DIFFERENCES FROM STRAIGHT BRIDGES Torsion effects Flange lateral bending Global stability CURVED STEEL I GIRDER BRIDGES RAMP GH OVER THE ERIE CANAL Constructability concerns INTRODUCTION COMPARISON WITH KINKED OR CHORDED BRIDGES Straight girders require less steel and have lower fabrication costs. Curved girders offer better appearance. Continuity over several spans allows simplified framing, efficient use of material, increased vertical clearance, use of longer spans, and fewer joints. Deck is easier to form with a constant deck overhang. The overall cost of curved bridges with curved girders may be similar to that of a curved bridge with straight members. CURVED STEEL BRIDGES Current practice usually fits bridges into the desired road alignment. Sometimes bridges are located in a curve. Curved bridges are commonly used at locations that require complex geometries and have limited rightof way, such as urban interchanges. Plate or box girders usually are the most suitable type of frame for such bridges. BEHAVIOR OF CURVED BRIDGES 1
ECCENTRICITY OF DEAD LOAD The structure is commonly subjected to significant torsion. The eccentricity of the total dead load increase the vertical reactions on the outside girders and decrease the vertical reactions on the inside girders. CROSS FRAMES Individual girders are relatively flexible and weak in torsion, so they need to be supported by crossframes or diaphragms at relatively close intervals along their lengths to avoid large torsional stresses and rotations between brace points. Cross frames are essential (primary) components in horizontally curved bridges. Cross frames transfer a large share of the V loads and provide torsional support to the girders. INTERNAL TORSIONAL RESISTANCE Internal torsional resistance is developed predominantly via the transfer of vertical shear forces between the girders by the cross frames and the slab. These vertical shears (V loads) increase the downward forces on the outside girders and decrease them on the inside girders. TWISTING AND WARPING TORSION OVERALL INTERNAL TORQUE The overall internal torque at any cross section is developed primarily by the differences in the girder shears across the width of the structure. The couples generated by each of the V loads on each of the girder free body diagrams also resist the tendency of the girders to twist about their axes. CHARACTERISTICS AND DETAILS TO AVOID Oversized or slotted holes: decrease the stability bracing efficiency of cross frames. Narrow bridges or bridge units: susceptible to large response amplifications due to global second order effects. V type cross frames without top chords: too flexible to stabilize girders prior of hardening of concr. deck. Long span I girder bridges without top flange lateral bracing systems: second order amplification effects more critical for longer spans. 2
SMALL CURVATURE ANALYSIS The effects of curvature may be ignored in the analysis when the girders are concentric and similar in stiffness, not skewed, and with an subtended angle of 3.5 degrees or less. In this case the girders may be analyzed individually with the span length equal to the arc length. Lateral flange bending effects still need to be considered in the design. VERTICAL LOADS Dead loads may be distributed in the same way as for straight girders. AASHTO LRFD live load distribution factors are not applicable to curved girders. The whole bridge has to be analyzed as a system. Influence surfaces or a 2D/3D analysis may be required to find the load applied to each girder. METHODS OF ANALYSIS Equilibrium of horizontally curved I girders is developed by the transfer of load between the girders. The analysis must recognize the integrated behavior of structural components. The entire superstructure, including bearings, is to be considered as an integral unit in the analysis. CENTRIFUGAL FORCES Centrifugal forces comprise a horizontal, radial loading on curved structures. These forces are determined as a percentage of the live load, without impact. Assumed to act 6 ft from the roadway surface. Induce shears and horizontal torques on the superstructure. Superelevation reduces the overturning effect. BEARING ORIENTATION In curved bridges, guide bars and slotted holes for expansion bearings are oriented along a chord that runs from the fixed point to the bearing under consideration. This allows the bridge to expand freely in both the longitudinal and transverse directions. 3
LEVELS OF ANALYSIS Approximate method: V load 2D grid: Girder and cross frames modeled using beam elements, nodes in a single horizontal plane. 2D plate and eccentric beam: similar to 2D grid, but the deck is modeled with plate elements offset a vertical distance from the steel elements. 3D: Girders and deck are modeled using plate elements, and cross frame elements are modeled using truss type elements. SPACING OF DIAPHRAGMS The spacing of diaphragms or cross frames shall not exceed: Lb Lr R/10 where Lr is a limiting unbraced length to achieve the onset of yielding in flanges with consideration of residual stress effects. These limits insure that the spacing of transverse stiffeners and the effect of amplified second order stresses are not excessive. PRELIMINARY DIAPHRAGM SPACING As a preliminary guide, the spacing of diaphragms should be less than: Lb =0.7 R bf DESIGN R = Girder radius bf = Flange width FLANGE LATERAL BENDING STRESS Flange lateral bending due to the curvature must be considered at all limit states and during construction. SHEAR CONNECTORS The shear connectors can be subjected to significant radial forces in addition to longitudinal shear forces. Fbu + fl / 3 Φf Fn Fbu = flange major axis bending stress fl = flange lateral bending stress (max. 0.6Fy) Φf Fn = factored flexural resistance 4
TRANSVERSE SECTION RAMP GH OVER THE ERIE CANAL PLAN GIRDER LAYOUT ELEVATION QUESTIONS? 5
Question 2 Name two characteristics or details to avoid in horizontally curved I girder bridges EVALUATION Question 1 Fill the blank: Cross frames or diaphragms are components in horizontally curved I girder bridges. a. Axial b. Secondary c. Narrow d. Primary Question 2 Name two characteristics or details to avoid in horizontally curved I girder bridges Oversized or slotted holes. Narrow bridges or bridge units. V type cross frames without top chords. Long span I girder bridges without top flange lateral bracing systems. Question 1 Fill the blank: Question 3 In curved bridges, how are guide bars and slotted holes for expansion bearings oriented? Cross frames or diaphragms are primary (essential) components in horizontally curved I girder bridges. 6
Question 3 In curved bridges, how are guide bars and slotted holes for expansion bearings oriented? Guide bars and slotted holes for expansion bearings are oriented along a chord that runs from the fixed point to the bearing under consideration. 7