# Exam Winter 2012

In this exercise a graphic statics-based approach to the design of an arch bridge is discussed. The requirements for the final bridge as well as the requirements during construction are considered. The task is to find the shape of a concrete arch bridge with a span of 60 m taking into account the dead load of the bridge and to calculate the supporting structure used during construction. The form of the arch is determined by the forces acting upon it. Diaphragm walls of different heights transmit the uniformly distributed load of the bridge deck to the arch.

## Design of an arch bridge

In this exercise a graphic statics-based approach to the design of an arch bridge is discussed. The requirements for the final bridge as well as the requirements during construction are considered.

The task is to find the shape of a concrete arch bridge with a span of 60 m taking into account the dead load of the bridge and to calculate the supporting structure used during construction. The form of the arch is determined by the forces acting upon it. Diaphragm walls of different heights transmit the uniformly distributed load of the bridge deck to the arch.

### Uniform loading

In order to find the funicular form of the arch, it is assumed that the dead load of the bridge is discretized into 8 equal point loads Pd = 1500 kN acting upon the arch. The arch spans between the supports 1 and 2 while passing through the apex 3. .

• Draw the form and force diagram of the arch for this loading. Since the loading is uniformly distributed, the parabola construction is used to find the funicular form (Steps 1-7).
• Indicate the maximum force in kN. Use the maximum force to dimension the arch in concrete C20/25. Calculate the required thickness t of the arch cross section at a width of 4.8 m in cm (Step 8).
• The different lengths of the diaphragm walls found can be used to determine the exasct dead load of each diaphragm wall. The magnitude of the horizontal force of the structure can be measured in the force diagram.

Click here to open the interactive drawing that shows the graphical solution. Use the slider in the bottom right corner to browse through the different steps.

### Non-uniform loading

Based on the form found for uniform loading, determine the exact point loads Pd1-Pd4 of the different diaphragm walls and find the form of the arch for non-uniform loading. The middle segment of the arch passing through the apex 3 is given as horizontal. The horizontal force in the arch is given as equivalent to the horizontal force in the arch found for uniform loading.

• Draw the form and force diagram of the arch for this loading (Steps 1-6).
• Indicate the maximum force in the arch (Step 7).

Click here to open the interactive drawing that shows the graphical solution. Use the slider in the bottom right corner to browse through the different steps.

### Supporting structure during construction

During construction, the bridge consists of two cantilevers. Temporary diagonals are required in the voids between the deck and the vertical diaphragm walls for   horizontal bracing. Thus, both parts of the structure become cantilevering trusses. The point loads Pd1-Pd4 are given. As seen in the second figure, both the diagonals and the top chord are realized as tension elements. The task is to determine the forces in one of the trusses and to dimension the diagonals.

• Draw the form and force diagram to determine the reaction forces of the overall system using a substitute system (Steps 1-5).
• Complete the force diagram and determine the magnitude of the member forces (Step 6-8).
• Indicate the direction of the truss members using red for tension and blue for compression (Steps 9-11).
• Determine the maximum force in compression and tension and give their values in kN. Dimension the most critical truss diagonal as a flat steel profile (S355) with a thickness of t = 2 x 20 mm. Calculate the height h in mm (Steps 12-13). Why are the diagonals oriented in this way?

Click here to open the interactive drawing that shows the graphical solution. Use the slider in the bottom right corner to browse through the different steps.