Seismic analysis of bridges with pile foundations
Introduction
Many bridges, especially long span bridges, are supported on pile foundations. A prototypal foundation is shown in Fig. 1. This consists of vertical and battered piles supporting a pile cap above the mudline. The piles may penetrate several layers of soil with varying strength and stiffness. The water table (not shown in the figure) may be found above or below the mudline, or above or below the pile cap. Older bridges are frequently founded on a large number—up to 50–100 or more—of small diameter piles per bent; 14 in deep HP sections were commonly used. New bridges, on the other hand, may utilize only a few, say 3–10, large diameter piles per bent. Concrete filled steel shells of 1–3 m diameter are coming in to frequent use as piles. An important, and somewhat surprising, characteristic of bridges with pile foundations is that the mass of the pile cap may be a significant fraction of the total mass of the structure tributary to a bent—it may be as much as 40% of the total mass. The mass of the pile cap is a largely unavoidable product of the plan area required to surround piles spaced at 2–3 times their diameter, times the depth of section required to distribute the column forces to the piles through shear and bending, times the density of concrete.
Because the mass of the pile caps may be a large fraction of the total mass of a bridge, the seismic response of its foundations to an earthquake may be an important factor in the response of the bridge as a whole. Pile foundations may do much more that simply transmit the ground motions to the bridge columns and superstructure. Pile foundations may have the following important effects:
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The stiffness of the piles may determine the ‘effective’ ground motions input to the structure, when the free-field motions vary with depth below the mudline. This ‘kinematic’ interaction [1] occurs because the stiffness of the piles constrains the free-field ground motions.
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The dynamic behavior of the foundations—considering the piles and pile caps as dynamic systems—may alter the ground motions transmitted to the bridge columns and superstructure. The resulting motions are often referred to as ‘inertia-interacted’ motions [2].
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In extreme cases, the fundamental period of the pile and pile cap systems may be comparable to that of the bridge columns and superstructure considered as fixed at the bottoms of the columns. There may then be a dynamic interaction between the bridge superstructure and the foundations.
Section snippets
Types of models
The seismic response of a pile foundation is the result of complex interactions between the pile cap, the piles and the soil. An analytical model intended to capture every aspect of this response would use solid elements to represent the components of the foundation and appropriate structural and geotechnical constitutive relationships.
This type of model is beyond the state of the practice in the structural engineering community, however. Practical models are built using beam elements to
Impedance matrix models
In the impedance matrix model the piles are substructured at the pile cap node; the substructure consists of stiffness and damping matrices that represent the properties of the whole pile group. This is shown in Fig. 3.
Individual pile models
Another approach to the simplification of foundation models is to substructure each pile independently, as shown schematically in Fig. 4. Here, the portion of each pile below the mudline is replaced with stiffness and damping impedance matrices. We show the case where a single beam element is used to model the free length of each pile. This will often be sufficient since, neglecting pile inertia forces, the variation of moment along the length of each pile is linear. The pile elements are
Detailed models
The most straightforward model is the detailed model, shown in Fig. 2. Here each pile is modeled with beam elements and the soil is modeled with linear or nonlinear, elastic or inelastic spring (or truss) elements. A model like this is not really necessary for the global analysis of most bridges, but it is useful for the analysis of special cases, or design studies.
Conclusions
We have discussed three approaches to the modeling of pile foundations for the seismic analysis of bridges. From simple to complex these models are the impedance matrix model, the individual pile model, and the detailed model. Of these models, we consider the individual pile model to be the most useful for the routine analysis of bridges. For a reasonable size of model, this model leaves all important aspects of the behavior of the structure in the global model where they can be readily
Acknowledgements
We wish to acknowledge the help of Ignatius (Po) Lam of Earth Mechanics Inc. who was the geotechnical engineer for both the San Diego–Coronado Bay Bridge and the new East Bay Bridge projects and who was very supportive of our efforts to develop the individual pile model for the former project. We also wish to thank Brian Maroney of the California Department of Transportation for his support of our efforts to develop a battered pile design for the new East Bay Bridge.
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