Transverse bending of a multi-beam bridge

Transverse bending of a multi-beam bridge

Author: B. Tritschler (Arcadis) – April 29th, 2020

1. Preamble

This document presents a solution for modeling a multi-beam deck with high longitudinal bending stiffnesses (pre‐tensioned or post-tensioned prestressed concrete beam types).

The purpose of this model is, more particularly, to study the transverse bending (study of the one-way joists and the intermediate cross girders) and to conclude on the relevance or not to model the one-way joist slab in an offset way, according to the efforts that we want to recover.

1.1 Need for a global study

The mechanical behavior of the hollow core element of the deck slab with intermediate cross girders is strongly conditioned by the transverse stiffness provided by the cross girders. In this case, the study of the slab is inseparable from the study of the intermediate cross girders.

In order to get as close as possible to the real state of stress of the slab and the cross girders, it is necessary to carry out a global study of the structure through a model using the stiffness of all the deck elements (beams - cross girders and slab).

The forces applied to the cross girders and the slab will be determined using a common finite element software to model the entire deck.

This model is the closest to reality because it takes into account the flexibility of the beams and the stiffness of the slab (when they exist).

The forces provided by the model directly cumulate the local forces and the general transverse bending of the deck.

In practice, there are two possible models to choose from:

• 3D modeling, with hanging beams and cross girders offset with respect to the plane (or level) of the slab.

• “Pseudo-plane" modeling, with beams and cross girders in the same plane (or level) as the slab.

Notes:

The different models will be set up with the interaxis distances of the beams and cross girders.

Transverse prestressing is not presented in this study.

1.2. PURPOSE OF THE NOTE

The purpose of this note is to compare two models of a same deck;

• a model taking into account the offset of the "bar" elements, i.e. the hanging beams and the cross girders.

• a model considering the "bar" elements that are the beams and the cross girders, without offset with respect to the plane of the hollow core element.

In both cases, the one-way joist slabs are modeled as shell elements of varying thickness.

The aim is to compare the results according to two criteria:

• validity of the results - direct comparison of the results provided by the two models / validation of the results.

• ease of exploitation of the results / analysis of the functioning of the models.

2. MODELING

2.1. REAL GEOMETRY

The geometry of the structure is detailed below.

2.1.1. Longitudinal section

Longitudinal section (cf. A009_0729-3_a1966_PL 05d_OUVRAGE_coffrage)

2.1.2. Cross section

2.1.3. Beam

2.1.4. Cross girders

2.2. GEOMETRY OF MODEL N°1

The first model consists of taking into account the offsets of the hanging beams and cross girders. For this model the geometry is visually very close to the real geometry.

2.2.1 Beams

Beam compression flanges are modeled in shell model as part of the slab. Only the hanging beams are modelled in a "bar" model, which is the most suitable hypothesis given their "linear" geometry.

The "complex" geometry of the hanging beams does not allow us to directly enter their dimensions in Robot; we will provide the software directly with the mechanical characteristics to be taken into account.

SECTION ANALYSIS

The variation in web thickness near the supports is neglected (little influence for the calculation of the bearing and the slab).

(hbeam = 1,90 m – hcompression flanges = 0,22 m – vpz = 0,642 m) => offset = 190 – 64,2 – 22/2 = 114,8 cm rounded to 1.15 m.

2.2.2 Cross girders

Cross girders compression flanges are modeled in shell model as part of the slab. Only the hanging beams are modelled in a "bar" model, which is the most suitable hypothesis given their "linear" geometry.

The geometry of the hanging beams is entered directly into the software.

=> offset. = 122/2 + 16/2 = 69cm.

2.2.3 One-way joist slab

The slab is made up of the elements between compression flanges of beams “hollow bricks of the joist slab " of constant thickness = 16cm, and the compression flanges, which will be considered variable from 16cm at the edge, to 22cm at the axis.

2.2.4 Span

The span is modeled on single supports (1 support/beam).

2.3. GEOMETRY OF MODEL N°2

For the second model, we consider the linear elements (beams and cross girders), in the middle plane of the slab. In order to model the longitudinal and transverse bending stiffnesses as accurately as possible, we will consider the real sections of the beams and cross girders.

2.3.1 Beams

The "complex" geometry of the hanging beams does not allow us to directly enter their dimensions in Robot; we will provide the software directly with the mechanical characteristics to be taken into account.

SECTION ANALYSIS

2.3.2 Cross girders

In accordance with the requirements of the standard post-tensioned prestressed beam file, the width of the slab taken into account on either side of the cross girders rib is equal to one tenth of the distance between the axes of the edge beams, i.e. 1.80m.

Flange total width = 1.80 + 0.25 + 1.80 = 3.85

2.3.3 Slab

The modeling of the slab is strictly identical to the one used for model n°1.

2.3.4 Span

The span is modeled on single supports (1 support/beam).

2.4. LOADING

2.4.1 Principle

We will compare the effect of identical loads on the two models.

To do so, we will consider cases of arbitrary charges:

• a distributed load over the entire slab (equivalent to loads brought by superstructures or to the loads of system A of instalment 61 part II)

• point loads, or system of point loads (corresponding to concentrated loads, reconciliation of the systems of loads of system B, of instalment 61 part II).

2.4.2 Distributed load

A distributed load of 1 t/m² (order of magnitude value of A(1)) will be considered.

2.4.3 Bc system (2 trucks of 30Ton per lane)

We will consider two Bc trucks, arranged side-by-side in order to obtain a maximum moment in the slab (resulting from the rear axles being centered on the plate).

Note: the central slab is not loaded (presence of a DBA).

2.4.4 Point load

We consider a point load of 10 Tons (equivalent to the Br wheel) centered on the plate.

2.5 SUPPORT CONDITIONS

Valid for both models: a simple support under each beam (except for one beam → pinned).

3. RESULTS

3.1 Central cross girders

3.1.1 Model n°1

3.1.2 Model n°2

3.1.3 One-way joist slab summary

The results provided by model n°1 are not directly usable: the effects only concern the hanging beams (normal effort + moment). It is necessary to recombine these effects with those of the one-way joist slab which correspond to the solicitations of the cross girder’s compression flanges (normal force and moment).

Reconstitution of the forces in the cross girder:

Hypotheses:

The normal effort of the compression flanges balances the one present in the hanging beams. The moment present in the one-way joist slab is negligible compared to the moment of the hanging beams (global effect).

MTotal=N∙Mhanging beam 

For information purposes, we will evaluate the width of the one-way joist slab which is necessary to balance the normal effort under a load case of type A(l).

Case 1 (distributed load)

(symmetrical load case)

We study several sections and integrate the normal stress. The integration is carried out according to the trapezoidal rule.

The integration of the normal stress on the sections studied above makes it possible to balance the normal effort existing in the hanging beams of the cross girder.

The required "real" width of the one-way joist slab is: (16.27 - 14.02)x2 = 4.50m.

That is to say a participating width of 4.50m (instead of the 3.85 recommended by the standard).

The participating width prescribed by the typical post-tensioned prestressed concrete beam standard is close to the one highlighted in this case: the geometry of the compression flange of the cross girders (T-beam) is validated.

The visible difference comes from the approximation made as to the width of the compression flange to be considered. The mapping on the following page shows the variations of the compression state in the slab: the participating width is variable: increasing from the edge towards the center. This difference remains small and thus allows the validation of both models.

The model which considers the offsets provides, after reconstitution of the efforts, moments in the cross girders which are very close to those provided by the model that does not consider offsets.

The increasing complexity of the model does not seem to bring any precision or visible advantage compared to a simpler model, brought back to the middle plane of the slab.

We recommend using a model similar to the model n°2 presented in this note.

3.2 One-way joist slab

3.2.1 Model n°1

3.2.2 Model n°2

3.2.3 One-way joist slab summary

Model No. 2, which does not consider offsets, provides very close moments in the one-way joist slab (within 3%) to those of the model considering offsets.

The increasing complexity of the model does not seem to bring any precision or visible advantage compared to a simpler model, brought back to the middle plane of the slab.

We recommend using a model similar to the model n°2 presented in this note.

4. Conclusion

A model with offset beams and cross girders does not provide directly usable results for the study of the beams and cross girders, contrary to a model brought back to the middle plane of the one-way joist slab.

Both models provide similar results with respect to the bending stresses of the slab.

We select a model similar to the model n°2 presented in this note, such model allowing us to obtain coherent results that can be directly exploited for the purpose of this study.

Note: The user must verify whether the approach presented in this note is applicable to his or her particular project.

5. APPENDIX - INFLUENCE OF BEAMS’ TORSIONAL INERTIA

For the selected model, we considered the non-cracked torsional inertia of the beams, a hypothesis which is not in conformity with the recommendations of the SETRA (operating society for transport and automobile repairs) guide PRP 75.

In order to assess the influence of the torsional inertia of the beams on the behavior of the intermediate cross girders and the hollow core element, we reran a calculation while considerably reducing the torsional inertia (so as to see an exaggerated effect).

We have divided by 10 the value of Ix retained in the previous model.

5.1 Cross girders

5.2 One-way joist slab

The moments show little variation relative to the torsional inertia retained.

Taking into account the torsional inertia in the cracked section should reduce the elastic value by about 30%, such a reduction will have no visible effect on the results.

It should be noted that we are, in almost all cases, in the presence of prestressed cross girders and hollow core elements which are therefore rarely cracked overall.

We suggest keeping the torsional inertia determined by an elastic calculation in uncracked section.

Note: The study of the influence of the torsional inertia explained above is only valid for structures with intermediate cross girders.