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 - multi-criteria 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 " =Guyon-Massonnet / 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 Guyon-Massonnet 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 Guyon-Massonnet 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:

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

2. If x≥4e the cross-sectional distribution obeys the GUYON-MASSONNET hypotheses.

3. If x<0<4e the cross-sectional distribution is as follows:

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

• the fraction x4eP is distributed following the GUYON-MASSONNET 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 Guyon-Massonnet 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…

G+Q support reactions 

• few differences between grill models

• more significant differences between Guyon-Massonnet 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 Guyon-Massonnet method

The forces in each beam calculated with the Guyon-Massonnet approach are reminded below (they are not weighted in this table).

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 Guyon-Massonnet approach and the other approaches.

For the shear as before, and logically, for the support reactions the differences are much more significant between the Guyon-Massonnet 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 Guyon-Massonnet distribution coefficients remain close to a value of 1.40 on the first 5 beams, i.e. the Guyon-Massonnet 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. 

3. If x<0<4e the cross-sectional distribution is as follows:

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

• the fraction x4eP is distributed following the GUYON-MASSONNET 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 Guyon-Massonnet, FE and manual calculation.

A point load of 100 kN/m has been placed on the edge of the deck.

• Guyon-Massonet-coefficient 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.