Expert system for the optimization of bridge orthotropic deck plates

Strength Traffic Load Model

Fatigue Traffic Load Model

Figure 4 - Traffic load models used in parametric FEM analysis.

Finite element modelling of orthotropic steel decks: A
parametric approach
Two other parametric finite element models were investigated to address the following needs:

• A local (system II) model whereas user defined Plate/Shell elements were employed to work out the longitudinal strength stresses and fatigue stress ranges using the traffic load models suggested by Eurocode 1. Note that this model does not give any information on transverse stresses related to cross section deformation, due to the fact that a static condensation procedure was employed to simulate the local stiffness.
• A parametric beam model was built where the upper plate and rib walls were simulated as 2D Beam elements. The latter model allows the local stress concentration under the loading axles to be taken into account, especially for fatigue life evaluation.

The following paragraphs provide further insight into how load and restraint conditions were considered for the above parametric models.

Local User - Defined Plate Model
The picture 4 illustrates traffic load models that were adopted for strength and fatigue limits states according to Eurocode 1.
The bridge deck was hence subdivided into traffic lanes, and loads were applied according to the following models:

a. Traffic load model 1 was applied to ultimate limit states. It includes both, a tandem load with four 150 kN concentrated forces (main lane) and a distributed force equal to 9 kN/m2 (main lane) acting both, within design lanes and between them. Many load conditions were investigated due to the fact that the tandem loads are allowed to move longitudinally on the bridge.
b. Fatigue model 3 was applied to the evaluation of stress ranges.

Figure 5 - Traffic loads as applied to the parametric FEM models.

The picture 5 illustrates how load conditions have been applied to the parametric local model.
A statically determinate restraint condition was applied to the in-plane stiffness. Fixed vertical restraints were placed at floor beam locations.

Figure 6 - Transverse Beam models used to work out local bending stresses and ranges.

Transverse Beam Model
Beam elements were used to evaluate the local transverse stress concentrations. The picture 6 illustrates the parametric 2D Beam model.
Concentrated springs were applied to simulate the vertical stiffness between floor beams. A rotational stiffness was also applied to each stiffener decoupled form the vertical one. A rigid frame was used to decouple vertical and rotational deformability for closed section stiffeners.

Tuning procedure for the Transverse
Beam Model
Once the parametric transverse beam model was simulated, the depth of beam elements had to be tuned to allow for meaningful results to be extracted. The depth was set in order to produce a bending stress at the rib web connection to the upper plate equal to the fatigue limit as extracted from Eurocode 3.

Figure 7 - Minimum elastic areas for positive and negative bending moments.

Figure 8 - Minimum areas for positive and negative bending moments checked according to EC3.

Figure 9 – Minimum cost geometries for positive bending moments including shear and welds.

Figure 10 - Minimum cost geometries for negative bending moments including shear and welds.

Orthotropic deck longitudinal
performance according to Eurocode 3
Once the complete set of internal forces and stresses have been determined for the orthotropic plate, code checks have been implemented according to Eurocode 3. Compact as well as slender sections have been addresses for in-plane and out-of-plane behaviour. Shear has been taken into account as well as welds connecting upper plate and rib.

Effective width
While traditional approaches (see reference [1]) evaluated the effective width as a function for the span of the stiffener and the longitudinal position along the static scheme only (i.e. negative or positive bending, cantilevered etc.), part 1-5 of Eurocode 3 allows to account for the ratio between the upper plate and stiffener cross sections.

Code check for class 1,2,3 and
4 sections
As highlighted in the following sections, all section classes had to be taken into account as the elastic optimum, for both opened and closed section ribs for a slender section type, is found precisely where the maximum increment of flexural inertia with minimum increase of cross section area can be obtained. In-plane and out-of-plane section classes were determined separately, the influence of shear was also considered for slender webs. Code checks were performed as an envelope of single force effects and combined actions.

Welding code checks
The shear stress produced by torsion was taken into account in the evaluation of weld throats.
Fatigue code checks
Fatigue checks were performed, both on longitudinal stresses coming from the global User-Defined Plate model and transverse stresses, obtained by the 2D Beam section analysis. The damage coming from parallel normal stresses was evaluated separately from the one given by other orthogonal shear and normal stress components. A coefficient of λmax=1.7 was assumed according to reference [4]. Stress ranges were also calculated assuming the only longitudinal traffic cycles act on the orthotropic plate deck.

Cost model for orthotropic steel decks
As highlighted in reference [3], the complete cost evaluation for an orthotropic deck plate can be subdivided into:

• Cost of material.
• Cost of fabrication.

In particular, the fabrication cost has to include cutting, positioning of steel plates, welding, post-welding treatments, application of paint, etc., thus leading to an expression similar to the following:

The above expression appears to be rather complicated and difficult to evaluate in normal case, because geographic location and technological know-how may have a significant influence on the above factors. A simplified approach was therefore preferred which takes into account both the cost of material and welding, divided by the transverse distance between stiffeners. The following expression was employed:

where:

• Ametal is the metallic area for the orthotropic deck plate.
• Aweld is the weld area in the deck plate cross section.
• K1 is the factor for the cost of base metal (€/kg)
• K2 is the factor for the time taken due to welding (€/hour)
• K3 is the welding efficiency of workmanship (kg/hour)
• dribs is the transverse distance between ribs.

Increasing complexity cost/weight Optimizations
As a preliminary analysis task, increasing complexity optimizations were performed to understand the contribution of single checks, i.e. the elastic or Eurocode behaviour of sections, influence of shear, welding and fatigue. The optimizations were performed by using the dedicated Excel link available in the modeFRONTIER workflow interface. Simplex and MOGA algorithms were used to work out the minimum cost/weight geometries.

Figure 15 - Eurocode checks for optimum Opened and Closed Section Plates

Pure flexure elastic weight optimization
The first and most simple task addressed is a weight optimization of opened and closed section shapes subject to a constant bending moment where elastic properties were used.

The following picture illustrates the minimum weight sections for both, negative and positive bending. The graph (picture 7) shown above gives rise to the following remarks:

a. The open section rib performs better elastically than the closed section one.
b. The optimal solutions are the same, both for positive and negative bending.
c. Optimal solutions can be achieved by using very slender sections, especially for open section shapes.

Figure 11 - Global optimization workflow.

Figure 12 - MOGA Iterations for the global optimization procedure – Opened Section Ribs.

Figure 13 - MOGA Iterations for the global optimization procedure - Closed Section Ribs.

Figure 14 - Cost Metric vs. MOGA Iterations.

Pure flexure weight optimization with Eurocode checks
The execution of code checks according

to Eurocode 3 does have a big impact on the minimum weight stiffener, as shown in the picture 8. Remarks:

a. For positive bending, the optimal section is open. For negative bending though, a closed section rib is convenient.
b. Negative bending solutions imply the maximum amount of cross sectional area, hence they govern the design.

Influence of out of plane shear and welding cost (longitudinal shear stress check only)

The influence of shear and welding cost was investigated assuming a simply supported static condition with a varying length. Minimum weight/cost solutions are shown in the following graph (picture 9 and 10). The following issues should be pointed out:

a. Out of plane shear has a significant influence, especially on positive bending, as it offers a limitation for the web slenderness.

b. For ordinary base metal to weld cost ratios, the size of the weld, deduced from using longitudinal shear stresses only, has no significant influence on the total cost. Local fatigue transverse checks are the ones which mostly influence the performance of the upper plate to rib weld.

Single objective optimization with multiple constraints
Global workflow
The following tasks were specified within the global modeFRONTIER workflow:

1. Assignment of geometric variables.
2. For closed section ribs, the torsional constant can be evaluated using the parametric finite element model.
3. Internal forces for ultimate limit state combinations can be calculated using the parametric User-Defined plate elements.
4. Fatigue ranges can be calculated using the parametric User-Defined plate elements.
5. The behaviour of a single stiffener is checked according to Eurocode 3 for the most severe positive and negative bending moments.
6. The weld throat is checked for strength.
7. Fatigue in the base metal and welds is verified according to Eurocode.
8. The cost function is evaluated for the specific design.

The picture 11 shows the global workflow as performed with the modeFRONTIER Interface. The above tasks are highlighted.
Runs
Global optimizations were run for both, the opened and closed section orthotropic plates. The iteration diagrams are shown in the pictures 12-13-14.

Global optimization result
Optimum designs were found with shapes very similar to the minimal Eurocode geometries. For a given distance between floor beams of 3000 mm, they respect the traditional sizing of stiffeners and upper plate. The importance of different code checks is displayed in the following picture.

It shows how the local fatigue checks strongly influence the design while longitudinal stresses are less important to the analyzed span length.

Bibliography
[1] P. Matildi, M. Mele – “Impalcati a Piastra Ortotropa” – Italsider.
[2] S. Timoshenko, S. Woinowsky Krieger – “Theory of Plates and Shells” – McGraw-Hill BOOK COMPANY, INC., Second Edition
[3] József Farkas, Károl Jármai – “Economic Design of Metal Structures” – MillPress, 2003
[4] Kiss K., Székely E., Dunai L.. – “Fatigue analysis of orthotropic highway bridge decks” – 2nd International PhD Symposium in Civil Engineering, Budapest 1998.
[5] J. Horn e N. Nafpliotis – “ Multiobjective Optimisation Using the Niched Pareto Genetic Algorithm” – Illinois Genetic Algorithm Laboratory, Urbana, USA, 1993.

For more information:
Ing. Daniele Schiavazzi
info@enginsoft.it