PART 3 – EXAMPLES AND COMPLETE CASE STUDIES

• EXAMPLES AND COMPLETE CASE STUDIES
• Example A - Modeling a complex high-rise building
• Example B - Mixed and Steel Girders
• Example C - Beam Grillage Modeling
• Example D - Simple example: modeling of a Br wheel
• Example E - Transverse bending of a prestressed concrete box girder
• Example F – Dynamic calculations of tanks
• Example G – Cable-stayed bridges
• Beam Grid Example
• Contribution to the beam grillage analysis
• Transverse bending of a multi-beam bridge
• Example of a beam grillage calculation according to different methods - Part 1
• Example of a beam grillage calculation according to different methods - Part 2
• Example of a beam grillage calculation according to different methods - Part 3
• Practical application of smoothing

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EXAMPLES AND COMPLETE CASE STUDIES

This part contains modeling examples for simple and more complex objects. They are presented as a complete or partial study or even a comparison of different models of the same structure. 

If you happen to have an example that you wish to share with us of a complexity or paradox noticed on a part of a model, please send it to the following address: elements.finis@afgc.fr. (It could be a note about a real project rendered anonymously.)

Example A – Modelling a complex high-rise building

Example A – Modelling a complex high-rise building

Example B – Modelling of composite bridges

Example B – Modelling of composite and steel bridges

Example C – Modelling of beam grids

Example C – Modelling of beam grids 

Example D – Simple example: modeling of a Br wheel 

Example D – Modelling of a Br wheel

Example E – Transverse bending of a prestressed concrete box girder

Example E – Transverse bending of a prestressed concrete box girder 

Example F – Dynamic calculations of tanks

Example F – Dynamic calculations of tanks

Example G – Cable-stayed bridges

Example G – Cable-stayed bridges

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Example A - Modeling a complex high-rise building

Example A - Modeling a complex high-rise building

• Franck DUBOIS - Structures Engineering

• Thierry RICHARD - Structures Engineering

Part A: Presentation of the example

Purpose of the example

This example focuses on the calculation of the general forces of a high-rise building using finite element global modeling.

This calculation takes place at the beginning of the execution studies, its objective is firstly to determine the reinforcement of the foundations (diaphragm walls and drilled shafts) of the base in order to start the drilling of the diaphragm walls and drilled shafts.

Then, in a second step, the modeling will be used to calculate the general forces in the walls and floors necessary to calculate the reinforcement of the different elements (walls, columns, floors).

This example studies in particular the sensitive points of the modeling of this building:

• soil-structure interaction;

• seismic calculations;

• non-linear calculations;

• as well as phased calculations.

Particular attention is paid to the management of the modeling, detailing its preparation, its integration in the general study, its organization and its validation.

Presentation of the project

This building is located in Monaco on a major urban site, in the middle of sloped soils. It is made up of 2 major areas:

• a base buried in its rear area on 13 levels consisting of rear parking lots and dwellings including a school on the sea side;

• a 25-storey tower sitting on part of the base.

Figure 2: general view of the model (3D view).

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Part B: Preparation and organization of the modeling

This is a very important step, because it will be very difficult to modify the model later on when it is advanced, so it is essential to have clarified all the characteristics of the calculation model before starting it.

A.1 Particularities and constraints of the project

They allow to identify the sensitive points of the modeling:

• The large excavation area, the school area and the tower form a single monolithic block, there are no expansion joints.

• The excavation is integrated in sloped terrains, its dimensions are significant (109m x 169m on 13 levels), its support consists of a peripheral diaphragm wall reinforced by buttresses and inner drilled shafts. 

• It is moreover asymmetrical with strong active earth pressure on the upstream zone and no passive earth pressure on the downstream zone.

• Diaphragm wall panels rest on buttresses (no embedding)

• The project is on an urban site in the middle of existing structures which are very close to the site

• Presence of the high-rise building on part of the base

• Taking into account a second tower (possibly built in the future) for the dimensioning of the base

• Very tight work schedule

A.2 PRO file studies
The study of the calculation notes of the PRO file allows to quantify a rough estimate of the forces, and to identify the sensitive points.

In our case, it clearly appears that the foundations are dimensioned by the earthquake cumulated to the earth pressure, in fact the foundations’ notes show very important shear rates in the diaphragm wall (5 Mpa) with reinforcement rates exceeding the usual uses.

A resizing of the foundations will certainly be necessary.

A.3 Main modeling features

The first findings for the modeling are:

• The size of the model will be very important because there is no expansion joint.

• Taking into account the active earth pressure as well as seismic calculations are the key elements of the modeling.

• The precision on deformations required due to the proximity of existing structures cannot be obtained through conventional structural modeling.

It is therefore necessary to carry out simultaneously a "3D Geotechnical” finite element modeling which will be used solely to calculate the deformations. It will take into account on the one hand all the soils around the project and on the other hand the structure of the project itself.

The active earth pressure applied to modeling will be derived from the "3D Geotechnical” model.

• Asymmetric active earth pressures and their high intensities require consideration of passive earth pressure zones, and modeling of horizontal wall friction on the ground where possible.

• The up and down excavation construction method without the use of tie rods, associated with active earth pressure asymmetry, has very important consequences on modeling. Indeed, during construction, the active earth pressures are taken up on the one hand by the peripheral diaphragm wall (which is classic), but also by the inner drilled shafts, which make the classical 2D calculations obsolete. It is therefore necessary to integrate the construction phases in the determination of the forces.

A.4 Input data:

It is necessary to list and validate all the input data that form the basis of the calculation:

• The PRO architectural file

• The PRO structural work file carried out by the project management (plans and calculation notes)

• The overview of the reservations which appear in the PRO files and which unfortunately will evolve throughout the studies

• The "2D" active earth pressure calculated by BET FONDATION in a first step, which will be used to develop the modeling

• The "3D" active earth pressure that results from the 3D geotechnical modeling that takes place in parallel with the structural modeling. They will make it possible to correct the classic "2D" active earth pressure caused by the geometry of both the soils and structure (vault effects in particular)

• Seismic active earth pressures that will be added to inertial seismic effects

• Ground stiffness and friction in statics

• Ground stiffness and friction in dynamics

• Construction phases

• Construction site methods

A.5 Output Data

They are quite simple as they are relative to the definition of the reinforcement of the diaphragm walls and drilled shafts, which are to be transmitted to BET Fondation so that it can carry out the reinforcement plans.

A.6 Interfaces between participants

The complexity of the input data and the ties between participants requires an interface flowchart to ensure consistency between the modeling and all other stakeholders. (see next page).

A.7 Assumptions Note and Modeling Methodology Note

The Assumptions Note describing the assumptions taken into account in the calculations (materials, definition of loads, etc.) must be accompanied by a Modeling Methodology Note describing how these assumptions are taken into account in the modeling (application of active earth pressure, calculation method, etc.).

The Methodology Note sets out the principles implemented in the modeling and allows all project participants to validate them before modeling them. This avoids the long and laborious modifications that often occur when the model is finished. To simplify, this is equivalent to writing the modeling note before it is completed.

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Part C: Modeling

A.1 Geometric modeling principle

A.1.1 Breakdown into three independent parts

The modeling has been broken down into three independent parts, assembled during the final phase:

Partial modeling

Complete model

A.1.2 Coordinate system

The general reference frame is taken in the direction of the rear wall, which corresponds to the axes of the tower and overall to the directions of the active earth pressures, in order to make the seismic calculation coherent as it will be carried out according to the main axes of the tower.

Global reference frame

Tower axes

A.1.3 Using the BIM model

It could have been interesting to use the PRO file including a BIM model to generate the geometry of the model, but it soon appeared it was much simpler and more rigorous to study each level separately from the AutoCAD files to define the average lines of the panels.

This helps understanding how the structure works and ensures the proper transfer of loads between levels or between the elements themselves.

This manual step is important because it simplifies the modeling by removing local details that have no effect on the overall distribution of the forces, which will then be the subject of local studies. It mainly deals with the alignment of walls, slabs, the removal of small reservations, the removal of secondary walls, etc.

A.2 Modeling of the rear zone of the "large excavation" base 

A.2.1 Model features

Model view of the rear area of the base: large excavation

It is the most delicate zone because it is subjected to earth pressure, and it is done from the top to the bottom with the following phasing:

• Diaphragm walls and drilled shafts

• Upper slab (tower support)

• First level earthworks levelling

• 1st floor concreting

• Lower levels according to the same principle

The active earth pressures asymmetry between upstream and downstream requires a partial active earth pressures resumption by the central drilled shafts during the levelling, such resumption not being calculable by conventional 2D methods – hence the need to proceed to a 3D phased calculation.

The main features of this modeling are:

• Modeling of the diaphragm wall in shell elements with the installation of springs at the junction between panels

• Modeling of the buttresses into shell elements, with springs modeling the support of the diaphragm wall on these buttresses

• Taking into account passive earth pressure zones in areas that do not affect the surrounding areas

• Taking lateral earth pressure into account in areas that don’t affect the surrounding areas

• Making 4 seismic calculations (+X -X +Y +Y directions)

• Phased calculation of earthworks phases

View of the large excavation without the upper slabs

A.2.2 Soil modeling by springs

The connection of the diaphragm wall to the soil depends on the direction of the force:

• Either the wall presses on the soil, the spring is then in passive earth pressure, and its stiffness is in passive earth pressure

• Either the wall pulls on the soil, there is then a detachment, the stiffness of the spring is thus null (the wall is in fact in an active earth pressure state)

Active and passive earth pressures of walls framing a floor

There are therefore active and passive earth pressure zones for each load case.

We have considered 2 methods:

• Either define manually by iterations the passive earth pressure zones for each load combination.

• Or use non-linear springs working only under passive earth pressure.

This non-linear second method was adopted, as the calculation times proved to be acceptable.

Two laws of physics are used for soil modeling springs:

   

Springs in passive earth pressure.   Friction springs (passive earth pressure at the bottom of the excavation).

The friction springs are horizontal: there is no friction in the vertical direction.

These springs are relative to frontal stiffnesses and friction stiffnesses, and they vary according to the soil layers.

Example of soil spring location

A.2.3 Connections between panels of molded walls

The connection between diaphragm wall panels and drilled shafts or between diaphragm wall and buttress is modeled by a gap between the elements: 0.20m between panels and 0.5m between wall and buttress. The 2 panels are then connected by a non-linear spring connection with the following behavior law:

Connection between panels of the diaphragm wall

A friction coefficient of 0.7 is added to model friction in the panel contact plane with a limit of 0.7 x normal stress to the surface.

A.2.4 Connections between the buttresses and the diaphragm wall panels

They are carried out according to the same principle than for the connection between diaphragm wall panels.

 

Connection between the buttresses and diaphragm wall panels

A.2.5 Loading

A.2.5.1 Weight loads

They are simple to describe: dead weight, additional permanent loads, and overloads.

A.2.5.2 Static active earth force

They are derived from 3D geotechnical modeling

A.3 Modeling of the front area of the base (school and dwelling)

This modeling is much more conventional and traditional.

It is made up of shell elements (floors and sails) and wire elements (posts and beams).

The foundations are modeled entirely as for the rear area.

This area is only subjected to weight loads.

General view of the school zone model

It should be noted that the lengths of the lower levels exceed 100 ml, thus exceeding the regulatory lengths for taking into account the shrinkage effect.

A calculation of this zone under the shrinkage effect will have to be carried out during the second phase of the study, in order to determine the longitudinal reinforcement of the walls and floors.

For homogeneity with the large excavation, the piles of the foundations are modeled entirely with the installation of horizontal and vertical springs according to the soil layers. 

A.4 Modeling of the tower

The modeling of the tower is conventional and traditional, consisting of shell elements for the sails and floors and wire elements for the beams and posts.

The main problem is the large size of the number of elements due to the large number of levels.

In the first step of this foundation calculation, the tower only interacts by its effects at its base (upper base level). The size of the meshes does not interact much, it will thus be large meshes.

View of a typical level of the tower

One of the peculiarities of the tower is the design of the slab edge which is very complex and variable at each level. There are no two identical slab edges.

The layout of the slab edge had to be worked on in order to remove many points from the architect's original DWG file, which had very small distances between two points – even down to the millimeter.

Meshing of the slab before purge the useless points on the slab edge

This example shows the problems encountered during the automatic recovery of DWG or REVIT files.

Although it does not initially study the effects of the wind, the modeling includes at each floor the definition of a "WIND" node positioned at the center of gravity of the floor and connected to the core walls by rigid links.

These nodes will then be used to introduce the wind load torsors calculated in the wind report for each level of the tower.

A.5 Modeling of the optional second tower 

This second tower is identical to the first one and may be built in the future.

Whereas for the first tower the need to model it entirely by shell elements quickly appeared, for this optional second tower, we studied solutions limiting the model size.

A.5.1 Solution 1: modeling of the 2nd tower by its forces torsor at its base

This is the simplest solution which consists in calculating the resulting torsors at the base (O2 point below) for each load case and applying them directly to the general model.

The resulting seismic torsors of the 2nd tower are then calculated on a model of the tower alone which is embedded at its base.

This method could not be implemented because it directly adds up the seismic forces of the two towers, thus generating inadmissible forces in the diaphragm wall. Indeed, it appeared during the PRO file that there was no seismic interference between the 2 towers.

A.5.2 Solution 2: modeling of the 2nd tower by a skewer model

We realized a skewer model of the isolated tower.

This model consists of a vertical bar modeling the core whose characteristics are defined from test cases of the tower model by arranging horizontal loads at the head (according to X and Y), then by studying the equivalent inertias producing the same deformations.

Given the asymmetries of the floors, it soon became clear that the conventional skewer modeling consisting in assigning masses at each level is not suitable, because torsion is then not taken into account.

Each floor has been cut into 4 parts to which the corresponding masses are assigned.

Visualization of the skewer model

We then compared and validated the resulting torsors between the 2 calculation models under the CQC seismic cases, and compared the main modes.

Mode 1: transverse according to X:

f= 0.68 Hz with 56% of the mass     f=0.63 Hz with 62 % of the mass

Mode 2: Longitudinal Y:

f= 0.77 Hz with 66% of the mass    f=0.70 Hz with 62 % of the mass

Torsion mode:

f=2.2 Hz f=1.3 Hz

Vertical seismic mode:

f= 5.0 Hz with 73% of the mass f=5.5 Hz with 82 % of the mass

The validated skewer model was implemented in the general modeling by having rigid joints between the base of the skewer, and the diaphragm walls and bearing drilled shafts of the large excavation.

Compared to the first solution, a decrease in the forces in the diaphragm walls appeared. On the other hand, the rigid connection has generated very significant not admissible forces in the diaphragm wall, which do not appear in the connection of the first tower. 

Therefore, the first level of the second tower should be modeled in shells to obtain consistent results.

This solution has therefore been abandoned in favor of the third solution.

A.5.3 Solution 3: modeling of the 2nd tower entirely by shells

It is the solution that we were trying to avoid that was used!!, the calculation times have increased while being reasonable.

Rear view of the model showing the anchoring of the 2nd tower in the diaphragm wall

Part D: Global modeling calculations

A.1 Global model features

The global model is meshed with elements of 1.50m size, except for the foundations (diaphragm wall, drilled shafts) with smaller meshes of 1m.

In the end, the model includes 168,000 nodes.

The total calculation with phasing and seismic calculations takes one night.

A.2 Phasing calculation

The calculation takes into account the 18 construction phases (earthworks) of the large excavation.

Then the school zone and the tower are activated.

Operating overloads can thus be applied to all floors of the model.

A.3 Seismic calculation

A.3.1 Modal spectral calculation

4 seismic calculations are performed in the directions +X, -X, +Y, and -Y, neutralizing the linear springs in tension in each case. These cases are then studied twice; with or without the 2nd tower.  There are therefore 8 modal spectral analyses.

Modal analyses are carried out on 100 modes, which allows to interest at least 70% of the participative mass. The residual mode is then added to reach 100% of the mass.

The 2 calculations with or without the second tower are quite close, the 1st mode is 0.471Hz with 2 towers, and 0.582Hz with only one tower.

First modes visualization (with 2 towers)

A.3.2 Seismic active earth pressure

Dynamic increments are applied in the 3 directions +X, -X, and -Y.

They are added to the static active earth pressure.

Cumulative dynamic increments with static active earth pressure

The seismic active earth pressures are then added to the inertial seismic forces from modal/spectral studies.

A.4 A few results

Cumulative phasing deformations

Deformations of dynamic increments

Seismic CQC deformations +X direction

A.5 Iron framework of the foundations: diaphragm walls, drilled shafts and buttresses

The global forces are calculated for the SLS and ULS limit states, and ULS seismic.

Cuts are made over the entire height of each panel to deduce the global resulting forces.

We visualize below the graphs of the forces in a buttress in the SLS state.

Normal stress in a buttress

Bending moment on high inertia

Bending moment on low inertia

The iron framework is then calculated by applying the usual and regulatory rules related to reinforced concrete.

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Part E: Modeling validation

This is the fundamental question of complex modeling: how can we show the validity of the results?

Several types of validation were carried out.

A.1 Comparison with PRO file studies

The main results were compared with those in the PRO file:

• Mass balance

• Torsor of support reactions for elementary load cases

• Deformations under permanent loads and earthquakes

• Specific modes

• Etc.

A.2 Internal validations during modeling

They are carried out during the modeling and during the verification of the main results

Moreover, they relate to:

• Geometry visualization, local frames, thicknesses, etc.

• Graphical visualization of the loads applied to the model,

• Visualization of steel mapping to show any problematic zones identified by large steel sections

• Visualization of the deformations of elementary load cases

A.3 Internal validations by a partner not involved in the modeling team 

A person external to the modeling team checked several points:

• Activation and deactivation of groups of elements (surface, wire, or spring elements) in the calculation phase

• Behavioral laws according to input data

• Behavior law assignment for all passive earth pressure or friction springs, depending on panel location and level.

• Orientation of non-linear springs which only work in compression

• Respect of the springs’ plastic zone during the different phases

• Loading of elements

A.4 Modeling validation meeting

Outside the modeling team, it is absolutely impossible for other project participants (project managers, technical controllers, construction site, other technical design offices, etc.) to understand the details of this "black box" and to be able to validate the results of the model.

A general meeting was therefore scheduled "live" in front of the modeling computer. Everyone was then able to request data visualizations, request additional results, understand the model structure, see all the parameters included in the calculation data, access intermediate results, etc.

The aim of such a meeting is to answer all the questions raised by the participants directly with the modeling computer.

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Part F: Calculations of the structure’s iron framework 

The modeling will be completed in a second phase by the calculation of wind and shrinkage forces.

The mesh will be refined for the tower’s walls by adopting 3 meshes on the height (i.e. a 1m mesh).

The walls will then be calculated directly from the results of the modeling by making cuts at their base.

The bending forces of the floors will be calculated "manually", i.e. with local modeling, which will be added to the membrane forces (N and FXY) determined by the modeling in order to determine their iron frameworks.

FXY membrane forces to be taken into account in the slab calculation

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Example B - Mixed and Steel Girders

Example B. Mixed and Steel Girders

The subgroup responsible for these examples is still working on the subject. Some patience will be required.

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Mixed Structure

This subgroup has studied the same structure in three different ways:

• A 2D wireframe approach (link) - Valentina Bruno (Setec tpi).
• A 2D beam grillage model - Hugues Somja (INSA de Rennes).
• A 3D model - Aymeric Perret du Cray (Setec tpi).

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Orthotropic Deck Steel Structure

• Setec pdf : 3 models;
• Arcadis: 1 models.

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Example C - Beam Grillage Modeling

Example C - Beam Grillage Modeling

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• Benjamin Tritschler (Arcadis) - Transverse Bending of a Multi-Beam Bridge

Link: https://cloud.afgc.asso.fr/s/9NK3MYEiYpGwsQ7

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• Jacques Combescure (Artes) - Beam Grillage

Lien: https://cloud.afgc.asso.fr/s/DcWCg2KWRLj2FMm

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• Pierre Perrin (Dir Est) - Contribution to the Study of Beam Grilles

Lien: https://cloud.afgc.asso.fr/s/yP33KYxSFjJPqmW

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• Didier Guth (diadès) - Modeling the Same Structure Using Various Approaches

Lien : https://cloud.afgc.asso.fr/s/KLJsLyZiC8pmmqY
Fichiers ST1 : https://wiki-gtef.frama.wiki/_media/fichiers_st1.zip

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Example D - Simple example: modeling of a Br wheel

Example D - Simple example: modeling of a Br wheel

The purpose of this example is to show the differences that can be obtained with several different software for the same calculation, which is very simple a priori, and to confirm the propositions of chapter D.5.2.

Authors: Franck Dubois, Valentina Bruno and Didier Guth.

Link: https://cloud.afgc.asso.fr/s/58c5AX359ePN92K

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Example E - Transverse bending of a prestressed concrete box girder

Example E - Transverse bending of a prestressed concrete box girder

Author : Jean-Paul Deveaud - Cerema est center

Link to the file: https://cloud.afgc.asso.fr/s/apaRyaTMbJpNtJg

Link to the file: https://cloud.afgc.asso.fr/s/XxmzPsxQjkpDTE7

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Example F – Dynamic calculations of tanks

Example F – Dynamic calculations of tanks

Coming soon... a little patience required.

Author: Gildas Potin & Co.

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Example G – Cable-stayed bridges

Example G – Cable-stayed bridges

Coming soon... a little patience required.

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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 mixed-frame 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 one-way joist slab) by cross girders (transverse beams).

1.1 The GUYON-MASSONNET method

In the 1950s, the construction of multi-beam 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:

1. Straight isostatic bridge deck with a 100 degree skew.

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

3. Straight isostatic bridge deck with a 70 degree skew.

4. Straight isostatic bridge deck with a 50 degree skew.

5. 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 one-way 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 1992-1-1 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+1m23.560-0.56m-1m+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 independent-span and post-tensioned 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 T-beams with a rib of 0.80x0.40m;

• A 25cm thick one-way 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:

In order to be able to compare the finite element method to the Guyon-Massonnet 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 mid-span 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 Guyon-Massonnet method

The mid-span moments per beam for a 100-degree isostatic deck are (for e=y):

It should be noted that the application of the Guyon-Massonnet 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 one-way 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 Guyon-Massonnet method to the two end beams with e=y:

The different torsional inertias used are listed below:

These results indicate the importance of the evaluation of torsional stiffness in the calculation of moments by the Guyon-Massonnet method. The presence of the one-way 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 mid-span moments per beam for a 100-degree isostatic deck are (for e=y):

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):

1.6.3 Complex model results

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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Contribution to the beam grillage analysis

Contribution to the beam grillage analysis

Author: Pierre PERRIN – Assistant director of the bridges department - east department

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 cross-sectional distribution calculation. In addition, the appearance of high-performance 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 non-exhaustive 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 open-webbed 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 hollow-core element. Transversally, the hollow-core element is represented by bars connecting these nodes. Longitudinally, bars representing each beam and its associated hollow-core 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) CHAMOA-P 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 hollow-core elements are represented by offset longitudinal bars. The intermediate and cantilever hollow-core elements are represented by longitudinal and transverse bars.

Cross section of existing deck

                      

   Figure 3 – Grillage model for the widening of a 2-ribbed bridge which is pre-stressed 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 cut-outs, whose morphology is similar to that of a multi-cellular 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 de-correlate 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 Guyon-Massonet, 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 Guyon-Massonet and Cart-Fauchart methods.

Guyon-Massonet 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 so-called "Guyon-Massonet" 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 Cart-Fauchart 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 November-December 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

Cart-Fauchart method

This method was described in the technical institute of public works (ITBTP) annals N° 271-272 of July-August 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 Guyon-Massonnet method: a more efficient analytical formulation, the possibility to take into account a non-zero 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 Cart-Fauchart 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 post-processing 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 Guyon-Massonet 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 multi-plate 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 mid-span (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 Guyon-Massonet 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 off-road 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 Guyon-Massonet 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 mid-span. The figure below illustrates this hypothesis on the case of the previous 26-beam 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 mid-span section. However, these coefficients apply at lower moments, which limits the oversizing of sections away from mid-span.

Figure 16 - Distribution coefficients according to longitudinal study section

Using the mid-span as the study section for the distribution of moments, a maximum coefficient of 2.55 is obtained in the most stressed beam at mid-span. 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 mid-span.

Figure 17 - Mid-span 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 mid-span. 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 mid-span (Km, orange curve) is 18%.

Figure 18 - Cross-sectional distribution of mid-span 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/CHAMOA-P/chamoa-p.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 GuyonMassonnet-Barès - 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 - Novembre-Décembre 1941

[6] - Méthode de calcul des ponts nervurés sans entretoise intermédiaire - Annales de l'ITBTP -Juillet-Aout 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°9-10 – 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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Transverse bending of a multi-beam bridge

Transverse bending of a multi-beam bridge

Author: B. Tritschler (Arcadis) – April 29th, 2020

1. Preamble

This document presents a solution for modeling a multi-beam deck with high longitudinal bending stiffnesses (pre‐tensioned or post-tensioned prestressed concrete beam types).

The purpose of this model is, more particularly, to study the transverse bending (study of the one-way joists and the intermediate cross girders) and to conclude on the relevance or not to model the one-way joist slab in an offset way, according to the efforts that we want to recover.

1.1 Need for a global study

The mechanical behavior of the hollow core element of the deck slab with intermediate cross girders is strongly conditioned by the transverse stiffness provided by the cross girders. In this case, the study of the slab is inseparable from the study of the intermediate cross girders.

In order to get as close as possible to the real state of stress of the slab and the cross girders, it is necessary to carry out a global study of the structure through a model using the stiffness of all the deck elements (beams - cross girders and slab).

The forces applied to the cross girders and the slab will be determined using a common finite element software to model the entire deck.

This model is the closest to reality because it takes into account the flexibility of the beams and the stiffness of the slab (when they exist).

The forces provided by the model directly cumulate the local forces and the general transverse bending of the deck.

In practice, there are two possible models to choose from:

• 3D modeling, with hanging beams and cross girders offset with respect to the plane (or level) of the slab.

• “Pseudo-plane" modeling, with beams and cross girders in the same plane (or level) as the slab.

Notes:

The different models will be set up with the interaxis distances of the beams and cross girders.

Transverse prestressing is not presented in this study.

1.2. PURPOSE OF THE NOTE

The purpose of this note is to compare two models of a same deck;

• a model taking into account the offset of the "bar" elements, i.e. the hanging beams and the cross girders.

• a model considering the "bar" elements that are the beams and the cross girders, without offset with respect to the plane of the hollow core element.

In both cases, the one-way joist slabs are modeled as shell elements of varying thickness.

The aim is to compare the results according to two criteria:

• validity of the results - direct comparison of the results provided by the two models / validation of the results.

• ease of exploitation of the results / analysis of the functioning of the models.

2. MODELING

2.1. REAL GEOMETRY

The geometry of the structure is detailed below.

2.1.1. Longitudinal section

Longitudinal section (cf. A009_0729-3_a1966_PL 05d_OUVRAGE_coffrage)

2.1.2. Cross section

2.1.3. Beam

2.1.4. Cross girders

2.2. GEOMETRY OF MODEL N°1

The first model consists of taking into account the offsets of the hanging beams and cross girders. For this model the geometry is visually very close to the real geometry.

2.2.1 Beams

Beam compression flanges are modeled in shell model as part of the slab. Only the hanging beams are modelled in a "bar" model, which is the most suitable hypothesis given their "linear" geometry.

The "complex" geometry of the hanging beams does not allow us to directly enter their dimensions in Robot; we will provide the software directly with the mechanical characteristics to be taken into account.

SECTION ANALYSIS

The variation in web thickness near the supports is neglected (little influence for the calculation of the bearing and the slab).

(hbeam = 1,90 m – hcompression flanges = 0,22 m – vpz = 0,642 m) => offset = 190 – 64,2 – 22/2 = 114,8 cm rounded to 1.15 m.

2.2.2 Cross girders

Cross girders compression flanges are modeled in shell model as part of the slab. Only the hanging beams are modelled in a "bar" model, which is the most suitable hypothesis given their "linear" geometry.

The geometry of the hanging beams is entered directly into the software.

=> offset. = 122/2 + 16/2 = 69cm.

2.2.3 One-way joist slab

The slab is made up of the elements between compression flanges of beams “hollow bricks of the joist slab " of constant thickness = 16cm, and the compression flanges, which will be considered variable from 16cm at the edge, to 22cm at the axis.

2.2.4 Span

The span is modeled on single supports (1 support/beam).

2.3. GEOMETRY OF MODEL N°2

For the second model, we consider the linear elements (beams and cross girders), in the middle plane of the slab. In order to model the longitudinal and transverse bending stiffnesses as accurately as possible, we will consider the real sections of the beams and cross girders.

2.3.1 Beams

The "complex" geometry of the hanging beams does not allow us to directly enter their dimensions in Robot; we will provide the software directly with the mechanical characteristics to be taken into account.

SECTION ANALYSIS

2.3.2 Cross girders

In accordance with the requirements of the standard post-tensioned prestressed beam file, the width of the slab taken into account on either side of the cross girders rib is equal to one tenth of the distance between the axes of the edge beams, i.e. 1.80m.

Flange total width = 1.80 + 0.25 + 1.80 = 3.85

2.3.3 Slab

The modeling of the slab is strictly identical to the one used for model n°1.

2.3.4 Span

The span is modeled on single supports (1 support/beam).

2.4. LOADING

2.4.1 Principle

We will compare the effect of identical loads on the two models.

To do so, we will consider cases of arbitrary charges:

• a distributed load over the entire slab (equivalent to loads brought by superstructures or to the loads of system A of instalment 61 part II)

• point loads, or system of point loads (corresponding to concentrated loads, reconciliation of the systems of loads of system B, of instalment 61 part II).

2.4.2 Distributed load

A distributed load of 1 t/m² (order of magnitude value of A(1)) will be considered.

2.4.3 Bc system (2 trucks of 30Ton per lane)

We will consider two Bc trucks, arranged side-by-side in order to obtain a maximum moment in the slab (resulting from the rear axles being centered on the plate).

Note: the central slab is not loaded (presence of a DBA).

2.4.4 Point load

We consider a point load of 10 Tons (equivalent to the Br wheel) centered on the plate.

2.5 SUPPORT CONDITIONS

Valid for both models: a simple support under each beam (except for one beam → pinned).

3. RESULTS

3.1 Central cross girders

3.1.1 Model n°1

3.1.2 Model n°2

3.1.3 One-way joist slab summary

The results provided by model n°1 are not directly usable: the effects only concern the hanging beams (normal effort + moment). It is necessary to recombine these effects with those of the one-way joist slab which correspond to the solicitations of the cross girder’s compression flanges (normal force and moment).

Reconstitution of the forces in the cross girder:

Hypotheses:

The normal effort of the compression flanges balances the one present in the hanging beams. The moment present in the one-way joist slab is negligible compared to the moment of the hanging beams (global effect).

MTotal=N∙Mhanging beam 

For information purposes, we will evaluate the width of the one-way joist slab which is necessary to balance the normal effort under a load case of type A(l).

Case 1 (distributed load)

(symmetrical load case)

We study several sections and integrate the normal stress. The integration is carried out according to the trapezoidal rule.

The integration of the normal stress on the sections studied above makes it possible to balance the normal effort existing in the hanging beams of the cross girder.

The required "real" width of the one-way joist slab is: (16.27 - 14.02)x2 = 4.50m.

That is to say a participating width of 4.50m (instead of the 3.85 recommended by the standard).

The participating width prescribed by the typical post-tensioned prestressed concrete beam standard is close to the one highlighted in this case: the geometry of the compression flange of the cross girders (T-beam) is validated.

The visible difference comes from the approximation made as to the width of the compression flange to be considered. The mapping on the following page shows the variations of the compression state in the slab: the participating width is variable: increasing from the edge towards the center. This difference remains small and thus allows the validation of both models.

The model which considers the offsets provides, after reconstitution of the efforts, moments in the cross girders which are very close to those provided by the model that does not consider offsets.

The increasing complexity of the model does not seem to bring any precision or visible advantage compared to a simpler model, brought back to the middle plane of the slab.

We recommend using a model similar to the model n°2 presented in this note.

3.2 One-way joist slab

3.2.1 Model n°1

3.2.2 Model n°2

3.2.3 One-way joist slab summary

Model No. 2, which does not consider offsets, provides very close moments in the one-way joist slab (within 3%) to those of the model considering offsets.

The increasing complexity of the model does not seem to bring any precision or visible advantage compared to a simpler model, brought back to the middle plane of the slab.

We recommend using a model similar to the model n°2 presented in this note.

4. Conclusion

A model with offset beams and cross girders does not provide directly usable results for the study of the beams and cross girders, contrary to a model brought back to the middle plane of the one-way joist slab.

Both models provide similar results with respect to the bending stresses of the slab.

We select a model similar to the model n°2 presented in this note, such model allowing us to obtain coherent results that can be directly exploited for the purpose of this study.

Note: The user must verify whether the approach presented in this note is applicable to his or her particular project.

5. APPENDIX - INFLUENCE OF BEAMS’ TORSIONAL INERTIA

For the selected model, we considered the non-cracked torsional inertia of the beams, a hypothesis which is not in conformity with the recommendations of the SETRA (operating society for transport and automobile repairs) guide PRP 75.

In order to assess the influence of the torsional inertia of the beams on the behavior of the intermediate cross girders and the hollow core element, we reran a calculation while considerably reducing the torsional inertia (so as to see an exaggerated effect).

We have divided by 10 the value of Ix retained in the previous model.

5.1 Cross girders

5.2 One-way joist slab

The moments show little variation relative to the torsional inertia retained.

Taking into account the torsional inertia in the cracked section should reduce the elastic value by about 30%, such a reduction will have no visible effect on the results.

It should be noted that we are, in almost all cases, in the presence of prestressed cross girders and hollow core elements which are therefore rarely cracked overall.

We suggest keeping the torsional inertia determined by an elastic calculation in uncracked section.

Note: The study of the influence of the torsional inertia explained above is only valid for structures with intermediate cross girders.

 

$translationBooks

Example of a beam grillage calculation according to different methods - Part 1

Example of a beam grillage calculation according to different methods 

Comparison of the results - multi-criteria analysis

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

1) Introduction

1.1 Objectives

In this document, we will model the same multi-beam deck using several approaches:

• the Guyon-Massonnet method,

• a plane grid approach,

• a 3D grillage (modeled as a ladder beam)

• a 3D model associating bars and shells, with two approaches. 

We will compare:

• Support reactions,

• Efforts, moments and shear.

In addition, we will perform two transverse bending calculations to highlight the differences and limitations of the methods.

We will test the incidence of a moderate skew (70 degrees), a more consequent skew (50 degrees) and the presence of cross girders.

A table, by way of conclusion, will attempt to give the reader some clues as to the advantages and disadvantages of each of the models.

The dimensions and applied loads are plausible for a structure deemed to be made of reinforced concrete.

We insist on the fact that each work is particular and that we find ourselves, in the context of this example, in a given configuration of flexural and torsional rigidity, and that consequently the conclusions cannot be generalized as such.

1.2 Bibliography

We invite the reader to refer to the following references:

[1] - Guide technique CHAMOA P CHaîne Algorithmique Modulaire Ouvrages d’Art – Apendix http://www.setra.fr/html/logicielsOA/Ponts_Types/CHAMOA-P/chamoa-p.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 – ITBTP annals - January 1962

[4] - Le calcul des grillages de poutres et dalles orthotropes selon la méthode Guyon-MassonnetBarès - 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 - November-December 1941

[6] - Méthode de calcul des ponts nervurés sans entretoise intermédiaire – ITBTP annals – July-August 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°9-10 – 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

[10]- Emploi des éléments finis en génie civil (Tome 1) : La modélisation des ouvrages – sous la direction de Michel Prat

[11] - Contribution à l’étude des grillages de poutres – Pierre Perrin – Dir Est – sur le wiki de l’AFGC [https://wiki-gtef.frama.wiki/accueil-gtef:partie-3:exemple-c]

[12] - Flexion transversale d'un pont multipoutre – Benjamin Tritschler – Arcadis - – sur le wiki de l’AFGC [https://wiki-gtef.frama.wiki/accueil-gtef:partie-3:exemple-c]

[13] Guide pour l’évaluation structurale et la réparation des Viaducs à travées Indépendantes à Poutres Préfabriquées précontraintes par post-tension (VIPP) – CEREMA – (à paraître)

[14] Dossier PRAD 73 – SETRA

[15] Dossier VIPP 67 – SETRA

1.3 Possible complements 

To complete the study, in a non-exhaustive way, in the end we could add:

• Modeling with 3D elements (see reference [11])

• The study of a longitudinal and/or transverse phasing, taking into account the creep shrinkage, either as at a fixed rate or by using a calculation with behavioral laws,

• A study using an improved "Guyon Massonnet" approach

• Tests to find the optimal width of the cross bands

• How to take into account geometric or material non-linearities,

And extend the study to cases of structures with flexible connections.

2) Description of the structure and loads 

2.1 Geometry

 It includes:

• Eleven ribs of 40 cm x 100 cm ht, spaced at 0.90 m intervals.

• Two 25 m spans,

• A 25 cm thick hollow-core element,

It is made of C35 concrete, E=36000 MPa, ν=0.2 and rests on simple supports.

2.1 Studied load cases

The applied loads are:

• Self-weight:

• A fictitious superstructure load of 3.00 kN/m² on the entire surface of the hollow-core element and 5 kN/ml on the edges at the end of the cantilever:

• Multiple operating loads

• A linear load of 9 kN/m in span 1, on beams 1 to 3:

• A linear load of 9 kN/m in span 1+2, on beams 1 to 3:

• A distributed load from the abscissa 10.00 to 11.25 m, astride beams 6 and 7

(will be used for the transverse bending study)

• A point load on the abscissa 1.25 m from an abutment on beam n°4:

(will be used to study support reactions)

• A linear load of 100 kN/ml at the edge of the deck:

(will be used for the transverse bending study)

3) Different modelling approaches

3.1 Approach using the Guyon Massonnet method

We will not come back to the method as such, it is largely explained in the texts cited in the bibliography of this document.

In practice, to apply the method, first we must calculate the efforts, case by case, on a 2D beam representing the whole width of the deck (if we are not interested in the deformations, this beam can have ordinary characteristics, as long as they are constant).

The cases of charges to be applied have been described in the previous §.

(For cases modeling a load distributed over two spans, we could have modeled a unit load and multiplied the results by the effective load).

Summary of charges:

• Self-weight (only one rib in our case) - case 1.

• Superstructure - distributed load - case 2:

• Cross beam - case 3 (covers both cases in the Guyon-Massonnet approach):

• Case span 1 and span 1+2 - cases 11 and 12

And the specific case for reactions:

Results:

• Self weight:

Superstructures, 3 kN/m²:

• Superstructures, cross beams:

• Envelope for cases span 1 and spans 1+2:

We use a program that automatically determines the parameters and calculates the lines of influence, also called "Guyon Massonnet coefficient" (KGM):

As an example, graphically, the KGM coefficients:

• for the left cross beam - (e.g. beam 2 takes up 2.50/11 of the load)

• for distributed load q (spans 1 or 1+2):

The second column contains the moment M, the shear V or the reaction Rmax, for the unit cases, for the whole deck, which are distributed according to the KGM of each beam:

Effort calculation according to the Guyon-Massonnet method

We ensure that the sum of the variable of interest of the complete deck (orange columns) is equal to the sum of the variable of interest of the 11 beams.

3.2 Plane grid approach - Model Grill1

This model is constituted of longitudinal bars representing the rib + the hollow core element (whose characteristics are defined in Appendix 1) and coplanar transversal bars of 2.50 m length.

Care is taken to apply the calculation method recommended by the SETRA (operating society for transport and automobile repairs)/CEREMA (center for studies and expertise on risks, environment, mobility and development) documents for the calculation of torsional inertias, in particular for the hollow core element, the torsional inertia is be3/6 - ref [13], [14] and [15].

Model view in plane grid

ULS envelope of the moment (1.35G+1.35Qspan), edge beam

ULS envelope of the shear (1.35G+1.35Qspan), edge beam

3.3 Ladder-beam approach - Model Grill2

The model is similar to the previous one, but the longitudinal beam is subdivided into two bars. A lower one which represents the rib alone (0.40*1.00 m²), an upper one which represents the hollow core element (0.90*0.25 m²). The characteristics are given in the appendix.

The two beams are connected by very rigid bars (S=100 m² and I=100 m4) to ensure mechanical continuity.

Special care is taken to make duplicate bars and rigid bars non-weighty.

It is not possible to read the moments directly (see also section C.8 of the wiki).

We have to start from a duplicate (N,M) in the rib, for example, and recalculate the moment from a stress line diagram).

Envelope (ULS N,M) (1.35G+1.35Q span), edge rib (beam 1)

Envelope (ULS N,M) (1.35G+1.35Qbay), second rib (beam n°2)

An advantage of this type of model is the possibility to define a construction phasing and elements with different materials without going through equivalence coefficients, if any.

3.4 Bar + shell approach

3.4.1 Model with offset hollow-core element- model EF1

This model is quite comparable to the model presented in the previous §, but the grillage analysis that models the hollow core element is replaced by shell-type finite elements.

3.4.2 Model with non-offset hollow-core element- model EF2

Reference is made to the "Guide pour l’évaluation structurale et la réparation des Viaducs à travées Indépendantes à Poutres Préfabriquées précontraintes par post-tension (VIPP)" - ref [13]: one possibility is to let the hollow core element joint with the neutral axis of longitudinal beams, whose cross section is composed of the hanging beams and the hollow core element (which brings this model close to the first grillage analysis).

It is evident that this approach slightly overestimates the overall longitudinal stiffness but offers a simplification in the results exploitation - The influence of this simplification remains to be verified on a case by case basis by comparing Ibeam(*) + Ihollow core element to Ibeam - see references [12] and [13].

In this case:

Ihollow core element=0.90*0.253/12=0.00117 m4

Ibeam=0.09076 m4 (see Apendix 1)

Relative difference Ibeam(*) + Ihollow core element / Ibeam: +1.29%, which is very small.

(*) beam= hanging beam + affected hollow core element.

$translationBooks

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.

$translationBooks

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.

 

$translationBooks

Practical application of smoothing

Practical application of smoothing

Example of the Br wheel in Handbook 61, title II

Finite Elements Working Group

Editor: D. Guth, with the support of V. Bruno and F. Dubois.

Introduction

1.1 Purpose 

The purpose of this document is, on the basis of an extremely simple example, to show the differences that can be obtained according to the modeling which is chosen, depending on a certain number of parameters such as: mesh size, modeling of the load, extent of smoothing, and on several software.

It is used as an example in §D.5.2.1 of the Wiki (https://wiki-gtef.frama.wiki/accueil-gtef:partie-2:chapitre-d:d5 ).

It does not claim to exhaustively represent the scenarios that the modeler may encounter, however it is easily replicable by everyone on their favorite software and can be used as a benchmark.

It aims at illustrating the common problems that engineers encounter in their daily work: how to model loads (point or distributed?), over which length to smooth (2h, 4h, other?), where to read the values (at the nodes, at the mesh centers?), what shape for the elements? (triangular? quadrilateral?), is the smoothing (or "averaging") performed by the software correct? etc.

If you want to share your results coming from other software than Robot (R), Sofistik (S) or Pythagoras (P), the right address is elements.finis@afgc.asso.fr .

1.2 Study configuration

We are going to calculate the forces resulting from the impact of a Br wheel (Handbook 61 title II), centered on a one-way joist slab bridge.

The calculations were carried out by SETRA with a Poisson's ratio of 0.15 for the concrete.

The guide does not specify whether the guide values have been smoothed or not.

→ We calculate the effect of a Br wheel, with a diffusion on E=10 cm.

The impact of 60cm (transverse) x 30 cm (direction of traffic) is, therefore, to be modeled as diffused on an impact of 80 cm x 50 cm².

The impact force is 10 tons, which we take equal to 100 kN, i.e. a pressure of 250 kPa.

The modeled slab has dimensions of b=40 m (b≈∞) and a=6 m:

The size of the meshes is 25x25 cm², where the thickness h of the slab is modeled as squared.

In addition, a calculation with 100x100 cm² meshes is also performed to evaluate the impact of increasing the mesh size on the results.

1.3 Studied cases 

We study, with 3 software (R, S and P), in whole or in a part, the following configurations:

• 25x25cm² mesh / distributed load / point load - quadrilateral elements (potentially triangular);

• 100x100cm² mesh / distributed load / point load - quadrilateral elements.

Synthesis and conclusion

All graphic outputs are provided in the Appendix.

We compare the calculated values to the BT1 values that serve as a reference point.

Moments in kN.m/m

Conclusion:

From these calculations, we can already extract the following conclusions:

• It is important to understand what the software does in terms of display and smoothing. Of course, we can always size with the worst case scenario, but even if we could justify it, this approach would not be economical. You should not hesitate to insist to the editor to ensure that the software does not remain a black box for the modeler;

• Avoid point forces, they are too unfavorable and, moreover, have no physical meaning. If describing impact surfaces for rolling loads is complex, for example, the single force must at least be transformed into several point forces;

• Given the differences observed, it is useless to provide results with 8 decimal places, 1 is enough;

• This example illustrates why, during a verification, an external or outside control of the studies, it is necessary to specify in the hypothesis/modeling note the mesh size, the type of element and the chosen smoothing logic. It is also possible to set the tolerance which will be accepted to consider the results as "correct". It is indeed unlikely that two software will provide the same results; the software must be able to converge with a maximum difference of 5% (given it is agreed on at the beginning of the study and in the case of a linear calculation).

• As indicated in the Wiki, the smoothing logic remains subject to the engineer's judgment, however the thickness of the slab and the load impact area must guide the choice.

Once the force is distributed over an impact area defined at the mean slab layer and the mesh size is in a reasonable ratio relative to the slab’s thickness and the dimensions of the load impact rectangle, then the peak value for reinforcement may be selected, where the value smoothed over 2 hours does not lead to too important deviations - which one can see in the table above.

It should be remembered that rolling loads will not stress the entire cross-sectional area at the same time, because they move by nature. The choice between a 2h or 4h smoothing must also be made on the basis of a reflection on the transversal redistributive capacity and/or the incidence of increased constraints by a few %; the consequences are not necessarily the same between a calculation in SLS characteristics and a calculation of crack opening or fatigue;

• If we consider that the SETRA values are correct values (although we have no indication about a possible smoothing), we realize that the approach consisting in smoothing for 2 hours seems to be the most valid;

• The triangular elements give worse values for the same mesh size, which confirms what the Wiki says in Part 2 - A2;

In any case, it is the responsibility of the engineer to get an idea of the impact of the mesh size or of how to model the loads, through tests on simplified parts of models and evaluating the sensibility of such tests.

In summary:

3) Appendix 1 - Model R1 - manual smoothing

This model has a node under the center of the impact rectangle as opposed to Model 2 which was built to have the center of a 25x25 mesh under the center of the impact rectangle - See Appendix 3.

This § calculates the smoothed forces, manually, as opposed to Appendix 2. We will see that the differences between the two approaches are minimal.

• Load impact area with mesh size

• Moment My (transversal)

Values taken every ≈12.5 cm at the central axis of the one-way joist slab:

• Average value on 2 meshes: 28.04 kN.m/m

• Average value on 4 meshes: 26.7 kN.m/m

• Moment Mx (longitudinal)

• Values taken every ≈12.5 cm at the transverse axis:

• Average value on 2 meshes: 23.3 kN.m/m

• Average value on 4 meshes: 21.9 kN.m/m

• Smoothing detail (from the integral):

We read with 2 meshes = 0.50 cm=2h and with 4 meshes=1 m=4h.

Reminder: effect of a load shift of 0.125 m in X (longitudinal): changes practically nothing, see below.

My, smoothed over 1.00 m -> My=26. kN.m/m

My, smoothed over 0.50 m ->My=28.0 kN.m/m

These values are almost identical to the case without shift.

4) Appendix 2 - Using the smoothing functions of a software - Model R1

Many tests are carried out to judge the sensibility of the mesh size.

A calculation with a point force is also computed.

4.1 Use of automated smoothing - distributed load - mesh size 25x25 cm².

On 0.50 m: My=14.03/0.50=28.06 kN.m/m

With a mesh refinement: My=13.96/0.50m=27.92≈28.06 kN.m/m

• For 1 m, My=26.73 kN.m/m

• With a mesh refinement: My=26.57 kN.m/m ≈26.73 kN.m/m

• Mx, smoothed on 0.50 m=11.71/0.50=23.42 kN.m/m

• Mx, smoothed on 1.00 m =21.95 kN.m/m

4.2 Use of automated smoothing - distributed load - mesh size 100x100 cm².

• My=13.97/0.5=27.94 kN.m/m

• My=26.57 kN.m/m

• Mx=10.95/0.5=21.90 kN.m/m

• Mx=20.53 kN.m/m

4.3 Smoothing with point force

Whether you smooth on 1.00 m or 50 cm, the smoothed value remains excessive.

• My=20.9/0.50 m= 41.80 kN.m/m

• My=34.85 kN.m/m, on 1 m.

• Mx=18.56/0.5=37.12 kN.m/m

• Mx=28.09 kN.m/m

4.3 Reminder: load distribution over several point loads

5) Appendix 3 - Model R2 - Diffused load centered mesh

(With centered mesh under the impact i.e. the center of a mesh is under the center of a load)

• Moment My

• Moment Mx

6) Appendix 4 - FE calculation data - models R1 and R2

Materials v=0.15

Sum of reactions:

7 Appendix 5 - Sofistik Model

7.1 Distributed load - mesh ≈25 x 25 cm².

(Curve generated from nodal values - method recommended by the editor).

• My=28.6 kN.m/m

Displayed on a grid with a mesh size of 25x25 cm:

Smoothing over 1 m would lead to (28.6+27.0)/2=27.8 kN.m/m

The same calculation with triangular elements: My=26.9 kN.m/m

• Mx=23.6 kN.m/m

Values displayed in 25x25 cm² meshes

Smoothing over 1 m would lead to (23.6+21.9)/2=22.8 kN.m/m

Same calculation with triangular elements: Mx=21 kN.m/m

Model view with triangular meshes

7.2 Point force- mesh ≈25 x 25 cm².

• My=52.6 kN.m/m

• Mx=45.7 kN.m/m

Data:

Loading (wheel impact):

Impact rectangle 80x50 with 250 Kpa load

• Materials (Poisson’s ratio for concrete)

8) Appendix 6 – Pythagore software

Material and load data:

• Poisson’s ratio for slab

• Distributed load

• Point load

Note on the smoothing performed by the software:

The node smoothing performed by the Pythagore software consists of an average, at a given node, of the results obtained for this node in the four finite elements having this node in common.

As a result, for the node located exactly in the middle of the plate (and load), these four values are symmetrically identical, so the smoothed and unsmoothed maximum values coincide, as shown in the following maps.

8.1 Distributed load - mesh size = 25 x 25 cm².

8.1.1 Without smoothing

My = 27.41 kN.m/m

Mx = 21.55 kN.m/m

8.1.2 With smoothing at the nodes

My = 27.41 kN.m /m (no change in peak value)

Mx = 21.55 kN.m/m (no change in peak value due to load symmetry)

8.2 Concentrated load - mesh size = 25 x 25 cm².

8.2.1 Without smoothing

My = 34.83 kN.m/m

Mx = 28.11 kN.m/m

8.2.2 With smoothing

Same peaks as without smoothing

8.3 Distributed load - mesh size = 100 x 100 cm2

8.3.1 Without smoothing

My = 19.69 kN.m/m

Mx = 13.48 kN.m/m

8.3.2 With smoothing

My = 19.69 kN.m/m (no change in peak value due to load symmetry)

Mx = 13.48 kN.m/m (no change in peak value)

8.4 Concentrated load - mesh size = 100 x 100 cm2

8.4.1 Without smoothing

Same peaks as with smoothing (see below).

8.4.2 With smoothing at the nodes

My = 21.71 kN.m/m

Mx = 15.21 kN.m/m

8.5 Distributed load - automatic meshing ≈ 25 x 25 cm2

8.5.1 Without smoothing

My = 27.54 kN.m/m (almost the same result as with the square mesh)

Mx = 22.45 kN.m/m

8.5.2 With smoothing at the nodes

Same peak values as without smoothing.

9) Appendix 7 – SETRA abacus (operating society for transport and automobile repairs) - Technical Bulletin n°1

Ma (transverse)=2870 t.m.

Mb=2350 t.m

10) Appendix 8 – Pücher abacus

M=Average value read/8/π*100 kN - this method includes a "certain" margin of error given how each person reads the contours.

Mxx=Mx+νMy et Myy=My+νMx

Numerical application:

We have Mx≈24.3 kN.m/m My≈19.9 kN.m/m

• Mxx=24.3+0.15x19.9=27.3 kN.m/m

• Myy=19.9+0.15x24.3=23.5 kN.m/m

Transverse moment (according to x)

Longitudinal moment (according to y)