Guidelines for Shielding Bridge Piers (2018)

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68 Verification and Validation This chapter presents four examples that illustrate the use of the proposed LRFD bridge design procedures and RDG procedures for shielding bridge piers. Please reference Appendix A: Proposed LRFD Bridge Design Pier Protection Specifications and Appendix B: Proposed RDG Occupant Protection Guidelines for the proposed procedures. The examples in the following sections illustrate both procedures. The processes described in the previous chapters of this report for the LRFD Bridge Design Specifications and the RDG were based on extracting portions of the RSAPv3 model and developing simplified statistical models to estimate the vari- ous conditional probabilities needed. In order to ensure that the procedure correctly replicates RSAPv3, the example prob- lems were analyzed directly using RSAPv3, and the results have been compared to the LRFD and RDG procedures in the following sections. 5.1 Example #1: Two-Lane Undivided Rural Collector with Three Pier Columns 5.1.1 Introduction The layout for Example #1 is shown in Figure 36, and the user-supplied input information is shown in Table 40. Example #1 represents a three-column pier system on the right side of the primary direction of an undivided two- lane rural collector with 10,000 vehicles/day, 5% trucks, and a PSL of 45 mph. The three pier columns are parallel to the roadway. The user wishes to evaluate the need for pier protection in order to protect the bridge from collapse. Based on the designer’s analysis, the bridge structure is not continuous, and the pier system is not redundant, so the risk assessment model will be used to assess the risk of bridge failure due to a pier collision. All three columns are 2 ft in diameter and spaced 10 ft on center. The designer has calculated that the lateral capacity is 250 kips for each column, well below the recommended lateral impact capacity of 600 kips. 5.1.2 Pier Protection Procedure 5.1.2.1 Find Site-Specific Adjustment Factor: Ni Direction #1 is arbitrarily defined as the northbound direction, and Direction #2 is the southbound direction. The pier is at risk from a collision from the primary right lane (Direction #1) but is also at risk of a collision from a vehicle crossing the centerline in the opposing direction (Direction #2), so both directions must be considered. Only the leading column in each direction needs to be evaluated since columns further downstream are inaccessible to heavy vehicles (i.e., these columns are essentially shielded by the leading column). Referring to Table 15 and the user-provided information in Table 40, the site-specific adjustments can be calculated; these are shown in Table 41. Since the highway characteristics are the same in each direction, the adjustment factors for each direction are also the same. In this example, the access density and PSL cause an increase in the total encroachment adjustment factor to 3.12. 5.1.2.2 Heavy-Vehicle Base Encroachment Frequency: HVEi Next the user must find the heavy-vehicle base encroach- ment frequency (HVEi) for each direction of travel. HVEi is an estimate of the annual number of heavy vehicles that leave the lane in the specified direction of travel and must be calculated for each direction of interest. In Example #1, the user goes to Table 13 (i.e., Appendix A, Table C3.6.5.1-2) with the highway type (undivided), the percentage of trucks (5%), and the two-way traffic volume (AADT = 10,000) and looks up the tabulated value. The expected average annual frequency of heavy-vehicle encroachments in each direction is 0.0019, as confirmed in Table 42. C H A P T E R 5

69 from the edge of travel as noted by the solid-white edge line (SWEL) of the primary lane. Column #3 is exposed to departures from the opposing lanes (i.e., Direction #2) since it is at the leading edge from that direction. The offset for Direction #2 is the 10-ft offset from the SWEL to the face of Column #3 plus the 12-ft lane width in Direction #1, or 22 ft. Recall from Table 40 that all three columns are 2 ft in diameter. Table 19 indicates a probability of striking an unshielded pier com ponent given an encroachment of 0.1432 for Direction #1 with a 10-ft offset. There is no entry for 22 ft, but interpolating between the value for 20 ft and 25 ft yields 0.0939 (see Table 43). However, interpolation is not necessary since taking the closest offset value would provide a slightly conservative probability. 5.1.2.4 Probability of the Worst-Case Collision Force Exceeding the Critical Pier Component Capacity Given a Collision: P(QCT > RCPC |C) The next step is to determine the probability that if a collision does occur between the critical pier component and a heavy vehicle, the resulting impact force will exceed the lateral resistance of the pier component. The roadway in this example is an undivided rural collector, which implies a certain distribution of heavy-vehicle mix as described Notes: Col = column, SWEL = solid-white edge line, DYCL = double yellow center line. Figure 36. Example #1 site layout. Bridge Characteristics Value Nominal resistance of critical pier component: RCPC (kip) 250 Critical pier component size (ft) 2 Number of columns in pier system 3 Pier redundancy? No Superstructure continuity? No Bridge type Typical Site and Traffic Characteristics Direction #1 Direction #2 Highway type Undivided Undivided Functional classification Rural Collector Rural Collector Two-way AADT (veh/day) 10,000 10,000 PT (%) 5 5 Offset to critical pier component: L3 (ft) 10 22 Major accesses (points) 2 2 Horizontal curve away from the pier? NA NA Horizontal curve radius Tangent Tangent Lanes in one direction 1 1 Lane width (ft) 12 12 PSL (mph) 45 45 Grade Flat Flat Table 40. User-input values for Example #1. Adjustment Factor Direction #1 Direction #2 Major accesses (fACC) 2.20 2.20 Lane width (fLW) 1.00 1.00 Horizontal curve radius (fHC) 1.00 1.00 Lanes in one direction (fLN) 1.00 1.00 Posted speed limit (fPSL) 1.42 1.42 Grade (fG) 1.00 1.00 Site-specific adjustment factor (Ni) 3.12 3.12 †Values are found by taking the user-supplied input data in Table 40 and calculating the appropriate adjustments from Table 15. Table 41. Site-specific adjustment factors for Example #1.† HVE1 HVE2 0.0019 0.0019 Table 42. Heavy-vehicle base encroachment frequency for Example #1. 5.1.2.3 Probability of a Collision with an Unshielded Pier Component Given a Heavy-Vehicle Encroachment: P(C|HVEi) Not all vehicles that leave the roadway will strike the pier system. The probability that an encroaching heavy vehi- cle will strike the pier system is a function of the offset of the pier component at the leading edge of each direction and the pier component’s size. Table 19 (i.e., Appendix A, Table C3.6.5.1-3) lists the probability of a collision with the pier by pier component offset and size. In this example, as shown in Figure 36, the face of Column #1 at the lead- ing edge of the pier system in the primary direction is 10 ft P(C|HVE1) P(C|HVE2) 0.1432 0.0939 Table 43. Probability of a collision with an unshielded pier component given a heavy-vehicle encroachment: P(C HVEi) for Example #1.

70 in Appendix F: Heavy-Vehicle Traffic Mix and Properties. Knowing that the functional classification is a rural collector, the user goes to the upper right section of Table 7 and selects the value corresponding to a 45-mph PSL and a critical pier component lateral resistance of 250 kips to find the value of 0.3710 (see Table 44). This value means that there is a 37% chance of the pier component failing if it is struck by a heavy vehicle on this type of roadway. Since all three piers are the same size, all three have the same probability of failure given a collision. 5.1.2.5 Annual Frequency of Bridge Collapse: AFBC Now the user is ready to calculate the expected annual frequency of bridge collapse, as follows, from the values previously determined in Table 41 (Ni), Table 42 (HVEi), Table 43 [P(C|HVEi)], and Table 44 [P(QCT > RCPC|C)]: i i i i i i i i i i i i i i AF HVE HVE AF HVE HVE HVE HVE AF 3.12 0.0019 0.1432 0.3710 3.12 0.0019 0.0939 0.3710 AF 0.0003 0.0002 AF 0.0005 BC 1 CT CPC BC 1 1 1 CT CPC 2 2 2 CT CPC BC BC BC N P C P Q R C N P C P Q R C N P C P Q R C i i m i i∑ [ ] [ ] [ ] [ ] [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) = > = > + > = + = + = = The annual expected frequency of collapse of the bridge is, therefore, 0.0005. Another way to view this is that if the agency owned 2,000 bridges that were identical to this one in terms of traffic, geometry, and structural characteristics, one of them would experience an impact that could cause failure each year. While this particular bridge-pier system has a relatively low lateral impact capacity (i.e., 250 kips for a 2-ft-diameter column), and the probability of failure given an impact is relatively high (0.3710), the percentage of trucks is small (5%) and the speed limit is modest (45 mph), resulting in a relatively small chance of a heavy-vehicle collision. This illustrates that for some bridges that are not designed for the 600-kip lateral impact load, the traffic characteristics at the site may make the chance of a heavy- vehicle collision small and, therefore, it is unnecessary to shield with a barrier. This particular bridge was defined as a “typical” bridge in Table 40 so, according to Article 3.6.5.1, it does not require shielding and need not be designed for impact loading because the probability of a failure-producing impact is below the 0.0010 threshold (i.e., 0.0005 < 0.0010). If this bridge were classified as a critical bridge, however, the pier system would require shielding since the expected number of potentially failure-producing impacts would be greater than the critical bridge threshold of 0.0001 (i.e., 0.0005 > 0.0001). 5.1.3 Occupant Protection Procedure The previous sections showed that the Example #1 pier system did not require shielding with a barrier because the probability of a truck collision leading to bridge collapse was sufficiently small. Even though the pier system does not need protection to guard against bridge collapse, the site condi- tions still must be examined to determine if the pier system needs to be shielded to minimize the chance of passenger vehicles striking the pier components and vehicle occupants becoming involved in a severe injury or fatal crash. Evaluat- ing the site for passenger-vehicle occupant protection is the objective of the next several sections. 5.1.3.1 Find Site-Specific Adjustment Factor: Ni The first step in both the LRFD and RDG procedures is identical, so the values in Table 41 for the site-specific adjust- ment factors and the procedures for determining them for the LRFD portion of the procedure are the same for this RDG portion of the procedure. 5.1.3.2 Passenger Vehicle Base Encroachment Frequency: PVEi Next, the user must find the passenger-vehicle base encroachment frequency (PVEi) for each direction of travel. PVEi is an estimate of the annual number of passenger vehi- cles that will leave the lane in the specified direction of travel and must be determined for each direction of interest. In Example #1, the user goes to Table 24 with the highway type, PT, and the two-way traffic volume and looks up the tabulated value. For an undivided highway with 10,000 vehicles/day and 5% trucks, the expected average annual frequency of passenger vehicle encroachments in each direction is 0.0358, as shown for reference in Table 45. P(QCT > RCPC|C)1 P(QCT > RCPC|C)2 0.3710 0.3710 Table 44. Probability of the worst-case collision force exceeding the critical pier component capacity given a collision: P (QCT > RCPCC ) for Example #1.

71 5.1.3.3 Probability of a Collision with an Unshielded Pier Component Given a Passenger Vehicle Encroachment: P(C|PVEi) The probability that an encroaching passenger vehicle will strike the pier system is a function of the offset of the pier component at the leading edge of each direction and the size of the bridge pier. Table 25 shows the probability of a passenger vehicle collision with the pier column by offset and size. In this example, as shown in Figure 36, the face of Column #1 at the leading edge of the pier system in the primary direction is 10 ft from the SWEL of the lane. Column #3 is the leading edge from the opposing lanes (i.e., Direction #2). The offset for Direction #2 is the 10-ft offset from the SWEL to the face of Column #3 plus the 12-ft lane width of the lane, or 22 ft. Recall from Table 40 that all three columns were 2 ft in diameter. Table 25 indicates that the probability of a pas- senger vehicle striking an unshielded pier component given an encroachment is 0.1004 for Direction #1 with a 10-ft offset and 0.0722 for Direction #2 with a 22-ft offset. 5.1.3.4 Probability of a Severe or Fatal Injury Given a Crash with an Unshielded Pier Component Occurs: P(KACUSP|C) The probability that a crash with an unshielded pier component will result in a severe or fatal injury is found by looking up the appropriate value based on the PSL in Table 26. For this 45-mph roadway, 2.18% of crashes with unshielded bridge-pier components are expected to result in severe or fatal injuries (see Table 47). As shown in Table 26, the percentage of severe and fatal crashes is about three times higher on a 65-mph roadway. The procedure, therefore, accounts for the lower risk of fatal or severe injury on lower- speed roadways. 5.1.3.5 Annual Frequency of Severe Injury or Fatal Crashes with an Unshielded Bridge Pier: AFKA CUSP Now the user is ready to calculate the expected annual frequency of severe and fatal passenger-vehicle bridge-pier crashes, as follows, from the values determined in Table 41 (Ni), Table 45 (PVEi), Table 46 [P(C|PVEi)], and Table 47 [P(KACUSP|C)]: i i i i i i i i i i i i ∑ [ ] [ ] ( ) ( ) ( )= +    = +        + +        = + = = AF 2 3 PVE PVE KA AF 3 2 3 3.12 0.0358 0.1004 0.0218 3 2 3 3.12 0.0358 0.0722 0.0218 AF 0.0004 0.0003 AF 0.0007 KA CUSP 1 CUSP BC KA CUSP KA CUSP n N P C P C i m i i i The annual expected frequency of severe or fatal injury crashes involving passenger vehicles and this pier system with these traffic and site characteristics is 0.0007. Another way to view this is that if traffic conditions remained the same forever, one severe injury or fatal crash could be expected every 1,429 years. Since the goal is to limit severe injury and fatal crashes to less than 0.0001 per pier system per year, this site requires shielding for occupant protection even though shielding is not required to protect the bridge from collapse. 5.1.4 Shielding Barrier Layout Since shielding is only required for vehicle occupant protection, a MASH TL-3 crash-tested strong-post w-beam guardrail can be used at the site. Vehicles can approach from either direction on this undivided roadway, so the guardrail will extend in Directions #1 and #2. For purposes of this example, it is assumed that the owner agency prefers a tangent rather than a flared installation. RDG Table 5-7 recommends a shy-line offset (LS) of 6 ft for a roadway with a 45-mph PSL. The offset to the nearest PVE1 PVE2 0.0358 0.0358 Table 45. Passenger vehicle base encroachment frequency for Example #1. P(C|PVE1) P(C|PVE2) 0.1004 0.0722 Table 46. Probability of a collision with an unshielded pier component given a passenger vehicle encroachment: P(C PVEi) for Example #1. P(KACUSP|C)1 P(KACUSP|C)2 0.0218 0.0218 Table 47. Probability of a severe or fatal injury given a crash with an unshielded pier component: P(KACUSPC ) for Example #1.

72 pier component is 10 ft from the edge of the lane, and a w-beam guardrail is a little less than 2 ft wide depending on the particular design. RDG Table 5-6 shows that in both finite element simulations and crash tests, MASH TL-3 strong- post w-beam guardrails [i.e., the Midwest Guardrail System (MGS) single w-beam with 6.25-ft post spacing] generally deflect about 3.5 ft, so there is not adequate deflection dis- tance behind the guardrail if it is placed at the edge of the 6-ft shy line (10 – 2 – 3.5 = 4.5 < 6). While placing the w-beam inside the shy line is acceptable, a better alternative would be to use the MGS with half-post spacings (i.e., 3.125 ft), which would have a deflection of less than 2 ft. The barrier should, therefore, be placed at the edge of the shy line, 6 ft from the edge of the lane (i.e., Ls = L2 = 6 ft) such that there is at least 3 ft of deflection space. RDG Table 5-10(b) recommends run-out lengths (LR) for roadways with traffic volumes of between 5,000 and 10,000 vehicles/day of 130 ft for 40 mph and 190 ft for 50 mph. Interpolating to the site condition of a 45-mph PSL results in a 160-ft run-out length (see Table 48). The usual RDG calculations needed to find the length of need (X) of the guardrail are summarized for both directions in Table 49, recognizing that the lateral extent of the area of concern (LA) is simply the lateral offset to the face of the pier column (Pi = 10 ft) plus the diameter of the pier column (Di = 2 ft). LA is the distance from the edge of lane to the back face of the pier column, so it is 10 ft plus the 2-ft diameter of the column (12 ft) in Direction #1 and the 12-ft lane width plus the 10-ft lateral offset to the face of the pier plus the 2-ft diameter of the column (24 ft) for Direction #2. RDG Equation 5-2 is used to determine the length of need for a tangent guardrail, as shown in Table 49. The values in Table 49 show the necessary length of need to shield the pier columns from passenger vehicles according to the RDG. These values are added to the length in front of the pier components themselves, which are spaced 10 ft on center and are 2 ft in diameter (20 ft + 2 ft = 22 ft). The total length of the TL-3 w-beam is therefore 142 ft (80 ft + 22 ft + 40 ft = 142 ft). These distances are measured to the length of need of the guardrail terminal, which is generally at post 3, so the end of the terminal would be another 12.5 ft up- and downstream of this w-beam guardrail. The layout of the guardrail needed to shield vehicle occupants from the pier is shown in Figure 37. 5.1.5 RSAPv3 Comparison The procedures described in the previous sections were developed by deconstructing the encroachment probabil- ity model programed in RSAPv3 into its constituent parts and developing tables and equations for the particular question of pier protection. Since the procedures are based on an encroachment probability approach, the example procedure should result in similar answers obtained using RSAPv3. Table 50 shows that where values from these pro- posed procedures and RSAPv3 can be checked, there is very good agreement, indicating that the procedures are verified by RSAPv3. RSAPv3 predicts the number of crashes with each bridge column from all directions, as shown in Table 51. The pier protection procedure only estimates the number of crashes with the pier components at the leading edge of each direction of concern, so impact with the interior columns in this case is neglected. Table 51 shows that each column downstream of the leading column in each direction experiences about one-quarter of the crashes of the leading-edge column. These interior crashes are much less likely to cause pier component failure since the outer columns shield the inner columns. Significantly, of the 18 heavy-vehicle bridge-pier collisions in Table 3 where a pier component either completely failed or was extensively damaged, the impact involved the leading column of the pier system; none of the cases involving failure appeared to have involved an initial collision with an interior pier column. RDG Table Parameter Direction #1 #2 5-7 LS Shy-line offset (ft) 6 6 5-9 a/b Flare rate – – 5-10(b) LR Run-out length (ft) 160 160 L1 Tangent length (ft) – – L2 Barrier offset (ft) 6 6 LA Lateral extent of area of concern (ft) 12 24 Table 48. Barrier layout parameters from the RDG for Example #1. Direction #1 Direction #2 Table 49. Required length of need (X) for tangent guardrail shielding pier for Example #1.

Figure 37. Occupant protection shielding barrier for Example #1.

74 RSAPv3 indicates a total of 0.0020 heavy-vehicle collisions per year with this three-column pier, which is 0.0007 heavy- vehicle collisions more than predicted by the LRFD procedure for direct impacts with Column #1 from Direction #1 or Col- umn #3 from Direction #2. As discussed in the previous para- graph, however, column failure is almost exclusively associated with impact with the leading column. If a more conservative approach were desired, the total num- ber of heavy-vehicle crashes could be estimated using the total number of columns (n), recognizing that interior columns experience one-quarter the impacts of leading-edge columns: AF 4 3 0.0006 0.0008 4 3 3 0.0021 HV CUSP n( ) ( )    + = +    + = This approach, however, is overly conservative and is not suggested. The situation for passenger-vehicle occupant protection is somewhat different since passenger vehicles have more maneu- verability than heavy vehicles. An interior column impact can result in severe or fatal injury crashes. As shown in Table 51, interior columns experience about one-third the number of the passenger vehicle crashes as the leading-edge column. The estimate of total passenger-vehicle crashes can be determined from the leading-edge collisions as follows, where n is the number of columns in the pier system: 2 3 AF 3 2 3 0.0112 0.0081 0.0322PV CUSP n ( ) +    = +    + = This value is 6% less than the number predicted by RSAPv3. 5.2 Example #2: Four-Lane Divided Rural Primary with Three Pier Columns on a Skew in the Median 5.2.1 Introduction The layout for Example #2 is shown in Figure 38, and the user-supplied input information is shown in Table 52. Example #2 represents a three-column pier system located in the center of a 42-ft-wide median of a rural divided Inter- state with 50,000 vehicles/day, 25% trucks, and a PSL of 65 mph. The three pier columns are placed in the median on a skew such that the leading pier in each direction is 10 ft from the edge of travel. The user wishes to evaluate the need for pier protection to protect the bridge from collapse. The bridge superstructure is not continuous, and the pier system is not redundant, based on the designer’s calculations, so the risk assessment model is used to determine if the pier system should be shielded to minimize the risk of bridge failure due to a pier collision. All three columns are 3 ft in diameter, and the designer has calculated that the lateral capacity of each of the columns is 900 kips. 5.2.2 Pier Protection Procedure 5.2.2.1 Find Site-Specific Adjustment Factor: Ni Direction #1 is arbitrarily defined as the northbound direction, and Direction #2 is the southbound direction. The pier is at risk of a collision emanating from a northbound left encroachment into the median from the primary lanes (i.e., Direction #1) but is also at risk of a collision from a left Parameter LRFD Procedure Direction RSAPv3 Direction #1 #2 #1 #2 Site-specific adjustment factor (Ni) 3.1240 3.1240 3.1240 3.1240 Base vehicle encroachment (ENCR) 2.6514 2.6514 2.6514 2.6514 Heavy-vehicle encroachment adjustment factor (fHV ENCR) 1.00 1.00 1.00 1.00 Annual unshielded pier collisions (AFHV CUSP) 0.0008 0.0006 0.0008 0.0005 Parameter RDG Procedure Direction RSAPv3 Direction #1 #2 #1 #2 Site-specific adjustment factor (Ni) 3.1240 3.1240 3.1240 3.1240 Base vehicle encroachment (ENCR) 2.6514 2.6514 2.6514 2.6514 Annual unshielded pier collisions with the lead column [Ni PVEi P(C|PVEi)] 0.0112 0.0081 0.0111 0.0078 Table 50. Comparison of RSAPv3 results and procedure results for Example #1. Column Heavy Vehicles Passenger Vehicles Direction Direction #1 #2 #1 #2 #3 0.0002 0.0005 0.0043 0.0078 #2 0.0002 0.0002 0.0041 0.0035 #1 0.0008 0.0001 0.0111 0.0036 Total 0.0012 0.0008 0.0195 0.0149 0.0020 0.0344 Table 51. Annual pier component collisions from RSAPv3 for Example #1.

75 Notes: COL = column, SYEL = solid-yellow edge line, BWLL = broken white lane line. Figure 38. Example #2 site layout. Bridge Characteristics Value Nominal resistance of critical pier component: RCPC (kip) 900 Critical pier component size (ft) 3 Number of columns in pier system 3 Pier redundancy? No Superstructure continuity? No Bridge type Typical Site and Traffic Characteristics Direction #1 Direction #2 Highway type Divided Divided Functional classification Rural Primary Rural Primary Two-way AADT (veh/day) 50,000 50,000 PT 25 25 Major accesses (points) 0 0 Horizontal curve away from the pier? NA NA Horizontal curve radius Tangent Tangent Lanes in one direction 2 2 Lane width (ft) 12 12 PSL (mph) 65 65 Grade (%) +4 -4 Table 52. User-input values for Example #2. Adjustment Factor Direction #1 Direction #2 Major accesses (fACC) 1.00 1.00 Lane width (fLW) 1.00 1.00 Horizontal curve radius (fHC) 1.00 1.00 Lanes in one direction (fLN) 1.00 1.00 PSL (fPSL) 1.00 1.00 Grade (fG) 1.00 1.50 Site-specific adjustment factor (Ni) 1.00 1.50 †Values are found by taking the user-supplied input data in Table 52 and calculating the appropriate adjustments from Table 15. Table 53. Site-specific adjustment factors for Example #2.† encroachment into the median from vehicles traveling in the southbound direction (i.e., Direction #2). Only the leading column in each direction needs to be evaluated since col- umns further downstream are shielded from heavy-vehicle impacts by the column at the leading edge. The user-provided information in Table 52 can be used in conjunction with Table 15 (i.e., Appendix A, Table C3.6.5.1-1) to calculate the site-specific adjustment, as illustrated in Table 53. All the adjustments are the same except for grade. In the primary direction, the grade is in the uphill direction, and in the oppos- ing direction, the grade is in the downhill direction. In this example, the site-specific adjustment factor in Direction #1 is 1.0 and 1.5 in Direction #2. 5.2.2.2 Heavy-Vehicle Base Encroachment Frequency: HVEi Next the user must estimate the annual number of heavy vehicles that will leave the lane in each direction of travel (HVEi). In Example #2, the user goes to Table 13 (i.e., Appen- dix A, Table C3.6.5.1-2) with the highway type (i.e., divided), the percentage of trucks (25%), and the two-way traffic vol- ume (AADT = 50,000) to find the expected average annual frequency of heavy-vehicle encroachments, which is 0.0065 in each direction, as shown in Table 54. 5.2.2.3 Probability of a Collision with an Unshielded Pier Component Given a Heavy-Vehicle Encroachment: P(C|HVEi) Larger pier components located closer to the traveled way are more likely to be struck by an errant heavy vehicle, so the probability that an encroaching heavy vehicle will strike the pier system given an encroachment is a function of

76 the offset of the pier component at the leading edge of each direction and the size of the bridge pier component, as listed in Table 19 (i.e., Appendix A, Table C3.6.5.1-3). In this example, as shown in Figure 38, the face of Column #1 at the leading edge of the pier system in the primary direction (i.e., Direction #1) is 10 ft from the solid-yellow edge line (SYEL). Column #3 is exposed to departures from the opposing lanes (i.e., Direction #2) since it is at the leading edge from that direction. The offset for Direction #2 is also 10 ft from the SYEL to the face of Column #3. Recall from Table 52 that all three columns are 3 ft in diameter, so the probability of striking an unshielded pier component given an encroachment based on Table 19 (i.e., Table C3.6.5.1-3) is 0.1521 for a 10-ft offset, as shown in Table 55. 5.2.2.4 Probability of the Worst-Case Collision Force Exceeding the Critical Pier Component Capacity Given a Collision: P(QCT > RCPC |C) The next step is to determine the probability that, if a collision does occur between the critical pier component and a heavy vehicle, the resulting impact force will exceed the lateral resistance of the pier component. The roadway in this example is a divided rural primary, which implies a certain distribution of heavy-vehicle mix, as described in Appendix F. Knowing that the functional classification is a rural primary, the user goes to the upper left section of Table 7 (i.e., Appendix A, Table C3.6.5.1-4) and selects the value corresponding to a 65-mph PSL and a critical pier component lateral resistance of 900 kips to find the value of 0.0594. This value means that there is a 5.94% chance of the pier component failing if it is struck by a heavy vehicle on this type of roadway (see Table 56). Notice that since the pier columns are relatively strong (900 kips), the probability of failure given an impact is relatively small (0.0594). 5.2.2.5 Annual Frequency of Bridge Collapse: AFBC Now the user is ready to calculate the expected annual frequency of bridge collapse, as follows, from the values previously determined in Table 53 (Ni), Table 54 (HVEi), Table 55 [P(C|HVEi)], and Table 56 [P(QCT > RCPC|C)]: i i i i i i i i i i i i i i AF HVE HVE AF HVE HVE HVE HVE AF 1.00 0.0065 0.1521 0.0594 1.5 0.0065 0.1521 0.0594 AF 0.0001 0.0001 AF 0.0002 BC 1 CT CPC BC 1 1 1 CT CPC 2 2 2 CT CPC BC BC BC N P C P Q R C N P C P Q R C N P C P Q R C i i m i i∑ [ ] [ ] [ ] [ ] [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) = > = > + > = + = + = = The annual expected frequency of collapse of the bridge is, therefore, 0.0002. Another way to view this is that if the agency owned 5,000 bridges that were identical to this one in terms of traffic, geometry, and structural characteristics, one of them would experience an impact that could cause failure each year. The columns in this particular bridge-pier system are well over the recommended lateral load capacity of 600 kips, so the probability of failure given an impact even on a high-speed, high-volume Interstate is relatively low (0.0594). This particular bridge was defined as a “typical” in Table 52, so according to Article 3.6.5.1, it does not require shielding to protect it from collapse because the probability of a failure-producing impact is well below the 0.001 threshold. 5.2.3 Occupant Protection Procedure The previous sections showed that the Example #2 pier system did not require shielding with a barrier because the probability of a failure-inducing truck collision was suf- ficiently small. Even though the pier system does not need protection to guard against bridge collapse, the site condi- tions still must be examined to determine if the pier system needs to be shielded to minimize the chance of passenger HVE1 HVE2 0.0065 0.0065 Table 54. Heavy-vehicle base encroachment frequency for Example #2. P(C|HVE1) P(C|HVE2) 0.1521 0.1521 Table 55. Probability of a collision with an unshielded pier component given a heavy-vehicle encroachment: P(C HVEi) for Example #2. P(QCT > RCPC|C)1 P(QCT > RCPC|C)2 0.0594 0.0594 Table 56. Probability of the worst-case collision force exceeding the critical pier component capacity given a collision: P(QCT > RCPCC ) for Example #2.

77 Table 57. Passenger vehicle base encroachment frequency for Example #2. PVE1 PVE2 0.0902 0.0902 P(KACUSP|C)1 P(KACUSP|C)2 0.0656 0.0656 Table 59. Probability of a severe or fatal injury given a crash with an unshielded pier component: P(KACUSPC ) for Example #2. vehicles striking the pier components and occupants becom- ing involved in a severe injury or fatal crash. Evaluating the site for passenger-vehicle occupant protection is the objective of the next several sections. 5.2.3.1 Find Site-Specific Adjustment Factor: Ni The first steps in both the LRFD and RDG procedures are identical, so the values in Table 53 representing the site characteristic adjustments and the procedures for determin- ing them for the LRFD portion of the procedure are the same for this RDG portion of the procedure. 5.2.3.2 Passenger Vehicle Base Encroachment Frequency: PVEi Next, the user must find the passenger-vehicle base encroachment frequency (PVEi) for each direction of travel. Table 24 shows that for a divided highway with 50,000 vehicles/day and 25% trucks, the expected average annual frequency of passenger vehicle encroachments in each direction is 0.0902, as shown for reference in Table 57. 5.2.3.3 Probability of a Collision with an Unshielded Pier Component Given a Passenger Vehicle Encroachment: P(C|PVEi) In this example, as shown in Figure 38, the face of Col- umn #1 at the leading edge of the pier system in the primary direction is 10 ft from the SYEL. Column #3 is exposed to departures from the opposing lanes (i.e., Direction #2) since it is at the leading edge from that direction. The offset for Direction #2 is also 10 ft from the SYEL to the face of Col- umn #3. As shown in Table 52, all three columns are 3 ft in diameter, so Table 25 indicates that the probability of a pas- senger vehicle striking an unshielded pier component given an encroachment is 0.1109 for Direction #1 and 0.1109 for Direc- tion #2. These values are tabulated for reference in Table 58. 5.2.3.4 Probability of a Severe or Fatal Injury Given a Crash with an Unshielded Pier Component Occurs: P(KACUSP|C) The probability that a crash with an unshielded pier com- ponent will result in a severe or fatal injury is found by looking up the appropriate values based on the PSL in Table 26. For this 65-mph roadway, 6.56% of the crashes with unshielded bridge piers are expected to result in severe or fatal injuries, as shown in Table 59. 5.2.3.5 Annual Frequency of Severe Injury or Fatal Crash with an Unshielded Bridge Pier: AFKA CUSP Now the user is ready to calculate the expected annual frequency of severe and fatal passenger-vehicle bridge-pier crashes, as follows, from the values previously determined in Table 53 (Ni), Table 57 (PVEi), Table 58 [P(C|PVEi)], and Table 59 [P(KACUSP|C)]: AF 2 3 PVE PVE KA AF 3 2 3 1.00 0.0902 0.1109 0.0656 3 2 3 1.50 0.0902 0.1109 0.0656 AF 0.0011 0.0016 AF 0.0027 KA CUSP 1 CUSP BC KA CUSP KA CUSP n N P C P C i m i i ii i i i i i i i i i i i ∑ [ ] [ ] ( ) ( ) ( ) ( ) ( ) = +    = +        + +        = + = = The annual expected frequency of severe or fatal injury crashes involving passenger vehicles and this pier system with these traffic and site characteristics is 0.0027. Another way to view this is that, if traffic conditions remained the same for- ever, one severe injury or fatal crash could be expected every 370 years. Since the goal is to limit annual severe injury and fatal crashes to less than 0.0001 per pier system per year, this site requires shielding with a MASH TL-3 w-beam guardrail P(C|PVE1) P(C|PVE2) 0.1109 0.1109 Table 58. Probability of a collision with an unshielded pier component given a passenger vehicle encroachment: P(C PVEi) for Example #2.

78 for occupant protection even though shielding is not required to protect the bridge from collapse. 5.2.4 Shielding Barrier Layout Shielding is only required for vehicle occupant protection in Example #2, so a MASH TL-3 strong-post w-beam guard- rail will be used. Vehicles can enter the median from either direction on this divided highway, so the guardrail will extend in Directions #1 and #2, as shown in Figure 39. For purposes of this example, it is assumed that the owner agency prefers to use a flared guardrail in this median situation. RDG Table 5-7 recommends a shy-line offset (LS) of 8.5 ft for a roadway with a 65-mph PSL. RDG Table 5-6 recommends about 3.5 ft of lateral space from the back of a MASH TL-3 w-beam barrier (i.e., MGS single w-beam at 6.25-ft post spacing) to the face of the hazard, and the barrier itself is about 2 ft wide, so the offset from the lane edge to the guardrail will be at least 10 – 2 – 3.5 = 4.5 ft, which is inside the shy-line distance. The user could use the MGS at half-post spacing, but it is also acceptable to place the barrier inside the shy line, especially in a median situation, as noted by the RDG. In this case, the designer can be satisfied with a 4.5-ft shoulder with the guardrail at the edge of shoulder since it is a median application, so the barrier offset used in L2 = 4.5 ft. The lateral extent of area of concern (LA) is the distance from SYEL to the back face of the farthest pier column from the road. The face of Column #1 is 10 ft from the edge of the left primary lane, and Column #3 is 42 – 10 = 32 ft from the left edge of the primary lane, so LA is 32 ft. Since the arrangement is symmetrical, LA is also 32 ft in Direction #2. RDG Table 5-9 indicates that the maximum flare rate for a rigid barrier inside the shy-line distance on a 65-mph roadway is 28:1. A tangent length in front of the columns of 24 ft is used. Interpolating from RDG Table 5-10(b) for a design speed of 65 mph results in a run-out length (LR) of 330 ft for a road- way with 50,000 vehicles/day. The values needed to use RDG Equation 5-1 to determine the length of need for a flared guardrail are shown in Table 60, and the calculation is shown in Table 61. The left column of Table 61 shows that a length of need of 246 ft is needed to shield the farthest column from the traveled lanes, so the 246 ft is measured from the face of Column #3 in Direction #1. Column #1 is closer to the traveled lanes, so the right column of Table 61 is used to make sure the length of need for Column #1 falls within that calculated for Column #3. Since Column #1 requires a length of need of 172 ft, and that is included within the length of need for Column #3, the arrangement shown in Figure 39 is sufficient. Direction #2 is a mirror image, so those values are not repeated. The values in Table 61 show the necessary length of need to shield the pier columns from passenger vehicles according to the RDG. Column # 3 in Direction #1 requires a longer length of MASH Tl-3 w-beam than Column #1 in Direc- tion #1; therefore, the 246-ft length governs. Both directions are symmetric; therefore, Direction #2 was not calculated. The MASH TL-3 w-beam guardrail needed to shield vehicle occupants from the pier system in Example #2 should extend 246 ft upstream of Column #3 in Direction #1 and 246 ft upstream of Column #1 in Direction #2. 5.2.5 RSAPv3 Comparison Table 62 compares the number of collisions from these suggested procedures to the values determined in an RSAPv3 simulation of the Example #2 conditions. As shown in Table 62, the suggested procedures are similar and slightly conservative to the values found from RSAPv3, indicating that the simpli- fied suggested procedures are verified by RSAPv3. As discussed for Example #1, the procedures are based on estimating the number of collisions with the leading column of the pier system. As shown in Table 63, the procedures accurately predict the number of crashes with the leading columns in Direction #1 and #2 for both heavy vehicles and passenger vehicles. The pier protection procedures are based on the heavy- vehicle crashes with the leading column, but the passenger- vehicle occupant protection procedures use an estimate of collisions with all the columns in the pier group as follows: 2 3 AF 3 2 3 0.0100 0.0150 0.0417PV CUSP n ( ) +    = +    + = In this example, the columns are at a 34-degree skew from the direction of the roadway. Table 63 shows that the passenger-vehicle occupant protection procedures under- predict the total number of passenger-vehicle pier column collisions by less than 2% even though the columns are arranged in a skew across the median. This illustrates that the interior columns need not be directly in a parallel line behind the lead columns for shielding to occur. 5.3 Example #3: Six-Lane Divided Urban Primary with Four Pier Columns Offset in the Median 5.3.1 Introduction The layout for Example #3 is shown in Figure 40, and the user-supplied input information is shown in Table 64. Exam- ple #3 represents a four-column pier system located offset toward the opposing side in a 47.5-ft-wide median of a six- lane divided urban Interstate with 80,000 vehicles/day, 20% trucks, and a PSL of 55 mph. The roadway curves to the right

Figure 39. Occupant protection shielding barrier for Example #2.

80 in the primary direction. The four pier columns are placed in the median tangent to the traveled way but 25 ft from the primary left edge and 20 ft from the opposing left edge. The bridge superstructure is not continuous, and the pier system is not redundant based on the designer’s calculations, so the risk assessment model is used to determine if the pier system RDG Table Parameter Direction #1 Direction #1 Column #3 Column #1 5-7 LS Shy-line offset (ft) 8.5 8.5 5-9 a/b Flare rate 28:1 28:1 5-10(b) LR Run-out length (ft) 330 330 L1 Tangent length (ft) 24 3 L2 Barrier offset (ft) 4.5 4.5 LA Lateral extent of area of concern (ft) 32 13 Table 60. Barrier layout parameters from the RDG for Example #2. = ( + − ) + Direction #1: Column #3 Direction #1: Column #1 = (32 + 1 28 24 − 4.5) 1 28 + 32 330 = 246 = (13 + 1 28 3 − 4.5) 1 28 + 13 330 = 172 Table 61. Required length of need for flared guardrail shielding pier for Example #2. Parameter LRFD Procedure Direction RSAPv3 Direction #1 #2 #1 #2 Site-specific adjustment factor (Ni) 1.00 1.50 1.00 1.50 Base vehicle encroachment (ENCR) 8.4673 8.4673 8.4673 8.4673 Heavy-vehicle encroachment adjustment factor (fHV ENCR) 0.2168 0.2168 0.2168 0.2168 Annual unshielded pier collisions (AFHV CUSP) 0.0010 0.0015 0.0009 0.0013 Parameter RDG Procedure Direction RSAPv3 Direction #1 #2 #1 #2 Site-specific adjustment factor (Ni) 1.00 1.50 1.00 1.50 Base vehicle encroachment (ENCR) 8.4673 8.4673 8.4673 8.4673 Annual unshielded pier collisions with the lead column (Ni PVEi P(C|PVEi)) 0.0100 0.0150 0.0095 0.0141 Table 62. Comparison of RSAPv3 results and procedure results for Example #2. Column Heavy Vehicles Passenger Vehicles Direction Direction #1 #2 #1 #2 #3 0.0002 0.0013 0.0031 0.0141 #2 0.0004 0.0006 0.0045 0.0066 #1 0.0009 0.0003 0.0095 0.0046 Total 0.0015 0.0022 0.0171 0.0253 0.0037 0.0424 Table 63. Annual pier component collisions from RSAPv3 for Example #2. should be shielded to minimize the risk of bridge failure due to a pier collision. All four columns are 2.5 ft in diameter, and the designer has calculated the lateral capacity of each of the columns is 500 kips. 5.3.2 Pier Protection Procedure 5.3.2.1 Find Site-Specific Adjustment Factor: Ni Direction #1 is arbitrarily defined as the northbound direction, and Direction #2 is the southbound direction. The pier system is at risk of a collision emanating from a left encroachment into the median from either the primary lanes (i.e., Direction #1) or the opposing lanes (i.e., Direction #2). Only the leading column in each direction needs to be eval- uated since columns further downstream are shielded by the column at the leading edge. The user-provided information

81 in Table 64 can be used in conjunction with Table 15 (i.e., Appendix A, Table C3.6.5.1-1) to calculate the site- specific adjustment, as illustrated in Table 65. In Direction #1, the horizontal curve is away from the pier columns, so the adjustment is 1.27. This indicates that vehi- cles will tend to continue straight on a curve and potentially exit toward the pier columns. In Direction #2, the horizontal curve is toward the columns, so the adjustment is 1.09, which is smaller because vehicles are less likely to exit the roadway. The number of lanes is accounted for in the “lanes in one direction” adjustment. For divided highways with two lanes in each direction, the adjustment is 1. For this highway, with three lanes in each direction, the adjustment for the number of lanes is 0.91. This highway has 11-ft-wide lanes rather than the more standard 12-ft-wide lanes, so the adjustment for lane width is 1.03, indicating a slight increase in the encroachment rate due to more constricted lanes. The resulting site-specific adjustments are 1.40 in Direc- tion #1 and 1.21 in Direction #2, indicating that there will be somewhat more departures from Direction #1, largely due to the horizontal curvature. 5.3.2.2 Heavy-Vehicle Base Encroachment Frequency: HVEi Next the user must estimate the annual number of heavy vehicles that will leave the lane in each direction of travel (HVEi). In Example #3, the user goes to Table 13 (i.e., Appen- dix A, Table C3.6.5.1-2) with the highway type (i.e., divided), the percentage of trucks (20%), and the two-way traffic volume (AADT = 80,000) to find that the expected average annual frequency of heavy-vehicle encroachments is 0.0103 in each direction, as shown in Table 66. 5.3.2.3 Probability of a Collision with an Unshielded Pier Component Given a Heavy-Vehicle Encroachment: P(C|HVEi) In this example, as shown in Figure 40, the face of Column #1 at the leading edge of the pier system in Direc- tion #1 is 25 ft from the SYEL. Column #4 is exposed to Bridge Characteristics Value Nominal resistance of critical pier component: RCPC (kip) 500 Critical pier component size (ft) 2.5 Number of columns in pier system 4 Pier redundancy? No Superstructure continuity? No Bridge type Typical Site and Traffic Characteristics Direction #1 Direction #2 Highway type Divided Divided Functional classification Urban Primary Urban Primary Two-way AADT (veh/day) 80,000 80,000 PT 20 20 Offset to critical pier component: L3 (ft) 25 20 Major accesses (points) 0 0 Horizontal curve away from the pier? Yes No Horizontal curve radius (ft) 2,000 2,000 Lanes in one direction 3 3 Lane width (ft) 11 11 PSL (mph) 55 55 Grade Flat Flat Table 64. User-input values for Example #3. Figure 40. Example #3 site layout. Adjustment Factor Direction #1 Direction #2 Major accesses (fACC) 1.00 1.00 Lane width (fLW) 1.03 1.03 Horizontal curve radius (fHC) 1.27 1.09 Lanes in one direction (fLN) 0.91 0.91 PSL (fPSL) 1.18 1.18 Grade (fG) 1.00 1.00 Site-specific adjustment factor (Ni) 1.40 1.21 †Values are found by taking the user-supplied input data in Table 64 and calculating the appropriate adjustments from Table 15. Table 65. Site-specific adjustment factors for Example #3.†

82 departures from Direction #2 since it is at the leading edge from that direction. The offset for Direction #2 is the 20-ft offset from the SYEL to the face of Column #4. All four columns are 2.5 ft in diameter, as shown in Table 64, so the probability of striking an unshielded pier component given an encroachment based on interpolation from Table 19 (i.e., Appendix A, Table C3.6.5.1-3) is 0.0870 for a 25-ft offset in Direction #1 and 0.1042 for a 20-ft offset in Direction #2, as shown in Table 67. 5.3.2.4 Probability of the Worst-Case Collision Force Exceeding the Critical Pier Component Capacity Given a Collision: P(QCT > RCPC |C) The roadway in this example is a divided urban Interstate, which implies a particular heavy-vehicle mix such as more single-unit trucks, as described in Appendix F. Knowing that the functional classification is an urban Interstate, the user goes to the lower left section of Table 7 (i.e., Appendix A, Table C3.6.5.1-4) and selects the value corresponding to a 55-mph PSL and a critical pier component lateral resistance of 500 kips to find the value of 0.6562, as listed in Table 68. There is a high chance (i.e., 65.62%) of one of these pier columns failing if it is struck by a heavy vehicle on this type of roadway with these traffic conditions. Since all the piers are the same size, all four have the same probability of failure given a collision. 5.3.2.5 Annual Frequency of Bridge Collapse: AFBC The user is now ready to calculate the expected annual frequency of bridge collapse, as follows, from the values previously determined in Table 65 (Ni), Table 66 (HVEi), Table 67 [P(C|HVEi)], and Table 68 [P(QCT > RCPC|C)]: i i i i i i i i i i i i i i ∑ [ ] [ ] [ ] [ ] [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) = > = > + > = + = + = = AF HVE HVE AF HVE HVE HVE HVE AF 1.40 0.0103 0.0870 0.6562 1.21 0.0103 0.1042 0.6562 AF 0.0008 0.0009 AF 0.0017 BC 1 CT CPC BC 1 1 1 CT CPC 2 2 2 CT CPC BC BC BC N P C P Q R C N P C P Q R C N P C P Q R C i i m i i The annual expected frequency of collapse of this bridge under these traffic conditions is 0.0017. If the agency owned 588 bridges that were identical to this one in terms of traffic, geometry, and structural characteristics, one of them could experience an impact that could cause failure each year. The columns in this particular bridge-pier system have a lateral resistance of 500 kips, which is under the recommended lateral load capacity of 600 kips, so the probability of fail- ure given an impact on this high-volume urban Interstate is relatively high (i.e., 0.6562). This particular bridge was defined as a “typical” bridge in Table 64 so, according to proposed Article 3.6.5.1, it requires shielding because the probability of a failure-producing impact is above the 0.001 threshold. The user could elect to shield the pier system in Direc- tion #2 and not in Direction #1 because the annual frequency of bridge collapse in that case would be just less than 0.001. If Direction #2 were shielded, then the annual frequency of bridge collapse would be entirely due to Direction #1 traffic, where AFBC 1 = 0.0008 < 0.0010. In this case, such a strategy is probably not wise since each direction is close to the criti- cal value by itself. This example does illustrate, however, that all directions need not be shielded as long as the annual fre- quency of bridge collapse for the entire pier system is less than the critical value. 5.3.3 Occupant Protection Procedure The previous section showed that the pier system in Example #3 requires shielding with a barrier because the HVE1 HVE2 0.0103 0.0103 Table 66. Heavy-vehicle base encroachment frequency for Example #3. P(C|HVE1) P(C|HVE2) 0.0870 0.1042 Table 67. Probability of a collision with an unshielded pier component given a heavy-vehicle encroachment: P(C HVEi) for Example #3. P(QCT > RCPC|C)1 P(QCT > RCPC|C)2 0.6562 0.6562 Table 68. Probability of the worst-case collision force exceeding the critical pier component capacity given a collision: P(QCT > RCPCC) for Example #3.

83 probability of a failure-inducing truck collision was suffi- ciently high. There is, therefore, no need to check the RDG occupant protection procedure because a MASH TL-5 con- crete barrier is already needed to shield the pier system from heavy-vehicle impacts. 5.3.4 Shielding Barrier Layout Since shielding is required for pier system protection from heavy-vehicle impacts, a MASH TL-5 rigid concrete barrier will be used in this example. Since vehicles can enter the median from either direction on this six-lane divided highway, the rigid concrete barrier will extend upstream of Column #1 in the primary direction and upstream of Col- umn #4 in the opposing direction, as shown in Figure 41. For the purpose of this example, it is assumed that the owner agency prefers to use flared barriers in this median situation. A longitudinal barrier layout procedure is provided in RDG Section 5.6.4. RDG Table 5-7 recommends a shy-line offset (LS) of 7.0 ft for a roadway with a 55-mph PSL. The barrier will be installed at the edge of the 6-ft median shoul- der, so there will be 19 ft from the traffic face of the barrier to the face of the pier columns from Direction #1 and 14 ft from Direction #2. This is much more than the 3.25 ft minimum clearance suggested by the proposed LRFD Bridge Specifica- tions procedure shown in Article 3.6.5.1, so the clearance is more than adequate. The suggested flare rate for a rigid barrier just inside the shy line on a 55-mph roadway is given in RDG Table 5-9 as 24:1. Interpolating RDG Table 5-10(b) for a design speed of 55 mph results in a run-out length (LR) of 265 for a roadway with more than 10,000 vehicles/day. The RDG barrier layout dimensions for this example are shown in Table 69. The lateral extent of area of concern (LA) is the distance from the SYEL to the back face of the pier column, so in Direction #1 it is the 25-ft offset from the primary left lane to the face of the pier plus the 2.5-ft diameter of the column, or 27.5 ft. In Direction #2, LA is the 20-ft offset from the left lane edge to the face of the pier plus the 2.5-ft column diameter, or 22.5 ft. RDG Equation 5-1 is used to determine the length of need for a flared guardrail, as shown in Table 70. The values in Table 70 show the necessary length of need to shield the pier columns from heavy vehicles according to Section 5.6.4 of the RDG. The rigid MASH TL-5 barrier needed to protect the pier columns from potentially failure- causing impacts should extend 187 ft upstream of Column #1 in Direction #1 and 176 ft upstream of Column #4 in Direc- tion #2, as shown in Figure 41. Both lengths are greater than the minimum 60 ft length of need. Additionally, the MASH TL-5 barrier must extend across the front of the columns, parallel to the road in both directions. The approach ends of these rigid MASH TL-5 barriers must be shielded with either an appropriate guardrail and guardrail terminal or a crash cushion. 5.3.5 RSAPv3 Comparison Table 71 shows the comparison values for the pier pro- tection and RSAPv3 results. The pier protection procedure results are similar though slightly conservative values com- pared to RSAPv3 for this four-column pier system. Even though the passenger-vehicle occupant protection procedure was not used in this example, these values were calculated to allow for a comparison with RSAPv3. Table 71 shows the comparison values between the occupant protec- tion procedures and RSAPv3. The values agree with the RDG procedure being somewhat conservative in comparison to the RSAPv3 estimates. While the passenger-vehicle occupant protection pro- cedures were not required in this example since a shielding barrier was already required for pier protection, it is none- theless interesting to determine if the number of predicted passenger-vehicle crashes with all the pier columns compares favorably with RSAPv3 (see Table 72). The passenger-vehicle protection procedures estimate a total of 0.0712 passenger vehicle crashes, whereas RSAPv3 predicts 0.0650, meaning the procedures under predict by about 6%. ( )+    = +    + =2 3 AF 4 2 3 0.0152 0.0152 0.0608PV CUSP n 5.4 Example #4: Six-Lane Rural Primary with Two Columns in a Gore of an Off-Ramp 5.4.1 Introduction Example #4 is a two-column pier system located in the gore of an off-ramp of a six-lane divided highway. The layout for Example #4 is shown in Figure 42, and the user-supplied input information is shown in Table 73. The median of the divided highway is not traversable, so there is no likelihood of a vehicle crossing over the median and striking the piers. The pier columns may be struck, however, by vehicles leaving the left side of the off-ramp (i.e., Direction #1) or leaving the right edge of the mainline (i.e., Direction #2). As shown in Table 73, the off-ramp is a one-way, one-lane roadway with a traffic volume of 5,000 vehicles per day, 5% of which are trucks. In contrast, the mainline divided high- way has 60,000 vehicles/day and 25% trucks. The face of Column #1 is 12 ft from the left edge of the one-way ramp, and Column #2 is 14 ft from the right edge of the mainline highway. The bridge superstructure is not continuous, and the pier system is not redundant, based on the designer’s

Figure 41. Pier protection shielding barrier for Example #3.

85 calculations, so the risk assessment model is used to deter- mine if the pier system should be shielded to minimize the risk of bridge failure due to a pier collision. Both columns are small, 2-ft-diameter circular columns, and the designer has determined that the lateral capacity of each is only 250 kips. 5.4.2 Pier Protection Procedure 5.4.2.1 Find Site-Specific Adjustment Factor: Ni Direction #1 is arbitrarily defined as the one-way off-ramp, and Direction #2 is the northbound mainline of the divided highway. As discussed previously, there is no need to include another direction for the opposing lanes of the divided high- way since the median is not traversable. The user-provided information in Table 73 can be used in conjunction with Table 15 (i.e., Appendix A, Table C3.6.5.1-1) to calculate the site-specific adjustment factors listed in Table 74. The adjustments are quite different for the two different directions since one direction is a one-way, low-speed, low-volume ramp, and the other is a high-speed, high-volume, six-lane highway. This example shows that the adjustments for each direction can be very different if the geometric and traffic characteristics are substantially different for the different potential impact directions. This allows the user to assess pier systems where the pier system is a risk from multiple direction with very different site and traffic characteristics. 5.4.2.2 Heavy-Vehicle Base Encroachment Frequency: HVEi The user estimates the annual number of heavy vehicles that encroach in each direction in Example #4 using Table 13 (i.e., Appendix A, Table C3.6.5.1-2). Direction #1 is a one-way ramp with a traffic volume of 5,000 vehicles/day. As stated in the note to Table 13, the encroachments for one-way roads are estimated by doubling the AADT of the one-way road and using the entries in Table 13 for divided highways. Entering Table 13 with an AADT of 2 • 5,000 = 10,000 vehi- cles and 5% trucks results in 0.0042 expected heavy-vehicle encroachments per edge in Direction #1, the one-way ramp. Direction #2 is the primary right edge of the six-lane mainline RDG Table Parameter Direction #1 #2 5-7 LS Shy-line offset (ft) 7 7 5-9 a/b Flare rate 24:1 24:1 5-10(b) LR Run-out length (ft) 265 265 L1 Tangent length (ft) 0 0 L2 Barrier offset (ft) 6 6 LA Lateral extent of area of concern (ft) 27.5 22.5 Table 69. Barrier layout parameters from the RDG for Example #3. = ( + − ) + Direction #1 Direction #2 = (27.5 + 1 24 0 − 6) 1 24 + 27.5 265 = 187 = (22.5 + 1 24 0 − 6) 1 24 + 22.5 265 = 176 Table 70. Required length of need for flared rigid barrier shielding pier for Example #3. Parameter LRFD Procedure Direction RSAPv3 Direction #1 #2 #1 #2 Site-specific adjustment factor (Ni) 1.40 1.21 1.40 1.21 Base vehicle encroachment (ENCR) 13.5477 13.5477 13.5477 13.5477 Heavy-vehicle encroachment adjustment factor (fHV 0.2682 0.2682 0.2682 0.2682 Annual unshielded pier collisions (AFHV CUSP) 0.0013 0.0013 0.0011 0.0012 Parameter RDG Procedure Direction RSAPv3 Direction #1 #2 #1 #2 Site-specific adjustment factor (Ni) 1.40 1.21 1.40 1.21 Base vehicle encroachment (ENCR) 13.5477 13.5477 13.5477 13.5477 Annual unshielded pier collisions with the lead column (Ni PVEi P(C|PVEi)) 0.0153 0.0152 0.0139 0.0138 ENCR) Table 71. Comparison of RSAPv3 results and procedure results for Example #3.

86 divided highway so the user enters the divided highway por- tion of Table 13 with an AADT of 60,000 vehicles/day and 25% trucks to find that 0.0078 heavy-vehicle encroachments can be expected annually, as shown for reference in Table 75. 5.4.2.3 Probability of a Collision with an Unshielded Pier Component Given a Heavy-Vehicle Encroachment: P(C|HVEi) In this example, as shown in Figure 42, the distance to the face of Column #1 from the edge of the one-way ramp (i.e., Direction #1) is 12 ft, and the distance from the right edge of the primary lanes of the mainline divided highway to Column #2 is 14 ft. Both columns are 2 ft in diameter with a lateral impact capacity of 250 kips, as shown in Table 73, so the probability of striking an unshielded pier component given an encroachment based on Table 19 (i.e., Appendix A, Table C3.6.5.1-3) is 0.1337 for a 12-ft offset in Direction #1 and 0.1247 for a 14-ft offset in Direction #2, as shown in Table 67. Table 19 only lists offsets of 10 and 15 ft, so these values were found using the equation at the bottom of Table 19, but the user could also interpolate the listed values and get essentially the same results. 5.4.2.4 Probability of the Worst-Case Collision Force Exceeding the Critical Pier Component Capacity Given a Collision: P(QCT > RCPC |C) The roadway in this example is a one-way off-ramp in Direction #1 and a divided rural primary highway in Direc- tion #2. Each functional class and highway type implies a particular heavy-vehicle mix. Knowing that the functional classification is rural divided, the user goes to the upper left Column Heavy Vehicles Passenger Vehicles Direction Direction #1 #2 #1 #2 #4 0.0003 0.0012 0.0066 0.0138 #3 0.0004 0.0003 0.0063 0.0060 #2 0.0003 0.0003 0.0066 0.0059 #1 0.0011 0.0003 0.0139 0.0059 Total 0.0021 0.0021 0.0334 0.0316 0.0042 0.0650 Table 72. Annual pier component collisions from RSAPv3 for Example #3. L3-1=12’ L3-2=14’ COL. #2 COL. #1 SY EL S W E L B W LL DI R. # 2 D IR . # 1 Figure 42. Example #4 site layout. Bridge Characteristics Value Nominal resistance of critical pier component: RCPC (kip) 250 Critical pier component size (ft) 2 Number of columns in pier system 2 Pier redundancy? No Superstructure continuity? No Bridge type Typical Site and Traffic Characteristics Direction #1 Direction #2 Highway type One-way ramp Divided Functional classification Rural primary Rural primary Two-way AADT (veh/day) 5,000 (one-way) 60,000 PT 5 25 Offset to critical pier component: L3 (ft) 12 14 Major accesses (points) 0 1 Horizontal curve away from the pier? Yes NA Horizontal curve radius (ft) 4,500 Tangent Lanes in one direction 1 3 Lane width (ft) 12 12 PSL (mph) 30 65 Grade (%) -5 Flat Table 73. User-input values for Example #4. Adjustment Factor Direction #1 Direction #2 Major accesses (fACC) 1.00 2.00 Lane width (fLW) 1.00 1.00 Horizontal curve radius (fHC) 1.11 1.00 Lanes in one direction (fLN) 1.00 0.91 PSL (fPSL) 1.18 1.00 Grade (fG) 1.75 1.00 Site-specific adjustment factor (Ni) 2.29 1.82 †Values are found by taking the user-supplied input data in Table 73 and calculating the appropriate adjustments from Table 15. Table 74. Site-specific adjustment factors for Example #4.†

87 section of Table 7 (i.e., Appendix A, Table C3.6.5.1-4). Similar to determining the AADT, one-way facilities are assumed to have the same vehicle mix characteristics as divided roadways so the user goes to Table 7 with the 30-mph PSL (i.e., less than or equal to 45 mph) of the one-way ramp to find that the probability of exceeding the 250-kip lat- eral capacity given that an impact occurs in Direction #1 is 0.8058. In Direction #2 on the mainline, the user goes to Table 7 with the 65-mph PSL to find that the probability of exceeding the 250-kip lateral capacity if an impact occurs is 0.9824, as listed for reference in Table 77. There is an extremely high chance of one of these pier columns failing if it is struck by a heavy vehicle from either direction due to the combination of the small lateral load capacity, the high speed and volume on the mainline, and the geometry on the ramp (e.g., curvature, grade). 5.4.2.5 Annual Frequency of Bridge Collapse: AFBC The user is now ready to calculate the expected annual frequency of bridge collapse, as follows, from the values previously determined in Table 74 (Ni), Table 75 (HVEi), Table 76 [P(C |HVEi)], and Table 77 [P(QCT > RCPC|C)]: i i i i i i i i i i i i i i ∑ [ ] [ ] [ ] [ ] [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) = > = > + > = + = + = = AF HVE HVE AF HVE HVE HVE HVE AF 2.29 0.0042 0.1337 0.8058 1.82 0.0078 0.1247 0.9824 AF 0.0010 0.0017 AF 0.0028 BC 1 CT CPC BC 1 1 1 CT CPC 2 2 2 CT CPC BC BC BC N P C P Q R C N P C P Q R C N P C P Q R C i i m i i The annual expected frequency of collapse of this bridge under these traffic conditions is 0.0028. This expected risk is almost three times higher than the critical risk of 0.001 for a typical bridge, so this pier system should be either shielded to protect the piers, or the pier columns could be redesigned so they are stronger. If this pier system was being contemplated for new con- struction, the designer could increase the lateral strength of the pier columns to 800 kips. In so doing, the probability of exceeding the impact load would decrease from 0.8058 to essentially 0 in Direction #1 and from 0.9824 to 0.2706 in Direction #2. The annual frequency of bridge collapse would then be 0.0005, which is less than the critical value of 0.001. On the other hand, if this bridge were already built and being evaluated for bridge collapse risk, then shielding the pier system would likely be the only feasible solution. 5.4.3 Occupant Protection Procedure The previous sections showed that the pier system in Example #4 requires shielding with a barrier because the probability of a failure-inducing truck collision is almost three times higher than the critical value of 0.001. A MASH TL-5 rigid concrete barrier would, therefore, be required to protect the pier system from heavy-vehicle impacts. Since the pier system must be shielded for pier protection reasons, there is no need to check for occupant protection from pier collisions since a barrier is already needed. 5.4.4 Shielding Barrier Layout Since shielding is required for pier component protection from heavy-vehicle collisions, a MASH TL-5 rigid concrete barrier would be used at the site. Since heavy vehicles are a risk in both directions, a shielding barrier should be used on the ramp approach as well as the approach on the mainline of the divided highway, as shown in Figure 43. For purposes of this example, it is assumed that the owner agency prefers to use a flared barrier in this ramp situation. RDG Table 5-7 recommends a shy-line offset (LS) of 4 ft for the 30-mph ramp, as shown in Table 78. On the ramp (i.e., Direction #1), the face of the pier is 12 ft from the left edge of the lane, and the pier protection procedures suggest HVE1 HVE2 0.0042 0.0078 Table 75. Heavy-vehicle base encroachment frequency for Example #4. P(C|HVE1) P(C|HVE2) 0.1337 0.1247 Table 76. Probability of a collision with an unshielded pier component given a heavy-vehicle encroachment: P(C HVEi) for Example #4. P(QCT > RCPC|C)1 P(QCT > RCPC|C)2 0.8058 0.9824 Table 77. Probability of the worst-case collision force exceeding the critical pier component capacity given a collision: P(QCT > RCPCC ) for Example #4.

88 3.25 ft of space from the back of the barrier to the face of the pier. A typical single-faced section of rigid concrete barrier is about 18 in. wide, so there is sufficient room to place the barrier at the edge of the 4-ft shy line. The RDG recommends an 8:1 flare rate for rigid barriers at or beyond the shy line on 30-mph roads, as shown in Table 78. The run-out length for 30-mph roadways with 5,000–10,000 vehicles per day is 90 ft, as also shown in Table 78. Table 79 shows that the shielding barrier should extend 36 ft upstream of Column #1 on the one-way ramp. RDG Table 5-7 recommends a shy-line offset (LS) of 7 ft for the 65-mph mainline divided highway, as shown in Table 78. The face of the pier is 14 ft from the right edge of the mainline highway, and the pier protection procedures suggest 3.25 ft of space from the back of the barrier to the face of the pier. A typical single-faced section of rigid concrete barrier is about 18 in. wide, so there is sufficient room to place the barrier at the edge of an 8-ft shoulder, which would be just beyond the 7-ft shy line. The RDG recommends a 20:1 flare rate for rigid barriers at or beyond the shy line on 65-mph roads, as shown in Table 78. The run-out length for 65-mph roadways with more than 10,000 vehicles/day is 330 ft, as is also shown in Table 78. Table 79 shows that the shielding barrier should extend 81 ft upstream of Column #2 on the mainline of the divided highway. In this case, the length of Direction #1 was calculated to require a length of need of 36 ft using the RDG procedure. This barrier, however, is being provided for pier protection. The minimum 60-ft upstream length recommended by the LRFD procedures should therefore be provided. Additionally, the ends of the two barriers are fairly close to one another and the intersecting roadways. The designer could, if this were new construction, provide stronger pier components. The design could also extend the barrier in Directions #1 and #2 to intersect and terminate both with a single crash cushion placed in front of the end of the rigid concrete barrier. This is shown in Figure 43. = ( + − ) + Direction #1 Direction #2 = (14 + 1 8 0 − 4) 1 8 + 14 90 = 35.6 36 = (16 + 1 20 0 − 8) 1 20 + 16 330 = 81 85 Table 79. Required length of need for tangent guardrail shielding pier for Example #4. Figure 43. Pier protection shielding barrier for Example #4. RDG Table Parameter Direction #1 #2 5-7 LS Shy-line offset (ft) 4 7 5-9 a/b Flare rate 8:1 20:1 5-10(b) LR Run-out length (ft) 90 330 L1 Tangent length (ft) 0 0 LA Lateral extent of area of concern (ft) 14 16 Table 78. Barrier layout parameters from the RDG for Example #4.

89 Parameter LRFD Procedure Direction RSAPv3 Direction #1 #2 #1 #2 Site-specific adjustment factor (Ni) 2.29 1.82 2.30 1.82 Base vehicle encroachment (ENCR) 5.8435 10.1608 5.8435 10.1608 Heavy-vehicle encroachment adjustment factor (fHV ENCR) 1.0000 0.2168 1.0000 0.2168 Annual unshielded pier collisions (AFHV CUSP) 0.0010 0.0017 0.0011 0.0017 Parameter RDG Procedure Direction RSAPv3 Direction #1 #2 #1 #2 Site-specific adjustment factor (Ni) 2.29 1.82 2.30 1.82 Base vehicle encroachment (ENCR) 5.8435 10.1608 5.8435 10.1608 Annual unshielded pier collisions with the lead column (Ni PVEi P(C|PVEi) ) 0.0218 0.0208 0.0164 0.0168 Table 80. Comparison of RSAPv3 results and procedure results for Example #4. Column Heavy Vehicles Passenger Vehicles Direction Direction #1 #2 #1 #2 #2 0.0011 0.0004 0.0164 0.0070 #1 0.0010 0.0017 0.0114 0.0168 Total 0.0021 0.0021 0.0278 0.0238 0.0042 0.0516 Table 81. Annual pier component collisions from RSAPv3 for Example #4. 5.4.5 RSAPv3 Comparison RSAPv3 is not able to analyze intersecting roadways, so this example was analyzed using two RSAPv3 simulations— one for each direction. Table 80 shows the comparison values for the LRFD pier protection procedure and RSAPv3 results. The pier protection procedure results are essentially identical to the RSAPv3 estimates for this two-column pier system. Even though the passenger-vehicle occupant protec- tion procedure was not used in this example, Table 80 also shows the comparison values between the occupant pro- tection procedures and RSAPv3. The values for the RDG procedure somewhat overpredict compared to the RSAPv3 estimates. While the passenger-vehicle occupant protection proce- dures were not required in this example since a shielding barrier was already required for pier protection, the num- ber of predicted passenger-vehicle crashes with all the pier columns compares favorably with RSAPv3. The passenger- vehicle protection procedures estimate a total of 0.0568, calculated as follows: ( )+    = +    + =2 3 AF 2 2 3 0.0218 0.0208 0.0568PV CUSP n RSAPv3 predicts 0.0516 (see Table 81), meaning that the RDG procedures overpredict by almost 10%. This is conser- vative but not unreasonably so.