Self-launching gantries and movable scaffolding systems (MSS) are heavy and flexible and interact with bridge piers and superstructure during bridge construction. Proper modeling of **bridge construction equipment** is indispensable to capture:

- The stiffness interactions with the bridge that govern staged application of span post-tensioning, stability of tall bridge piers, distribution of thermal loads and other primary forms of structure-equipment interaction.
- The stress distributions within the gantry to be used for the design review of the machine.
- The general buckling modes of the gantry and the local buckling modes of its members. The buckling modes depend on load geometry and combinations (there is no natural set of buckling modes as there is for the natural vibration modes) and multiple load cases and combinations are tested to determine the design-governing one. Boundary conditions and internal releases of degrees-of-freedom may have a marked impact on buckling analysis as well. The figure shows the first lateral torsion-flexure mode in a 4600-element model of a 1340t self-launching gantry for macro-segmental construction during handling of an 820t pier segment.
- The buckling factor (factored load multiplier) for the different buckling modes.

Because of the geometric complexity of these machines and the additional complexity of varying geometry during self-launch and operations, the results of analysis and the reliability of the design-review process depend on the level of detail used for numerical modeling.

The optimal level of detail for the numerical model of self-launching gantries and MSS depends on numerous factors. The more refined the model, the more accurate the results of analysis and the deeper the understanding of the behavior of the machine. On the other hand, the complexity and quantity of load cases, load combinations and support conditions to analyze suggest the use of simple models to rapidly investigate all the possible combinations.

Simple beam models facilitate the research of the design-governing conditions, provide the stress magnitude to be expected from more refined analyses and the boundary conditions to assign to local models, and are often accurate enough for **underslung box girders** supported on stiff pier brackets.

Three-dimensional truss models are necessary with machines comprising **overhead trusses supported on flexible pier crossbeams** or when only some of the trusses sustain the load of segments, casting cell or traveling cranes. Modelling the supports is also necessary when the machine is propped from foundations.

The front support legs of a **single-girder telescopic gantry** are closely spaced to take support on pier-cap and underbridge, while the rear C-frame is wider to feed precast segments longitudinally from the completed deck. In some **single-girder overhead MSS**, and in most **span launchers**, the rear C-frame is wider to insert full-width prefabricated cages or precast spans. Most support legs are equipped with multiple hydraulic systems for geometry adjustment. In the presence of so many technological constraints, the structural nodes are so complex that local three-dimensional models with shell or solid finite-elements are often necessary to investigate stress distribution. Beam models are used to analyze span erection, structure-equipment interaction at the application of post-tensioning, and self-launching, as well as to identify the boundary conditions to apply to local three-dimensional solid models.

Most **self-launching gantries and MSS** use trusses as main longitudinal load-carrying members. Lateral bracing is used between the trusses and when possible also outside the trusses (this is the typical case of single-truss overhead MSS) to provide lateral stiffness and increase the lateral vibration frequencies of the machine. An ideal truss should meet three conditions:

- The members of diagonals, verticals and chords are perfectly hinged at the truss nodes. This condition is never respected in the bridge erection machines. Long continuous chord segments are used for fast site assembly, and the field splices between the segments are located at the center of the panel to use the same node design throughout the truss. The nodes are continuous, and when the field splices of diagonals and verticals are designed with shear pins, the pins are located far from the geometric node of the panel to simplify truss fabrication and site assembly.
- The loads are applied at the truss nodes. Also this condition is never respected. Technological requirements dictate the position of the support points of precast segments or casting cells, the loads applied by traveling cranes travel along the chords, the support reactions travel along the bottom chords during self-launch, and the trusses are often supported with out-of-node eccentricity also during span fabrication.
- The gravity axes of all members converging into a node cross at the geometric panel node. This condition could actually be respected although in most cases the convergence points of diagonals and verticals into the chords are staggered to minimize welding.

Depicting stresses generated by out-of-node eccentricity, geometry imperfections and flexibility of supports requires accurate modelling. The numerical models often describe the entire machine: trusses, winch-trolleys if capable to exert a lateral restraint action on the compression chords, launch saddles, and tower-crossbeam assemblies. Out-of-node eccentricity is modelled at permanent connections, field splices and support points.

The flanges of top and bottom chords are kept flush full-length for smooth travelling of winch-trolleys during operations and of the truss during repositioning on the launch bogies. The chords often use different shapes and vertical steps arise in the chord gravity axis at section changes, which is another cause of local load eccentricity.

Special joints are modelled at points of discontinuity such as structural boundaries, internal releases of degrees-of-freedom, changes in cross-section, field splices, support points, points of application of localized loads, and points where the deflections are to be determined. Auxiliary joints are used in the chords to investigate load and support conditions where the loads are applied far from the geometric nodes of the panel.

Additional loads and masses are applied to represent non-structural attachments such as hydraulic systems, mechanical components, power-pack units and generators. Scale factors are used to modify the cross-sectional properties to allow for distributed masses or additional stiffness. End offsets are used to account for the finite size of diagonals, verticals and chords at the connections.

The reliability of internal releases of degrees-of-freedom should always be critically reviewed. For example, many overhead trusses are supported on tower-crossbeam assemblies. One tower restrains the trusses longitudinally, and the launch bogies on top of the other tower allow thermal displacements. Before winning the breakaway friction of the launch bogies, the thermal deformations of the trusses generate lateral bending in the pier crossbeams and P-delta effects in the support towers. If the towers are flexible, the longitudinal thermal load at the launch bogies may be insufficient to win the breakaway friction. Before releasing translational degrees-of-freedom in the model, therefore, it is necessary to check that the longitudinal stiffness of the tower-crossbeam assemblies does exceed the breakaway resistance. Breakaway frictional loads are then applied to the crossbeams to investigate the P-delta effects in the support systems of the gantry.

Similar considerations apply for the releases of rotational degrees-of-freedom. When **launch bogies** or **friction launchers** are used to launch the gantry, the truss is supported on articulated beams that may be modelled with eccentric cylindrical hinges. When the pier towers have four distant legs and use two sets of crossbeams, the crossbeams are supported on interconnected hydraulic cylinders to generate a flexural hinge during launch. Similar solutions are also used in the W-frames on through girders. The cylinders are equipped with mechanical locknuts that are released prior to lowering the MSS for span release, and are tightened at the end of repositioning. Different restraint conditions are therefore modelled for the supports during self-launch and operations.

During self-launch, pins and equalizing beams of the launch bogies are modelled to equalize the support reactions applied to the truss despite different deflections of the support points. During span casting, the support cylinders are modelled with simpler frame-elements, and the rotational degrees-of-freedom on top of the jacks are released to model the tilt saddles of the jacks. Analysis with tightened locknuts will show that inner crossbeams and W-frames are more loaded than the outer ones due to the flexural rotations in the trusses under the load of an entire span of segments or the casting cell of an MSS. When one line of cylinders supports PTFE saddles for thermal sliding, additional bending results from breakaway friction at the sliders.