finite element analysis | cooper river bridge
Spring 2014
CEE 3710: Structural Modeling and Behavior
Faculty: Anthony Ingraffea
Cornell University | College of Engineering
Models are idealized to be the answer to questions persisting in nature. However, the truth lies in the fact that natural systems are never closed, and their results are always non-unique. Models are only relative, approximate, and heuristically designed. All finite models and predictions are wrong, and all structures are built on the approximation of how wrong you are. Yet they can still be of use to engineers and designers.
Using the example of the Cooper River Bridge, I tested multiple versions of mathematical, finite analysis modeling (Visual FEA and MASTAN), which unlike physical modeling, establishes legitimacy, does not contain detectable or known errors, and is internally consistent. However, it does not always admit correct answers because although mathematically the model is consistent, it does not mean the model is based off correct assumptions, calculations, and boundary conditions. Mathematical modeling can hypothetically test infinite numbers of manipulations and produce correct data, yet even though the data is correct, the equations they are derived from could be erroneous.
The design
Opened in 1992, the Cooper River Bridge represented a new generation of truss bridges that remain aesthetically pleasing while extending a bridge’s lifespan and easing maintenance requirements. The three span continuous modified Warren truss, supported by reinforced concrete piers, foundations, and steel friction piles, was designed to achieve the structural engineering objectives within safety, serviceability, feasibility, sustainability, durability, aesthetics, and economic achievability.
fea modeling
In the instance of the FEA Model of the Cooper River Bridge, the linear span of the bridge was simplified as a beam. A truss is a beam shot full of holes. By treating each side of the bridge as a beam, using top and bottom chords as ‘flanges’ and diagonals as ‘webs’, the parallel axis theorem can equate an equivalent Young’s Modulus. By ignoring the web of the beam because it doesn’t transmit shear stress, the load is mostly distributed along the top and bottom chords. Idealizing top and bottom chords introduce error by assuming the chords are the same size, the centroid is exactly in the center of the chords, and the area of top and bottom chords are averaged from values provided by the Cooper River Bridge Case Study.
The foundation elements were similarly idealized as the piers, but rather than a column, they were modeled as a solid, reinforced concrete beam. Unlike the piers, the foundation elements had a solid, reinforced concrete cross section, but didn’t drastically change the ratio of area of concrete to area of steel- on average a 90:10 ratio.
Due to the size ratios of different foundations, the number of piles attached to each foundation differed. Based off the AutoCAD dimensions, there were 6 piles on Foundation 32, and 10 piles on Foundation 33. Based off research, the typical pile design used in substructure foundations was A36 steel, H-piles. By calculating the reaction forces, the amount of force applied to each pile and required surface area was determined. These minimum requirements were then idealized using the ‘best’ fit H-pile designed chosen from a construction company’s H-pile designs.
mastan modeling
In comparison to the FEA Model, the MASTAN Model of the Cooper River Bridge was simplified even more into a simple frame supported by three reaction forces. In the ideal model, the axial forces within a frame should be less than the axial forces on the same element in a truss because a ratio of the frame’s axial forces is transmitted into element moments. In comparison, a beam transmits all axial forces into moments, and a truss only has axial forces- no bending moments. Using the frame idealization, the modeled demonstrated where there was the ‘most bending’ within the idealized bridge. Ultimately, the frame model acts behaves like a hybrid of the extreme truss and beam idealizations.
There were also different idealized load conditions, supplied by previous assignments. Although model the FEA Model and MASTAN Model both experienced linear, uniformly distributed loads, the MASTAN Model had a 0.7641 kips/in load, whereas the FEA Model had a 2.375 kips/in load based off the idealized values of dead and live loads provided by the Case Study.
Further more, the MASTAN model had only frame elements to consider, whereas the FEA Model had to create an interface between multiple elements (bridge, piers, foundation, piles, soil). The MASTAN Model was simplified to have supports in the y-direction at nodes L0 and L7, and supports in the x-direction at nodes U15 and L14.
deflection differences
The maximum deflection at the midspan on the Visual FEA Model was -86.85 inches using a mesh combination of both triangular and quadrilateral mesh screens.
In comparison, using hand calculations, the bridge was modeled as a frame in MASTAN, the maximum vertical displacement at the midspan was -42.6 inches. The deflection on the Visual FEA Model is approximately twice the deflection of the MASTAN model. This could be a result of reducing the amount/type of error approximations that accrued throughout the finite element analysis. Also, the Visual FEA Model accounts for many of the structure’s elements, yet models the bridge as a beam, thus transmitting most of the axial forces into moments. More application of bending moments could transmit into a larger deflection. In comparison, the MASTAN Model models the bridge as a frame, such that it that transmits axial forces and ideally acts as a combination of a beam and truss model. The smaller deflection in the MASTAN Model validates the execution of the frame analysis because the -42.6 inches deflection is more ‘expected’ than the -86.85 inches.