Contribution to the beam grillage analysis
Contribution to the beam grillage analysis
Date: Wednesday, April 29th, 2020
1) Introduction
The modeling of structures by a grillage model is frequently used in the field of structural engineering. This technique consists in establishing systems of 2 or more beam families, most often parallel, which are assembled at nodes and able to make various angles between them. It has the advantage of being declined to many typologies of works, and allows to perform the modeling of a work with a simple bar calculation software, while retaining the possibility of local refinement. The exploitation of the results is also advantageous in that the forces are directly recovered in the bars. On the other hand, the implementation of a grillage model, depending on the type of work and the load to be studied, can become tedious. Indeed, the multiplicity of beam elements to be defined, the precautions to be taken in the definition of the mechanical characteristics, or the need to discretize the applied loads can considerably lengthen the time required for the model. Moreover, if this work is not carried out with sufficient precautions and checks, the accuracy of the obtained results can be affected.
Grillage modeling competes with other methods or tools to evaluate the mechanical behavior of structures. Thus, we find in the literature several analytical methods or charts for the crosssectional distribution calculation. In addition, the appearance of highperformance finite element calculation software, with the possibility of modeling in beam elements, but also in plate elements or in 3D, offers multiple alternatives to the classic grillage modeling.
This chapter details through examples the different structures that can be modeled, as well as the multiplicity of finite element models that can be considered, for a given structure, as an alternative to a conventional grillage model. We are particularly interested in the problem of structures such as bridges with girders under pavement, including intermediate spacers or not, and possibly with a skew. The examples are intended to present the advantages and disadvantages in the development or operation of the results of each type of model.
2) Modeling
2.1 Models in classic beam grillage
2.1.1 Types of bridges which can be modeled by grillage models
In a classic way, the principle of a grillage model consists in modeling a structure in the 2 dimensions of space, the structure being able to receive loads perpendicularly to its plane. By extension, a grillage model can also be realized in the 3 dimensions of space to take into account offsets between the elements. A grillage is generally used to model structures or parts of structures having a behavior similar to that of a slab, with possibly a preferred direction (orthotropy, as is the case for girder bridges under pavement).
The structures concerned are numerous, and we can list in a nonexhaustive way:

orthotropic bridges, reinforced concrete or prestressed concrete bridges, possibly of variable inertia longitudinally, and with or without cantilevers.

ribbed slab bridges, with wide or narrow ribs

bridges with girders under pavement, with reinforced concrete girders, or prestressed concrete girders, by post tension or by pretensioning. These structures may possibly include cross girders on supports or intermediate cross girders (intermediate spacers are excluded for pretensioned).

bridges with coated girders and possibly with openwebbed and transversely prestressed girders.

Lightweight concrete bridges

etc.
Structures can have several spans and may present a skew.
2.1.2 Classic grill examples
We give here a few examples to illustrate the diversity of structures that can be modeled using classic grills. We draw the attention to the fact that this type of model requires being careful about the characteristics to be assigned to the bars in order to avoid redundancy of bending and torsional stiffness (see on this subject the recommendations of the SETRA (operating society for transport and automobile repairs) guide PRP 75 [2]).
Figure 1 below shows a beam grid made with the ST1 software for the recalculation of a pretensioned type structure. This structure has been reinforced at the edge by a coated beam structure that appears in the model with a negative offset. The structure has a skew of 80 grades, a width of 16.20m and has 2 isostatic spans of 11.36m each.
Grillage model of reinforced Cross section
pretensioned type structure Detail of the area reinforced with coated girders
Figure 1 – Grillage model of a pretensioned type structure reinforced by an adjoining structure
The modeling is based on nodes defined at the center of gravity of the hollowcore element. Transversally, the hollowcore element is represented by bars connecting these nodes. Longitudinally, bars representing each beam and its associated hollowcore element are defined by leaning on the same nodes through rigid offsets: the effective distance between the centers of gravity of the elements is therefore taken into account. The specific offset of the reinforced zone in coated beams is taken into account with the same principle.
Figure 2 below shows a grillage model generated by Cerema's (center for studies and expertise on risks, the environment, mobility and development) CHAMOAP bridge design software. This model is made specifically for the calculation of stresses in the joist trimmer beams of a reinforced concrete or prestressed concrete type slab bridge. The structure here comprises 3 spans of 11.5m  15m  11.5m for a width of 17.50m and has a skew of 76 degrees. The model makes it possible to reproduce the behavior of the 3D structure in particular at the supports to dimension the reinforcement in these areas. It takes into account the positioning and elasticity of each support and includes a refined discretization on supports to allow the precise evaluation of longitudinal and transverse local moments.
Figure 2 – Grillage model of a reinforced concrete type slab bridge
Figure 3 shows a project for the widening of a concrete ribbed bridge which is prestressed by a mixed box girder type structure. The initial structure has 2 continuous spans of 33.40m and 36.30m respectively, for a total width of 12.70m. The 2 ribs, the box girder and their associated hollowcore elements are represented by offset longitudinal bars. The intermediate and cantilever hollowcore elements are represented by longitudinal and transverse bars.
Cross section of existing deck
Figure 3 – Grillage model for the widening of a 2ribbed bridge which is prestressed by connection with a mixed box girder type structure.
The examples in Figure 4 and Figure 5 below are taken from the SETRA (operating society for transport and automobile repairs) Guide PRP 75 [2]. Figure 4 concerns the beam grid modeling of a 2 wide rib structure with a span of 39.50m, consisting of an isostatic span with a skew of 51 degrees.
Figure 4 – Grillage model of a ribbed slab bridge
Figure 5 shows a slab with cutouts, whose morphology is similar to that of a multicellular box girder bridge. The layout of the supports leads to an atypical trapezoidal geometry that requires a thickening of the webs on the high span side.
Grillage modelling
Cross section
Figure 5 – Grillage model of a hollow core slab
The examples presented illustrate the versatility of beam grid modeling. However, these models present some difficulties. On the one hand, the discretization into beam elements can be tedious to implement and make the application of moving loads on the structure more complex. On the other hand, the recovery of results in forces or constraints may require a reconstruction for justification. Depending on the case studied and the tools available, it may be appropriate to use a model comprising plates or shells elements to represent the slabs or the portions of the hollow core deck.
2.2 Finite element, plate and 3D modeling
2.2.1Beams and plates modeling
Plate (or shell) type finite elements allow to represent either directly the slab or only the hollow core part of a girder bridge. The use of plates and shells has a double advantage: on the one hand the implementation of the model and the loads is easy, and on the other hand the representation of the behavior of the structure is more faithful to reality (plate effect). The constitution of the plate element mesh is managed by the software’s algorithm, which simplifies the construction compared to a grillage model. The geometry of the mesh can be performed in a single plane by defining if necessary, and if the software allows it, an offset of the plates or beams with respect to the mesh. This offset doesn't appear in the mesh but simply as a characteristic of the elements: it is therefore advisable to check that the software actually takes into account the offsets. This verification can be done at least by comparing the average deflection calculated with the finite element model under uniform loading and that calculated with the beam model affected by the characteristics of the total deck width.
Figure 6  Modeling beams and plates, plane geometry
If the software used doesn't offer the possibility to offset plates or beams, it is necessary to build a mesh with rigid offsets (Vierendeel girder modeling). The realization of the model is then more complex but the facilities provided by the plates are kept.
Figure 7  Modeling beams and plates, geometry with rigid offset bars
The presence of intermediate cross girders can also be taken into account by adding crossbeam elements. The use, from a plane geometry, of plate or beam offsets then remains possible. The implementation of the cross girders can also be carried out by integrating offset bars into the mesh (Figure 8).
Figure 8  Beams and plates modeling, geometry with rigid offset bars and intermediate cross girders
Based on these main modeling principles, several variants can be established to be adapted according to the morphology of the deck. For example, the rigid offset bars can also be defined as skewed as in the example defined in Figure 3.
The support conditions must include a fixed support for the torsional effect on beams. This assumption is valid whatever the model used (beam grid, or volumetric finite elements), and is also found in the basic assumptions of the analytical methods for calculating the transverse distribution described below.
To apply forces to the model, it is necessary to provide either a geometry with groups of meshes dedicated to loading, or a sufficiently fine and regular mesh of the slab (generally quadrangles) to allow the description of the impact zones a posteriori by selection of meshes.
The exploitation of the results is carried out from the forces coming from the different elements, which must be recombined for the justification of the composite beam (or cross girders) sections and hollow core elements.
2.2.2 3D modeling
The realization of a complete 3D model of the structure can be relevant in certain cases, for example in order to check local stresses, or when the geometry has a particular configuration that is difficult to model using only plate and beam elements. The construction of the geometry in 3D, if it appears complex at first glance, can be done efficiently with current 3D design tools that offer advanced extrusion functions. The 3D approach is valuable as it allows to avoid the calculation of the mechanical characteristics of the beam elements. In addition, the location of the material is precise, allowing in particular to define the transverse span of the hollow core elements between beam ends and not between axes, or to avoid material redundancies at the intersections between elements that are found in other types of modeling. Finally, this type of model allows a direct graphic visualization of the geometry and deformations.
Figure 9  3D volumetric modeling of a post tensioned type structure with cross girders
The definition of the study sections, in order to be justified, must be thought out from the beginning if we want to obtain clean cuts with nodes located in the cutting plane. In the previous example, since the model was built by successive extrusions between cross girders, additional study sections were provided to facilitate the exploitation of the results (Figure 10) while reducing the size of the output files.
Figure 10  3D volumetric modeling: study sections
The volume model must also include a fixed support for the torsional effect on beams. It is possible to create this embedding using boundary conditions, or by modeling the volumes of the supported beams as in Figure 9.
The application of forces on the model must, as with plate models, be anticipated, and can also be done in 2 different ways. In the case of considering groups of meshes dedicated to load impacts, it is possible to decorrelate the load impacts of the intermediate study sections (longitudinal or transverse) by providing a horizontal separation of the upper part of the hollow core element, as shown in Figure 11. This arrangement allows multiple loading configurations to be managed without interfering with the deck construction elements.
Figure 11  Volumetric modeling: definition of loading impacts
The results from a volumetric model are given in constraints: depending on the justifications to be carried out it is necessary to integrate these constraints to deduce the torsor resulting in forces.
2.3 Analytical methods for calculating transverse distribution
Independently of finite element methods, the search for alternatives to classic grillage modeling has already been the subject of numerous works. As early as the 1920s, conservative methods based on simplified hypotheses made it possible to treat the simplest grillage systems using charts. Subsequently, the three main analytical methods still used today are those of GuyonMassonet, Courbon and CartFauchart. We recall here the principles and the fields of validity. Recent improvements have been made to these methods, such as the analytical formulation of Millan which generalizes the GuyonMassonet and CartFauchart methods.
GuyonMassonet method
A major evolution in transverse distribution calculations was provided in the 1960s by the theories of Guyon, Massonet and Barès, who developed a calculation method [3] [4] taking into account the torsional rigidity of the elements. A novelty of this method is the assimilation of the structure to an orthotropic slab, governed by an equation with partial derivatives of the form;
with:
P Longitudinal bending stiffness per unit length
E Transverse bending stiffness per unit length
P Torsional rigidity around the longitudinal axis per unit length
E Torsional rigidity around the transverse axis per unit length
This socalled "GuyonMassonet" method is based on the calculation of the transverse distribution coefficients which simplify the resolution of equation (1) in order to obtain the deformation and then the forces in the structure. It was then improved and extended by other authors, for example to extend it to the calculation of structures with edge girders different from standard girders.
This method is recommended for the calculation of slab or girder bridge type decks with a high number of girders and with or without intermediate cross girders. Due to the slab analogy, the method is well adapted to decks with more than four girders. For a 3 or 4 girder structure, it is preferable to use the CartFauchart method if there are no intermediate cross girders.
Courbon Method
The principle of this method was set out in the Annales des Ponts et Chaussées (Bridges and roads annals) of NovemberDecember 1941 [5]. 5] It applies to structures with intermediate cross girders which are considered as infinitely rigid. In practice, it is used when the span of the structure is about twice its width, and the height of the spacers is close to that of the beams. Figure 12 illustrates the principle of the method: an infinitely rigid section of deck transversely rests on springs whose stiffness is proportional to the deflection at half span of the beams. The reactions on each spring give the transverse distribution of the applied load.
Figure 12  Courbon method principle
CartFauchart method
This method was described in the technical institute of public works (ITBTP) annals N° 271272 of JulyAugust 1970 [6]. It is applicable to decks without cross girders (except on supports) and made of beams, ribs of constant section, or multiple box girders if the deformation of the box girders can be disregarded. The hollow core element is assimilated to a series of infinitely thin cross girders fixed on the beams. The resolution of the equation system of deformation is carried out by development in Fourier series.
This method can be used for bridges with girders or ribs without cross girders, as well as for multiple box girders, if, as a first approximation, the inherent deformation of the box girders is disregarded.
Innovative Millan method
This method is also based on the theory of orthotropic slabs and was presented in the newsletter Construction Métallique (steel construction) n°2 in 2004 [7]. It brings improvements to the GuyonMassonnet method: a more efficient analytical formulation, the possibility to take into account a nonzero Poisson’s ratio for orthotropic slabs, or to impose edge conditions on slab elements. This last possibility allows the study of the local bending of the hollow core element by fixing the free edges or the modeling of several contiguous slab elements. It is thus possible to finely model girder or ribbed bridges  which is a generalization of the CartFauchart method  without necessarily having all these elements being identical. Practical elements concerning the application of the Millan method to the modeling of girder bridges of any geometry are given in article [9] of the Bulletin Ouvrages d'Art (Bridges newsletter) n°71.
2.4 Criteria for choosing a model or calculation method
2.4.1 Geometrical specificities of the work
The analytical methods presented above are only valid if they remain within their scope. The criteria related to the cross girders’ stiffness have been recalled for each method and make it possible to orient oneself towards the method that seems best adapted to the structure, subject to the condition that the structure presents a relatively regular geometry. On the other hand, if the studied structure is more atypical  with asymmetries, thickness variations, strong skew (> 70 degrees), local reinforcements...  it becomes necessary to build a specific calculation model such as a grillage model or finite element model.
2.4.2 Loading
Loading the model can be more or less complex. In particular, the discretization of loads on a conventional grillage model is often tedious and requires approximations. From this point of view, finite element modeling with plate or 3D elements is preferable in terms of accuracy of load impacts.
The choice of the model can also be determined by the automatic load displacement facilities that may be provided by the software used.
Whichever model is chosen, it may be judicious to go through the construction of the lines or influence surfaces of the studied effect to perform the resolution of the equilibrium of the model only once, and to quickly deduce the forces obtained under any loading position.
2.4.3 Global or local justification
Depending on the element to be justified, the model can be global or local. For example, to study the local stress path in assemblies, or in parts of specific geometry, a 2D or 3D finite element modeling is the most appropriate. On the other hand, other local effects do not systematically require the use of 2D or 3D finite element modeling. For example, the calculation of local forces in the joist trimmer beam of slab bridges, as shown in Figure 2, is carried out with a simple refinement of the model in a grillage model at the support.
2.4.4 Post processing
Depending on the nature of the justifications to be conducted, the results from the model may require postprocessing which can be managed either by the software or by an external component. The most frequent problems are those of combinations or envelopes of forces. This functionality should not be neglected for studies with multiple load cases.
3) Specific points and examples
3.1 Transverse moment distribution
Results obtained with different models
The different models or calculation methods presented above give similar results, but with generally safe values for the GuyonMassonet method. Below is an example of a pretensioned structure with 26 beams, a span of 25m, a width of 21m, with identical beams, subjected to the centered loading of an regulatory E3F1 type exceptional convoy (400 tons). The structure is studied with several different models: Guyon Massonet model, Millan multiplate model (contiguous slabs), volumetric model (Figure 13). and model plates and beams.
Figure 13  Volumetric model: displacements under E3F1 convoy
The curves of the transverse distribution coefficients of moments at midspan (Figure 14) are consistent between all the models, except for the Guyon Massonet method, which remains safe (16% increase in bending forces in the most stressed beam).
Figure 14  Comparison of distribution coefficients on the different models
Case of structures with reinforced edge beams
The GuyonMassonet method, which comes down to an equivalent orthotropic slab, does not allow the direct processing of the excess stress generated on the edge beams when these are more solid or twinned (a layout commonly adopted to deal with the problem of offroad vehicle impacts). An adaptation of the method to this specific case was therefore proposed in the European Journal of Civil Engineering [8]. Otherwise, if no adaptation is carried out, the GuyonMassonet method becomes false for structures with solid or twin edge beams: the example of a structure with twin edge beams (Figure 15) shows that the forces obtained in the edge beams are largely underestimated and those in the central beams overestimated compared to other models.
Transversal distribution of moments. Structure with twin edge beams
Figure 15  Comparison of the distribution coefficients on a structure with twin edge beams (source [9] Bulletin Ouvrages d'Art (Bridges newsletter) n°71)
Evolution of the transverse distribution according to the load position
No matter which model is used, it is customary to dimension the structures by considering the transverse distribution of moments as identical all along the length of the structure, by safely using the distribution coefficients obtained at midspan. The figure below illustrates this hypothesis on the case of the previous 26beam structure subjected to a uniform surface load of 3m wide over the entire deck length. The transverse distribution of moments on each beam is given at different study sections. It can be seen that the maximum coefficients decrease as the study section deviates from the midspan section. However, these coefficients apply at lower moments, which limits the oversizing of sections away from midspan.
Figure 16  Distribution coefficients according to longitudinal study section
Using the midspan as the study section for the distribution of moments, a maximum coefficient of 2.55 is obtained in the most stressed beam at midspan. The influence of the load position on this distribution is then studied. The 3m wide load, previously applied over the whole length, is reduced to a length of 1m and then positioned at different distances from the support. Figure 17 shows the decrease in the coefficients when we move away from midspan.
Figure 17  Midspan moment distribution coefficients for different load positions
3.2 Transverse distribution of shear force
The study of the transverse distribution of the shear force near the supports is often carried out by simplification using the coefficients of transverse distribution of the moments at midspan. It should be noted that this hypothesis can lead to deviations of around 20% from the actual distribution of the shear force. As an example, Figure 18 shows the values of the transverse distribution coefficients under the effect of an E3F1 convoy running centrally. For the most stressed beam, the ratio between the distribution coefficients of the shear force on support (Kq, blue curve) and the moment at midspan (Km, orange curve) is 18%.
Figure 18  Crosssectional distribution of midspan moment and shear force on support
The previous curve of shear force distribution on support is obtained for the E3F1 uniform load which is applied over the entire length of the structure and over a 5.15m width. Actually, the transverse distribution depends on the longitudinal positioning of the applied load, for a given transverse width of the load. The study of the effect of the longitudinal position of the load on the distribution is carried out by assuming a 3m wide and 1m long load, positioned at different distances from the support (Figure 19). It can be seen that near the support the loads don't have the effect of distribution: only the beams located immediately under the load take up the loads. This underlines the fact that the given order of magnitude of 20% difference between Kq and Km applies to a longitudinally uniformly distributed load: if the load is punctual and depending on its position on the deck, the deviations can be more significant.
Figure 19  Distribution of shear force on support for different load positions
Furthermore, it should be noted that near the support, the shear forces are actually reduced because the efforts are transferred directly to the support. This favorable effect can only be taken into account in a 3D model in which a diffusion of forces (direct transmission connecting rod) takes place in the material near the support.
4) References
[1]  Guide technique CHAMOA P CHaîne Algorithmique Modulaire Ouvrages d’Art – Annexes http://www.setra.fr/html/logicielsOA/Ponts_Types/CHAMOAP/chamoap.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  Annales de l’ITBTP  janvier 1962.
[4]  Le calcul des grillages de poutres et dalles orthotropes selon la méthode GuyonMassonnetBarès  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  NovembreDécembre 1941
[6]  Méthode de calcul des ponts nervurés sans entretoise intermédiaire  Annales de l'ITBTP JuilletAout 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°910 – 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
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