Example of a beam grillage calculation according to different methods  Part 2
Example of a beam grillage calculation according to different methods
Comparison of the results  multicriteria analysis
Editor : Didier GUTH  Arcadis  June 14th, 2020
4) Results comparison
4.1 Support reactions
We calculate the "SLS" reactions, i.e., we cumulate G+Q (span 1 or 1+2). The unit cases are given in Appendix 1.
Comparison logic:

Column " %1 " = max of (Grill1, Grill2, EF1, EF2) / min (Grill1, Grill2, EF1, EF2) – 1

Column " %2 " =GuyonMassonnet / Average (Grill1, Grill2, EF1, EF2) – 1
It should be noted that the edge of the deck has been deliberately loaded to amplify some phenomena.

The differences are small between the reactions of the models in bar grillage and bar grillage + shells,

On the other hand, between the GuyonMassonnet approach and the grillage approach, the differences are relatively big, up to 33%, on the first beams. The studies carried out at CEREMA (center for studies and expertise on risks, environment, mobility and development) led to the same observations ref [11].

There are two main reasons for these differences:

partly due to the fact that the effect of the 5 kN/m load at the edge of the corbel, and to a lesser extent the weight of the corbel itself, is not correctly apprehended by the GuyonMassonnet calculation, which does not take into account the real effect of the load offset,

the most important fact is because the distribution coefficients are quite reliable for the distribution of moments (ref [11]), but are quite erroneous to estimate the part of the loads close to the supports that actually passes through a beam.
It is useful to refer to SETRA’s PRAD73 file which specifies the correction to be made:
Let X be the longitudinal abscissa of the load P, measured from the support axis and e the spacing between beams:

if x=0 the transverse distribution is done by assuming the hollow core element articulated on the beams

If x≥4e the crosssectional distribution obeys the GUYONMASSONNET hypotheses.

If x<0<4e the crosssectional distribution is as follows:

the fraction 1x4eP is distributed by assuming the hollow core element articulated on the beams

the fraction x4eP is distributed following the GUYONMASSONNET model.
It should be noted that despite these differences, the automatic calculation software considered that distributed loads of type A(l) (uniformly distributed load UDL nowadays) could be legitimately distributed with the coefficients of the GuyonMassonnet method.
Identifying the support nodes:
101 = beam 1 on abutment
111 = beam 1 central support
201 = beam 2 on abutment
211 = beam 2 on central support
Etc…
SLS Max 

Support 
GM 
Grill1 
Grill2 
EF1 
EF2 
%1 
%2 
101 
247.5 
366.5 
373.8 
361.3 
365.8 
3% 
33% 
111 
793.5 
1023.0 
1028.3 
1019.1 
1015.3 
1% 
22% 
201 
236.0 
199.3 
197.4 
223.6 
215.0 
13% 
13% 
211 
759.2 
763.4 
758.9 
804.2 
786.4 
6% 
2% 
301 
224.5 
238.4 
238.7 
224.7 
223.2 
7% 
3% 
311 
725.1 
751.5 
748.6 
743.5 
743.5 
1% 
3% 
401 
213.4 
196.3 
195.0 
196.2 
197.3 
1% 
9% 
411 
692.6 
652.6 
650.7 
645.5 
651.3 
1% 
7% 
501 
203.7 
185.3 
183.2 
181.3 
183.5 
2% 
11% 
511 
664.4 
611.1 
608.4 
598.7 
607.1 
2% 
10% 
601 
196.0 
177.0 
175.2 
173.8 
176.1 
2% 
12% 
611 
642.0 
587.7 
585.2 
575.7 
584.7 
2% 
10% 
701 
190.3 
173.1 
171.7 
169.7 
172.0 
2% 
11% 
711 
625.9 
575.7 
573.8 
564.2 
572.9 
2% 
9% 
801 
186.6 
170.7 
170.0 
168.3 
170.1 
1% 
10% 
811 
615.8 
569.9 
569.0 
561.4 
568.8 
2% 
9% 
901 
184.6 
173.7 
173.4 
170.0 
170.3 
2% 
7% 
911 
610.9 
576.5 
576.1 
569.6 
572.4 
1% 
6% 
1001 
183.9 
165.3 
167.8 
171.0 
166.0 
3% 
10% 
1011 
610.0 
564.2 
568.3 
592.6 
583.5 
5% 
6% 
1101 
184.1 
210.6 
212.3 
215.4 
215.7 
2% 
14% 
1111 
612.1 
691.0 
693.1 
690.9 
681.8 
2% 
11% 
max: 
max: 

∑ 
9602 
9623 
9619 
9621 
9622 
13% 
33% 
G+Q support reactions

few differences between grill models

more significant differences between GuyonMassonnet and grills, but they are reasonable, except for the edge beam (33%).
4.2 Moments and shear
The moments and shear by Guyon Massonnet's method are calculated by assigning to these forces the same distribution coefficient as before.
The Grid1 and EF2 models allow for the recovery of forces directly in the bars, while the Grid2 and EF1 models require recalculation using a data conversion tool.
4.2.1 GuyonMassonnet method
The forces in each beam calculated with the GuyonMassonnet approach are reminded below (they are not weighted in this table).
GuyonMassonnet coefficients > to divide by 11 

Beam 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 

Y= 
4.5 
3.6 
2.7 
1.8 
0.9 
0 
0.9 
1.8 
2.7 
3.6 
4.5 

Distributed superstructure 
1.000 
1.000 
1.000 
1.000 
1.000 
1.000 
1.000 
1.000 
1.000 
1.000 
1.000 

Left cross beam 
3.287 
2.523 
1.860 
1.317 
0.893 
0.577 
0.348 
0.186 
0.072 
0.013 
0.081 

Right cross beam 
0.081 
0.013 
0.072 
0.186 
0.348 
0.577 
0.893 
1.317 
1.860 
2.523 
3.287 

>∑ cross beam 
3.206 
2.510 
1.932 
1.503 
1.241 
1.154 
1.241 
1.503 
1.932 
2.510 
3.206 

Q span 
2.407 
2.089 
1.751 
1.407 
1.087 
0.812 
0.586 
0.406 
0.262 
0.144 
0.042 

Support 
1.347 
1.412 
1.459 
1.445 
1.322 
1.136 
0.933 
0.738 
0.560 
0.401 
0.257 
Beam 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 

Moment on support 

Selfweight 
1198.0 
1198.0 
1198.0 
1198.0 
1198.0 
1198.0 
1198.0 
1198.0 
1198.0 
1198.0 
1198.0 
1198.0 
Superstructure 
2320.0 
210.9 
210.9 
210.9 
210.9 
210.9 
210.9 
210.9 
210.9 
210.9 
210.9 
210.9 
Cross beam 
391.0 
114.0 
89.2 
68.7 
53.4 
44.1 
41.0 
44.1 
53.4 
68.7 
89.2 
114.0 
Span 
2109.0 
461.5 
400.5 
335.7 
269.8 
208.4 
155.7 
112.4 
77.8 
50.2 
27.6 
8.1 
M= 
18389.0 
1984.4 
1898.6 
1813.3 
1732.1 
1661.4 
1605.6 
1565.4 
1540.2 
1527.8 
1525.7 
1530.9 
Moment Span 1 

Selfweight 
674.0 
674.0 
674.0 
674.0 
674.0 
674.0 
674.0 
674.0 
674.0 
674.0 
674.0 
674.0 
Superstructure 
1305.0 
118.6 
118.6 
118.6 
118.6 
118.6 
118.6 
118.6 
118.6 
118.6 
118.6 
118.6 
Cross beam 
220.0 
64.1 
50.2 
38.6 
30.1 
24.8 
23.1 
24.8 
30.1 
38.6 
50.2 
64.1 
Span 
1615.0 
353.4 
306.7 
257.1 
206.6 
159.6 
119.2 
86.0 
59.6 
38.5 
21.1 
6.2 
M= 
10774.0 
1210.1 
1149.5 
1088.4 
1029.3 
977.0 
934.9 
903.5 
882.3 
869.7 
864.0 
862.9 
Shear C0 

Selfweight 
144.0 
144.0 
144.0 
144.0 
144.0 
144.0 
144.0 
144.0 
144.0 
144.0 
144.0 
144.0 
Superstructure 
278.0 
25.3 
25.3 
25.3 
25.3 
25.3 
25.3 
25.3 
25.3 
25.3 
25.3 
25.3 
Cross beam 
47.0 
13.7 
10.7 
8.3 
6.4 
5.3 
4.9 
5.3 
6.4 
8.3 
10.7 
13.7 
Span 
295.0 
64.6 
56.0 
47.0 
37.7 
29.2 
21.8 
15.7 
10.9 
7.0 
3.9 
1.1 
V= 
2251.0 
247.5 
236.0 
224.5 
213.4 
203.7 
196.0 
190.3 
186.6 
184.6 
183.9 
184.1 
Shear P1, left 

Selfweight 
240.0 
240.0 
240.0 
240.0 
240.0 
240.0 
240.0 
240.0 
240.0 
240.0 
240.0 
240.0 
Superstructure 
464.0 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
Cross beam 
78.0 
22.7 
17.8 
13.7 
10.7 
8.8 
8.2 
8.8 
10.7 
13.7 
17.8 
22.7 
Span 
422 
92.3 
80.1 
67.2 
54.0 
41.7 
31.2 
22.5 
15.6 
10.1 
5.5 
1.6 
V= 
3682.0 
397.3 
380.1 
363.1 
346.8 
332.7 
321.5 
313.5 
308.4 
305.9 
305.5 
306.5 
4.2.2 Model Grill1

ULS transverse shear force:

ULS moment:
4.2.3 Model Grill2

ULS transverse shear force:

Transverse shear force hollow core element and rib, ULS:
If we load the cross beams, we end up with the same amount but the distribution is different.
When using meshes, do not forget to sum the vertical shear in the two bars of the ladder beam to obtain the shear in the compound section.

N and M forces, ULS
Hence the evaluated torsor:
N= 1604.0 kN M=622.0 kN.M
Table 0.25 x 0.84 m2 (the participating width is calculated)
Hanging beam 0.40 x 1.00 m2
lower fiber upper fiber
σ(N) 4.01 MPa 4.01 MPa
σ(M) 9.33 MPa 9.33 MPa
σ(N+M) 13.44 MPa 5.32 MPa
it is deduced, for the rib:
Vupper= 0.285m and σupper, table= 9.99 MPa
Vlower= 0.715m
Constraints integration:
Calculated moment= 1645 kN.m
Torsor:
N= 2169.0 kN M=1494.0 kN.M
Table 0.25 x 0.38 m2 (the participating width is calculated)
Hanging beam 0.40 x 1.00 m2
lower fiber upper fiber
σ(N) 5.42 MPa 5.42 MPa
σ(M) 22.41 MPa 22.41 MPa
σ(N+M) 27.83 MPa 16.99 MPa
it is deduced, for the rib:
Vupper= 0.379m and σupper, table= 28.19 MPa
Vlower= 0.621m
Constraints integration:
Calculated moment= 2872 kN.m
4.2.4 Model EF1

Span M ULS=601 and N ULS=1783 kN

Support M ULS=1177 and N ULS=3035 kN
Torsor:
N= 1783.0 kN M=601.0kN.M
Table 0.25 x 1.05 m2 (the participating width is calculated)
Hanging beam 0.40 x 1.00 m2
lower fiber upper fiber
σ(N) 4.46 MPa 4.46 MPa
σ(M) 9.02 MPa 9.02MPa
σ(N+M) 13.47 MPa 4.56 MPa
it is deduced, for the rib:
Vupper= 0.253m and σupper, table= 9.07 MPa
Vlower= 0.747m
Constraints integration:
Calculated moment= 1740 kN.m
(Ed: paradoxically, the width of the hollow core element is 1.05 while in theory we can only have 0.90 m…)
Torsor:
N= 3035.0 kN M=1177.0 kN.M
Table 0.25 x 0.84 m2 (the participating width is calculated)
Hanging beam 0.40 x 1.00 m2
lower fiber upper fiber
σ(N) 7.59 MPa 7.59 MPa
σ(M) 17.66 MPa 17.66 MPa
σ(N+M) 25.24 MPa 10.07 MPa
it is deduced, for the rib:
Vupper= 0.715m and σupper, table= 18.90 MPa
Vlower= 0.285m
Constraints integration:
Calculated moment= 3112 kN.m
V=405 and 672 kN respectively.
4.2.5 Model EF2
We understand the interest of this modeling, since the efforts can be obtained by direct reading:

Mspan=1604 kN.m, Mmin=2932 kN.m

V=404 and 671 kN respectively
4.2.6 Summary
We note that the differences in moments are "relatively" small, both with the GuyonMassonnet approach and the other approaches.
For the shear as before, and logically, for the support reactions the differences are much more significant between the GuyonMassonnet method and the other models (from 15 to 20%)  for the edge beam and for a distributed load.
4.3 Load close to a support
We will study the impact of a load placed close to a support:
Global support reactions
It is immediately noticeable that the GuyonMassonnet distribution coefficients remain close to a value of 1.40 on the first 5 beams, i.e. the GuyonMassonnet approach will assume that this load is distributed almost uniformly in these 5 beams (as a reminder, the implementation of the method assumes sinusoidal loads distributed over the entire length of the beam, which is obviously not the case here).
Results (reminder, only load):
We note a difference of 80% on this reaction alone! Node 401 is the support of the 4th beam on the abutment of the loaded span:
Table of support reactions
However, if we apply the method of PRAD Folder 73.

If x<0<4e the crosssectional distribution is as follows:

the fraction 1x4eP is distributed by assuming the hollow core element articulated on the beams

the fraction x4eP is distributed following the GUYONMASSONNET model.
The load has been applied on the beam to simplify the calculations, so we will have:
R=(1 1.25/4/0.90)*200+1.25/4/0.90*1.445/11*200=140kN, value completely in accordance with the reactions of the grillage calculation models.
If we are looking for relatively specific efforts on support devices, we cannot do without the PRAD 73 approach.
This complicates the calculations that we could try to automate, but remains essential.
4.4 Corbel load
This short chapter aims to show the differences that one could have between a GuyonMassonnet, FE and manual calculation.
A point load of 100 kN/m has been placed on the edge of the deck.

GuyonMassonetcoefficient μ:
The calculation requires a Fourier series development and the taking into account of at least 3 harmonics. We end up with: 23.3 kN.m/m:

Manual calculation: M=100 kN/m*0.45 m= 45 kN.m/m

FE calculation
M≈30 kN.m/m
Conclusion:
There are significant differences between the 3 approaches. The manual approach remains safe, especially since it does not take into account the beneficial "equalizer" effect of a cross beam, which exists in many cases.
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