FULL SCALE NONLINEAR SEISMIC BRIDGE DESIGN ANALYSIS … · or displacements often imported manually into spreadsheets. ... design changes, ... FULL SCALE NONLINEAR SEISMIC BRIDGE

FULL SCALE NONLINEAR SEISMIC BRIDGE DESIGN ANALYSIS ON A DESKTOP COMPUTER


Peter R. Barrett

Vice President, CAE Associates, USA

Eduardo Salete

Professor, UNED, Spain

THEME

Structural Analysis and Design

KEYWORDS

Finite element analysis, FEA, Seismic analysis, Concrete design, HPC Computing, Base isolation, AASHTO, Nonlinear, Contact, Time History

SUMMARY

With advances in software and desktop computer performance it is now possible to perform automated design calculations using full scale, real time nonlinear simulations. This paper presents an example of a detailed bridge seismic analysis that simultaneously performs analysis and design calculations automatically. The model includes nonlinear hysteresis base isolation supports which enable the predictions of local and global deformation histories. The fidelity of the model is sufficient to capture time-marching stresses in the columns and concrete deck where automated code checking is applied. The analyses are run using shared memory parallel processing to reduce computational resources.

In most design projects high end analysis and local concrete design are preformed independently with data transfer limited to isolated forces, moments or displacements often imported manually into spreadsheets. The more complex analyses are limited to a validation role as opposed to being implemented early in the design cycle. If these analyses discover necessary design changes, these modifications can lead to time and cost overruns.

The modeling technique presented herein demonstrates the use of integrated commercial FEA software that automates the design of concrete reinforcement during the analysis simulation meeting the extreme loading conditions of the seismic event. Bending, shear and torsional reinforcement are designed on an element-by-element basis where the enveloping rebar requirements can be


graphically displayed. Displacement, stress and reaction time history data is also captured.

Nonlinear effects include large deflections and surface-to-surface contact elements used to model expansion joints as well as the hysteretic response of the base isolation supports. The model includes a combination of beam, shell and 3-d solids, where the post processing of the 3d solid elements is performed using a unique method of transforming these results into design based forces and moments.

These calculations are done on a desktop computer utilizing parallel processing algorithms that leverage its multiple cores. Benchmark results on the computational efficiencies are reported.

1: Example Arch Bridge Finite Element Model

In order to highlight the advances in the seamless integration of analysis and design and the efficiency of HPC solutions, reinforcement design of a concrete arch bridge under seismic loading is presented. The example finite element calculation provides an illustration of various modeling techniques integrated into a single analysis. An overview of the analytical model is presented via a description of each element type with its corresponding application:

3-d brick finite elements are used to mode the concrete bridge deck. Multiple elements through the thickness are utilized to accurately capture the in-plane forces, out-of-plane bending and shear loads. Complicated bridge deck geometry can only be modeled with bricks. Software and hardware now make it possible to model either partial or even complete bridges using 3-d continuum elements. Explicit concrete cracking simulation can also be added to these elements if material damage is an analysis requirement.

3-d shell elements are utilized to model the concrete arch section of the example bridge. The span to thickness ratio of the arch is such that a linear through wall thickness stress distribution is sufficient, hence the use of shell elements is warranted to simplify this analysis model.

3-d beam finite elements are used to model the bridge piers and foundation piles. The piles provide axial and flexural support that can be simulated with beam elements; the forces, moments and shears calculated in these elements are used directly for design purposes. Reinforcement design is performed to determine the quantity of rebar required or if reinforcement is prescribed it is checked to determine factors of safety or utilization ratios.


3-d composite layered elements [1] are used to model the base isolation pads. These nonlinear composite components provide critical energy dissipation during a seismic event. The layered elements deployed in the model explicitly simulate this composite response. These elements utilize a highbred of steel and hyperelastic rubber material properties. Additional beam elements with elastic-plastic material properties could be added to the model to simulate a lead core if applicable.

3-d surface-to-surface contact elements are used to simulate the expansion joints on either end of the bridge that allow separation and relative sliding between the bridge and abutments. The contact elements also introduce frictional resistance damping into the finite element model.

A large lumped mass element is utilized to simulate the foundation support to which the seismic accelerations are applied. This large mass element eliminates the need to explicitly model the soil. The ground motion applied to this mass is simultaneously applied to all support points of the bridge via coupling equations.

Solid-to-Shell design elements [2] are "pseudo shell" elements that are used strictly for design purposes. These shell elements have no active degrees of freedom, but are used to store forces and moments computed automatically via numerical integration of the through thickness stresses in the 3-d solid elements. Nonlinear response of the solid elements is transformed to equivalent forces and moments necessary for code design evaluations. These variable thickness pseudo shells are used explicitly to design the deck reinforcement according to the prescribed code (AASHTO, ACI, Eurocode, etc). Required user input prior to the analysis is limited to the reinforcing material and cover depth.

Figure 1 illustrates the concrete arch bridge finite element model where the shell and beam elements have been expanded from surfaces and lines into three-dimensional solids to illustrate their cross-section properties.


Figure 1: Arch Bridge Finite Element Model

In order to correctly characterize the true bridge deformations, the nonlinear material behavior of the base isolation supports must be modeled. There are several options for modeling these devices depending upon their design including nonlinear springs and/or explicit modeling using layered elements. Figure 2 illustrates a single layered element along with its force-deflection behavior. A lead core modeled with a plastic beam would increase the hysteresis loop and damping effects if it were added to the design.

Figure 2: Layered element simulating an elastomeric bearing with force-displacement response

Boundary conditions applied to the finite element model consist of coupling equations used to tie the foundation supports to the central large mass where


the seismic loading is applied. A fixed displacement of the large mass is applied and then removed during the analysis and replaced with the equivalent reaction force. Figure 3 illustrates these coupling equations and the reaction force (red arrow) applied to the large mass.

Figure 3: Boundary Conditions Applied to the Arch Bridge Finite Element Model

If multiple ground motion input is required, separate groups of coupling equations can be associated with each large mass. For example the ground motion on one side of the valley could be defined out-of-phase with the opposite side of the valley. The model could also be expanded to include a column of soil with continuum or equivalent nonlinear spring elements to model soil-structure interaction. Independent or an offset of the vertical acceleration histories can be applied in the horizontal directions.

2: Analysis Sequence

A sequential analysis process is used to simulate the seismic response of the arch bridge. The analysis procedure includes a static pre-load followed by a nonlinear transient dynamic analysis. The step-by-step loading sequence deployed to represent the bridge seismic response is summarized as follows:

1. First, a static g-load representing the gravitational loading on the bridge and supporting structure is defined. This acceleration load is applied to all elements except the large point mass representing the foundation. The mass of the bridge deck is increased to represent the effects of the


non-structural dead loads. The analysis type is prescribed as transient dynamic for the entire simulation; however, for the static step the effects of time integration are turned off. The large mass displacements are fixed at this point in the solution.

2. A second static load step is analyzed with no loading changes; therefore, when the transient analysis is activated no initial velocities are imposed on the bridge (two results sets with the same displacements creates zero delta displacements and hence, zero velocity)

3. The time integration transient response is turned on and the fixed displacement is replaced with the equivalent reaction force to allow ground motion to occur.

4. The transient time history acceleration loading is imposed on the large mass element which in turn is coupled to the foundation supports. The first 20 seconds of the 1940 El Centro (Imperial Valley) quake [3] is used as input for the example analysis (see Figure 4) where 2/3 of this vertical acceleration is applied along the length and transverse to the span. A time step size of .02 seconds, driven by the acceleration time history captures the inertial response of the bridge. Local damping is provided locally via the contact and base isolation elements. Global damping is input via stiffness proportional material damping linked to the fundamental frequency of the bridge.

‐3.00E‐01

‐2.00E‐01

‐1.00E‐01

0.00E+00

1.00E‐01

2.00E‐01

3.00E‐01

4.00E‐01

0 2 4 6 8 10 12 14 16 18 20

Accleration (g's)

Time (seconds)

20 Second Earthquake Input of the 1940 Imperial Valley Earthquake

El Centro 1940 North ‐ South Record

Figure 4: Initial 20 seconds of El Centro Earthquake Record Analyzed


3: Analysis Results

Running a full transient dynamic nonlinear analysis allows the analyst the flexibility of storing a limited or full suite of solution data. Time history displacements, stresses, strains, forces, moments and reactions are available for all applicable nodes and elements in the model. However, without automation in the postprocessing, the quantity of data quickly becomes overwhelming since one can easily create a terabyte of data if all results in all elements were stored at every point of the time history solution. It is critical to automate the postprocessing since the value of the detailed analysis quickly diminishes if the postprocessing steps are more time consuming than the analysis. Contour plotting, sorting and enveloping algorithms allow the analyst to integrate these routines directly into the analysis sequence. Figures 5 through 7 provide examples of results available as part of the postprocessing that can be automatically generated including displacement histories, model displacement contours and member forces. Animations are also a great resource for interpreting results from a time history solution.

‐0.3

‐0.25

‐0.2

‐0.15

‐0.1

‐0.05

0

0.05

0.1

0.15

0.2

0.00E+00 2.00E+00 4.00E+00 6.00E+00 8.00E+00 1.00E+01 1.20E+01 1.40E+01 1.60E+01 1.80E+01 2.00E+01

Displacement (meters)

Earthquake Time History (Seconds)

Bridge seismic displacment at deck center and abutment

Deck center ‐ vertical

Deck center ‐ along span

Deck center ‐ transverse to span

Abutment ‐ vertical

Abutment ‐ along span

Abutment ‐ transverse to span

Figure 5: Arch Bridge Time History Displ. at Bridge Centerline and Abutment


Figure 6: Arch Bridge Displ. Scaled by 10x @ time = 5 seconds

Figure 7: Arch Bridge Example Pier Moments


4: Design Evaluation

Section 5.4 of the AASHTO Guide Specifications for LRFD Seismic Bridge Design manual [4] provides specifications for seismic bridge analytical procedures. Three methods are described:

1. Equivalent Static Analysis

2. Elastic Dynamic Analysis

3. Nonlinear Time History Method

The static method is limited to short to medium spanned, regular configurations only. The code commentary states that the elastic dynamic analysis procedure does not provide an adequate level of sophistication to estimate the dynamic behavior of the bridge. The nonlinear time history method is recommended for the case where seismic isolation is present. In the code commentary to Section 5.4.4, the nonlinear dynamic analysis value is stated as the more comprehensive analysis method because the effect of the inelastic behavior is included directly in the simulation. Seismically isolated structures with long periods or large damping ratios require a nonlinear dynamic analysis because the simulations using effective stiffness and damping may not properly represent the effect of isolation units or the response of the structure.

Since the nonlinear method provides the most accurate analysis and is acceptable practice by the code authority, it is suggested herein to take these analytical results one step further. Utilize the model typically used for computing accurate global displacements and apply these results directly to the bridge design process. Using the force and moment histories stored in the pseudo deck shell elements, the user can automate design operations to determine reinforcement requirements over the entire deck surface. Figure 8 illustrates an example contour plot of required bottom surface axial reinforcement.

Figure 8: Arch Bridge Roadway Required Slab Bottom Lateral Reinforcement


An axial load-bending moment interaction diagram is internally computed for each shell integration point, according to the design code. This interaction diagram determines the critical force and moment design limits, taking into account the reinforcement distribution. From these interaction diagrams, the needed reinforcement amount is computed. Figure 9 illustrates a design check interaction diagram.

Figure 9: Arch Bridge Example Interaction Diagram

Different reinforcement amounts are obtained for each seismic step. The envelope of reinforcement required to guarantee the safety of the bridge is easily computed as a direct postprocessing option that can be integrated into the analysis input.

5: HPC Computing

High-performance computing (HPC) is a term that arose after the expression "supercomputing" and now typically means taking advantage of multi-core and multi-processor computers. Desktop workstation and even laptops are now


all sold with multiple cores. Engineering software that takes advantage of this widespread availability of multi-core processors can result in impressive performance gains on medium to large problems. Historically these large performance gains were limited to explicit structural dynamics and CFD calculations. However these economical machines such as a dual processor quad or hexa-core machine can provide significant computational advantages for implicit dynamic solutions of even a medium sized model. Table 1 presents some benchmarks for the example concrete arch bridge seismic solution. The model has 34,000 active degrees of freedom and required 2006 cumulative iterations to solve. The seven core solution cut the run time by over a factor of 2 which can be significant for large seismic analyses.

HPC Solution Performance

Number of Cores CPU Time Wall Clock Time(Hours) (Hours)

1 3.55 3.642 4.29 3.564 5.08 2.537 5.13 1.67

11 6.69 1.59

Table 1 - HPC Performance for Concrete Arch Bridge Seismic Design Analysis

6: Conclusions

This paper provides an example of how engineers can leverage current software and hardware to generate faster and more robust seismic bridge design evaluations. Key points include:

Utilize integrated code checking in a high fidelity finite element simulation including concrete design from 3-d continuum finite element solutions.

Reduce solution times using desk top parallel processing for even medium size finite element models.

Automate the interpretation of detailed response data including graphical representation of forces, moments, displacements, stresses, and strains through enveloping of seismic time history results.


REFERENCES

1. ANSYS®, Release 12.1, Help System, Solid Shell 190 Element Documentation, ANSYS, Inc.

2. CivilFEM®, Release 12.1, Theory Manual , Solid to Shell Design Procedure, Ingeciber, SA

3. "May 1940 Time History Data Files from El Centro Site Imperial Valley Irrigation District", Pacific Earthquake Engineering Research Center http://peer.berkeley.edu/research/motions/

4. AASHTO Guide Specifications for LRFD Seismic Bridge Design, American Association of state highway and transportation officials. 2009

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FULL SCALE NONLINEAR SEISMIC BRIDGE DESIGN ANALYSIS … · or displacements often imported manually into spreadsheets. ... design changes, ... FULL SCALE NONLINEAR SEISMIC BRIDGE