less significant for higher skewed bridges. Again, with higher skews, the load sharing between girders is non-uniform and thus the tolerance to TD support movements is lower. Figure 6-13 - Interaction plot of girder spacing and span length.

NCHRP Project 12-103 153 6.4 Two-Span Continuous Pre-Stressed Concrete Bridges 6.4.1 Controlling Limit State The figures in this section present the controlling limit states for each type/location of support movement. For LD and TD movements occurring at the abutment, the Strength I Flexure and Shear limit states generally controlled tolerance. The Service III limit state controls for LD and TD movements occurring at the pier 99.5% and 88% of the time, respectively. Figure 6-14 - Controlling limit state for a LD support movement occurring at the abutment of a two-span continuous PS concrete multi-girder bridge.

NCHRP Project 12-103 154 Figure 6-15 - Controlling limit state for a TD support movement occurring at the abutment of a two-span continuous PS concrete multi-girder bridge. Figure 6-16 - Controlling limit state for a LD support movement occurring at the pier of a two-span continuous PS concrete multi-girder bridge.

NCHRP Project 12-103 155 Figure 6-17 - Controlling limit state for a TD support movement occurring at the pier of a two-span continuous PS concrete multi-girder bridge. 6.4.2 Service Tolerance to LD Movements Occurring at the Abutment The Service limit states controlled the tolerable LD support movement at the abutment in approximately 3% of the population (see Figure 6-14). Figure 6-18 gives the plot of span length versus tolerable LD support movement based on the Service I/III limit state. As evident in this figure, the current AASHTO LRFD criterion gives a conservative estimate of tolerance for the entire population. That is, the current criterion estimated lower values of tolerable support movement than what was observed. Given the high levels of tolerable support movement, rigorous analysis was not conducted. Only the parameters that influence Service tolerance were identified. Span length was identified as the most influential parameter. The influence of girder spacing and skew were less significant, but not negligible. The effect of the influential parameters is described by the effects plot below (Figure 6-19).

NCHRP Project 12-103 156 Figure 6-18 – Service tolerance to a LD support movement as a function of span length occurring at the abutment of a two- span continuous PS concrete multi-girder bridge. Figure 6-19 - Effects plot for tolerance to LD movements occurring at the abutment of a two-span continuous PS concrete multi-girder bridge. 6.4.3 Strength I Flexure Tolerance to LD Movements Occurring at the Abutment The Strength I Flexure limit state controlled tolerable LD support movement at the abutment in approximately 55% of the population (see Figure 6-14). Figure 6-20 gives the plot of span length versus tolerable LD support movement based on the Strength I Flexure limit state. As with the Service limit

NCHRP Project 12-103 157 state, a considerable number of bridges have high Strength I Flexure tolerance to LD movements occurring at the abutment. Again, the current AASHTO LRFD criterion gives a conservative estimate of tolerance for nearly the entire population. That is, the current criterion estimated lower values of tolerable support movement than what was observed. Given the high levels of tolerable support movement, rigorous analysis was not conducted. Only the parameters that influence Strength I Flexure tolerance were identified. Span length was identified as the most influential parameter. The influence of girder spacing and skew were less significant, but not negligible. The effect of the influential parameters is described by the effects plot below (Figure 6-21). Figure 6-20 – Strength I Flexure tolerance to a LD support movement as a function of span length occurring at the abutment of a two-span continuous PS concrete multi-girder bridge.

NCHRP Project 12-103 158 Figure 6-21 - Effects plot for tolerance to LD movements occurring at the abutment of a two-span continuous PS concrete multi-girder bridge. 6.4.4 Strength I Shear Tolerance to LD Movements Occurring at the Abutment The Strength I Shear limit state controlled the tolerable LD support movement at the abutment for approximately 40% of the population (see Figure 6-14). When Strength I for shear controlled, the controlling location occurred at an exterior girder (Figure 6-22), and always over the pier. Since the Strength I Shear limit state controlled for bridges with higher skew, it appears that the relatively small tolerance to support movement is due to the issue of load distribution within highly-skewed bridges (see Section 6.1.2). Larger girder spacing was associated with observations that displayed controlling tolerance at an interior girder.

NCHRP Project 12-103 159 Figure 6-22 - Controlling location of Strength I Shear tolerance to a LD support movement occurring at the abutment of a two-span continuous PS concrete multi-girder bridge. Figure 6-23 gives the plot of span length versus tolerable LD support movement based on the Strength I Shear limit state. A considerable number of observations (approximately 10%) fall below the current AASHTO LRFD criterion. Further investigation found that these observations were all for highly skewed bridges (with an average skew of 45⁰). This is evident in the plot of skew and Strength I Shear tolerance given by Figure 6-24. Figure 6-23 – Strength I Shear tolerance to a LD support movement as a function of span length occurring at the abutment of a two-span continuous PS concrete multi-girder bridge.

NCHRP Project 12-103 160 Figure 6-24 – Strength I Shear tolerance to a LD support movement as a function of skew occurring at the abutment of a two- span continuous PS concrete multi-girder bridge. Span length (L), girder spacing (S), and skew were identified as predictor variables (influential parameters) affecting Strength I Shear tolerance. The effects plot below (Figure 6-25) illustrates the effect of each variable on Strength I Shear tolerance. Tolerance increases with length, and girder spacing, but decreases with increasing skew. Figure 6-25 - Effects plot for tolerance to LD movements occurring at the abutment of a two-span continuous PS concrete multi-girder bridge.

NCHRP Project 12-103 161 The following interaction plots describe the interactions between span length/girder spacing (Figure 6-26) and span length/skew (Figure 6-27), respectively. The accompanying scatter plots are given by Figure 6-23 and Figure 6-24, respectively. Smaller girder spacing has less of an influence on tolerance for all values of span length. That is, as girder spacing increases, its effect on tolerance becomes more significant, especially for longer spans. This is indicated by the slope of the interaction lines in Figure 6-26. The interaction observed in Figure 6-27 shows that span length has greater influence on tolerance for lower values of skew. Girder spacing and skew appear to explain the bridges that exhibited little tolerance to LD support movement based on the Strength I Shear limit state. Again, this is consistent with the discussion of load distribution within highly skewed bridges (See Section 6.1.2). Figure 6-26 - Interaction plot of span length and girder spacing.

NCHRP Project 12-103 162 Figure 6-27 - Interaction plot of span length and skew. 6.4.5 Service Tolerance to TD Movements Occurring at the Abutment The Service I and III limit states controlled the tolerable TD support movement for approximately 21% of the population (see Figure 6-15). The location of the controlling member included both interior and exterior girders. Service III controlled in the positive moment region for approximately 62% of the population. For these bridges, the TD movement induced positive bending and effectively increased the tension at the bottom of the girders. Further investigation found this to be associated with higher- skewed bridges when the movement occurred at the obtuse corner of the bridge. Service I controlled for 28% of the population. For these bridges, the compression over the pier due to the negative moment induced by the TD movement exceeded the Service I limit state before the Service III limit was exceeded in the positive moment region. Bridges that were controlled by Service I were more likely to be shorter, lower skewed bridges with the TD movement occurring at the acute corner of the bridge.

NCHRP Project 12-103 163 Figure 6-28 - Controlling location of Service tolerance to a TD support movement occurring at the abutment of a two-span continuous PS concrete multi-girder bridge. Figure 6-29 gives the plot of span length versus tolerable TD support movement based on the Service limit states. The current AASHTO LRFD criterion (0.004L) is plotted as a point of comparison. Approximately 25% of observations fall below the current criterion. These observations were controlled by the Service III limit state. Tolerance is much higher for bridges when Service I controls. Figure 6-30 gives the plot of skew versus tolerable TD support movement and shows that tolerance decreases with higher skewed bridges, and that the Service III limit state controls for these bridges.

NCHRP Project 12-103 164 Figure 6-29 – Service tolerance to a TD support movement as a function of span length occurring at the abutment of a two- span continuous PS concrete multi-girder bridge. Figure 6-30 – Service tolerance to a TD support movement as a function of skew occurring at the abutment of a two-span continuous PS concrete multi-girder bridge.0 Span length (L), girder spacing (S), and skew were identified as the most influential parameters that affect tolerance to TD movements occurring at the abutment. Width was also identified to influence tolerance but to a much lesser extent. The effects plot below (Figure 6-31) shows the effect of each predictor variable (influential parameter) on Service tolerance. Higher tolerance is associated with

NCHRP Project 12-103 165 increasing span length, girder spacing, and width, while lower tolerance is associated with increasing skew. Figure 6-31 - Effects plot for tolerance to TD movements occurring at the abutment of a two-span continuous PS concrete multi-girder bridge. The interactions between span length/girder spacing, and span length/skew were found to affect Service tolerance. Figure 6-31 gives the interaction plot of span length and girder spacing. The accompanying scatter plot is given by Figure 6-29. Lower tolerance is associated with smaller girder spacing for all span lengths. As span length increases, the effect of girder spacing becomes more significant as indicated by the slope of the interaction lines for larger girder spacing.

NCHRP Project 12-103 166 Figure 6-32 - Interaction plot for span length and girder spacing. Similar behavior exists for the interaction of skew and span length. Figure 6-33 gives the interaction plot of skew and span length. The accompanying scatter plot is given by Figure 6-29. As span length increases, the effect of skew becomes more significant. Tolerance will generally increase with span length for bridges with less skew. Lower tolerance is associated with higher skew for all values of span length. Figure 6-33 - Interaction plot for span length and skew.

NCHRP Project 12-103 167 6.4.6 Strength I Flexure Tolerance to TD Movements Occurring at the Abutment The Strength I Flexure limit state controlled the tolerance to TD support movement for only 9% of the population (see Figure 6-15). Figure 6-34 gives the plot of span length versus tolerable TD support movement. The current AASHTO LRFD criterion (0.004L) is plotted as a point of comparison. As evident in this figure, the current criterion is conservative for approximately 98% of the population. That is, the current criterion estimated tolerable support movements less than what was observed. Given the high levels of tolerable support movement, rigorous analysis was not conducted. Only the parameters that influence Strength I Flexure tolerance were identified. Span length (L), girder spacing (S), and skew were all determined to be predictor variables (influential parameters) for Strength I Flexure tolerance to TD movement. The effect of the influential parameters is described by the effects plot below (Figure 6-35). Figure 6-34 – Strength I Flexure tolerance to a TD support movement as a function of span length occurring at the abutment of a two-span continuous PS concrete multi-girder bridge.

NCHRP Project 12-103 168 Figure 6-35 - Effects plot for tolerance to TD movements occurring at the abutment of a two-span continuous PS concrete multi-girder bridge. 6.4.7 Strength I Shear Tolerance to TD Movements Occurring at the Abutment The Strength I Shear limit state controlled the TD support movement at an abutment for approximately 70% of the population (see Figure 6-15). When shear controls, the movement was a maximum at the acute side of the abutment. The controlling member was primarily at the obtuse-side exterior girder, opposite the girder that displayed the maximum movement (i.e. the stationary girder). Figure 6-37 gives the plot of span length versus tolerable TD support movement for the Strength I Shear limit state. The current AASHTO LRFD criterion is plotted as a point of comparison. Approximately 45% of the population falls below the current criterion. Further investigation found that this low level of tolerance was associated with bridges that have smaller girder spacing and higher skew, which is consistent with the discussion in Section 6.1.2.

NCHRP Project 12-103 169 Figure 6-36 - Controlling location of Strength I Shear tolerance to a TD support movement occurring at the abutment of a two-span continuous PS concrete multi-girder bridge. Figure 6-37 Strength I Shear tolerance to a TD support movement as a function of span length occurring at the abutment of a two-span continuous PS concrete multi-girder bridge. In Figure 6-37, it appears that the use of length as an explanatory variable for tolerable TD support movement may be inappropriate for the Strength I Shear limit state as no clear trend is observable. In fact, girder spacing may be the appropriate explanatory variable, as indicated in Figure 6-38, which shows girder spacing versus tolerable TD support movement based on the Strength I Shear limit state.

NCHRP Project 12-103 170 Figure 6-38 – Strength I Shear tolerance to a TD support movement as a function of girder spacing occurring at the abutment of a two-span continuous PS concrete multi-girder bridge. Linear regression analysis identified girder spacing (S) as the most influential predictor variable for tolerable TD support movement (Figure 6-39). Span length (L), skew, and width (W) were also identified as predictor variables but their effect on tolerance is much less significant. Figure 6-39 - Effects plot for tolerance to TD movements occurring at the abutment of a two-span continuous PS concrete multi-girder bridge.

NCHRP Project 12-103 171 6.4.8 Service Tolerance to LD Movements Occurring at the Pier The Service III limit state controlled tolerance to LD movements occurring at the pier for nearly the entire population (see Figure 6-16). For LD movements controlled by the Service III limit state, both interior and exterior girders controlled, and the controlling location was always in the positive moment region. Figure 6-40 gives the plot of span length versus the tolerable LD support movement based on the Service III limit state. This plot indicates that two-span continuous PS concrete bridges exhibit very little tolerance to LD movements occurring at the pier. In fact, the current AASHTO criteria is unconservative for the entire population. That is, the current criterion predicts larger values of tolerable support movement than what was observed. The reason for the difference in behavior compared with steel multi-girder bridges was traced to the lack of additional capacity in the positive moment region provided fatigue limit state as discussed in Section 6.1.4. Figure 6-40 – Service tolerance to a LD support movement as a function of span length occurring at the pier of a two-span continuous PS concrete multi-girder bridge. 6.4.9 Strength I Flexure Tolerance to LD Movements Occurring at the Pier The Strength I Flexure limit state never controlled the tolerable LD support movement at the pier (see Figure 6-16). Given the high levels of tolerable support movement (compared to the Service III limit state), rigorous analysis was not conducted. Figure 6-41 gives the plot of span length versus tolerable LD support movement based on the Strength I Flexure limit state.

NCHRP Project 12-103 172 Figure 6-41 – Strength I Flexure tolerance to a LD support movement as a function of span length occurring at the pier of a two-span continuous PS concrete multi-girder bridge. 6.4.10 Strength I Shear Tolerance to LD Movements Occurring at the Pier The Strength I Shear limit state controlled the tolerable LD support movement at the pier for a single bridge (see Figure 6-16). Further investigation found that for this one case, the level of tolerable support movement based on the Strength I Shear limit state was within a tenth of an inch of Service III tolerance. Since Service III nearly always controls, rigorous analysis was not conducted for the Strength I Shear limit state. Figure 6-42 gives the plot of span length versus the tolerable LD support movement based on the Strength I Flexure limit state.

NCHRP Project 12-103 173 Figure 6-42 – Strength I Shear tolerance to a LD support movement as a function of span length occurring at the pier of a two-span continuous PS concrete multi-girder bridge. 6.4.11 Service Tolerance to TD Movements Occurring at the Pier The Service III limit state controlled the tolerable TD support movement at the pier for 88% of the population (see Figure 6-17). For TD movements controlled by the Service III limit state, the controlling location occurred at both interior and exterior girders, but was always within the positive moment region. Figure 6-43 gives the plot of span length versus tolerable TD support movement based on the Service III limit state. The current AASHTO LRFD criterion (0.004L) is plotted for a point of comparison. Similar to LD support movements at a pier, two-span continuous PS concrete bridges exhibit very little tolerance to TD movements occurring at the pier. The current AASHTO LRFD criterion is unconservative for the entire population. That is, the current criterion predicts larger values of tolerable support movement than what was observed. This sensitivity to pier support movements is different than what was observed for steel bridges and it was traced to a lack of a fatigue limit state, which provides additional capacity to steel bridges (for the Strength I Flexure and Service II limit states) in the positive moment region, as discussed in Section 6.1.4.

NCHRP Project 12-103 174 Figure 6-43 – Service tolerance to a TD support movement as a function of span length occurring at the pier of a two-span continuous PS concrete multi-girder bridge. 6.4.12 Strength I Flexure Tolerance to TD Movements Occurring at the Pier The Strength I Flexure limit state never controlled the tolerable TD support movement at the pier (see Figure 6-17). For this reason, rigorous analysis was not conducted. Figure 6-44 gives the plot of span length versus the tolerable TD support movement based on the Strength I Flexure limit state. Figure 6-44 – Strength I Flexure tolerance to a TD support movement as a function of span length occurring at the pier of a two-span continuous PS concrete multi-girder bridge.

NCHRP Project 12-103 175 6.4.13 Strength I Shear Tolerance to TD Movements Occurring at the Pier The Strength I Shear limit state controlled the tolerable TD support movements at the pier for 12% of the population (see Figure 6-17). Further investigation found that the average level of tolerable support movement for these observations was approximately one inch (to the order of the levels observed for the Service III limit state). Since the Service III limit state primarily controls, rigorous analysis was not conducted for the Strength I Shear limit state. Figure 6-45 gives the plot of span length versus tolerable TD support movement based on the Strength I Shear limit state. Figure 6-45 – Controlling location of Strength I Shear tolerance to a TD support movement as a function of span length occurring at the pier of a two-span continuous PS concrete multi-girder bridge. 6.5 Three-Span Continuous Pre-Stressed Concrete Bridges 6.5.1 Controlling Limit State The figures in this section show the percentage of bridges that controlled the level of tolerable support movement for each limit state and for each type/location of support movement. For LD and TD movements occurring at the abutment, each of the limit states were found to control for a considerable portion of the population. Which limit state controlled depended on the type/location of the support movement as well as the bridge configuration. Similar to the results for two-span continuous bridges, the Service III limit state generally controlled for all types of support movement occurring at the pier.

NCHRP Project 12-103 176 Figure 6-46 - Controlling limit state for a LD support movement occurring at the abutment of a three-span continuous PS concrete multi-girder bridge. Figure 6-47 - Controlling limit state for a TD support movement occurring at the abutment of a three-span continuous PS concrete multi-girder bridge.

NCHRP Project 12-103 177 Figure 6-48 - Controlling limit state for a LD support movement occurring at the pier of a three-span continuous PS concrete multi-girder bridge. Figure 6-49 - Controlling limit state for a TD support movement occurring at the pier of a three-span continuous PS concrete multi-girder bridge.

NCHRP Project 12-103 178 6.5.2 Service Tolerance to LD Movements Occurring at the Abutment The Service limit states controlled the tolerable LD support movement at the abutment for 54% of the population (see Figure 6-46). Service III controlled in the positive bending region for 52% of the population, while Service I controlled for compression over the pier for the remaining bridges. When Service controlled, the controlling location included both interior and exterior girders. Figure 6-50 gives the plot of span length versus tolerable LD support movement based on the Service limit states. As apparent in this figure, a considerable number of bridges have a high tolerance to LD support movements occurring at the abutment for the Service limit states. In fact, the current AASHTO LRFD criteria is conservative for the entire population. That is, the current criterion estimates levels of tolerable support movement less than what was observed. Given the high levels of tolerable support movement, rigorous analysis was not conducted. Only the parameters that influence Service tolerance were identified. Span length (L) was identified as the most influential parameter. The influence of girder spacing (S), skew, and width (W) were less significant. The effect of the influential parameters is described by the effects plot below (Figure 6-19). Figure 6-50 - Service tolerance to a LD support movement occurring at the abutment of a three-span continuous PS concrete multi-girder bridge.

NCHRP Project 12-103 179 Figure 6-51 - Effects plot for tolerance to LD movements occurring at the abutment of a three-span continuous PS concrete multi-girder bridge. 6.5.3 Strength I Flexure Tolerance to LD Movements Occurring at the Abutment The Strength I Flexure limit state controlled the level of tolerable support movement at the abutment for approximately 26% of the population (see Figure 6-46). Figure 6-52 gives the plot of span length versus tolerable LD support movement based on the Strength I Flexure limit state. As with the Service limit states, a considerable number of bridges have high Strength I Flexure tolerance to LD movements occurring at the abutment. Given the high levels of tolerable support movement, rigorous analysis was not conducted. Only the parameters that influence Strength I Flexure tolerance were identified. Span length was identified as the most influential parameter. The influence of girder spacing, skew, and width were less significant. The effect of the influential parameters is described by the effects plot below (Figure 6-21).

NCHRP Project 12-103 180 Figure 6-52 – Strength I Flexure tolerance to a LD support movement occurring at the abutment of a three-span continuous PS concrete multi-girder bridge. Figure 6-53 – Strength I Flexure tolerance to a LD support movement occurring at the abutment of a three-span continuous PS concrete multi-girder bridge. 6.5.4 Strength I Shear Tolerance to LD Movements Occurring at the Abutment The Strength I Shear limit state controlled the tolerable LD support movement at the abutment for approximately 20% of the population (see Figure 6-46). Figure 6-54 gives the plot of span length versus tolerable support movement based on the Strength I Shear limit states. The current AASHTO LRFD

NCHRP Project 12-103 181 criterion (0.004l) is conservative for nearly the entire population. That is, the current criterion estimates levels of tolerable support movement lower than what was observed. Figure 6-54 – Strength I Shear tolerance to a LD support movement occurring at the abutment of a three-span continuous PS concrete multi-girder bridge. When Strength I Shear limit state controlled, the controlling location included both exterior and first interior girders, but it was always over the nearest pier to the abutment that displayed the movement. For bridges with high skew angles, the exterior or first interior girder generally controlled indicating that the load distribution of highly-skewed bridges had an influence (see Section 6.1.2). Figure 6-55 gives the plot of skew versus tolerable support movement based on the Strength I Shear limit state. Bridges that were controlled by an interior girder had larger girder spacing.

NCHRP Project 12-103 182 Figure 6-55 – Strength I Shear tolerance to a LD support movement occurring at the abutment of a three-span continuous PS concrete multi-girder bridge. Linear regression analysis identified span length (L), girder spacing (S), and skew as predictor variables (influential parameters) of the Strength I Shear limit state. The effect of each influential parameter is described by the effects plot below (Figure 6-56). Higher tolerance is associated with increasing span length and girder spacing, while lower tolerance is associated with increasing skew. Figure 6-56 – Effects plot for tolerance to LD movements occurring at the abutment of a three-span continuous PS concrete multi-girder bridge.

NCHRP Project 12-103 183 The interactions of skew and girder spacing with span length, were found to affect the Strength I Shear tolerance. Figure 6-57 gives the interaction plot of span length and girder spacing. Lower tolerance is associated with bridges with smaller girder spacing. As span length increases, the effect of girder spacing becomes more significant. Similar behavior was observed for the interaction of skew with span length, as shown in Figure 6-58. Lower tolerance is associated with bridges that have higher skew, and as span length increases, the effect of skew becomes more significant. The accompanying scatter plot for these interactions is given by Figure 6-54. Figure 6-57 - Interaction plot of span length and girder spacing.

NCHRP Project 12-103 184 Figure 6-58 - Interaction plot of span length and skew. 6.5.5 Service Tolerance to TD Movements Occurring at the Abutment The Service I and III limit states controlled the tolerable TD support at the abutment for approximately 41% of the population (see Figure 6-47). For these limit states both interior and exterior girder controlled. Similar to two-span continuous bridges, the Service III limit state controlled in the positive moment region for approximately 70% of the population. For these observations, the TD movement induced positive bending in the interior girders effectively increasing the tension at the bottom of the girders. Further investigation found these observations to be associated with higher-skewed bridges when the movement occurred at the obtuse corner of the bridge. Service I controlled for 30% of the population. For these observations, the compression over the pier due to the negative moment induced by the movement exceeded the Service I limit state before the Service III limit was exceeded in the positive moment region. Further investigation found that these observations were associated with shorter, lower skewed bridges when the TD movement occurred at the acute corner of the bridge.

NCHRP Project 12-103 185 Figure 6-59 - Controlling location of Service tolerance to a TD support movement occurring at the abutment of a three-span continuous PS concrete multi-girder bridge. Figure 6-60 gives the plot of span length versus tolerable TD support movement based on the Service limit states. The current AASHTO LRFD criterion (0.004L) is plotted as a point of comparison. Approximately 10% of the population falls below the current criterion. Further investigation found that the observations that fall below the current criterion were associated with bridges that have higher skew (an average skew greater than 45⁰). Figure 6-60 – Service tolerance to a TD support movement occurring at the abutment of a three-span continuous PS concrete multi-girder bridge.

NCHRP Project 12-103 186 6.5.6 Strength I Flexure Tolerance to TD Movements Occurring at the Abutment The Strength I Flexure limit state controlled tolerable TD support movement at the abutment for 14% of the population (see Figure 6-47). A considerable number of bridges were found to have a high Strength I Flexure tolerance to TD support movements occurring at the abutment. Figure 6-41 gives the plot of span length versus tolerable TD support movement based on the Strength I Flexure limit state. The current AASHTO LRFD criterion (0.004L) is plotted again, for comparison. Given the high levels of tolerable support movement, rigorous analysis was not conducted. Figure 6-61 – Strength I Flexure tolerance to a TD support movement occurring at the abutment of a three-span continuous PS concrete multi-girder bridge. 6.5.7 Strength I Shear Tolerance to TD Movements Occurring at the Abutment The Strength I Shear limit state controlled the tolerable TD support movement at the abutment for 45% of the population (see Figure 6-47). When Strength I for shear controlled, the controlling location was generally an exterior girder (Figure 6-62), and generally at the exterior girder opposite to where the TD movement took place (i.e. the stationary girder). Tolerance commonly controlled at the exterior girder for bridges with higher skew, indicating that Strength I Shear tolerance was driven by the load distribution of highly-skewed bridges (see Section 6.1.2). Larger girder spacing was associated with larger tolerance to TD support movements. This is likely due to the uncoupling of strength and stiffness discussed in Section 6.1.3.

NCHRP Project 12-103 187 Figure 6-62 - Controlling location of Strength I Shear tolerance to a TD support movement occurring at the abutment of a three-span continuous PS concrete multi-girder bridge. Figure 6-63 gives the plot of span length versus tolerable TD support movement based on the Strength I Shear limit state. The current AASHTO LRFD criterion (0.004L) is plotted again, for a comparison. Approximately 15% of the population falls below the current criterion. Given this fact, and the rather large variability evident in Figure 6-63, span length may not be the most appropriate explanatory variable for estimating tolerable support movement. In contrast, a strong correlation was observed between tolerance and girder spacing. This is evident in Figure 6-64 that gives the plot of girder spacing versus tolerable TD support movement based on the Strength I Shear limit state.

NCHRP Project 12-103 188 Figure 6-63 – Strength I Shear tolerance to a TD support movement occurring at the abutment of a three-span continuous PS concrete multi-girder bridge. Figure 6-64 – Strength I Shear tolerance to a TD support movement occurring at the abutment of a three-span continuous PS concrete multi-girder bridge. Linear regression analysis identified girder spacing (S) as the most influential predictor variable for tolerable TD support movement (see Figure 6-65). Span length (L), skew, and width (W) were also identified as predictor variables but their effect on tolerance was much less significant.

NCHRP Project 12-103 189 Figure 6-65 - Effects plot for tolerance to TD movements occurring at the abutment of a three-span continuous PS concrete multi-girder bridge. 6.5.8 Service Tolerance to LD Movements Occurring at the Pier The Service III limit state controlled the tolerable LD support movement at the pier for the entire population (see Figure 6-48). For LD movements controlled by the Service III limit state, the controlling location can occur at both interior and exterior girders, but it always occurs within the positive moment region. Figure 6-66 gives the plot of span length versus tolerable LD support movement based on the Service III limit state. The plot indicates that three-span continuous PS concrete bridges exhibit very little tolerance to LD movements occurring at the pier. Also, apparent in this figure is that the current AASHTO LRFD criterion is unconservative for the entire population. That is, the current criterion estimates levels of tolerable support movement greater than what was observed. Due to the nature of LD movements occurring at the pier, and the lack of available capacity for the Service III limit state (see Section 6.1.4), parameters that influence superstructure tolerance cannot be identified.

NCHRP Project 12-103 190 Figure 6-66 – Service tolerance to a LD support movement occurring at the pier of a three-span continuous PS concrete multi- girder bridge. 6.5.9 Strength I Flexure Tolerance to LD Movements Occurring at the Pier The Strength I Flexure limit state never controlled the tolerable LD support movement at the pier (see Figure 6-48). For this reason, rigorous analysis was not conducted for the Strength I Flexure limit state. Figure 6-67 gives the plot of span length versus the tolerable LD support movement based on the Strength I Flexure limit state.

NCHRP Project 12-103 191 Figure 6-67 – Strength I Flexure tolerance to a LD support movement occurring at the pier of a three-span continuous PS concrete multi-girder bridge. 6.5.10 Strength I Shear Tolerance to LD Movements Occurring at the Pier The Strength I Shear limit state never controlled the tolerable LD support movement at the pier (see Figure 6-48). For this reason, rigorous analysis was not conducted for the Strength I Shear limit state. Figure 6-67 gives the plot of span length versus tolerable LD support movement based on the Strength I Shear limit state.

NCHRP Project 12-103 192 Figure 6-68 – Strength I Shear tolerance to a LD support movement occurring at the pier of a three-span continuous PS concrete multi-girder bridge. 6.5.11 Service Tolerance to TD Movements Occurring at the Pier The Service III limit state controlled the tolerable TD support movement at the pier for 91% of the population (see Figure 6-49). For TD movements controlled by the Service III limit state, the controlling location included both interior and exterior girders, but it always occurred in the positive moment region. Figure 6-69 gives the plot of span length versus tolerable TD support movement based on the Service III limit state. The plot indicates that three-span continuous PS concrete bridges exhibit very little tolerance to TD support movements at the pier. The current AASHTO LRFD criterion (0.004L) is plotted as a point of comparison. Due to the nature of the TD movements occurring at the pier, and the lack of available capacity for the Service III limit state, parameters that influence superstructure tolerance cannot be identified.

NCHRP Project 12-103 193 Figure 6-69 – Service tolerance to a TD support movement occurring at the pier of a three-span continuous PS concrete multi- girder bridge. 6.5.12 Strength I Flexure Tolerance to TD Movements Occurring at the Pier The Strength I Flexure limit state never controlled the TD support movement at the pier (see Figure 6-49). For this reason, rigorous analysis was not conducted for the Strength I Flexure limit state. Figure 6-70 gives the plot of span length versus tolerable TD support movement based on the Strength I Flexure limit state.

NCHRP Project 12-103 194 Figure 6-70 – Strength I Flexure tolerance to a TD support movement occurring at the pier of a three-span continuous PS concrete multi-girder bridge. 6.5.13 Strength I Shear Tolerance to TD Movements Occurring at the Pier The Strength I Shear limit state controlled the tolerable TD support movement at the pier for 9% of the population (see Figure 6-49). Further investigation found that the average level of tolerable support movement for these observations was approximately 1.5 inches. For this reason, rigorous analysis was not conducted, since it was, on average, much larger than the tolerable TD support movement associated with the Service III limit state. Figure 6-71 gives the plot of span length versus tolerable TD support movement based on the Strength I Shear limit state.

NCHRP Project 12-103 195 Figure 6-71 – Strength I Shear tolerance to a TD support movement occurring at the pier of a three-span continuous PS concrete multi-girder bridge. 6.6 Summary of Results As discussed in Section 5 and throughout this section, the parameters that influence superstructure tolerance to support movements will vary depending on the type and location of support movement. In the study of PS concrete multi-girder bridges, span length, girder spacing, and skew were identified as the most influential parameters. The levels of tolerable support movement observed for the various different bridge configurations, under each of the limit states, were compared to current AASHTO guidance on what constitutes as a “tolerable” support movement. The current guidance listed in the AASHTO LRFD specifications (0.008L for simply supported bridges, 0.004L for continuous) was only intended for LD movements however it is used for comparison purposes when examining TD movements. The current criterion was found to be conservative for many bridges, however, there were a considerable number of bridges where the current guidance was found to be unconservative. Note: guidance was deemed conservative if the observed level of tolerable support movement was found to be greater than what is suggested by AASHTO LRFD. In contrast, the guidance was deemed unconservative if the observed level of tolerable support movement was found to be less than what is suggested by AASHTO LRFD. Table 6-3 compares the results of this study to the current AASHTO guidance for all bridges. PS concrete multi-girder bridges exhibit very low tolerance to LD and TD support movements due to the limits of the

NCHRP Project 12-103 196 Strength I Shear and Service III limit states. This is evident in Table 6-3 that shows that current AASHTO guidance is unconservative for a considerably large number of bridges. In fact, Table 6-3 - Summary of results for prestressed concrete multi-girder bridges. Continuity Type of Support Movement Limit State Comparison with Current AASHTO LRFD Guidance (% Failing) Comments Simple- Span LD Support Movement at Abutment Strength I Flexure 0% - Strength I Shear 0% - Service I & III 0% - TD Support Movement at Abutment* Strength I Flexure 6% - Strength I Shear 68% Due to inability of SLG model to properly account for dead load distribution of skewed bridges Service I & III 68% Due to the lack of additional capacity for the Service III limit state Two-Span Continuous LD Support Movement at Abutment Strength I Flexure 3% - Strength I Shear 0% - Service I & III 0% - TD Support Movement at Abutment* Strength I Flexure 1% - Strength I Shear 18% Due to inability of SLG model to properly account for dead load distribution

NCHRP Project 12-103 197 of skewed bridges Service I & III 10% - LD Support Movement at Pier Strength I Flexure 31% Due to increase in positive moment in positive moment region Strength I Shear 10% - Service I & III 100% Due to the lack of additional capacity for the Service III limit state TD Support Movement at Pier* Strength I Flexure 28% Due to increase in positive moment in positive moment region Strength I Shear 56% Due to inability of SLG model to properly account for dead load distribution of skewed bridges Service I & III 100% Due to the lack of additional capacity for the Service III limit state Three-Span Continuous LD Support Movement at Abutment Strength I Flexure 2.5% - Strength I Shear 0% - Service I & III 0% - TD Support Movement at Abutment* Strength I Flexure 1% - Strength I Shear 18% Due to inability of SLG model to properly account for dead load distribution of skewed bridges

NCHRP Project 12-103 198 Service I & III 10% - LD Support Movement at Pier Strength I Flexure 31% Due to increase in positive moment in positive moment region Strength I Shear 10% - Service I & III 100% Due to the lack of additional capacity for the Service III limit state TD Support Movement at Pier* Strength I Flexure 28% Due to increase in positive moment in positive moment region Strength I Shear 56% Due to inability of SLG model to properly account for dead load distribution of skewed bridges Service I & III 100% Due to the lack of additional capacity for the Service III limit state ** Although current AASHTO LRFD guidance is not intended for TD movements, these movements are still considered for comparison purposes. Due to the unconservative nature of the current AASHTO guidance the Research Team proposes three potential options for predicting tolerable support movements: 1. Specify constant level of tolerable support movement to account for low tolerance under the Service III limit state 2. Retain the current model and specify range of applicability (i.e. for specific movements or limit states) 3. Develop a new model for predicting tolerable support movements and specify range of applicability (i.e. for specific movements or limit states) After discussion with the project panel, the Research Team decided to develop two expressions for estimating tolerable support movement: (1) an expression for estimating tolerance under the Strength limit state, and (2) a separate expression for estimating tolerance under the Service III limit state.

NCHRP Project 12-103 199 Service III will control tolerance for most prestressed concrete bridges, however a designer may wish to allow a service limit state to be exceeded. For this reason, a separate expression was developed. In developing the expression to estimate maximum tolerable support movement, a traditional line of best fit through the mean of the data was not desirable as this line would overestimate tolerable support movement for half of the dataset. Instead, what is desired is the lower bound envelope of the dataset. In mathematics, this is called a convex hull or convex envelope. To avoid additional conservatism in the already conservative estimates of maximum tolerable support movement, a curve was fit through the 5th percentile of the data. That is, this curve was fit as an envelope to 95% of the data. The curve and its expression are provided in Figure 6-72 below for Strength I limits. This expression is valid for simple span and multiple-span continuous steel multi-girder bridges under the Strength I limits of flexure and shear, for the ranges of applicability noted in Table 6-4. Figure 6-73 gives the expression developed for estimating tolerance under the Service III limit state. This expression is valid for simple span and multiple-span continuous steel multi-girder bridges under the Service III limits of tension in the bottom of the prestressed concrete girder, for the ranges of applicability noted in Table 6-4. Table 6-4 - Expressions for estimating maximum tolerable support movements of prestressed concrete multi-girder bridges. Type of Superstructure Applicable Cross-Section from Table 4.6.2.2.1-1 Tolerance Estimate Range of Applicability Concrete Deck, Reinforced Concrete Slab on Prestressed Concrete Beams k (also d through j, however this expression may provide more conservative estimates for these bridge types as these types are typically constructed outside the range of applicability, or will have a lower cross-sectional stiffness than the bridges studied in the research) Service III 40ft ≤ L ≤ 160ft 5ft ≤ S ≤ 12ft 0 ≤ Skew ≤ 45° 36ft ≤ Width ≤ 72ft 20 ≤ L/d ≤ 30 Strength I 0.13 ܮܵ 0.0005 ܮ

NCHRP Project 12-103 200 Figure 6-72 – Scatter plot of tolerance for the Service III limit state with the expression developed for estimating maximum tolerable support movement.

NCHRP Project 12-103 201 Figure 6-73 – Scatter plot of tolerance for Strength I limit states with the expression developed for estimating maximum tolerable support movement.