## Client:

Sintagma S.r.l. - Perugia (IT)

Bridges with steel-concrete deck from 1 to 6 spans, 30 – 90 m, 2 longitudinal beams, Abutments and piers in reinforced concrete. Seismic isolation. Calculation report for executive project.

Alhambra srl and Sintagma srl have developed the executive design of 5 bridges for the modernization to 4 lanes of the italian national roads SS514 and SS194. The decks, with a steel-concrete composite structure, are one for each carriageway, seismically isolated with high damping rubber devices and have from 1 to 6 spans, spans from 30 to 90 m, 2 beams with or without an intermediate beam, the abutments and piers are in reinforced concrete. Four decks have a rectangular plan and have the 2 longitudinal beams connected by I transverse beams without the presence of an intermediate beam, the only deck curved in plan, called “Piano delle Rose”, is the one with the largest spans and in this case the intermediate beam is present, supported by K-shaped reticular transverse and lower torsion braces are present as well.

Decks have the following characteristics:

– 1 skew deck, single span 46.5 m (Buonafede)

– 1 deck, 2 spans 40-41 m (Barbaianni)

– 1 deck, 3 spans 30-50-30 m (Margi)

– 1 deck, 5 spans 48-66-66-66-48 (S. Leonardo)

– 1 deck curved in plan, 4 spans 75-87.5-87.5-75 m (Piano delle Rose)

In the following, reference will be made to the Piano delle Rose viaduct which is the most interesting. The first part of the case study mainly highlights the characteristics of the LUSAS finite element software, the second part those of the PontiEC4 software.

The analysis of the structure is performed by modeling with the finite element method, adopting the LUSAS system.

The structure was modeled by discretizing the slab and the beams webs with 4-node shell elements QTS4 type, the flanges with beam elements BMI21 type. This allowed the modeling of the real position of the reticular transverse trusses and of the torsion braces, also modeled with BMI21 beam elements and with the release of the bending moments at the ends. The intermediate beam is also inserted in the model, modeled with BMI21 beam elements. The portion of the slab straddling the supports, for an extension equal to 15% of the span of the respective spans, was assumed to be cracked.

Below is a rendered three-dimensional view of the complete model and one of the elements in steel.

The identification of the traffic load positions suitable to give the maximum / minimum design actions on the various elements of interest is carried out through an automated procedure, typical of the LUSAS finite element system, which involves the processing of the surfaces of influence of the various stress characteristics, in the points of interest through a “Direct Method Influence” (DMI) analysis, preliminar to the search for the most unfavorable traffic load positions, through a “Vehicle Load Optimization” (VLO) analysis.

The stresses N, T, M in the points of interest of a beam-shell model are obtained by integrating the results on “slices” of the model placed independently of the mesh; the influence surfaces, defined in order to evaluate the maximum / minimum bending and shear stress on the most stressed longitudinal beam, can be calculated, either in correspondence of the mesh nodes or of the slices.

The following figure shows, through red rectangles, the schematic positioning of the various “slices” on which the surfaces of influence have been made (only 2 spans of the deck has been studied for symmetry).

*“Slices” positions for influence surfaces (red rectanguls)*

The “Vehicle Load Optimization” analysis generates as many loadcases as there are maximized / minimized stresses at a given slice.

The following figure shows the load pattern, generated by the VLO, which maximizes the bending moment on the girder placed on the outside curve, in correspondence with the support of the pile P1; the arrows in red represent the tandem loads while those in magenta the distributed loads, segmented on the roadway according to the indications of the surface of influence.

*Traffic load pattern generated automatically by the “Vehicle Load Optimization” analysis – Loadcase that maximizes My of the girder placed on the outside of the curve on the P1 pile*

The “slices” allow us to have the stresses in significant and a priori chosen sections according to our needs, making the visualization phase of the results concise and fast without losing the precision of the representation of the diagrams of the stress even with a small number of sections, as shown by the following diagram of the moment My.

*My trend along the external and intermediate girder for the fundamental ULS*

The definition of the slices in the Composite Design Member of LUSAS is useful because it allows you to automatically create the verification sections in PontiEC4, nesting each one in the corresponding segment and load the corresponding design stresses, otherwise the sections and segments must be defined directly in PontiEC4 and the stresses imported from an Excel folder.

*PontiEC4 Geometry window – Section definition*

Based on the abscissa of the section, PontiEC4 software associates with the section the width of the slab actually collaborating, to be considered in the checks, due to the shear lag effect.

The effect of shear lag was evaluated in the following form where a reduction in the collaborating width of the slab is noted, when you get closer to the supports.

*PontiEC4 shear lag data window*

The variation of the collaborating width of the slab is represented in the graph generated by PontiEC4 below.

*Collaborating width of the slab along the whole viaduct*

U.L.S. Fundamental: Resistance checks

U.L.S. resistance check is performed considering combination with Mmax/Mmin/Vmax/Vmin,and relative correspondent stresses.

The main steps of the checks are summarized below.

– Pre-classification of the section carried out on the basis of the geometric characteristics of the individual sub-components.

– Effective classification of the section made on the basis of the effective value of NEd, MEd.

– Plastic bending check (sections cl. 1 and 2): Evaluation of the maximum plastic utilization ratio η1; carried out with reference to NEd, MEd acting separately and combined.

*Classification and plastic check – η 1 of the section on pile support axis P1*

*Plastic utilization ratio *η_{ 1}* along 2 spans of the viaduct *

– Plastic analysis: resistance domains of N / Mrd and N / Mf, rd section (domain of the section without the web).

*Plastic domain of the section at pier P1*

Elastic bending check (sections cl. 3-4): evaluation of the maximum elastic utilization ratio η1, carried out for the sections in class 3-4 with reference to the gross / effective geometric characteristics. The effective geometric characteristics are deduced in an iterative manner, taking into account the parasitic deflections that arise as a result of the eccentricity assumed by the axial design action caused by the progressive “shift” of the elastic neutral axis.

The stresses are evaluated in correspondence with the 8 fibers indicated in the following diagram.

In the context of the stress calculation, the slab is considered “cracked” (non-reactive) when the compression stress evaluated in correspondence with the medium fiber is null. Simultaneously with the cancellation of the slab, the stresses from primary shrinkage are also canceled.

*Stresses and elastic check *η_{1}* of the section at pier 1*

*Elastic utilization ratio η1 along the first 2 spans*

– Shear check – sections not subject to “shear buckling”: The calculation of the plastic resistant shear, and the calculation of the shear utilization ratio are carried out.

– Shear check – for sections subjected to “shear buckling” the reduction η_{w, }coefficient is evaluated and then is subsequentely evaluated the shear resistance V_{b,Rd} as sum of the web contribution V_{bw,Rd} and, if applicable, of the flanges contribution V_{bf,Rd}.

Check considers also Bending/shear interaction if present.

*Shear check of the section at pier 1*

*Shear utilization ratio η3 along the first 2 spans*

S.L.s. Characteristic: stress limit check

The check is carried out with reference to the Von Mises stresses evaluated under the S.L.S.

*Utilization ratio SigEd/fyd along the first 2 spans*

*Utilization ratio SigEd/fyd along the first 2 spans*

S.L.S. Frequent: web breathing check

The check is carried out using the rigorous method, consisting in the direct check of the stability of the subpanels with reference to the stresses of the S.L.S. frequent.

The stress induced by the S.L.S. frequent combination., represented by σx,Ed,ser and τx,Ed,ser, is therefore compared with the normal and critical tangential stresses of the panel, using the relationship (1993-2 clause 7.4.(3)).

*Web Breathing check on section at pier 1*

*Web Breathing check along the first 2 spans*

Fatigue L.S.

Fatigue checks are carried out using the λ coefficient method, associated with the FLM3 fatigue vehicle (EN 1993-2 chap. 9)

The method consists in the extraction, for the various points of interest, of the stress range Δσp due to the single transit of a specific load model (FLM3).

*Fatigue L.S. details check on the section at pier 1*

*Delta Sigma for details at the web-flange connection along the first 2 spans*

Shear connectors check

The calculation of the slip in the various sections and the comparison with the resistance of the design connectors is carried out automatically by PontiEC4 within the conditions considered. (Mmax/min and Vmax/min).

*ULS utilization ratio for shear connectors on section at pier 1*

*ULS utilization ratio for shear connectors along the first 2 spans*

*“The possibility of obtaining the resulting stresses N, T, M in a series of sections of a beam-shell model, coupled with the availability of generating influencing surfaces for the same sections, allowed to model the bridge with transverse beams and torsion braces in their real position, fully considering the effects of the curved plan, and allowed to have the stress characteristics diagrams, the values of which have been directly used for the checks with PontiEC4. The analysis software allowed a fast and excellent optimization of the structural steel and of reinforcement steel.”.*

* **Federico Durastanti – Technical Manager Sintagma srl (Perugia)*