Design Guide for Bridges for Service Life (2013)

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596 F.1 bAckground This section contains a procedure developed to extend the application of jointless bridges to curved steel I-girder bridges (Doust 2011). Several limitations must be ob- served when using the suggested approach that reflect the range of parameters consid- ered in its development. The study considered several bridge configurations for which detailed finite element analyses were conducted. These analyses were then used to (1) comprehend the performance of jointless curved girder bridges and (2) develop approximate solutions that are in reasonable agreement with the results of detailed finite element analysis. Various assumptions and cases were considered during the devel opment of the suggested approach: 1. Steel I-girder superstructure made composite with concrete deck; 2. Concrete integral abutments at the bridge ends supported on steel H-piles; 3. One or more intermediate piers isolated from the bridge superstructure by elasto- meric bearings; 4. Concrete parapets integrally connected to the concrete deck; 5. Superstructure superelevation ranging between 0% and 6%; 6. Abutment wall height ranging between 9 and 13 ft; 7. Wingwalls separated from the abutment wall by means of joints; 8. Approach slab connected to abutment wall using a pinned connection detail; 9. Bridge plan symmetric with respect to the midlength of the bridge; 10. Radial piers and abutments (i.e., the lines of all abutments and piers intersect at the bridge center of curvature); f CURvED GiRDER BRiDGES

597 Appendix F. CURvED GiRDER BRiDGES 11. Bridge arc length-over-width ratio larger than 3.0; 12. Ratio of the lengths of end spans to interior spans approximately equal to 0.8; and 13. All intermediate spans of approximately equal length. The following two sections present step-by-step procedures to calculate the magni- tude and direction of bridge end displacements and determine the optimum abutment pile orientation. F.2 cALcuLAting mAgnitude And direction oF end diSPLAcement For curved integral abutment bridges meeting the limitations described earlier, the following procedure can be employed to calculate the magnitude and direction of end displacements: 1. Determine the point of zero movement for the bridge and consequently the bridge length along the centerline of the bridge (L0) that should be used in calculating the end displacement. For symmetric bridges supported on a substructure with rela- tively symmetrical stiffness, it can be assumed that L0 is equal to half the bridge total arc length. Otherwise, a more detailed approach that takes into account the relative stiffnesses of the supports should be used to calculate the point of zero movement. 2. Determine the effective coefficient of thermal expansion by using Equation F.1: α α α( ) ( ) ( ) ( ) = + + EA EA EA EAequivalent deck girder deck girder (F.1) 3. Calculate the bridge shortening due to contraction by using Equation F.2: T Lconstruction equivalent 0α∆ = ⋅∆ ⋅ (F.2) 4. Find the modification factor for bridge shortening due to contraction by using the information provided in Figure F.1, which provides the relationship between the radius of curvature and the modification factor used in Equation F.5. 5. Determine the equivalent shrinkage strain by using Equation F.3: ε ε ε ε( ) ( )( ) ( )= + − + EA EA EAsh sh sh sh,equivalent ,girder ,deck ,girder deck deck girder (F.3) 6. Calculate the bridge shortening due to shrinkage by using Equation F.4: Lshshrinkage ,equivalent 0ε∆ = ⋅ (F.4) 7. Find the modification factor for bridge shortening due to shrinkage by using Figure F.2.

598 DESiGN GUiDE FOR BRiDGES FOR SERviCE LiFE 0.9 0.95 1 1.05 1.1 1.15 100 1000 10000 M od ifi ca ti on F ac to r, g T U c Radius (ft) Outer Arc Inner Arc Figure F.1. Modification factor for bridge contraction. Figure F.2. Modification factor for bridge shrinkage. 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 000010001001 M od ifi ca ti on F ac to r, g S h Radius (ft) Outer Arc Inner Arc

599 Appendix F. CURvED GiRDER BRiDGES 8. Calculate the total factored bridge shortening by using Equation F.5: 1.3 TUc Shtotal thermal shrinkageγ γ( )∆ = ∆ + ∆ (F.5) 9. Calculate the bridge width effect factor with the following equations. These fac- tors are calculated for the inner (Equation F.6) and outer (Equation F.7) corners of the bridge separately. The purpose of these factors is to determine the direction of end displacement. k W L 1 0.84 C in = + (F.6) k W L 1 0.84 C out = − (F.7) 10. Find the direction of the bridge corner displacements by using Equations F.8 and F.9 for inner and outer displacements, respectively: k L R 90 11 in degreesin inα = −           (F.8) k L R 90 11 in degreesout outα = −           (F.9) 11. Knowing the total bridge shortening found in Step 8 and the direction found in Step 10, solve Equations F.10 through F.16 to find the new location of the bridge corner. The corner of the bridge is assumed to be originally located at the coor- dinates xA = RA and yA = 0, in which RA is the radius of the bridge at that specific corner. x ab a b b R a a1 / 1A 2 2 2 2 2 2( )( )( ) ( )= − + − − ′ + +′ (F.10) y ax bA A= +′ ′ (F.11) where a tanα= − (F.12) b R tanα= (F.13) y x tan A A 1γ =     − ′ ′ (F.14) L R2 β γ( )′ = ′ − (F.15) in which

600 DESiGN GUiDE FOR BRiDGES FOR SERviCE LiFE L R2 β = (F.16) 12. Using the new coordinates of the bridge corner xA′ and yA′, the components of bridge corner displacement are found as shown by Equations F.17 and F.18: x Rx A A∆ = −′ (F.17) yy A∆ = ′ (F.18) F.3 oPtimum PiLe orientAtion In curved bridges, the optimum orientation of the piles depends mainly on the bridge geometry. In contrast to straight bridges, the optimum direction is not the same for all curved bridges. This section presents a method to find the optimum pile orientation in a curved bridge that is based on finite element simulation of several curved integral steel I-girder bridges (Doust 2011). The same concept employed for straight bridges is also used for curved girder bridges; namely, the piles should be oriented so that the strong axis of their sections is perpendicular to the direction of bridge maximum displacement. The following steps should be used to obtain the optimal abutment pile orientation: 1. The critical load combination for the design of the piles should be determined to be either expansion based or contraction based. Figure F.3 may help for determining the controlling load combination. 100 1000 10000 100 400 700 1000 Ra di us (ft ) Bridge Length (ft) Expansion Control Contraction Control Figure F.3. Controlling type of load combination.

601 Appendix F. CURvED GiRDER BRiDGES 2. The direction of bridge maximum end displacement, as defined in Figure F.4, should be determined using the curves presented in Figure F.5. 3. The strong axis of the abutment piles perpendicular to the displacement direction found in Step 2 should be oriented. If the type of critical load combination can- not be distinguished for a specific bridge, the bridge should be analyzed for both expansion-control and contraction-control pile orientations from Step 2, and then the optimum orientation should be chosen. x y ux uyu A A' -100 -75 -50 -25 0 25 50 75 100 125 150 0.00 1.00 2.00 3.00 An gl e of D ir ec ti on (d eg .) L/R Contraction-Control Expansion-Control Figure F.5. Angle of direction of bridge end displacement. Figure F.4. Direction of bridge maximum and displacement.