Beam Grid Example
Beam Grid Example
1. Beam grid
The beam grid is a generally horizontal structure consisting of longitudinal beams, called "main beams", associated with transverse beams. Beam grid modeling allows the calculation of a large number of structures in civil engineering, it is mainly used for the following works:

the bridge deck with prefabricated beams made of prestressed concrete with bonded wires, or reinforced concrete girders;

the mixedframe deck when it has more than two beams;

the ribbed slab deck;

by extrapolation, the slab type deck.
This modeling has the advantage of imposing the main directions of the reduction elements. Indeed, the plate calculation produces principal directions of moments that often deviate from the directions of the reinforcement, and the complex system of deviatoric moments or mxy torsional moments must be taken into account in the strength calculation.
Whereas, for a wireframe model, when the directions of the reinforcement are fixed, according to the geometry of the structure and the recommendations, we will be able to deduce the directions of the beams system in our model. The beam grid remains a simple model to model and above all simple to use. Consequently, we are often led to model a concrete slab (bridge oneway joist slab) by cross girders (transverse beams).
1.1 The GUYONMASSONNET method
In the 1950s, the construction of multibeam bridges required calculations tools. The method developed by Yves Guyon [Calculation of Wide Bridges with Multiple Girders Joints with Cross Girders], then improved by Charles Massonnet [Contribution and Complements to the Calculation of Multi Girder Bridges] is the most widely used, it has been compared to model tests or measurements on real bridges and has always been found to be in good or very good agreement with experimental measurements. The method uses two fundamental parameters whose data can be used in a FEM.
The data:
1. the flexural stiffnesses of the beams and cross girders:
B_{P}=E.I_{P}
B_{E}=E.I_{E}
2. the torsional stiffnesses of the beams and cross girders:
C_{P}=E.K_{P}
C_{E}=E.K_{E}
The method assumes that all the beams are identical. However, in order to make building easier, the edge beams can be different!
The deck jacking operations require the construction of an end cross girder:
Moreover, this is an isostatic bridge, and even if the method was extended to hyperstatic continuous bridges by the rule of thumb which consists in calculating a fictitious span with the same elastic deflection under a concentrated load applied in the middle of the span, what about the stresses on supports? Can the moments on the piles be calculated using a method based on the deformation calculation?
Modern calculation devices have favored geometrical and architectural criteria in construction and have taken many works out of the method's application field. The FEMs seem to abolish the perimeter of calculation of the structures, and the difficulty often lies in the choice of the data and the model and the exploitation of the results.
Note.  In reinforced concrete it is preferable to use reducing elements rather than stress tensors.
1.2 Purpose of this chapter
The simplicity of exploiting the results makes the beam grid a widely used calculation model. In this chapter, we will discuss data elaboration and the transformation of the results for the effects of the skew and the implementation of hyperstatic spans. These computational difficulties are discussed through the following examples:

Straight isostatic bridge deck with a 100 degree skew.

Bridge deck with 2 hyperstatic straight spans with a 100 degree skew.

Straight isostatic bridge deck with a 70 degree skew.

Straight isostatic bridge deck with a 50 degree skew.

Straight 2 span hyperstatic bridge deck with a 50 degree skew.
1.3 Beam characteristics
As for many statically indeterminate structures, the results are subject to the relative stiffness of each beam grid. The load distribution on the main beams is favored when the cross girders are stiff in relation to the main beams.
1.3.1 Flexural beam stiffness
The main beams are usually surmounted by a continuous oneway joist slab and the beam grid is made up of a single plan, therefore the sections of each structural element should be well defined.
The geometry of the compression flanges is defined in EN 199211 art. 5.3.2.1.
with:
1.3.2 Torsional beam stiffness
Due to the lack of discontinuity between the beams, we cannot strictly speak of torsion, it is rather a plate bending. However, in order to remain consistent with the beam logic, a torsional inertia is determined for each element. By considering the beam free to rotate around its center of torsion, the stiffness is underestimated. Therefore, we will use for rectangular elements a shape coefficient K = 1/3, whatever the slenderness of the part is: J=ab33(*).
Note.  K is a shape coefficient studied by Mr. Caquot. He indicates that K can be evaluated with a very good approximation by the formula 1k=1+1m23.5600.56m1m+12 where m is the ratio ab with a≥b… according to “Reinforced Concrete Form” by R. Chambaud et P. Lebelle.
(*) Important: The SETRA (operating society for transport and automobile repairs)/CEREMA (center for studies and expertise on risks, environment, mobility and development) guides, the PRP 75, the guide relating to independentspan and posttensioned prestressed beam viaducts and the guide relating to viaducts with prefabricated beams made of prestressed concrete with bonded wires all offer a more sophisticated approach  it will be necessary to refer to them in the context of a real project.
1.4 Model geometry
1.4.1 Model characteristics
The model characteristics are:

Span: 15.0m;

Width: 9.4m;

9 main Tbeams with a rib of 0.80x0.40m;

A 25cm thick oneway joist slab.
Cross section of the deck
Longitudinal section of the isostatic deck
1.4.2 Geometrical characteristics of the sections
The characteristics of the sections are:

Common main beams:

Section:

Flexural inertia:

Torsional inertia:

Common cross girders:
The model is made with a cross girder every 150cm:

Section:

Flexural inertia:

Torsional inertia:
The FE model presented below is made with ROBOT v2019 Autodesk:
1.4.3 Note on torsional inertia
By default, the software calculates the torsional inertia. The table below compares the different calculations:
J[m4] 
Software 
J=∑ab3/3 
J=∑ J_Caquot 
Beam 
0.0191936 
0.0261760 
0.0199764 
Cross girder 
0.0069919 
0.0078125 
0.0069646 
In order to be able to compare the finite element method to the GuyonMassonnet method we will use the torsional inertias from the software.
1.5 Loading of the model
Each beam of the different geometries is loaded with:

A midspan point load of 100kN:

A 10kN/m distributed load along each beam:
1.6 Model results
1.6.1 Comparison of simple geometry with the GuyonMassonnet method
The midspan moments per beam for a 100degree isostatic deck are (for e=y):
Beams y (m) 
GuyonMassonnet method 
FEM model 

K 
Point load Miso=P∙l4 
Distributed load Miso=q∙l28 
Point load 
Distributed load 

Edge 
0.405 
150 kN.m 
112 kN.m 
169 kN.m 
99 kN.m 
3.375 
0.244 
91 kN.m 
68 kN.m 
121 kN.m 
65 kN.m 
2.250 
0.188 
70 kN.m 
52 kN.m 
107 kN.m 
52 kN.m 
1.125 
0.170 
63 kN.m 
47 kN.m 
102 kN.m 
48 kN.m 
Center 
0.166 
62 kN.m 
46 kN.m 
101 kN.m 
47 kN.m 
It should be noted that the application of the GuyonMassonnet method results in greater stresses on the edges than the FE model.
This can be explained by the fact that the torsional inertia of the beams joined by the oneway joist slab is underestimated. Indeed, if the torsional inertia of the main beams is higher, the strain of the edge beam will be taken off, the load being better distributed over the entire deck.
Below is a study in which the torsional inertia of the main beams Kp is affected by a coefficient (noted f_Kp) varying from 0.10 to 3.00. The graph presents the comparison of the moments obtained by applying the GuyonMassonnet method to the two end beams with e=y:
The different torsional inertias used are listed below:
[m4] 
J = Σab3/3 
J=Σ 
Software 
FEM = GM [y=b; e=b] 
FEM = GM [y=3b/4; e=3b/4] 
Kp 
0.0261760 
0.0199764 
0.0191936 
0.0282102 
0.0258026 
These results indicate the importance of the evaluation of torsional stiffness in the calculation of moments by the GuyonMassonnet method. The presence of the oneway joist slab tends to increase the main beams’ torsional inertia.
1.6.2 Consideration of the geometry of the edge beams and end cross girders
The midspan moments per beam for a 100degree isostatic deck are (for e=y):
Beams 
Moment  Bar model: beams of identical inertia = basic case 
Moment  Bar model: variable inertia beams (edge) and end cross girders 

y(m) 
Point load 
Distributed load 
Point load 
Impact 
Distributed load 
Impact 
Edge 
169 kN.m 
99 kN.m 
154 kN.m 
9.81% 
86 kN.m 
15.24% 
3.375 
121 kN.m 
65 kN.m 
123 kN.m 
1.31% 
66 kN.m 
1.62% 
2.250 
107 kN.m 
52 kN.m 
107 kN.m 
0.60% 
53 kN.m 
1.15% 
1.125 
102 kN.m 
48 kN.m 
102 kN.m 
0.01% 
48 kN.m 
0.02% 
Center 
101 kN.m 
47 kN.m 
100 kN.m 
0.23% 
46 kN.m 
0.47% 
These results, which are far from calling into question the method’s validity, indicate the importance of the decrease in the stiffness of the edge beams and the influence of the end cross girder.
Note: the torsional inertias retained are given below (software column):
J[m4] 
Software 
J = Σab3/3 
J=ΣJ_Caquot 
Beam 
0.0191936 
0.0261760 
0.0199764 
Cross girder 
0.0069919 
0.0078125 
0.0069646 
End cross girder 
0.0232905 
0.0318938 
0.0232925 
Edge beam 
J=ΣJ_Caquot 
0.0242880 
0.0181062 
1.6.3 Complex model results
Max. span moment 
Beams – y(m) 

edge 
3.375 
2.250 
1.125 
Center 

Isostatic deck with a 70 degree skew 
Point load 
147 kN.m 
120 kN.m 
107 kN.m 
101 kN.m 
99 kN.m 
Distributed load 
77 kN.m 
61 kN.m 
51 kN.m 
44 kN.m 
43 kN.m 

Note 
The influence of the skew is very small and the common cross girders (hollow core elements) are arranged skewed. 

Isostatic deck with a 50 degree skew 
Point load 
138 kN.m 
105 kN.m 
98 kN.m 
88 kN.m 
93 kN.m 
Distributed load 
72 kN.m 
54 kN.m 
43 kN.m 
38 kN.m 
37 kN.m 

Note 
The common cross girders are laid out straight. The end cross girder, on the obtuse angle side, is in flexion rather than torsion, therefore it reduces the span moments to its detriment. 

Statically indeterminate deck (2 spans) with a 100 degree skew 
Point load 
130 kN.m 
105 kN.m 
93 kN.m 
90 kN.m 
88 kN.m 
Distributed load 
65 kN.m 
51 kN.m 
41 kN.m 
37 kN.m 
38 kN.m 

Note 
There is no variation on the load distribution between beams caused by the deck being statically indeterminate 

Statically indeterminate deck (2 spans) with a 50 degree skew 
Point load 
132 kN.m 
97 kN.m 
91 kN.m 
82 kN.m 
87 kN.m 
Distributed load 
66 kN.m 
48 kN.m 
36 kN.m 
33 kN.m 
31 kN.m 

Note 
The fact that the deck is statically indeterminate causes little variation in the span moments. 

Max. pile moment 
Beams – y(m) 

edge 
3.375 
2.250 
1.125 
Center 

Statically indeterminate deck (2 spans) with a 100 degree skew 
Point load 
63 kN.m 
40 kN.m 
33 kN.m 
31 kN.m 
30 kN.m 
Distributed load 
70 kN.m 
46 kN.m 
41 kN.m 
39 kN.m 
38 kN.m 

Note 
No variation on the load distribution between beams caused by the deck being statically indeterminate 

Statically indeterminate deck (2 spans) with a 50 degree skew 
Point load 
84 kN.m 
53 kN.m 
31 kN.m 
28 kN.m 
27 kN.m 
Distributed load 
92 kN.m 
46 kN.m 
37 kN.m 
36 kN.m 
36 kN.m 

Note 
There is a concentration of moments close to the obtuse angle of the load. 
1.7 Conclusion
For a classical bridge deck, i.e. a high stiffness of the beams compared to the cross girders, the difference of the finite element model beam grid compared to other methods is mainly expressed through the consideration of a lower stiffness of the edge beams and the consideration of the end cross girders.
When the skew of the deck is pronounced, only a finite element model can accurately value the reduction elements of the beams.
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