CHAPTER 5 MODELING OF THE BRIDGE

Transcription

1 62 CHAPTER 5 MODELING OF THE BRIDGE 5.1 MODELING SAP2000, a nonlinear software package was used for modeling and analysing the study bridge. The following list provides details about the element type used for modeling the components of the bridge structure in SAP2000. a) Superstructure - Spine element b) Bent cap beams - Inelastic Beam-Column element c) Bent Columns - Inelastic Beam-Column element d) Elastomeric bearing - Link element e) Expansion joint - Gap element Superstructure In this study, the structural component model (SCM) was used to resemble the geometry of the bridge prototype. Since the in-plane and flexural stiffness of the bridge superstructure were large compared to the lateral stiffness of the supporting bents, a rigid-body dynamic system for the superstructure was assumed. The super-structure was represented by a single line of multiple three-dimensional frame elements (i.e., a spine-type configuration), which passes through the centroid of the superstructure and remains elastic for lateral loadings.

2 63 The eight equal simply supported spans of the bridge were separated by a gap, equal to the width of the expansion joint provided between the abutments and decks, and the multi-column bents and decks. Each span was discretized into four elements. This discretization was made at cross beam locations. The effective stiffness property of the longitudinal members evaluated from the experimental investigation was considered, and is given in Equation 5.1. EI eff = 0.8EI g (5.1) where, I eff = Effective moment of inertia I g = Gross moment of inertia EI eff EI g = Effective flexural stiffness = Gross flexural stiffness A transverse rigid bar of length equal to the deck width was connected to the tip of the spine element of each span, where the columns were located and where the abutment was located (Murat and Bruneau 1995). The rigid bar was oriented in the transverse direction. Thus, the rigid bar was used to model the interaction between the translation of the columns in the transverse and longitudinal directions and the translation as well as the rotation of the deck (column location) about the vertical axis. One end of the bearing elements was connected to this rigid bar at the column locations and the other end of the bearing elements was connected to the bridge columns. To measure the relative longitudinal displacement between the adjacent decks at the expansion joint, another rigid bar was connected to the tip of the spine element at the adjacent spans (column location).

3 Multi-column Bent The bridge consists of seven multi-column bents, and every bent was modeled as a plane frame (Figure 5.1). The bent cap beams were modeled as beam-column elements, and the effective flexural stiffness was taken as EI eff = 0.5EI g. The lumped plastic approach was used in this study to model the nonlinear behaviour of the column. In the lumped plastic model, the effective sectional properties were used to reflect the concrete cracking and reinforcement yielding. The bases of the columns were assumed to be fixed, as the foundations were deep (well foundation), resting on hard strata. A A Section A-A Inelastic Beam-Column element Rigid zone Inelastic Beam-Column element Confined concrete 150mm c/c 10# - 28mm Un-confined concrete Figure 5.1 Modeling of a multi-column bent The effective moment of inertia of the column was calculated, based on the cracked section using the moment-curvature (M- ) curve (Figure 5.2). The moment curvature values are shown in Table 5.1.

4 65 The effective stiffness was calculated from the Equation (5.2) at the first theoretical yield of reinforcement. The effective moment of inertia was found to be 0.55I g. Effective stiffness, EI eff M y (5.2) y where, M y and y represent the yield moment and curvature Curvature, rad/m Figure 5.2 Moment-curvature curve Table 5.1 Moment curvature values Sl. No. Moment (knm) Curvature (rad/m) Yield Ultimate Yield Ultimate The flexural hinges on the frame elements were assigned with default properties prescribed in SAP2000 described in FEMA-356 (2000). The multi-columns in all the bridge bents fall under the category of long column. As long columns are vulnerable to axial-flexure failure, PMM hinges

5 66 (P - Axial, M - M2 moment, and M - M3 moment) were assigned to the ends of the columns. Bent cap beams were assigned with Moment (M3) hinges at the ends. The default hinge properties are section dependent and based on the nonlinear modeling parameter as given in Table 9-6 of the ATC-40 (1996) document. The hinge properties were assigned through the definition of the moment-rotation relation, the interaction surface, and acceptance criteria. From the moment-rotation relation, the yield rotation was automatically determined, based on the defined material properties and the post-yielding stiffness (5% of the elastic stiffness). The acceptance criteria were deformations that were normalized by the yield deformation for Immediate Occupancy (IO), Life Safety (LS), and Collapse Prevention (CP) performance levels. For the PMM hinge, the acceptance criterion is the plastic rotation normalized by the yield rotation, which was calculated based on the section properties. The force-deformation behaviour of the hinge is shown in Figure 3.1. A, B, C, D, E are the points defining the moment-rotation relationship. Immediate Occupancy (IO), Life Safety (LS) and Collapse Prevention (CP) are the performance levels of the hinges at different steps of the pushover analysis. The values which were assigned at the respective points are shown in Table 5.2. Table 5.2 Hinge values of the columns (assigned) Sl.No Points M/M y y 1. A B IO LS CP C D E

6 Elastomeric Bearing The horizontal sliding behaviour of the interface between the bearing and the girder or cap beam was modeled, using the linear spring element or link element, as shown in Figure 5.3. A link element is composed of lateral, vertical and rotational stiffness components (Muthukumar 2003, Akogul and Celik 2008). Translational or effective stiffness is used to consider the nonlinear behaviour of elastomeric bearing. Shear modulus values for elastomers in bridge bearings range between 0.8 MPa and 1.20 MPa (IRC83-(part III)-1987) depending on their hardness. The effective shear modulus of pads was taken as 0.9 MPa. The bridge is longitudinally free up to the maximum elastomer flexibility. The initial stiffness of the spring was calculated from the geometric properties of the pad using the Equations (5.3, 5.4 and 5.5). Translational stiffness, K H (5.3) Vertical stiffness, (5.4) K V Rotational stiffness, (5.5) K where, G - Rigidity Modulus, E - Young s Modulus, A - Cross sectional area of the bearing pad, h e - Height of the bearing pad and I - Moment of Inertia. The properties of the elastomeric bearing pad are shown in Table 5.3. GA h e EA h e EI h e Elastomeric bearing KV = Vertical stiffness KH = Horizontal stiffness K = Rotational stiffness Link element Figure 5.3 Modeling of the elastomeric bearing pad

7 68 Table 5.3 Properties of the elastomeric bearing pad Sl.No. Properties Value 1. Elastomer Size Length 0.320m Width 0.500m Height m 2. Gross plan area 0.16m 2 3. Moment of inertia x10-6 m 4 4. Shape factor Elastic modulus kn/m 2 6. Effective Shear modulus 900 kn/m Expansion Joint In the study bridge, the shear key or doweled bearings, which prevent transverse displacement, were not provided, allowing translational movement which depends on the elastomer flexibility. The expansion joints between the adjacent deck slabs, the abutment and the deck slab were modeled using gap elements (Muthukumar 2003). The mathematical modeling of the expansion joint as the gap element is shown in Figure 5.4. The gap element is a compression only element, such that it will contribute resistance when the relative displacement between the adjacent spans is more than the initial gap of 25.40mm. When the gap closes, pounding occurs, and the gap element offers infinite stiffness.

8 69 A Edge of deck Gap element Mathematical modeling of expansion joint (Detail A) as gap element Figure 5.4 Modeling of the expansion joint The effective stiffness, K eff was calculated using the Equation 5.6. K Effective stiffness, (5.6) eff m where, k h - Impact stiffness parameter with the typical value of 25,000 kin -3/2 k h m - Maximum penetration of deck (25.4mm) Abutment Bridge abutments are only effective in compression, and the analytical response of bridges is significantly affected by the modeling characteristics of the abutment stiffness and capacity. Due to the large soil mass that interacts with the abutment and the abutment geometry, which exhibits higher stiffness values than other bridge bents, the abutment is assumed to have a perfectly plastic behaviour after reaching its ultimate

9 70 strength. These reasons result in more seismic forces to be attracted to the abutment. The resistance offered by the abutment to the inertial force is based on its structural capacity and soil resistance. In this study, the support provided by the abutment was assumed as fixed against vertical translation and the stiffness properties of the translational spring in the longitudinal and transverse directions are given in Equations (5.7) and (5.8). The seat type abutment is shown in Figure 5.5. kt kl All dimensions are in mm Figure 5.5 Seat type abutment with longitudinal and transverse stiffness where, w- width of the abutment back wall, b width of wing wall, h height of wing wall. Longitudinally, the soil behind the back wall was assumed to have a stiffness (k L ), which is related to the area of the back wall and transversely, the stiffness was considered 2/3 effective per length of the inside wing wall (assuming the wing wall is designed to take the load), and the outside wing wall is only 1/3 effective per wing wall length for a resultant stiffness, k T.

10 71 The modeling of the bridge is shown in Figure 5.6. The stiffness properties of the abutment in transverse and longitudinal directions are shown in Table 5.4. k L wh kn / m (5.7) kt b kn / m (5.8) Table 5.4 Stiffness properties of the abutment Sl. No. Longitudinal stiffness, k L (kn/m) Transverse stiffness, k T (kn/m) DETAIL A Y Z X Figure 5.6 (Continued)

11 72 Bent column (Inelastic beam-column element) Bent cap beam (Inelastic beam-column element) Edge of the deck (Rigid end) Superstructure (Spine element) Rigid link Elastomeric bearing (Link element) Expansion joint (Gap element) DETAIL A Figure 5.6 Modeling of the bridge using SAP2000