Example of a beam grillage calculation according to different methods - Part 3

Example of a beam grillage calculation according to different methods 

Comparison of the results - multi-criteria analysis

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

5) Automated reinforcement calculation

5.1 (Automated) bending reinforcement

Section C8. of the wiki discusses an example of a ribbed floor - we repeat the same approach below.

• Calculation on T-sections (the moments are taken from the previous §).

• Calculation on isolated ribs (values that would be recovered from an automated calculation).

• Reinforcement of the hollow core elements (we loaded at the right of the beams to avoid cumulating global and local effects):

SUMMARY:

Again, as in the body of the Wiki (section C8), using the automatic reinforcement features on a ladder beam model, separately on the upper element and on the lower element does not lead to a reinforcement that conforms to the reinforced concrete design assumptions. It is therefore necessary to recalculate a global torsor to determine the reinforcement (without forgetting the splice reinforcement).

Caution: the ladder beam model can lead to underestimating the shear on the last section if the hollow core element is loaded. (see 4.3.2)

5.2 Reinforcement under local load (20 kN/m² over 1.25 x 0.90 m²

(Under this one isolated case)

• FE approach global + local effects:

• Myy, hollow core element span, smoothed=1.34 /0.625 m=2.14 kN.m/m (stretched lower fiber).

Myy, rib support: 0.47/0.625=0.75 kN.m/m (lower fiber also stretched).

Axis cut (unsmoothed):

• ST1 approach, crossbar, global effects:

We recover the My moments in the concrete strips of the hollow core element:

Slightly enveloped approach: Mon ribs≈3.38 kN.m/2.5m=1.35 kN.m/m (stretched lower fiber).

Do not forget to take into account the stresses of fixed ended plates, as the stresses from the bar model only reflect the overall effects:

Results of the calculation in a fixed ended plate - Calculation with 0.90 m spans, to remain homogeneous with the global FE model calculation

• Above Mplate on ribs=-1.29 kN.m/m, Mplate in span=0.64 kN.m/m

Total support≈1.35-1.29 ≈+0.06 kN.m/m (under this load case, in the middle axis)

(Nevertheless, it can be seen that the moment decreases very quickly at the loading end -> it would be possible to calculate the reinforcements with the perfect fixed end moment which in this case represents few reinforcements M=-1.29 kN.m/m -> As≈1.29/0.85/0.22/30=0.24 cm²/m).

Total hollow core element span = 1.35 global +0.64 local = 1.99 kN.m/m.

• Guyon-Massonnet approach :

• Myy=1.20 kN/m/m

In this case too, we have to add the forces of fixed ended plates, so we would have 1.20+0.64=1.84 kN.m/m.

Conclusion: the 3 approaches lead to values that do not show significant differences. The Guyon-Massonnet method will be extremely fast, if an automated software is available.

For a local bending calculation, the bar model should be refined (the strips here are 2.50m wide for a load of 1.25 m long).

6) Specific points

6.1 Skew

We will study the incidence of the skew in two configurations, 70 degrees which is assumed to be the limit for not taking into account the skew, and 50 degrees. We start from the second FE model because it is very easy to modify.

The bar grillage requires a specific joint layout of the transverse strips since these, in spite of the skew, must remain perpendicular to the ribs, as can be seen in this example:

© EGIS

6.1.1 Skew model with 70 degrees

6.1.2 Skew model with 50 degrees

6.1.3 Comparison

• Moments and transverse shear in the most stressed beam:

1.35* [G+(Span 1, Span 2, Span 1+2)

"%2" is the maximum relative difference between Guyon-Massonnet/FE efforts (%=GM/max(FE)-1)

• Supporting Reactions:

Reminder of the support nodes numbering for reactions on the next page.

And for reactions, G+Q (kN) - denomination "FEi", i is the skew in degrees:

6.2 Presence of cross girders

Considering that the efforts without cross girders are as follows:

6.2.1 Cross girders only on supports 

The cross girders contribute to the redistribution of transverse shear forces on supports:

6.2.2 Intermediate cross girders

There too...

• The cross girders make it possible to redistribute the forces between the beams more efficiently.

6.2.3 Guyon-Massonnet approach

With this method, there is no choice but to "spread" the stiffness of the cross girder - these values will be reduced to 1 linear meter, thus redivided by 8,333 m.

Impact on parameters α and θ:

Without cross girders With cross girders

α=0.4599 α=0.1244

θ=0.6422 θ=0.3203

With the cross girders, the distribution curves become almost rectilinear, we are getting closer to a calculation according to the Courbon method.

This means that offset beams always take up more loads than central beams (see Courbon's formula).

We don’t redo the whole calculation process, but as a comparison, the coefficients in both cases:

• Without cross girders

• With cross girders

• Graphic for cross beams (in orange, with cross girders):

• Graph for q load in span:

• Graph for the 4th loaded beam ("Support" case):

6.3 The crossfall

In order to follow "reality", we could be tempted to model the structures with the crossfall. It should be noted that this is of little interest and that, instead of stresses, vertical axis moments (Mz) in particular will appear and the question arises as to whether or not to use these stresses in the dimensioning. Illustration...

6.3.1 Model Grill1

• Model without crossfall

Grillage n°1 (plan), moment Mz under ULS envelope (-> Mz=0 everywhere)

• Applying a crossfall

Grillage n°1, with a roof profile at 2.5 (mechanical characteristics unchanged for the beams).

Vertical axis moment on ribs

Associated transverse shear

Likewise, in the hollow core element

For horizontal axis moments in the edge beam:

and the vertical axis transverse shear:

• Insignificant differences compared to the plan calculation.

For support reactions: we compared SLS reactions (G+Qtr 1 or G+(Tr1and2)) and the maximum difference is less than 4%).

It does not seem wise to model the crossfall as long as it remains reasonable to avoid having to manage "parasitic" efforts.

6.3.2 Model Grill2 (ladder beam)

Finally, it can be noted that by the structure of the model grillage in ladder beam (with 3D frame), with or without crossfall makes vertical axis moments appear due to the nature of the model.

Crossfall cross section

3D view

Moment Mz (vertical axis) in the longitudinal hollow core element bars

(An automatic calculation would lead to a reinforcement cross-section of ≈63.2/0.9/0.85/43.5=1.9 cm², but would not correspond to any mechanical continuity logic of the hollow core element).

Moment Mz (vertical axis) in longitudinal hanging beam

An automatic calculation would lead to a reinforcement cross-section of ≈27.6/0.9/0.35/43.5=2.0 cm² on the vertical side of the rib.

Moment Mz (vertical axis) on transverse straps

An automatic calculation would lead to a reinforcement section of ≈279.4/0.9/2.45/43.5=2.91 cm², on the vertical face, but would not correspond to any logic of mechanical continuity of the hollow core element).

6.4 Small discussion about torsion

Charles Massonnet in [4] wrote: "It is acceptable to neglect the effect of torsion in the case of open-section steel girder bridges without slabs. But it can never be neglected in monolithic girder bridges or in prestressed bridges (...)".

6.4.1 Torsional inertia calculation

The first question for the practitioner is how to calculate the torsional inertia. The PRAD 73 and VIPP67 documents, references [13], [14], [15] give rules to be applied. In particular, remember that the inertia of the hollow core element must be divided by 2 (be3/3 → be3/6) to remain compatible with the plate theory.

In addition, the PRP 75 guide recommended reducing this inertia by 10%.

Finally, it should be noted that this calculation of torsional inertias assumes that compressed concrete is a perfectly elastic material; it may be prudent to reduce them by 10% to take into account the fact that concrete is not, in reality, perfectly elastic.

6.4.2 What to do with all the calculated efforts?

Law of equilibrium

• For the hollow-core element, a reinforcement can be calculated from a reinforced concrete sizing method (Capra, Wood and Armer, ...) based on bending and torsional moments (see WIKI - part 1 – E,

• For the rib, the situation is a little more complex...

The PRP 75 guide should be read in detail to understand the method to be applied. The method consists in taking a quarter of the sum of the torsional moments (/lm), in algebraic value, arriving at the same node.

• We apply it to the model Grill1

Torsional moment in the structure - longitudinal beams

Torsional moment in the structure – crossbeams

• We perform the calculation according to the PRP 75 method for nodes 301 and 401.

Effort Mx

From the above sign convention, we apply the method of the PRP 75 guide and calculate Mxy (kN.m/m).

Efforts at the end of bar 301

Efforts at the end of bar 401

• In the hollow core element, this Mxy effort would have to be taken into account via one of the known methods: Capra, Wood and Armer, ...

• In the beam, this moment would have to be taken into account to justify the straight section and the reinforcement... Let us note that the VIPP and PRAD guides do not address this subject of torsion very much. (to be confirmed)

Effect of a ULS torsion of 1.35*11.20 kN.m/m*0.90 m=13.6 kN.m (BAEL calculation).

For massive sections such as those in this example, the torsion is absolutely not dimensional.

Model EF2:

Torsional moments in the hollow core element

Torsional moments in the ribs

The forces are almost identical to those of the bar grillage. Again, the logic of the PRP 75 guide should be applied.

• Guyon-Massonnet method

With a Guyon-Massonnet type calculation, we arrive to 8.2 kN.m/m and 8.8 kN.m/m, we remain in the rough estimate, but... with the paradox that the Guyon-Massonnet method will give zero torsional moments on supports (respectively maximal at mid-span), whereas this is precisely where they are maximal in a grid (respectively zero, NB: for this load case).

7) Conclusion

This example showed (if the reader was not convinced) that the way of modeling will lead to different results.

We propose below some guidelines to guide the choice of modeling. The author's assessments.

[0] the need to automate calculations for a certain number of similar structures may influence the choice of modeling and lead to the application of the Guyon-Massonnet method while the calculation of a single bridge may make this method less profitable.

[1] subject to having automated the calculation of the coefficients of the Guyon-Massonnet method.

[2] only one wire beam to be modeled.

[3a] the calculation, per load case, of the part taken up by each beam can be tedious - a preliminary calculation of influence surfaces may be necessary.

[3b] easier if the software knows how to make convoys run on a surface - a preliminary calculation of influence surfaces may be necessary.

[4] may require significant work on spreadsheet unless the software (e.g. ST1) knows how to incorporate user variables, variables assigned to each beam and each load case.

[4a] requires a global model in parallel or a good spreadsheet to manage concomitances.

[4b] on some software, it is complicated to have concomitant reactions for envelopes.

[5] relative precision and tedious to compute in the absence of prior programming of the decompositions into series and coefficient ν and τ. Nevertheless, the rough estimate is quickly given in this case by the GM method.

[6] requires the calculation of additional fixed end slab forces because these models only provide the overall transverse bending.

[7] the cumulation of general and local transverse bending gives an advantage, but does not allow one to easily distinguish the part of each one. It seems permissible to keep a somewhat enveloping approach.

[8] See CEREMA's CHAMOA guide.

[9] the geometrical setting of the transverse strips (perpendicular to the ribs) requires a small geometrical study beforehand.

[10] their stiffness is spread out and the recovery of the efforts in the cross girders seems complex from the.

[11] joint layout of the crossbars to be planned + study under local load to be planned - same logic as for the transversal hollow core element.

There are several things to think about before embarking on the model:

• What degree of accuracy is expected?

• Will it be necessary to justify all sections or just check a few sections?

• Is it necessary to change the calculation method if we check a design from the 1960s, for example?

It should be noted that the different models will lead to results with differences of up to a few tens of % on the calculated efforts. It is interesting to know that when performing a counter-calculation ... it is impossible to obtain a perfect convergence of the results.

On the other hand, it is important to understand the reasons for the discrepancies. This document hopes to contribute to that understanding.

Finally, it should be noted that none of the models will be able to automate 100% of the justifications and reinforcement.

Anything else?

8) Appendix - Guyon-Massonnet calculation data

Pi, bar model - Rib features:

Potential abscissa of charges:

We calculate the distribution coefficients using a software that implements the Guyon-Massonnet method.

• Fictitious isostatic calculation span: 22.867 m

• Bending and torsional inertias of the main beam (calculated according to the method recommended in documents [13] to [15]).
  

Main ribs:   Hollow core element:

Detail for the rib: the characteristics are the same as for the FE software, nothing to report.

Sx=0.6250 m², Iy=0.09076, Iz=(0.90^3*0.25+0.40^3*1.00)/12=0.0205 m4

Ix= ?

K(hollow core element) ? 0.9/0.25=3.6 -> kcalc=0.2722 to be divided by 2 for the hollow core element k=0.1361,

K(rib)=k(2*1/0.4)=k(5.0)=0.292

Ix=0.1361*0.90*0.25^3+0.292*1.00*0.4^3=0.02060191 m4 = 2,060,191 cm4, not much different from the bar software: 2,238,759 cm4 (previous page).

Flexion and torsion inertia of the hollow core element:

NB: b=25 m to force a calculation with 1/6 be3 (does not pose a problem, since the method spreads out the inertias and reduces them to 1 lm of hollow core element).

Reduced to 2.50 m for the grillage calculation (1/10 of the above value).

Sx=0.625 m², Iy=0.003255 m4 , Iz=2.50^3*0.25/12=0.3256 m4, Ix=0.00651 m4

9) Model ST1

• Plan model features:

Main ribs:       Hollow core element:

We reduce the hollow core element to b=2.50 m for the grillage calculation (1/10 of the above value).

Sx=0.625 m², Iy=0.003255 m4 , Iz=2.50^3*0.25/12=0.3256 m4 et Ix=0.00651 m4

• Characteristics for the ladder beam: ribs and longitudinal hollow core element

Rib Superstructure hollow core element

• Loading

NB: do not forget not to activate the self weight of the straps, otherwise it will be counted twice.

10) Appendix - Bars + Shells model

10.1 3D Model

• Hanging beams features

• Hollow core element features

10.2 2D Model

11) Appendix - Results tables

Support reactions

Beam 1: nodes 101, 111, 121

Beam 2: nodes 201, 211, 221, etc…

Little difference, but we can see that the models with shells overload the edge beams (c effect).

Superstructures:

Large difference between the Guyon Massonnet approach and all the approaches due to offset loads in the corbel, in particular.

Span loading -span 1:

Span loading – span 1+2

12) Appendix - ST1 files

Text files can be downloaded from the website.