Analysis of reinforced concrete with FEM/fr

Contexte
L'atelier FEM offre la possibilité d'estimer le niveau de ferraillage requis dans une structure en béton afin d'éviter une rupture fragile sous tension ou par cisaillement.



Ceci est effectué avec la méthode décrite dans "Computation of reinforcement for solid concrete", P.C.J. Hoogenboom and A. de Boer, HERON Vol. 53 (2008) No. 4. Il s'agit essentiellement d'une routine de post-traitement pour Calculix, qui calcule les principales contraintes de traction dans le béton à partir d'une analyse élastique et les utilise pour déterminer le ferraillage minimal requis dans les trois directions de coordonnées pour éviter les ruptures. Dans l'analyse, il est supposé que le béton ne peut pas supporter de contraintes de traction, alors que l'acier est utilisé à sa capacité maximale (c'est-à-dire qu'il prend en compte toute la contrainte).

Le renforcement requis est exprimé en termes de rapport de renforcement. C'est le rapport de l'acier à la surface du béton. Par exemple, un rapport de ferraillage de 0,01 dans la direction x (rx = 0,01) signifie que la section totale des barres de ferraillage s'étendant dans la direction x doit être égale à 1% de la surface de la section en béton traversée. Une section hypothétique de 1 m x 1 m devrait dans ce cas contenir 0,01 m2 d'acier, ce qui pourrait être obtenu en utilisant 90 barres d'armature de 12 mm de diamètre chacune (surface d'acier = 90 * PI * (0,012)^2 / 4 = 0,0102 m^2). Si le rapport de renforcement requis sur cette section transversale en béton est uniforme, les barres pourraient être placées sur une grille d'équidistance de 9 x 10 avec un entraxe d'environ 10 cm. Il s’agit toujours d’un chiffre pratique dans lequel il reste suffisamment d’espace entre les barres pour permettre au béton de passer et d’assurer un remplissage de haute qualité. Des valeurs beaucoup plus élevées conduiraient à une grille de renforcement très dense avec des problèmes potentiels de qualité, tandis que des valeurs beaucoup plus basses pourraient entraîner de grandes fissures de tension dans la section transversale entre les barres. Dans la pratique, le ratio typique va de 0,002 à 0,02 (= 0,2% à 2%). Vous trouverez des informations supplémentaires dans les codes de conception.

Si le rapport de renforcement requis n'est pas uniforme sur toute la section, la section peut être divisée pragmatiquement en sous-sections avec un rapport plus ou moins uniforme et un renforcement appliqué à ces sections. Un exemple sera donné plus tard.

En guise de mise en garde, la conception d'une structure en béton sûre et durable requiert bien plus que ce que l'atelier FEM peut actuellement fournir. Par exemple, la méthode ne calcule pas la largeur des fissures (importante pour la durabilité et la fonctionnalité), ni les déformations précises (les résultats FEM pour le béton sont simplement linéaires-élastiques), et ne tient pas compte des exigences en matière d'ancrage des armatures (qui entraîneraient une augmentation des taux de renforcement requis dans les zones d'ancrage). De plus, il ne prévoit pas non plus l'écrasement du béton (bien que l'indication de cela puisse être obtenue en traçant la contrainte de Mohr-Coulomb - voir plus loin), ce qui pourrait signifier que le béton se ruine avant que le renfort ne cède, entraînant une défaillance fragile de la structure globale. Cette limitation et d'autres signifient que la fonctionnalité béton de FEM ne peut être utilisée que pour évaluer des conceptions conceptuelles, tandis que les décisions de conception détaillées critiques pour la sécurité et les performances devraient être laissées à des professionnels qualifiés.

Géométrie du modèle, charges et supports
Bien que la routine béton de FEM n'exige aucun critère supplémentaire en matière de géométrie, de charges et de supports, il convient de garder à l’esprit que des angles vifs et un support sur une arête ou un sommet peuvent introduire des concentrations de contraintes qui conduiront à des taux de renforcement extrêmement élevés et irréalistes à ces endroits ou à leur proximité.

Paramètres des matériaux
L'atelier FEM dispose d'un matériau spécifique pour les matériaux renforcés, qui combine un matériau de matrice (par exemple le béton) et un matériau de renforcement (par exemple de l'acier). Pour l'analyse du béton armé avec FEM, les paramètres suivants doivent être spécifiés au minimum :

for concrete:

- Young’s modulus (used in the Calculix analysis to calculate elastic deformations and stresses) - Poisson ratio (same) - uniaxial compressive strength (used during post-processing in FEM to calculate the Mohr Coulomb stress as an indicator for crushing or shear failure in concrete) - friction angle (same)

for steel:

- yield strength (used during post-processing in FEM to calculate reinforcement ratios)

Please note that three types of analysis are performed: 1) An elastic analysis using Calculix (only utilising the elastic parameters for concrete); 2) A post processing step to analyse the required reinforcement (only utilising the yield strength of steel) and 3) Calculation of the Mohr Coulomb stress (only using the strength parameters of concrete, i.e. uniaxial compressive strength and friction angle). The Mohr Coulomb stress can be reviewed in the VTK pipeline.

Application
In the remainder of this article a few practical cases will be analysed to discuss application of the method.

Simply supported beam with uniform load
A 4.0x0.1x0.3m concrete beam is loaded by self weight and a 100kN (25kN/m) distributed load.

The material parameters are as follows:

for concrete:

- Young’s modulus = 32 GPa (as per Calculix default for concrete) - Poisson ratio = 0.17 (as per Calculix default for concrete) - uniaxial compressive strength = 30 MPa (concrete type C30/37) - friction angle = 30 degrees

for steel:

- yield strength = 500 MPa

The specific weight of the concrete is taken as 24kN/m^3

The required reinforcement in x-direction is very high (5.4%) and exceeds typical maximum percentages allowed by code to prevent brittle failure. The high shear stresses at the supports also lead to a requirement of high reinforcement:



The Mohr Coulomb plot shows that beam is indeed prone to crushing on the compression side (Mohr Coulomb stress > 0.0), as would be expected with a very high reinforcement percentage:



Both the reinforcement ratio and Mohr Coulomb stress indicate that we have an issue and that we need to rethink our conceptual design. Potential solutions are to increase the beam dimensions or use pre-stressed concrete. Further details can be found in the following post: https://forum.freecadweb.org/viewtopic.php?f=18&t=28821&start=10#p235003

Beam with mid-span support
A 8.0x0.2x0.4m concrete beam is loaded by self weight and a 160kN (20kN/m) distributed load.

The material parameters are as follows:

for concrete:

- Young’s modulus = 32 GPa (as per Calculix default for concrete) - Poisson ratio = 0.17 (as per Calculix default for concrete) - uniaxial compressive strength = 25 MPa (concrete type B25) - friction angle = 30 degrees

for steel:

- yield strength = 286 MPa (reduced from 500 MPa to account for a safety factor of 1.75)

The specific weight of the concrete is taken as 24 kN/m^3

The ParaView plot of the exported VTK file shows that the reinforcement requirement is largest at the top of the beam near the central support. Here the highest bending moment occurs. The maximum reinforcement ratio is with 0.02 at the high end of the practical range quoted earlier:



The required area of steel at the central support can be obtained with a ParaView integration filter applied to the mid section of the beam:



The panel at the bottom of this picture shows that the total required steel area at this cross section is 389.6 mm^2. As one reinforcement bar of diameter 12mm has a cross sectional area of 113mm^2, it means that 4 bars would be required, giving a cross sectional area of 452 mm^2. These would need to be placed near the top of the beam, while maintaining sufficient concrete cover. The theoretical center of gravity for the reinforcement can be found by integration:

CoG_y = Integrate (rx * y dy dz) / Integrate (rx dy dz) CoG_z = Integrate (rx * z dy dz) / Integrate (rx dy dz)

These integrals can also be determined with ParaView and give for the present case (see bottom panels in the above picture):

CoG_y = 38961 / 389.6 = 100.0 mm   CoG_z = 134917 / 389.6 = 346.3 mm

which is, as expected, center-width and near the top.

The reinforcement requirement found above agrees well with that obtained using traditional methods:

https://forum.freecadweb.org/viewtopic.php?f=18&t=28821&start=20#p235063

Finally, a Mohr Coulomb stress check should be performed to check potential crushing of the concrete. For this check, the characteristic compressive strength of concrete (25MPa) should be divided by an appropriate material factor (>1.0).

Shear wall with uniform load
A 4.0x2.0x0.15m wall is supported by two 0.5m wide columns. The wall is loaded by self weight and a 1.0MN distributed load at the top.

The material parameters are as follows:

for concrete:

- Young’s modulus = 32 GPa (as per Calculix default for concrete) - Poisson ratio = 0.17 (as per Calculix default for concrete) - uniaxial compressive strength = 20 MPa - friction angle = 30 degrees

for steel:

- yield strength = 286 MPa

The specific weight of the concrete is taken as 24 kN/m^3

The horizontal reinforcement ratio peaks at 0.014 (1.4%) near the bottom center section of the wall and the vertical reinforcement ratio is at a maximum 0.008 (0.8%) near the corners of the wall with the columns, where the shear stresses are highest:



The above picture shows possible zones of constant reinforcement ratio for the design of reinforcement. Although a minimum reinforcement percentage of 0.2% is chosen, it will be hard to achieve such a low value in practice, given that the the spacing should not exceed a practical limit (say 300mm). Even with a light reinforcement grid of 10mm bars (cross sectional area = 78mm^2), the reinforcement ratio would then be 2 * 78 / (150 * 300) = 0.0035 (0.35%). (Note: the factor 2 stems from the fact that the grid will be placed at both faces of the wall). If we add one more bar to the grid (halving the distance) the reinforcement ratio would double to 0.7% and one more would give approximately 1%. So most of the reinforcement requirement could be achieved by starting off with a grid of d=10mm at 300x300mm spacing and adding bars in horizontal or vertical direction, as required. This would cover all but the requirement at the bottom of the wall, where we could add 3 bars d=12mm, giving a horizontal reinforcement ratio of 3 * 113mm^2 / (150mm * 150mm) = 0.015 (1.5%). Here it is assumed that the height of the bottom zone is 150mm. Alternatively, we could chose 2 bars of 16mm diamter, which would achieve the same reinforcement ratio for a zone of 180mm height.

Finally, a review of the Mohr Coulomb stress shows that no concrete crushing is expected in the wall. https://forum.freecadweb.org/viewtopic.php?f=18&t=28821&start=10#p234673

Deep beam with opening
The FIB Practitioners' Guide to Finite Element Modelling of Reinforced Concrete Structures contains a design example of a deep concrete beam with opening. The example is used in that report to demonstrate the "Strut-and-Tie" method. Here the results will be compared to those obtained with the FreeCAD FEM workbench.

The beam dimensions are 11.0x4.0x0.6m and it is loaded at the top by a distributed load of 120kN/m and a load of 5000kN introduced by a 1m wide column. The factored compressive strength of the concrete is 0.75 x 0.6 x fc = 0.45 * 35 = 15.8MPa and the factored yield strength of the reinforcement steel is 315MPa.

The reinforcement ratios and principal concrete stresses (compression only) derived with FreeCAD are shown below:



The required horizontal reinforcement (below in red) is determined by integration of the horizontal reinforcement ratio over the vertical cuts of interest (below in black). This is done using a Paraview integration filter.



The insert to the above figure shows a comparison of reinforcement requirements (in mm^2 of steel) determined with FreeCAD to those in the FIB report.

The following shows how the integration over lines of interest works in Paraview:



Finally a plot of compressive and tensile principal stresses to demonstrate how stresses flow through the beam.



The tensile stress pattern suggests an alternative design concept using pre-stressing cables (superimposed in white). This concept is further elaborated in the following post:

https://forum.freecadweb.org/viewtopic.php?f=18&t=33049