Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc. Development of a Tamar Bridge Finite Element Model R. J. Westgate, J. M. W. Brownjohn The University of Sheffield Department of Civil and Structural Engineering, Sheffield, United Kingdom, S1 3JD cip08rjw@sheffield.ac.uk ABSTRACT As part of an ongoing long-term monitoring research project, a finite element model has been developed to model the behavior of the Tamar Bridge in Plymouth, UK. The structure is a suspension bridge which underwent significant remedial and expansion work, including the addition of stay cables to the support structure. This paper presents the production of the model, with holistic static and modal responses of the two cable structural systems acting together. Keywords: structural health monitoring, hybrid, environmental, finite element, non-linear 1 INTRODUCTION Finite element (FE) analyses are used in conjunction with long-term monitoring of structures, to predict static and dynamic responses under various loads. In the context of Structural Health Monitoring (SHM) research, FE models are often used to simulate conditions that the researcher may wish to emulate with data recorded from the real-life structure. The structural systems of suspension bridges are complicated compared to other bridges, since they rely on the equilibrium of the deck with the tensions in the cables. The behaviour of the bridge is non-linear, due to the non- compressive strength of cables and hangers, and the movements at the bearings. Suspensions bridges that are a hybrid of suspension and stay cable support systems may be created as a result of improvements to support increased traffic levels. The number of papers concerning hybrid suspension bridges is limited, and only touch upon the theory. A more detailed approach to model these two systems together, which indi- vidually are self-configuring and non-linear processes, would be a welcome contribution to the theoretical modelling of tension structures. This paper presents a methodology for creating a FE model for this type of suspension bridge, and how the static deflection of the bridge was optimised to get as close to zero displacement as possible. This paper follows a different method compared to other published works [1] and considers the cable forces are all contributing simultaneously. A simple parametric sensitivity study has also been presented, to demonstrate how the components of the hybrid suspension bridge contributes to its natural frequencies. 2 BRIDGE DESCRIPTION AND HISTORY The Tamar Bridge shown in figure 1 carries the A38 trunk road across the River Tamar from Saltash in Cornwall to Plymouth in Devon, and is financed by its toll income for traffic usage. It was first opened in 1961, with a design by Mott Hay and Anderson, as a bridge supported by suspension cables and a truss girder. It has a main span of 335m and symmetrical side spans of 114m. Accompanied by anchorage and support spans, it achieves a span of Figure 1: Photograph of the Tamar Suspension Bridge, Plymouth, UK 642m. The main towers are constructed out of reinforced concrete, seated on caisson foundations and are 73m tall, with the deck suspended halfway up. The truss is 15.2m wide and 4.9m deep, consisting of rectangular hollow sections, and originally supported a concrete deck. The suspension bridge cables are 0.38m thick, consisting of 31 steel locked cable ropes of 60mm diameter. Vertical locked coil hangers are at 9.2m centers and are 50mm in diameter. During its life it has supported vehicles loads in excess of its design capacity. The bridge subsequently underwent a strengthening and widening (upgrading) scheme that was completed in December 2001. This involved adding a lane each side of its truss via cantilevers, and replacing the concrete deck with an orthotropic steel deck with shear boxes. Additionally, there was an installation of sixteen 100mm diameter stay cables, spanning from the main towers to either the truss or the side towers. The bridge is continuous at the Plymouth tower, whilst a lateral thrust girder forms an expansion joint at the Saltash tower. 3 FINITE ELEMENT MODELING A high resolution 3D FE model was developed for the studies of a long term health monitoring project, intended as a numerical representation of the suspension bridge, and a research tool for investigating its performance under varying environmental conditions, such as ambient temperature and wind. The program of choice was ANSYS. Truss elements were designed as beam elements with six degrees of freedom at each node, shell elements were used for the plates on the deck and the towers, and cables were modelled as tension-only spar elements. Boundary conditions and movement joints are a necessary evil for the analyst; they have to imitate the behaviour of the actual structure correctly, whilst at a resolution agreeable for the whole model. Vertical hinges connecting the truss to a seat on each tower were treated as beam elements with rotational resistance released in the span-wise direction, and the lateral thrust girder was implemented in a similar way. The bridge model is fully fixed at the bases of the towers, and at the road-adjoined side-spans. 4 INTERNAL FORCE EQUILIBRIUM The following section documents the various steps followed to ensure that the forces acting within the suspension and stay cables would provide a neutral profile under dead loading conditions. Modal solutions from the FE analyses are compared to previously obtained results [2] by their frequencies and order of mode shapes, as another measure for the model’s accuracy. 4.1 ACCEPTABLE PROFILE SHAPE The first step was to ensure that the finite element model was distorting into a reasonable shape when vertical loads were applied to it. This involved iterating the strains in the suspension cables until the displacements in the deck were minimised. For a suspension bridge, this involves adjusting the main-span and side-span suspension cables, and the stay cables, at every iterative step, to ensure that the deflections in the spans and towers remained small. The secant iteration method was used to find the cable tension at dead load to reach a suitable profile, since a derivative of the target function is not a requirement within its formula. This is useful when the combination of two variables in the structural system makes linear interpretation impossible, and the equation defining a suitable structural stiffness is indeterminate. The bi-variable form is: − − −1 F (xn ,yn) F (xn−1,yn−1) F (xn,yn) F (xn−1,yn−1) x + x − − F (x , y ) n 1 = n − xn xn−1 yn yn−1 n n G x ,y −G x ,y G x ,y −G x ,y (4.1) y y ( n n) ( n−1 n−1) ( n n) ( n−1 n−1) G(x , y ) n+1 n − − n n " xn xn−1 yn yn−1 # where x and y are variables, F (x, y) and G(x, y) are functions of the variables and n is the present iterative step number. The first pass minimised the vertical deflection on the main span, and the span-wise deflection of the tips of the towers, using the forces in the main and side span suspension as the variables. This setup provided an ‘M’ shaped sinusoidal deflected shape on the main span, as shown in figure 2, which are a result of the stay cable forces. The side spans had inadequately large vertical deflections, and a simple study showed that this behaviour still existed when there was zero initial forces in the cables. This precluded a reconsideration of the side-cable profile. Figure 2: Initial deflected shape. The gradient of the side-span suspension cables was made at an 11 degree angle to the horizontal at the side towers, which was taken from a past construction paper of the Tamar Bridge by Anderson [3]. The reformed profile produced deflections of 0.1m, which were acceptable. Due to the ‘M’ deflected shape the minimised outcome was modified to be the mean result of the mid-span and quarterly deflections, along with the deflections of the towers. Least squares method was attempted, but this lead to divergence in the iteration process. 4.2 CABLE FORCE ASSEMBLIES The forces applied in the cables were adjusted to provide a reasonable performance of the bridge. This was achieved by analysing the deflections and modal properties of the suspension bridge, in the structural configurations it was in at 1961 and 2001. These two scenarios were used to check that values of the cable forces could be translated to both stages in the bridge’s life comfortably. Modal properties from 1961 were taken from a paper by Williams [4], written before the expansion work. Results from 2001 are in a paper by Brownjohn [2]. Two candidates for possible assemblies were determined: 1. Zeroed mass and initial forces in 2001 elements, and added a concrete deck with iteration of suspension cable strains. • Followed by non-zeroing previous elements, and iterated stay-cable forces. 2. Used stay cable forces found from site tests, and iterated main suspension cable only. These arrangements were applied in the finite element model, and static and modal solutions were obtained. The results are presented in tables 1 and 2. Table 1: Test arrangements for the Tamar Bridge - Deflections. Applied Forces(×104kN) Displacement (m) Year Arrangement Main susp. Side susp. Stay cable Mid-span Side-span cable cable (quarter span) 1 2.4787 2.6048 0 -0.0008 -0.0060 1961 2 2.0296 2.0597 2.529 -0.4239 -0.2849 1 2.4787 2.6048 0 0.3104 0.2716 2001 2 2.0296 2.0597 2.529 -0.0397 0.0739 Table 2: Final arrangements for the Tamar Bridge - Frequencies.