Expert system for the optimization of bridge orthotropic deck plates

Orthotropic plate steel decks offer an efficient solution for the design of long span bridges subject to weight constraints as well as for smaller spans when a fast erection is to be provided. The present work describes a cost optimization design procedure where force effects were evaluated using parametric finite element technology. Eurocode 3 was considered for the longitudinal verification of compact or slender ribs (taking into account open
and closed section shapes), while traffic models were assumed according to Eurocode 1, for strength, service and fatigue. The simplified definition of a cost metric allows a single objective optimization to be run using the modeFRONTIER environment. Results are described with reference to the commonly adopted designs for continuous bridge girders.

Torsional stiffness calculated by FEM and parametric response surface based on analytical formulae.

Practical design approach for orthotropic decks: Rules and limitations.
Common orthotropic deck design practice
Simplified approaches are commonly used for the verification of orthotropic steel decks, mainly due to the following facts:

• The torsional stiffness of the deck is of importance as it influences the diffusion of traffic load to the stiffeners underneath, especially for close shaped longitudinal ribs. For opened section ribs a decoupled approach can usually produce higher accuracy.
  
• The evaluation of transverse stress concentrations between the web of stiffeners and the upper plate can be difficult to estimate, especially when considering the high redistribution obtained from experimental tests.

Hence, the design practice is influenced by commonly accepted geometry solutions, that in most cases, comply with the minimum size requirements according to international standards.

Geometry and weld throat limitations according to international standards
The Eurocode 3 Part 2 - UNI ENV 1993-2:2002 provides the following limitations for highway steel bridges.
For the upper plate thickness, depending on the wearing surface thickness, the following limitations hold:

• Thicker than 12 mm for wearing surface more than 70 mm thickness, where traffic has to be expected.
  
• Thicker than 14 mm for wearing surface less than 70 mm thickness but more that 40 mm thick, where traffic has to be expected.
  
• The upper plate should be thicker than 10 mm for the remaining areas.

The space between the web of longitudinal ribs should be limited to:

• Less than 300 mm for areas subject to traffic loads.
  
• Less than 400 mm for the remaining areas.

The plate thickness for longitudinal stiffeners should be limited to:

• 6 mm for close shaped stiffeners.
  
• 10 mm for open shaped stiffeners.

The weld throat connecting the upper plate and the web of stiffeners should be:

• Greater than or equal to the stiffener web thickness.

Other geometrical requirements are addressed by the Eurocode regarding the areas where stiffners are connected to the cross beams.
Following is another example of shape constraints for orthotropic plates according to AASHTO LRFD Bridge Design Specifications (Third edition) at section 9 “Deck and Deck System”:
Section 9.8.3.7.1 States that the upper plate thickness should be not less than:

• 14.0 mm.
  
• 4 percent of the larger spacing of rib webs.

Furthermore, the rib web thickness for close shaped stiffeners should be not less then 6.0 mm.

An equation is then reported in section 9.8.3.7.2 which has to be fulfilled by the geometric orthotropic plate dimensions.
As the above paragraphs show, similar requirements are in place for European and American standards.

Modelling assumptions and structural systems for orthotropic deck stress evaluation
The resultant stresses acting on the orthotropic deck can be considered as the superposition of three different structural systems, as follows:

1. System I is the deck plate supported by longitudinal ribs.
   
2. System II is the deck plate with ribs as (elastically) supported by the transverse floor beams.
   
3. System III is the deck plate as part of the bridge cross section subject to longitudinal stresses induced by the global static behaviour.

The following paragraphs highlight the assumptions which have been considered in the present article.

Vertical Stiffness of floor beams
A rigid behaviour was assumed for the vertical stiffness of the cross floorbeams. This assumption is supported by the AASHTO sections that explain how an approximate analysis of the orthotropic deck system can be worked out, in particular:

• Section 9.8.3.5.2 “Decks with Open Ribs” states: “For rib spans smaller than 3000 mm for decks with shallow floor beams, the flexibility of the floor beams shall be considered when calculating the force effects in the ribs”. Owing to the fact that 3000 mm is regarded as a typical distance between floor beams, the current assumption does not appear to be heavily restrictive.

Stress field in the region of intersection between stiffeners and floor beams
Stress concentrations and fatigue life in intersection areas between the ribs and floor beams were not evaluated. Our opinion is that a slight change in thickness distribution could be accommodated to build the orthotropic plate at these locations, where the stress field is significantly influenced by transverse stresses.

System III stress distribution
The stress field owing to global bridge behaviour was neglected in the present work for the following reasons:

1. This study was focused on the optimum design of a general orthotropic plate geometry, as such the full bridge sections and internal action distributions were of limited interest.
   
2. According to reference [1], a reduction of the stress distributions coming respectively from system I, II and III can be performed due to the usual reserve experienced during experimental tests, compared to linear calculations. A typical reduction factor employed for system II stresses is 0.45. By noting that, for usual span to depth ratios of longitudinal bridge girders, the ratio between local stresses and global ones is around 0.5 (maximum benefit for main girders and orthotropic plate design). The results show that sufficient accuracy can be obtained without reduction of both system II stresses and admissible strength values, and without taking system III stresses into account.
   
3. It is worth noticing that local fatigue (local transverse bending of rib webs connected to the upper plate) is the most important factor that influences the design. A reasonable approximation introduced on linear longitudinal stresses does not have a big impact on the optimum plate geometry.

Calculation of force effects on orthotropic decks: Traditional analytic and numerical approaches.
Theoretical model: Huber’s equation
The most common theoretical model describing the out-of-plane behaviour of an orthotropic deck (structural system II) is the Huber approach, which results in the following bi-harmonic equation:
If the direction x is assumed as the longitudinal reference, the following meaning can be adopted for the constants:

• Kx is the longitudinal stiffness coefficient for out of plane (constant curvature) bending.
  
• H is the torsional stiffness coefficient for the deck plate.
  
• Ky is the transverse stiffness coefficient for out of plane (constant curvature) bending.

Evaluation of stiffness coefficients for most common rib geometries

Orthotropic deck plate dimensions.

For open section rib geometries, the following assumptions hold:

• The transverse bending stiffness Ky is small compared to Kx and can be neglected.
  
• The torsional stiffness H is small and can be neglected.

The Huber equation reduces to the equation of a simple beam, and the flexural behaviour of two adjacent stiffeners is fully decoupled.

For closed section rib geometries, only Ky could be neglected, while a significant torsional-flexural coupling exists as torsional stiffness plays an important role.
Stiffness coefficients for bending (constant curvature) behaviour can be easily evaluated by hand or analytic formulae available in literature (see reference [2]). Torsional stiffness of open section orthotropic plates can be deduced from analytical formulae as well, while closed section torsion requires a deeper investigation.
The solution of the partial differential equation is, for closed section ribs, more complicated and a tailored analytical or numerical method is required. The following approaches are usually adopted:

Analytical methods

Final parametric closed section model and applied rigid links.

An analytical method was developed to address the above stress evaluation. One of the most common approaches is the Pelikan-Engesser method which evaluates the internal action distribution in the longitudinal ribs through a two-step procedure that neglects the flexibility of the floor beams in the first instance but subsequently includes it.

Numerical methods
In recent years, the finite element method has achieved widespread popularity for the solution of structural linear and non linear problems. Furthermore, the evolution of finite element codes and optimization software has been such that the effort of parametric finite element modelling is becoming simpler and simpler. In this article, the Straus7 Finite Element Analysis System was used for structural calculations. The main features implemented in the Straus7 environment and extensively used in the current study are the following:

1. A user defined Plate/Shell to simulate the correct stiffness of a general shape rib and to work out integrated moments, out of plane forces. The latter elements also allow the mesh to be independent from the actual deck shape, to account for moving traffic load models required to be applied by international codes.
   
2. The Straus7 API interface allows finite element calculations to be performed without any interaction with the graphical interface. Hence, a computer program can be written to build parametric models and to extract results.

Torsional stiffness evaluation for closed section stiffeners orthotropic decks
The first issue addressed is the correct evaluation of the torsional stiffness of orthotropic plates with closed section ribs.
Parametric modelling
A parametric finite element model has been programmed with the following geometric variables:

• Cross distance between ribs (A).
  
• Total number of ribs.
  
• Upper plate thickness (B).
  
• Rib plate thickness (C).
  
• Upper rib dimension (D).
  
• Lower rib dimension (E).
  
• Height for the stiffener (F).

Figure 1 illustrates the physical meaning of the above values.

Restraint condition
Once the parametric model geometry is specified, a restraint condition has to be applied to obtain a constant cross curvature (torsional) behaviour from which the proper constant can be retrieved. This was enforced by using rigid constraints acting on the section plane and applying a specified rotation to the longitudinal ends of the model. The following picture illustrates the final parametric model.

Comparison with an open section-like analytical formulation
To confirm that an analytical formulation cannot be obtained for the torsional behaviour of a closed section rib plate, a simple Design of Experiment modeFRONTIER simulation was performed to explore the potential values as a result of changing the input variables (dimensions and thickness). The resultant values were compared to the ones obtained by using the following formula:

where:

• E is the elastic modulus of steel.
  
• h is the upper plate thickness.
  
• υ is the steel poisson ratio.
  
• G is the shear modulus of steel.
  
• J is the torsional constant for the longitudinal rib.
  
• a is the distance between two adjacent stiffeners.

The value of J was calculated with the following expression, typical for close section members (Bredt approach):

where:

• Ω is the mid plane area for the stiffener.
  
• c is the line describing the closed section perimeter.
  
• s is the thickness of each steel plate which forms the rib.

A parametric linear user response surface was used to work out the best correction to the above formulation to fit the obtained torsional constants. Figure 3 illustrates the obtained values, it may be deduced that no significant correlation can be drawn between the numerical and analytical results.

This paper summarizes the thesis work carried out by P. Locardi, with the supervision by D. Schiavazzi and S. Odorizzi (EnginSoft).

For further info, please contact:
Daniele Schiavazzi
info@enginsoft.it

Article published in the Magazine: EnginSoft Newsletter Year 4 n.3