Reference: Rawlings, M.R., Balling, R.J., “ Collaborative Optimization with Disciplinary Conceptual Design”, 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St.Louis, MO, Sep 2-4,1998, pp1580-1587.

Description: The collaborative optimization approach is applied to the design of a long span cable-supported bridge. The work of conceptually designing the bridge is divided among two teams of designers assigned to different “disciplines” or parts of the complex bridge structure. One team is to design the deck and the other to design the superstructure. Each team is responsible for the conception, formulation, search and analysis of its part of the bridge design. The work of these disciplinary teams is coordinated at the system level to achieve system objectives.

System Representation:

Design Structure Matrix:

Analysis: The analysis is performed for each set of design variable values for a specific concept to calculate constraints and objectives. Each analysis calculates the structural response due to its own weight, the specified traffic loads and the forces that are transferred through cables.

Deck Analysis

The deck was modeled as shown in the Fig. The deck is supported by a series of roller supports representing the cable attachments. Input to the analysis of the deck includes the horizontal forces transferred through the cables and the values of each variable pertaining to the specific concept. Each analysis calculates the cost and the vertical force the deck exerts on the cable at each attachment. Because the loading is uniform, the vertical forces at each cable attachments are equal.

where d_{deck} is the discrepancy value for the deck concept in consideration, C_{deck} is the actual cost of the deck, and V is the actual vertical force the deck exerts at each cable attachment.

Superstructure Analysis

The superstructure was modeled as shown in Fig. Input to the analysis of the superstructure consists of the vertical force the deck exerts on the end of each cable and the variable values pertaining to the specific concept. For each concept, the cables are rigidly attached to the tower at each attachment point. The analysis considers two loading conditions. The first condition considers the response of the structure with the entire bridge loaded with traffic. This produces the maximum axial force in the towers. The vertical force at the end of each cable is the same. The second condition considers the effects when only the center span is loaded with traffic. This will produce the maximum moment in the towers. The vertical force at each attachment located between the towers is greater than the force on the cables on the outside span to simulate this loading condition. Each analysis calculates the cost and the horizontal force transferred through each cable. The V’ is used as the input to the analyses of the superstructure concepts and tries to minimize the following discrepancy function.

where Hj is the actual horizontal force value transferred in the jth cable and N is the total number of the cables. The lowest value of dsuper is then returned to the system level.

Design Variables: The design concepts that have been considered include the concrete box girder and the steel girder and cross beams for the deck.
Concrete box girder d.v’s

Depth

S_{girder}

Width of girder t_{girder}

Width of slab t_{slab}

Width of soffit t_{soffit}

Steel girder and cross beam d.v’s

Width of slab t_{slab}

Width of the web t_{web}

Width of the flange t_{flange}

Base length of the flange b_{flange}

Depth of the web D_{web}

The superstructure d.v’s

Depth of the beam d_{beam}

Height of the tower h_{tower}

Inside length l_{inside}

Outside length l_{outside}

Inside width w_{inside}

Outside width w_{outside}

Optimization: Collaborative optimization is used as the optimization method. The system optimizer coordinates the coupling effects between disciplines while allowing the disciplines to work in parallel. It also ensures that the system objectives are met. The system starts the process by assuming the values for target vertical (V’) and horizontal (H’) forces and target costs for each discipline (C’deck and C’super). These target values are mere predictions of what the actual values are mere predictions of what the actual values calculated at the discipline level will be. These target values are sent to each discipline. The disciplines then perform design in parallel.

Dicipline Optimization

The goal of the design at the discipline level is to find a concept and its associated constraints and minimize the discrepancy with the target values sent down from the system. In the deck design discipline, it searches over its two concepts and optimizes the design variables for each. The deck discipline must satisfy all the associated constraints.

System Optimization

The goal of the design at the system level is to find target values that satisfy system constraints and minimize the overall cost of the bridge. The two constraints the system must satisfy correspond to the discrepancies returned from the disciplines. Each discrepancy must be equal to zero. The concept design is being performed in both disciplines. Each discipline searches over discrete concepts and optimizes the continuous valued variables associated with each concept.