A Simplified Method for Modeling Soli-Structure Interaction for Rigid Frame Structures

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A Simplified Method for Modeling Soil-Structure Interaction

for Rigid Frame Structures

Hari Aamidala, PE, M.ASCE1;and John Kim, Ph.D., PE2

1Project Manager and Lead Structural Engineer, Parsons Brinckerhoff, Inc., 465

Springpark Place, Herndon, VA 20170. E-mail: [email protected] Professional Associate and Supervising Bridge Engineer, Parsons

Brinckerhoff, Inc., 530 East Main Street, Suite 701, Richmond, VA 23219. E-mail:

[email protected]

Abstract

A structure can be modeled using a multitude of methods, ranging from simple

mathematical models with assumptions based on engineering judgment, to complex

3D models that attempt to capture every single variable that affects the behavior ofthe system. This paper, utilizing analysis of Rigid Frame Structures, explores

different methods of modeling structures in an effort to understand the benefits, or

lack thereof, of introducing more variables and complexity into a model. The focus ison a simplified method for the modeling of soil-structure interaction and analyzing a

rigid frame structure. Through this effort it was found that, rather than use an

expensive 3D modeling software, commonly available structural analysis software

like LARSA can be used to create a simple soil-structure interaction model. Thissimple method allowed us to quickly create numerous models to efficiently analyze

different alternatives.

INTRODUCTION

In developing models, an engineers primary goal is to understand the behavior of a

simple or complex structure, as best as possible. A model is designed to represent the

behavior of a system by using a set of variables and a set of mathematical equationsthat establish relationshipsbetween those variables. Ultimately, models are used to

help us study the effects of different variables and make better decisions. This study

was primarily focused on evaluating different methods to model soil-structure

interaction for rigid frame structures.

Rigid Frame type construction has been utilized for cut-and-cover tunnels and bridges

for more than 100 years (PCA 1936). Some of these structures have utilized archconcept while others are simply designed as frames. This concept was originally

developed as a cost effective and attractive alternative to conventional arch bridges;

however, it is extensively used today in top down construction.

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Prior to computer software, numerous manual methods like column analogy (Cross

1930) and moment distribution methods (Timoshenko et al 1965) were utilized tomodel rigid frame structures, which involved complex time-consuming analyses. In

recent years this process has been replaced with computer analysis that involves 3D

modeling, using software specifically intended for soil-structure interaction. Though

3D modeling using these complex software for soil-structure interaction could yieldgood results, it can be clumsy to work with and could inadvertently introduce errors

into the analysis due to the use of large number of variables with inherent

assumptions, and it could also be cost prohibitive.

In an effort to simplify the modeling and to improve the performance of the analysis,

a simplified structural model was created, which would only use the basic structuralelements, and grounded linear and non-linear spring elements to model soil-structure

interaction. To evaluate this simple method and to compare with other modeling

methods, a couple of case studies are being utilized Load Rating of Rigid FrameBridges and Design of a Rigid Frame Rail Structure using top-down construction.

MODELS

A model is a physical or mathematical representation of a real world system. A

mathematicalmodel is designed to represent the behavior of a system by using a setof variables and a set of mathematical equations that establish relationships between

those variables. A physical model can especially be useful in validating a

mathematicalmodel when Structural Engineers are trying to understand the behaviorof a real world structure (Gomez-Rivas et al 2012). However, models cannot be

expected to represent the behavior of a real world system with 100% accuracy. Agood model is the one that provides enough accuracy to make sound decisions. A

model cannot ever be totally complete; therefore, there is no need to refine the model

unless the quality of the decisions made is affected by the inaccuracy of the model.When credible observations conflict with the predictions made by the model, either

the data/assumptions are incorrect or the model itself is incapable of providing

enough accuracy. Understanding what needs refinement is important to improve the

accuracy of the results.

As an example, Figure 1 shows a mathematical model that is represented by an

equation, which helps predict compressive strength of a particular concrete at a givenpoint of time, based on past concrete compressive strength test data. If the model

represented by the equation in Figure 1 cannot accurately predict the compressive

strength of this particular concrete, it could either be because the concrete test datawas incorrect or the mathematical equation used to best fit the data was incorrect. So

in this particular situation rather than only focus on the equation, the modeler should

also look at the data and the strength tests used to collect that data, to improve theaccuracy of the model.


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RIGID FRAME STRUCTURES

A rigid frame concrete bridge is a bridge structure where the deck is integrally or

rigidly connected to the abutments and piers of the structure. It is typically simpler

and more economical to build a rigid frame structure for top-down construction. Oneof the direct benefits from the continuity or rigidity is that the moments are small in

the sections near the center of the deck of the rigid frame structure compared with

corresponding moments in a simply supported deck of the same length (PCA 1936).The other significant benefit is reduced maintenance cost by eliminating bearings and

joints at the abutments compared to a conventional beam and deck type bridge.

Therefore, reducing the life cycle costs significantly. Also Rigid Frame ConcreteBridges may be widened with little alterations to the existing bridge, without

interfering with normal traffic.

Figure 2 shows a 2-span continuous Concrete Rigid Frame Bridge for I-395 SB over

George Washington Memorial Parkway. This type of construction was typical forbridges over George Washington Memorial Parkway, which was designed for

recreational driving built in 1950s thru 1960s, in the Virginia suburb of DC. All RigidFrame Bridges on GW Pkwy utilize the arch concept resulting in economical and

aesthetically pleasing bridge structures.

Figure 2. 2-Span continuous Rigid Frame Bridge, I-395 SB over GW Pkwy


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MANUAL ANALYSIS METHODS

Before the advent of computers, all structural analysis was performed manually by

solving equations developed using calculus or using graphical methods developed to

simplify these analyses. Some of the methods used to manually analyze are discussed

below.

Column Analogy Method

This was a method proposed by Hardy Cross where bending moments in simpleframes may be computed by a procedure analogous to the computation of fiber

stresses in short columns subject to bending, and that slopes and deflections in these

structures may be computed as shears and bending moments, respectively, onlongitudinal sections through such columns (Cross 1930). Professor Cross analyzed

24 frames by this method and presented a chart, shown in Figure 3, which could be

used for estimates and preliminary design of typical rigid frame structures that werebuilt during that time (PCA 1936).

Figure 3. Chart for Column Analogy Method (PCA 1936)

Table 1. Crown Moments created by uniformly distributed load (PCA 1936)


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Moment-Distribution MethodThis method first introduced by Hardy Cross was widely used for the analysis ofstatically indeterminate frame structures (Patil 2013). Typically analysis of frames

required solving a system of linear algebraic equations, which can be readily solved.

However, in many cases, it required solving more complicated system of equations,

and their solution was cumbersome. Moment-Distribution Method was one of themethods where successive approximations were used to analyze these structures.

In this iterative method, all joints are first assumed to be restrained against rotation sothat fixed-end momentsare calculated. The joints are then successively released. The

unbalanced moment, which is the algebraic sum of all the fixed-end moments at a

joint is calculated, this represents the total turning effort exerted on the joint(Timoshenko 1965). Releasing a joint is equivalent to distributing the unbalanced

moment among the members meeting at the joint based on the relative stiffness of the

members, using a factor called distribution factor. The final end moments at the jointare then obtained by superimposing the distributed moments on the previously

calculated fixed-end moments. One half of the distributed moment is carried over tothe far end of the member. The released joint is again restrained before moving to the

next joint. The same sets of operations are carried out at each joint till all the jointsare completed. This completes one cycle of operations. The process is repeated for a

number of cycles till the values obtained are within the desired accuracy (Patil 2013).

CASE STUDY 1: LOAD RATING OF RIGID FRAME BRIDGES

As part of a Maintenance and Rehabilitation effort, I-395 Bridges over George

Washington Memorial Parkway were evaluated for structural capacity. These bridgesshowed cracks, delamination and significant efflorescence on the underside of the

deck, which possibly stemmed from drainage issues on the top of the deck.

I-395 NB Bridge over George Washington Memorial Parkway was concrete rigid

frame structure that was composed of a two single span rigid frames with a

continuous deck, which was built in 1950. It utilized an arch concept with a thin deck

at the crown. I-395 SB Bridge over George Washington Memorial Parkway was atwo span continuous concrete rigid frame structure, which was built in 1960. It had a

similar thin deck at the crown utilizing an arch concept. See Figure 2 for the picture

of the I-395 SB Bridge.

To load rate these structures, we utilized LARSA for modeling the structure. The

tapered leg elements of the frame were modeled as single members while the deck ofthe frame was divided into several tapered members to model the parabolic underside

of the deck, thereby allowing us to model the arch effect of the parabolic underside.

Also as part of this effort, several different manual analysis methods were looked at,which were possibly used during the original construction of these structures.


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In an effort to compare the LARSAstructural model with Column Analogy Chart, a

simple 1 kip/ft. uniformly distributed load was applied on a single span rigid frameLARSA model, shown in Figure 4, and the results were compared with the results

obtained from the Chart, shown in Figure 3. Moments at the crown and the knees,

comparing Column Analogy Chart vs. LARSA model, are shown in Table 2.

Figure 4. Model for comparison of Column Analogy Chart vs. LARSA model

Chart LARSA

Mcrown 3.81 k-ft 3.68 k-ft

Mcorner -10.87 k-ft -10.65 k-ft

Table 2. Results of Comparison of Column Analogy Chart vs. LARSA model

It is important to note that in our load rating model, the effect of horizontal earthpressure was ignored for these structures. In comparison with the effect of the other

loads, earth pressure moments are generally of little importance in stress

determination (PCA 1936). At the crown, both vertical dead and live loads create

tension at the bottom of the deck, while the earth pressure creates tension at the top ofthe deck; it is therefore conservative to ignore the earth pressure at the crown. The

corner momentdue to earth pressure, for this type of concrete rigid frame structure,

would be negligible compared to the moment due to dead plus live loads thereforecan be ignored. However, it is important to note that soil stiffnessdefinitely affects

the fixity at the foundation. See Figure 5 for the illustration of foundation fixity.

Figure 5. Foundation Fixity, which is to be assumed based on soil stiffness

If the soil behind the abutments is too stiff then regardless of the foundation type,

supports will need to be modeled as fixed. On the other hand if the soil is too soft

then supports will need to be modeled as hinged. Actually, the supports are rarely


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hinged or fixed, but the foundation conditions lie somewhere in the range between

these two extremes (PCA 1936). Therefore, it is important to model it correctly.

CASE STUDY 2: DESIGN OF A RAIL STRUCTURE RIGID FRAME

As part of Dulles Corridor Metrorail Project Phase 2, WMATA Metrorail System is

being extended through a busy Dulles Toll Road (DTR) Corridor. Based on an early

2012 study by Fairfax County Department of Transportation, extending Town CenterParkway across DTR Corridor was identified as a crucial connection between Reston

downtown on the north of DTR, and businesses on the south. See Figure 6 for the

aerial view of the project location. After evaluating several overpass and underpassalternatives, an underpass alternative was identified as the most feasible solution.

However, this connection was only in the Countys long-term plan. WMATA was

concerned that once the Metrorail was built through the DTR corridor, construction ofa future underpass would lead to severe service disruptions.

To minimize future service disruptions, through a multi-agency and consultant effort,

it was decided that a rail structure should be constructed as part of the DullesCorridor Metrorail Phase 2 Project that could allow for a future underpass with

minimal service disruptions to the Metrorail. This was only possible with a top-down

construction concept that could be achieved by a 2-Span Rigid Frame ConcreteStructure that is composed of Secant Pile WallAbutments and Piers, and Reinforced

Concrete Slab, cast integral with the walls. This would have an added advantage of

eliminating bearings, and abutment and pier joints, which would significantly reducethe future life cycle costs. Also it was important to build a structure that would not

require bridge inspections in the interim condition before the underpass isconstructed.

Top-Down ConstructionTo utilize top-down construction concept for this project, Secant PileAbutments and

Pier, from the existing ground, will be constructed first. Therefore, the support of

excavation walls will serve as permanent substructure units, in the future. Next the

roof slab is constructed as a slab-on-grade and is rigidly connected to the Secant Pilewalls. Secant Pile Walls were preferred over Tangent Pile Walls because the superior

performance of the Secant Pile Walls was expected in this shallow ground water

condition. Once the Roof Slab is constructed, the Metrorail tracks will be placed onthe top. Until the future underpass needs to be constructed, this structure will be

hidden from plain sight. See Figure 7 for the illustration of Rail Structure elevation

prior to construction of the underpass. When we are ready to construct the underpass,in the future, bridge structures will need to be constructed for Dulles Toll Road and

Airport Access Road, and the underpass can be extended by excavating under the roof

slab of the Rail Structure. Apart from any architectural treatment to the wall face,Secant Pile Walls, without any additional work, should be capable of serving as

Bridge Substructures. See Figure 8 for the illustration of Rail Structure elevation with

the underpass.


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Soil-Structure Interaction Modeling

One of the key concerns with the analysis of this structure was that we needed to

model the Structure and the Soil-Structure Interaction using a method that would

allow us to quickly modify the span lengths, under-clearance, different soil

parameters, etc.

After evaluating different modeling software and different methods of modeling soil-

structure interaction, it was decided that a complex finite element model, withStructure and Interface elements, and hardening soil model, would be too clumsy to

allow us to evaluate different scenarios for this project. See Figure 9 for an

illustration of a possible 3D finite element soil model for a similar structure. It isimportant to note that though finite element software offer flexibility to create

complex soil models, they would create a model with too many elements based on too

many assumptions, creating an unwieldy model.

Figure 9. Illustration of a 3D Finite Element Soil Model

Creating a simple model that can provide the desired level of accuracy is really

important to any Structural Engineer. With that in mind a LARSA model was created

using simple structural elements, and linear and non-linear grounded spring elementsto model soil stiffness (Kim 2009). This significantly reduced the number of variables

in the model, thereby providing the Structural Engineer with good control of the

analysis without sacrificing the quality of the results. The model was simple and

allowed us to pre and post process data using excel spreadsheets. See Figure 10 forthe LARSA model created for this project.

Figure 10. Illustration of the LARSA model


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Soil Stiffness for the grounded spring elements can be calculated using modulus of

subgrade reaction for the different soil material present at different elevations.Modulus of subgrade reaction can be calculated using triaxial compression tests on

compacted soil samples, plate load tests, or estimated from Standard Penetration Test

blow counts, based on the soil type. For a structure width of 7 feet, if the grounded

spring elements are placed at every 1 foot vertically, for a modulus of subgradereaction of 75pci, the Spring Stiffness (K)can be calculated as shown in the equation

below.

Grounded spring element is defined using a single joint, in LARSA, with no length

associated with the element. The element represents a spring connecting a joint in thestructure to the ground at its undeformed location. Linear Grounded Spring Elements,

with same stiffness in tension and compression, are used to model the foundation.Non-Linear Grounded Spring Elements, when soil has stiffness in compression but

not in tension or vice versa, are used to model the soil fill behind the abutments.

Figure 11. Illustration of Spring Elements in the LARSA model

CONCLUSIONS

The key takeaway from this effort is that, no model is totally complete, it is important

to simplify the model to only include whats needed to make sound decisions. It is

important for an engineer to understand all the variables used and assumptions madewith all the variables in the model. Also based on the level of accuracy that can

reasonably be achieved, Structural Engineers should consider increasing the safety

factors, beyond code requirements, to account for the model uncertainty (Alvi, 2013).

The use of linear and non-linear grounded spring elements is an efficient method to

model soil stiffness, even when there is a significant variation in the soil profile,compared to creating an unwieldy model built using hardening soil model and

interface elements. This simplified approach can allow engineers to make quickchanges to the model and evaluate different alternatives with relative ease.


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REFERENCES

Alvi, I. (2013). Engineers Need to Get Real, But Cant: The Role of Models.

Proceedings of 2013 ASCE Structures Congress, Pittsburg, PA.

Cross, H. (1930). The Column Analogy, University of Illinois, Urbana, IL.

Gomez-Rivas, A., Pincus, G., and Tito, J.A. (2012) Models, Computers andStructural Analysis Proceedings of 10th

Latin American and Caribbean

Conference for Engineering and Technology, Panama City, Panama.

Kim, J.S., Garro, R., and Doty, D.A. (2009). A simplified Load Rating Method forMasonry and Reinforced Concrete Arch Bridges. Proceedings of AREMA

Annual Conference, Chicago, IL, September, 2009.

PCA (1936).Analysis of Rigid Frame Concrete Bridges without Higher Mathematics,4

thEd., Portland Cement Association, Chicago, IL.

Patil, P.R., Pidurkar, M.D., and Mohankar, R.H. (2013) Comparative Study of End

Moments Regarding Application of Rotation Contribution Method (KanisMethod) & Moment Distribution Method for the Analysis of Portal Frame.

IOSR Journal of Mechanical and Civil Engineering, 7(1), 20-25.Stern, D.P. (2007). From Stargazers to Starships. NASA, Greenbelt, Maryland

http://www-istp.gsfc.nasa.gov/stargaze/Scolumb.htm(Jan. 10, 2015)Timoshenko, S.P., and Young, D.H. (1965). Theory of Structures, 2

ndEd., McGraw-

Hill, New York.


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A Simplified Method for Modeling Soli-Structure Interaction for Rigid Frame Structures

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