Example of a beam grillage calculation according to different methods - Part 1

Example of a beam grillage calculation according to different methods 

Comparison of the results - multi-criteria analysis

Editor : Didier GUTH - Arcadis - June 14th, 2020

1) Introduction

1.1 Objectives

In this document, we will model the same multi-beam deck using several approaches:

• the Guyon-Massonnet method,

• a plane grid approach,

• a 3D grillage (modeled as a ladder beam)

• a 3D model associating bars and shells, with two approaches. 

We will compare:

• Support reactions,

• Efforts, moments and shear.

In addition, we will perform two transverse bending calculations to highlight the differences and limitations of the methods.

We will test the incidence of a moderate skew (70 degrees), a more consequent skew (50 degrees) and the presence of cross girders.

A table, by way of conclusion, will attempt to give the reader some clues as to the advantages and disadvantages of each of the models.

The dimensions and applied loads are plausible for a structure deemed to be made of reinforced concrete.

We insist on the fact that each work is particular and that we find ourselves, in the context of this example, in a given configuration of flexural and torsional rigidity, and that consequently the conclusions cannot be generalized as such.

1.2 Bibliography

We invite the reader to refer to the following references:

[1] - Guide technique CHAMOA P CHaîne Algorithmique Modulaire Ouvrages d’Art – Apendix http://www.setra.fr/html/logicielsOA/Ponts_Types/CHAMOA-P/chamoa-p.html 

[2] - Guide pour l’utilisation des programmes de réseaux de poutres - PRP 75 - SETRA -1975

[3] - Compléments à la méthode de calcul des ponts à poutres multiples - C. Massonnet – ITBTP annals - January 1962

[4] - Le calcul des grillages de poutres et dalles orthotropes selon la méthode Guyon-MassonnetBarès - R. Barès et C. Massonnet - Dunod – 1966

[5] - Calcul des ponts à poutres multiples solidarisées par des entretoises – J. Courbon - Annales des ponts et chaussées - November-December 1941

[6] - Méthode de calcul des ponts nervurés sans entretoise intermédiaire – ITBTP annals – July-August 1970

[7] - Nouvelle formulation analytique de la flexion transversale d'une dalle orthotrope - A.L. Millan - Construction Métallique n°2 – 2004

[8] - Méthode de Guyon Massonnet Barès appliquée aux ouvrages à poutres d'inertie distincte - G. Bondonet et P. Corfdir - Revue Européenne de Génie Civil - Volume 9, n°9-10 – 2005

[9] – Calcul analytique de flexion des ponts à poutres de géométrie quelconque, calage des inerties de torsion transversale par comparaison à des calculs aux éléments finis – P. Perrin et G. Bondonet – Bulletin Ouvrages d’Art - n°71 – 2015

[10]- Emploi des éléments finis en génie civil (Tome 1) : La modélisation des ouvrages – sous la direction de Michel Prat

[11] - Contribution à l’étude des grillages de poutres – Pierre Perrin – Dir Est – sur le wiki de l’AFGC [https://wiki-gtef.frama.wiki/accueil-gtef:partie-3:exemple-c]

[12] - Flexion transversale d'un pont multipoutre – Benjamin Tritschler – Arcadis - – sur le wiki de l’AFGC [https://wiki-gtef.frama.wiki/accueil-gtef:partie-3:exemple-c]

[13] Guide pour l’évaluation structurale et la réparation des Viaducs à travées Indépendantes à Poutres Préfabriquées précontraintes par post-tension (VIPP) – CEREMA – (à paraître)

[14] Dossier PRAD 73 – SETRA

[15] Dossier VIPP 67 – SETRA

1.3 Possible complements 

To complete the study, in a non-exhaustive way, in the end we could add:

• Modeling with 3D elements (see reference [11])

• The study of a longitudinal and/or transverse phasing, taking into account the creep shrinkage, either as at a fixed rate or by using a calculation with behavioral laws,

• A study using an improved "Guyon Massonnet" approach

• Tests to find the optimal width of the cross bands

• How to take into account geometric or material non-linearities,

And extend the study to cases of structures with flexible connections.

2) Description of the structure and loads 

2.1 Geometry

 It includes:

• Eleven ribs of 40 cm x 100 cm ht, spaced at 0.90 m intervals.

• Two 25 m spans,

• A 25 cm thick hollow-core element,

It is made of C35 concrete, E=36000 MPa, ν=0.2 and rests on simple supports.

2.1 Studied load cases

The applied loads are:

• Self-weight:

• A fictitious superstructure load of 3.00 kN/m² on the entire surface of the hollow-core element and 5 kN/ml on the edges at the end of the cantilever:

• Multiple operating loads

• A linear load of 9 kN/m in span 1, on beams 1 to 3:

• A linear load of 9 kN/m in span 1+2, on beams 1 to 3:

• A distributed load from the abscissa 10.00 to 11.25 m, astride beams 6 and 7

(will be used for the transverse bending study)

• A point load on the abscissa 1.25 m from an abutment on beam n°4:

(will be used to study support reactions)

• A linear load of 100 kN/ml at the edge of the deck:

(will be used for the transverse bending study)

3) Different modelling approaches

3.1 Approach using the Guyon Massonnet method

We will not come back to the method as such, it is largely explained in the texts cited in the bibliography of this document.

In practice, to apply the method, first we must calculate the efforts, case by case, on a 2D beam representing the whole width of the deck (if we are not interested in the deformations, this beam can have ordinary characteristics, as long as they are constant).

The cases of charges to be applied have been described in the previous §.

(For cases modeling a load distributed over two spans, we could have modeled a unit load and multiplied the results by the effective load).

Summary of charges:

• Self-weight (only one rib in our case) - case 1.

• Superstructure - distributed load - case 2:

• Cross beam - case 3 (covers both cases in the Guyon-Massonnet approach):

• Case span 1 and span 1+2 - cases 11 and 12

And the specific case for reactions:

Results:

• Self weight:

Superstructures, 3 kN/m²:

• Superstructures, cross beams:

• Envelope for cases span 1 and spans 1+2:

We use a program that automatically determines the parameters and calculates the lines of influence, also called "Guyon Massonnet coefficient" (KGM):

As an example, graphically, the KGM coefficients:

• for the left cross beam - (e.g. beam 2 takes up 2.50/11 of the load)

• for distributed load q (spans 1 or 1+2):

The second column contains the moment M, the shear V or the reaction Rmax, for the unit cases, for the whole deck, which are distributed according to the KGM of each beam:

Effort calculation according to the Guyon-Massonnet method

We ensure that the sum of the variable of interest of the complete deck (orange columns) is equal to the sum of the variable of interest of the 11 beams.

3.2 Plane grid approach - Model Grill1

This model is constituted of longitudinal bars representing the rib + the hollow core element (whose characteristics are defined in Appendix 1) and coplanar transversal bars of 2.50 m length.

Care is taken to apply the calculation method recommended by the SETRA (operating society for transport and automobile repairs)/CEREMA (center for studies and expertise on risks, environment, mobility and development) documents for the calculation of torsional inertias, in particular for the hollow core element, the torsional inertia is be3/6 - ref [13], [14] and [15].

Model view in plane grid

ULS envelope of the moment (1.35G+1.35Qspan), edge beam

ULS envelope of the shear (1.35G+1.35Qspan), edge beam

3.3 Ladder-beam approach - Model Grill2

The model is similar to the previous one, but the longitudinal beam is subdivided into two bars. A lower one which represents the rib alone (0.40*1.00 m²), an upper one which represents the hollow core element (0.90*0.25 m²). The characteristics are given in the appendix.

The two beams are connected by very rigid bars (S=100 m² and I=100 m4) to ensure mechanical continuity.

Special care is taken to make duplicate bars and rigid bars non-weighty.

It is not possible to read the moments directly (see also section C.8 of the wiki).

We have to start from a duplicate (N,M) in the rib, for example, and recalculate the moment from a stress line diagram).

Envelope (ULS N,M) (1.35G+1.35Q span), edge rib (beam 1)

Envelope (ULS N,M) (1.35G+1.35Qbay), second rib (beam n°2)

An advantage of this type of model is the possibility to define a construction phasing and elements with different materials without going through equivalence coefficients, if any.

3.4 Bar + shell approach

3.4.1 Model with offset hollow-core element- model EF1

This model is quite comparable to the model presented in the previous §, but the grillage analysis that models the hollow core element is replaced by shell-type finite elements.

3.4.2 Model with non-offset hollow-core element- model EF2

Reference is made to the "Guide pour l’évaluation structurale et la réparation des Viaducs à travées Indépendantes à Poutres Préfabriquées précontraintes par post-tension (VIPP)" - ref [13]: one possibility is to let the hollow core element joint with the neutral axis of longitudinal beams, whose cross section is composed of the hanging beams and the hollow core element (which brings this model close to the first grillage analysis).

It is evident that this approach slightly overestimates the overall longitudinal stiffness but offers a simplification in the results exploitation - The influence of this simplification remains to be verified on a case by case basis by comparing Ibeam(*) + Ihollow core element to Ibeam - see references [12] and [13].

In this case:

Ihollow core element=0.90*0.253/12=0.00117 m4

Ibeam=0.09076 m4 (see Apendix 1)

Relative difference Ibeam(*) + Ihollow core element / Ibeam: +1.29%, which is very small.

(*) beam= hanging beam + affected hollow core element.