Choix de la forme de la section transversale
Forme idéale théorique d’une structure enterrée
The ideal shape of a structure is defined as one that supports loads solely through the deviation of normal forces, without subjecting the structure to bending In this scenario, the line of pressure aligns perfectly with the axis of the structure.
Hypothèse sur l’action du sol
Determining the curve for an underground vault requires precise knowledge of the pressures exerted by the soil on the structure These pressures depend on the deformation of the structure due to soil-structure interaction and the method of backfill placement, making them unknown in advance.
The determination of the ideal shape is conducted for a horizontal free surface under the sole influence of the soil, assuming that the stress state within the soil is unaffected by the presence of the structure The accepted stress state is described by equations 2.1 and 2.2, where the principal stresses are vertical and horizontal The vertical stress corresponds to the weight of the soil column located between the examined point and the free surface, while the horizontal stress is considered proportional to the vertical stress and is defined by the thrust coefficient K, as indicated in equation 2.2.
K z e x = ⋅σ = ⋅γ ⋅ σ (2.2) avec σ x = contrainte horizontale, σ z = contrainte verticale, K = coefficient de poussée, γ e = poids volumique du sol et z = profondeur mesurée depuis la surface libre
On admet de plus un massif sans eau, donc sans pressions interstitielles Les contraintes totales sont égales aux contraintes effectives
Cette hypothèse très simplificatrice suffit cependant pour comprendre le fonctionnement statique général de la structure Le comportement du système complet sera abordé au chapitre 4
Equation différentielle de la forme idéale
The calculation of the ideal shape relies solely on equilibrium conditions The resulting shape is accurately defined for an isostatic arch with three joints (at the key and each base), which is statically determinate However, this calculation does not hold true for a hyperstatic arch, as the shortening induced by normal forces introduces bending stresses within the structure.
La forme idéale de la vỏte est décrite par la relation : z = f(x) (2.3) avec x la distance horizontale mesurée depuis l’axe de symétrie de la vỏte
A noter que la courbe recherchée passe par les trois articulations (Moment M = 0) et présente une pente horizontale en clef de vỏte, conséquence de la symétrie de la situation examinée
The geometric parameters of the studied case are illustrated in Figure 2.1 The span of the vault, defined as the distance between the two supports, is denoted as l_b, while the height of the vault is represented as l_h, and the soil cover on the vault is indicated as ∆h.
Figure 2.1: Etat de contraintes dans le sol pour la détermination de la forme idéale ds dz dx
Figure 2.2: Elément infinitésimal et polygones des forces
L’expression des conditions d’équilibre de l’élément infinitésimal de la figure 2.2 conduit à l’équation différentielle de la forme idéale de la vỏte enterrée (voir annexe) donnée par :
( H γ e (2.4) avec H 0 = effort normal en clef de vỏte par mètre courant
H γ e (2.5) et la solution à cette équation est une caténọde appelée aussi projection de chaỵnette, définie par :
Si la portée et la hauteur de la vỏte sont fixées, il est possible de déterminer l’effort normal en clef :
La forme de la caténọde est attribuée à Denfert-Rochereau (1859, voir Legay 1900)
In general cases where K ≠ 0, equation 2.4 must be solved numerically If H0 is known, the curve can be determined by starting the calculation from the key of the arch The horizontal force is gradually deflected by the applied loads To find the ideal shape of the arch with height l_h and span l_b, it is necessary to iteratively adjust H0 until the curve aligns with the supports.
Figure 2.3: Tracé idéal des vỏtes enterrées
La figure 2.3 présente des courbes théoriques calculées selon cette démarche pour différents coefficients de poussée et pour deux efforts normaux en clef de vỏte H 0,A et
H 0,B Le fonctionnement statique de la vỏte peut être expliqué qualitativement par le polygone des forces de la figure 2.2
Lorsque K = 0, seules les contraintes verticales dévient la résultante alors que lorsque
When K is not equal to zero, horizontal constraints also play a significant role For a constant initial height H0, the curve deviates more rapidly as K increases, resulting in a notable decrease in the range achieved for a selected height lh Therefore, to ensure that the curve passes through identical supports regardless of the value of K, it is essential to increase H0.
Selon l’équation 2.4, la forme idéale dépend de la couverture de terre, de la poussée des terres et des dimensions de la vỏte
Figure 2.4 illustrates that the span and height of the vault significantly impact its ideal shape In a given project, these dimensions are typically fixed, which means that the practical implications of these geometric parameters are somewhat limited.
Figure 2.4: Influence des propriétés géométriques sur la forme idéale théorique
Soil properties that determine stress conditions and loads are often not accurately known during the design phase of a structure, and soil coverage can vary along the construction Therefore, understanding how the ideal shape evolves in relation to these parameters is of particular importance.
La figure 2.5 montre que les tracés obtenus peuvent varier fortement selon les paramètres choisis z- ∆ h
Figure 2.5: Influence du coefficient de poussée K et de la couverture de terre ∆ h sur la forme idéale théorique
The earth pressure, represented by the coefficient of earth pressure, significantly influences the ideal shape This influence is particularly evident when the coefficient varies while other parameters remain constant As the coefficient K increases, the trajectory deviates further from the axis of symmetry In some instances, it may even extend beyond the defined area of the targeted span (x > lb / 2), necessitating a change in the slope's sign on the curve In such cases, the curvature is typically pronounced near the supports.
The depth of the tunnel significantly impacts the design of the ideal arch, with its influence largely determined by the thrust coefficient When K equals 0, this influence is minimal, but it becomes substantial when K is 1.0.
The relative variation of stresses between the base and the apex of the vault is the key factor behind this phenomenon This variation is significant at shallow depths but approaches zero at greater depths The ideal shape evolves based on the relative distribution of these stresses At very deep levels, stresses can be considered uniform, resulting in geometric shapes Specifically, when K = 0, the curve is a parabola; for K = 1, it forms a circle, while other configurations yield an ellipse.
Figure 2.5 illustrates the geometry of a standard cross-section for a two-lane road without a shoulder, which may significantly deviate from the ideal layout The actual eccentricity varies from this basic comparison because contact pressures are influenced by the shape of the arch Additionally, the hyperstatic nature of monolithic structures causes the eccentricity to be distributed on either side of the structure's axis.
In the 19th century, determining the ideal design of arches was crucial for the construction of masonry bridges, given the low tensile strength of the materials used To ensure stability, it was essential for the line of pressure to remain within the structure, necessitating careful selection of the arch shape.
Les travaux de plusieurs scientifiques et constructeurs (Rankine 1876, Legay 1900, Séjourné 1913) témoignent parmi d’autres de cet intérêt et représentent l’état des connaissances à l’époque
In his work, Séjourné (1913) catalogs and describes the major masonry arch bridges of his time, providing a snapshot of the state of the art in bridge construction at the beginning of the 20th century He notably discusses the various forms of vaults, including the use of the catenary shape, exemplified by the Pont d’Avignon This type of bridge, featuring continuous facings, shares significant similarities with buried vaults The fill material placed between the facings serves a role akin to that of the soil, yet it is regarded merely as dead weight, contributing only vertical stress Notably, the horizontal thrust was not considered in the assumptions made during that period.
Sections transversales réelles
The ideal shape may not be cost-effective due to construction constraints, leading to its limited practical application Furthermore, soil pressures rarely align with the simplified distributions previously discussed.
Les sections construites aujourd’hui visent généralement à réduire l’emprise de l’ouvrage en fonction du gabarit d’espace libre à respecter, de faỗon à minimiser le volume de terrain à excaver
Sections can be categorized into two main families: frames, which consist of vertical walls topped with a horizontal slab, and vaults Additionally, these sections may include one or more tubes.
The general functioning of these cross-sections can be assessed by determining the pressure line In this case, the problem differs from the previous one, as the shape of the section and the loads are assumed to be known It is sufficient to determine the internal forces and represent the pressure line This is accomplished using calculation software (Statik 4, Cubus AG 2002), assuming a linear elastic behavior of the structure and loads defined from equations 2.1 and 2.2 The fixings in the foundations and the foundation soil are considered infinitely rigid.
Figure 2.9 illustrates the calculated pressure lines for two standard geometry covered trenches: one frame type (a and b) and one vault type (c and d), subjected to simplified earth pressures For comparison purposes, both structures share the same height and span, despite differing clear space profiles The geometry and loading conditions are perfectly symmetrical.
La comparaison entre les figures 2.9a et 2.9c, respectivement 2.9b et 2.9d met en évidence les différences fondamentales entre le comportement d’une section de type cadre et celui d’une section de type vỏte
Pour une vỏte, la reprise des charges est assurée principalement par la déviation de l’effort normal L’excentricité de la ligne des pressions est généralement assez faible
The structure primarily experiences normal compressive forces, with bending moments and shear forces remaining relatively low This design is particularly well-suited for this type of loading.
Pour un cadre, la ligne des pressions s’éloigne fortement de la géométrie de la structure
La reprise des charges est alors assurée en grande partie par la flexion de la dalle supérieure et des parois
Figure 2.9: Lignes des pressions pour une section de type cadre (a et b) et une section de type vỏte (c et d)
The impact of soil thrust and land cover is not influenced by the shape of the structure These trends align with the observations discussed in the chapter.
Par contre, une variation de ces paramètres peut avoir des conséquences différentes selon la forme de la section
As the soil cover increases, the eccentricity of the pressure line rises rapidly at mid-span and at the supports of the upper slab, leading to significant bending moments Additionally, the slope of the pressure line steepens considerably near the supports of the upper slab within the walls Consequently, the shear forces that the slab must withstand become increasingly problematic The capacity of the reinforced concrete structure under these types of loads is quickly reached, making it necessary to generally abandon this geometry when the soil cover is substantial.
Compared to other types, the vault section can support significantly larger earth covers with highly satisfactory performance It is statically more efficient in handling symmetrical earth loads.
Asymmetrical loads are significantly less advantageous for a symmetrical arch Figure 2.10 illustrates the pressure line calculated for a scenario resulting from poor backfilling planning, where the fill levels on the left and right sides of the tunnel differ greatly This discrepancy renders the arch shape ineffective in efficiently bearing the applied loads, leading to substantial bending moments within the structure The limitations discussed for symmetrical loading frames also apply by analogy Therefore, highly asymmetrical situations should be avoided for a symmetrical arch.
Figure 2.10: Chargement asymétrique d'une vỏte symétrique (état de construction)
The impact of water on pressure lines is demonstrated through two sections, focusing on a static water table that reaches the surface (case ∆h = 2 m) Water increases lateral thrust, which consists of hydrostatic pressure and earth pressure calculated from effective vertical stresses, using the accepted lateral thrust coefficient.
La figure 2.11 montre que la prộsence de l’eau conduit à un ô aplatissement ằ de la ligne des pressions L’effet est similaire à une augmentation du coefficient de poussée
This case is theoretical, as the section must be sealed and watertight to withstand hydrostatic pressure Additionally, it is essential to assess the risk of structural uplift due to Archimedes' principle, considering scenarios both without water and with water up to a height of 2 meters, resulting in a total height of 11 meters for length and breadth.
K = 0.5 sans eau avec eau (jusqu'à la surface) h = 2 m = 11 m l b = 7 m l h
Figure 2.11: Influence d’une nappe phréatique statique sur la ligne des pressions: a) cadre et b) vỏte
The static behavior of a structure is influenced by its geometry and the type of loads it must bear Structures can be categorized based on how they handle these loads: those primarily subjected to normal compressive forces tend to align with the pressure lines, while structures that experience significant bending do not follow these pressure lines.
The parametric study reveals that soil surcharge and earth pressure significantly impact the theoretical ideal shape of the vault, consequently affecting its overall behavior Therefore, accurately assessing the distribution and intensity of contact pressures acting on the structure's perimeter is essential This evaluation is the focus of Chapter 4.
Optimisation de la forme
From a static perspective, it is usually beneficial to approach the pressure line as closely as possible However, reinforced concrete allows for some deviation from this line Additionally, frame-type sections can perform satisfactorily and may be more economical in certain situations.
It is essential to remember that it is always possible to significantly enhance the performance of a structure by locally adjusting its geometry, while still adhering to the project's other constraints and without complicating the construction process.
Certain construction details can be critical for sizing, and local geometric adaptations that align with pressure lines can often address these issues This approach typically reduces the amount of concrete and reinforcement needed while enhancing the structure's quality by minimizing cracking and deformation and increasing ductility.
Eléments de structure rectilignes principalement fléchis
Solution arcs-et-câbles
Figure 2.14a illustrates an isolated subsystem featuring an upper slab span, where resultant forces exit the structure at points a and b, which is unacceptable To address this, the strut can be redirected to remain within the structure, as depicted in Figure 2.14b, by incorporating vertical and horizontal reinforcements in the upper slab The vertical reinforcement transmits tensile forces, counterbalanced by a funicular arch formed by loads acting between points a and b The vertical reaction components at points a' and b' are managed by vertical reinforcements, while horizontal components necessitate the addition of a lower horizontal reinforcement between points a' and b' This equilibrium solution comprises three thrust-compensated funicular structures: a central arch with a lower tie and two lateral arches with upper ties This method, termed the arches-and-cables solution, is a viable option according to plasticity theory and serves as a pedagogical tool for developing models that better reflect actual behavior.
Figure 2.14: Dalle supérieure: a) ligne des pressions, b) solution d'équilibre arcs-et-câbles, c) champ de contraintes pour la solution avec armature d'effort tranchant répartie et d) modèle bielles-et-tirant équivalent
The tensile and compressive forces generated at the connections have the same impact as their resultant shifted along the pressure line These forces must then be transferred to the trench walls supported by the frame angles, as discussed in section 2.3.
This solution requires the installation of a concentrated shear force framework near points a and b, which may lead to certain construction challenges Additionally, the resistance of the compressed arches supported by the walls can be diminished by cracking if their inclination to the horizontal becomes too shallow, potentially impacting the overall behavior of the slab.
Solution avec armature d’effort tranchant répartie
Mode de reprise des charges
La figure 2.14c propose une alternative avec la mise en place d’une armature d’effort tranchant répartie sur la totalité de la dalle Le champ de contraintes de la figure 2.14c
The generally accepted model for sizing reinforced concrete beams involves the indirect transmission of loads to supports through inclined compression fields, supported by stirrups The inclination angle of these compression fields, denoted as α, must adhere to specific criteria to ensure the structure exhibits ductile behavior, as highlighted by Muttoni et al (1997) The recommended limits set by SIA 262 (SIA 2003b) are outlined in inequality 2.8.
L’effort normal de compression a un effet favorable et permet de réduire cette inclinaison
Near the embedded elements, compression fields converge to create a fan shape that effectively transfers loads to a minimally sized support area The angle of the struts forming the fan gradually changes from α to 90 degrees.
Le dimensionnement des armatures et la vérification des éléments en béton peuvent ensuite être effectués en résolvant le champ de contraintes par équilibre
La figure 2.15a montre deux sous-systèmes permettant d’effectuer le dimensionnement à l’effort tranchant de la dalle à proximité de la paroi centrale
L’équilibre des efforts verticaux agissant sur le premier sous-système conduit à l’expression des étriers nécessaires à la reprise de l’effort tranchant α
The total surface area of the stirrups (A sw) in the slab width (b w) is arranged regularly with a spacing (s) of 2.9 The angle of inclination of the compression fields (α) and the distance (z) between the resultant forces of tension and compression are critical for ensuring the effective transfer of bending moments.
M, V = effort tranchant à une distance zcotα de l’encastrement (section critique), f s = limite d’écoulement de l’armature q α
Figure 2.15: Reprise de l'effort tranchant: a) dimensionnement des étriers et b) vérification de la contrainte dans le champ de compression déterminant
The balance of vertical forces in the second subsystem allows for the determination of stress within the compression field, which can then be compared to the effective strength of concrete, denoted as f'c.
La résistance effective des champs de compression est diminuée par la propagation de fissures inclinées dont l’ouverture est contrôlée par les étriers D’après Muttoni et al
1997, la résistance effective du béton peut être déterminée par :
Cette équation est prise en compte par la norme SIA 262 (SIA 2003b) en considérant f c0 = 30 MPa
The shear resistance can be enhanced by increasing the thickness of the slab near the embedded connections in the walls The operation of the slab with an inclined lower member is illustrated, where the compression strut necessary to counteract the bending moment at the connection is angled at δ The vertical component of this strut, as defined in the relevant equation, plays a crucial role in the vertical force equilibrium of the subsystem This mechanism allows for the transfer of a portion of the shear force to the support without relying on the stirrups.
Figure 2.16: Adaptation locale de la forme: a) épaissement local de la dalle b) fonctionnement statique et c) contribution de la membrure comprimée à la reprise de l'effort tranchant
The ductility of a reinforced concrete structure refers to its ability to withstand deformations without experiencing sudden failure This capacity for deformation is essential for redistributing forces, which is crucial for enhancing the structure's strength and accommodating any imposed deformations.
La ductilité de la structure est aussi une condition de base à l’application de la théorie de la plasticité et des méthodes de dimensionnement et de vérification qui en découlent
La capacité de déformation (en particularité pour les structures linéaires, la capacité de rotation des rotules plastiques) est donc d’un intérêt prépondérant
According to the CEB model code of 1990, plastic rotation is defined as the rotation resulting from plastic deformations observed at failure Mathematically, it is expressed by the integral equation involving parameters such as the plastic strain and the corresponding variables.
1 (2.14) avec l pl = longueur de la zone plastique, d = hauteur statique, x = hauteur de la zone comprimée, ε sm = déformation moyenne de l’armature, ε sm,y = déformation moyenne de l’armature à l’écoulement
The rotational capacity of reinforced concrete structures has been the focus of extensive research over the past few decades A comprehensive overview of the current state of knowledge on this topic is detailed in the CEB 1998 report.
Several theoretical models have been proposed to determine the plastic rotation capacity, including those by Langer (1997), Sigrist (1995), Bigaj and Walraven (2002), and Bachmann (1967) These models focus on accurately assessing and integrating the deformations of a reinforced concrete beam at failure A realistic estimation of this deformation state requires precise consideration of the behavior of the reinforcing steel, concrete, and the bond between the reinforcement and concrete Generally, these models align with experimental results and effectively replicate the influences of key parameters.
The rotational capacity of a reinforced concrete beam is reached when the most stressed section attains its ultimate curvature Two failure modes can be identified based on the ultimate deformations of materials, the rate of reinforcement, and the normal force: failure due to anchor pullout of the reinforcement and failure due to crushing of the compressed concrete.
The regime change is illustrated in Figure 2.17, which shows the total rotation at failure, both experimentally measured and calculated using one of the models This rotation includes contributions from both elastic and plastic deformations.
Langer 1997 l 0 /d = 13.0 f t / f y = 1.24 écrasement du béton comprimé arrachement de l'armature
Figure 2.17: Capacité de rotation θ en fonction du taux d'armature ρ Comparaison entre les simulations de Langer (1997) et les essais de Eifler et Plauk (1974), d'après CEB
La rotation plastique (équation 2.14) peut aussi être exprimée par : m pl pl χ θ = l ⋅ (2.15) ó χ m représente la courbure moyenne sur la zone plastique
The capacity for plastic rotation is influenced by factors that affect both the extent of the plastic zone and the curvature within it, particularly the maximum curvature at failure A detailed description of the factors impacting rotation capacity can be found in CEB 1998.
Ces facteurs ne sont pas forcément connus lors du dimensionnement d’une structure
L’évaluation de la capacité de rotation d’une rotule plastique doit par conséquent être faite de manière suffisamment prudente pour couvrir les incertitudes existantes.
Solution sans armature d’effort tranchant
Mode de reprise des charges
The load transmission models discussed so far require the installation of shear reinforcement However, most covered trenches today are constructed without such reinforcements, as they complicate the execution process.
In the absence of this framework, the mechanisms for transferring shear forces to the supports significantly change The behavior of a beam without stirrups is detailed by Muttoni et al (Muttoni, Fernández Ruiz 2006; see also Muttoni).
En stade non fissuré, l’état de contraintes peut être évalué par la théorie de l’élasticité
As loads increase, flexural cracks gradually develop within the structure The load transfer mechanism, according to elasticity theory, becomes invalid, as the transmission of tensile stresses through the edges of the cracks becomes negligible.
La transmission des efforts doit alors être assurée par d’autres modes Selon Muttoni et
Schwartz (Muttoni, Schwartz 1991 voir aussi Muttoni, Fernández Ruiz 2006), les modes de transfert représentés à la figure 2.18 sont possibles Le comportement réel résulte d’une combinaison de ces trois modes
Figure 2.18: Mode de reprise de l’effort tranchant sans étriers: effet porte-à-faux, effet d’engrènement et effet goujon (d’après Muttoni, Schwartz 1991, figure tirée de Guandalini 2005)
Figure 2.19a illustrates a potential model for load transmission in the upper slab, qualitatively considering the presence of bending cracks in highly stressed areas Load transfer to the supports occurs indirectly, following a combination of the modes depicted in Figure 2.18.
Experimental observations indicate that cracks near supports propagate into the compressed zone, tilting toward the point of reaction introduction These cracks intersect a portion of the accepted concrete ties, rendering them ineffective A direct transmission from the point of zero moment to the support can be conceptualized based on plasticity theory However, tests reveal that a significant opening crack, known as a critical crack, traverses the compression strut The presence of such a crack severely impacts the strength of a compressed strut, affecting the ultimate load capacity as determined by plasticity theory.
2.19b ne peut pas être atteinte La rupture se produit alors généralement par effort tranchant et non par flexion q 1 a)
Figure 2.19: Mode de transfert des charges dans la dalle supérieure sans armature d’effort tranchant pour différents niveaux de charge avec a) appui indirect et b) appui direct
According to Muttoni (2003, 2003a; Muttoni & Fernández Ruiz, 2006), the resistance is significantly influenced by the position and opening of the critical crack, as well as the size of the aggregates, which relates to the roughness of the crack.
Le critère de rupture semi-empirique donné par l’équation 2.16 est proposé par Muttoni
According to Muttoni (2003), the initiation of the critical crack is correlated with the longitudinal deformation ε in the critical zone, which is located 0.6d from the compressed area of the concrete and 0.5d from the point of application of a concentrated force.
= + (2.18) ó d g est la taille maximale des granulats La déformation ε peut être estimée par un calcul élastique en stade fissuré
Numerous comparisons with various tests indicate that this criterion effectively assesses the shear resistance of elements without stirrups across a wide range of parameter variations Notably, the significant scale effect observed experimentally is accurately replicated by the model.
Figure 2.20: Critère de rupture et comparaison avec des essais (d’après Muttoni 2003): a) poutre simple sous charge ponctuelle et b) sous charge répartie
La déformation dans la zone critique peut aussi être liée à la déformation longitudinale de l’armature de flexion εs et donc aux sollicitations de flexion La norme SIA 262 (SIA
2003b) propose un modèle de rupture simplifié basé sur les moments de flexion
According to SIA 262, the shear resistance is influenced by potential plastic deformations of the reinforcement or its arrangement These factors are accounted for by selecting a fixed value of k v = 3 and increasing k v by 50% compared to the elastically calculated value.
Ces deux effets sont confirmés par des recherches expérimentales
A study by Baron highlights that a staggered arrangement of reinforcement significantly reduces strength (Baron 1966) This weakening is due to adhesion mechanisms between the reinforcement and concrete, specifically the anchoring effect of stopped bars, which leads to localized cracking and weakens the element This negative influence can be particularly critical in covered trenches, especially vaulted types, which require numerous overlapping joints for practical reasons related to geometry and bar transport.
The crucial role of bond in the shear resistance mechanisms of stirrup-less elements has been experimentally demonstrated by Kani et al (1979) Tests on identical elements with artificially reduced bond show that the shear resistance of traditional ribbed reinforcement elements is decreased by 40% compared to those with smooth reinforcement This reduction is attributed to the lack of bond, which hinders crack formation in the critical zone, ultimately affecting the strength of the compressed struts responsible for transferring shear forces to the supports (as illustrated in Figure 2.19b, see Muttoni, Schwartz 1991).
Vaz Rodrigues et Muttoni ont étudié expérimentalement l’influence des déformations plastiques de l’armature sur la résistance à l’effort tranchant (Vaz Rodrigues, Muttoni
In a study conducted by Vaz Rodrigues et al (2004, 2005), tested elements consisted of slab strips supported on two points, featuring a main span and an overhang These elements were subjected to a central load Q on the main span and an additional load αQ at the end of the overhang, allowing for a variable level of fixity at the intermediate support The beams had a rectangular cross-section measuring 450 mm in height and 250 mm in width, with reinforcement in both the lower and upper layers at a ratio of ρ = 0.79% The reinforcing steel used included cold drawn steel (Topar R for eight beams) and hot rolled steel (Topar S for three beams).
Figures 2.21d and 2.21e illustrate the evolution of the rotation ψ in the zone ó at the point of rupture, as defined by figures 2.21b and 2.21c, which is comparable to the zone ó where plastic deformation occurs This evolution is analyzed in relation to the normalized shear stress in the critical section, also detailed in the figures The observed mode of failure is noted, with shear failure being prevalent in nearly all tests, even after the reinforcement has undergone plasticization.
This experimental study confirms the validity of the rupture criterion established by equation 2.16 when the reinforcement deformations are elastic Furthermore, it clearly demonstrates a reduction in resistance due to the plasticization of the bending reinforcement This decrease in resistance is attributed to the propagation of the critical crack and the increase in its opening as the beam deforms under a constant load, entering the plastic plateau.
Figure 2.22 illustrates that SIA 262 (k v = 3, SIA 2003b) predicts excessively high resistance in the plastic regime, as the design resistance (level d) closely aligns with the laboratory-measured values Consequently, the defined safety margin is inadequate when significant plastic deformations occur.
Angles de cadre et autres détails constructifs
Angle de cadre sous moment négatif
In most cases, the frame angles of covered trenches experience negative moments, where the resultant force passes inside and the tension member is on the outside As illustrated in Figure 2.12, the transfer of moment from the slab to the wall is facilitated by a compressed strut formed in the diagonal, which redirects the compressive and tensile forces acting on the inner and outer faces of the structure, respectively (refer to Figure 2.29a).
Figure 2.29: Fonctionnement statique de l’angle de cadre soumis à un moment négatif
(cas h paroi = h dalle ): a) transmission du moment seul et b) prise en compte de l’effort normal et de l’effort tranchant
The model illustrated in Figure 2.29a can be easily extended to incorporate the effects of normal and shear forces In this scenario, the shear force and normal force acting within the slab are balanced by the corresponding normal and shear forces in the wall, as depicted in Figure 2.29b.
When the thicknesses of the slab and wall are identical, the reinforcement scheme must facilitate the consistent transmission of tensile forces within the tensioned member If this reinforcement is not continuous due to construction-related factors, such as adhering to concrete pouring stages, it is essential to ensure that the anchorage length of the various reinforcements allows for the complete transfer of forces across the frame angle.
Angle de cadre sous moment positif
In specific situations, such as under a strong asymmetric force, the frame angle may experience positive moments A stress field can be achieved by replacing the links with tension members and vice versa, as illustrated in Figure 2.30a However, this solution necessitates the installation of an inclined brace within the angle's diagonal, significantly complicating the construction process.
Une alternative consiste à dévier la bielle comprimée à deux reprises en prolongeant les armatures principales jusqu’à la face extérieure de l’angle de cadre, comme le montre la figure 2.30b
Figure 2.30: Angle de cadre soumis à un moment positif: modèles possibles selon la théorie de la plasticité: a) déviation simple, b) déviation double et c) déviation supplémentaire par une armature inclinée
Both solutions require an external anchoring system (anchor plate) to fully divert the compressed rod An experimental study conducted by Nilsson demonstrated that structural details failing to meet this requirement do not enable the full activation of the resistance of adjacent elements.
(voir Nilsson 1973) Dans ces cas, la rupture a généralement lieu brutalement par la formation et l’ouverture d’une fissure diagonale dans l’angle de cadre
Figure 2.31 illustrates the effectiveness of the structural details tested by Nilsson, defined as the ratio of the moment resistance of the frame corner to the minimum moment resistance of adjacent elements The anchoring of reinforcements in the compressed zone of adjacent elements (Detail A) leads to very early failure of the frame corner The solution with loops arranged in the plane of the main reinforcements (Detail B) is less detrimental but still inadequate The use of a continuous reinforcement bent at 270° provides a more effective alternative.
(détail C) ou de boucles s’ancrant dans la zone comprimée (détail D) sont plus efficaces mais ne permettent pas d’atteindre une résistance supérieure à celle des éléments adjacents
One alternative is to partially divert the compressed connecting rod outside the frame angle by implementing an inclined framework, which helps relieve the main structures This construction detail has been proposed and tested by experts in the field.
Nilsson's Figure 2.31 (detail E) illustrates that this solution achieves an efficiency greater than one This indicates that failure occurs in the adjacent elements following the hardening of the reinforcements, rather than at the frame corner Furthermore, the ductility of the structure remains unaffected by the behavior of the frame corner.
The practical application of this detail can be challenging due to the concrete pouring stages, which complicate the installation of the inclined reinforcement Consequently, the use of Detail D may be acceptable However, it is essential to enhance the strength of the corner frame by approximately 20% for elements with dimensions similar to those tested.
Nilsson) pour exclure une rupture fragile dans l’angle de cadre La résistance des éléments adjacents devient déterminante
La conception d’un tel détail doit ainsi être effectuée avec soins sous peine d’affecter le comportement de l’ensemble de la structure
Figure 2.31: Comportement des angles de cadre soumis à un moment positif pour différents détails d'armature (d'après Nilsson 1973)
Eléments de liaison
When constructing a covered trench made up of two adjacent vaults, one option is to utilize the geometry of the structure to create a space for technical ducts by connecting the two vault keys with a horizontal slab.
The behavior of this element requires careful evaluation Initially, it can be viewed as a unidirectional slab fixed within the walls and subjected to earth surcharge However, depending on the material properties, it may also experience normal tensile stress that adds to the bending moments This is particularly true in cases of differential settlements occurring between the vertical supports of the side walls and the central wall.
According to SIA 262 (SIA 2003b), tensile forces have a significantly adverse effect on the shear strength of elements without stirrups This tensile effort leads to increased cracking in critical areas, which negatively impacts shear resistance as outlined in equation 2.16.
Une augmentation locale de l’épaisseur de cette dalle aux encastrements ou la disposition d’étriers permettent d’améliorer sensiblement sa résistance
Figure 2.32: Elément de liaison entre deux vỏtes
Particularités des éléments courbes fléchis
Essais existants
La résistance à l’éclatement du béton d’enrobage a fait l’objet d’une recherche assez limitée Les travaux expérimentaux de Franz et Fein (Franz, Fein 1971), de Neuner et
Stửckl (Neuner, Stửckl 1981) et Intichar et al (Intichar 2002, Intichar et al 2004) ont ainsi contribué aux connaissances actuelles sur le phénomène
Les essais réalisés à ce jour ont permis de mettre en évidence les deux modes de rupture discutés plus haut et l’influence des paramètres géométriques sur la résistance
L’occurrence de l’un ou l’autre mode de rupture dépend principalement de critères géométriques, en particulier de l’espacement entre les barres et de l’épaisseur de l’enrobage
In cases of small cover or large spacing between bars, failure occurs through the detachment of a corner, as illustrated in Figure 2.34b Conversely, for small spacing between bars or large cover, failure is characterized by the formation of a crack in the plane of the reinforcements, leading to the block detachment of the cover.
The thickness of the coating has increased significantly, making the fracture mode that leads to crack formation along the reinforcement bars a crucial factor in determining the typical spacing used.
Figure 2.35a illustrates the average rupture stress (indicated by the coefficient k) measured during various tests, focusing on those characterized by crack formation along the reinforcement plane The tests conducted by Franz and Fein, as well as Intichar et al., show an average rupture stress equivalent to 30-40% of the tensile strength calculated based on measured compressive strength (Franz, Fein 1971; Intichar 2002) In contrast, the tests by Neuner and Stürckl demonstrate rupture at higher stress values (Neuner, Stürckl).
Figure 2.35b illustrates the coefficient k as a function of the longitudinal deformation of reinforcement bars, calculated from rupture loads with a lever arm estimated based on plastic calculations The figure reveals a correlation between the deformation of the reinforcements and the spalling resistance of the surrounding material, indicating that spalling resistance tends to decrease as deformation increases This observation suggests an interaction between the phenomenon and adhesion mechanisms.
Stürckl specimens, characterized by a null longitudinal deformation under different loading modes, exhibit higher resistances This hypothesis is supported by both theoretical and experimental research, which indicates that the failure of the coating can occur during a pull-out test, as discussed in Schenkel and Vogel's studies from 1997 and 1998.
Franz et Fein 1971 Neuner et Stửckl 1981 Intichar et al
Franz et Fein 1971 Neuner et Stửckl 1981 Intichar et al
Figure 2.35 illustrates a comparison of tests conducted on the bursting of the coating, highlighting two key aspects: a) the reduction coefficient relative to the ratio of the spacing between the bars and the thickness of the coating, and b) the longitudinal deformation observed in the reinforcements, as calculated.
Modèles théoriques et empiriques existants
La plupart des modèles connus à ce jour ont été développés durant les années 1970’s et
In the 1980s, models were developed based on the constraints surrounding reinforcement bars, assuming a linear elastic behavior of concrete These models are typically semi-empirical, as they are adjusted to incorporate experimental results.
Une description des modèles de Franz et Fein (Franz, Fein 1971), de Fein et Zwissler
(Fein, Zwissler 1974), Neuner (Neuner 1983) et Intichar et al (Intichar et al 2004) est donnée en annexe
Current models explain the low measured resistances compared to the resistance determined by plastic calculation (k = 1) due to stress concentrations near the reinforcement bars, which lead to the failure of the element once the tensile strength of the concrete is reached These models assume that no redistribution of stresses is possible at this state, overlooking the post-peak tensile behavior of concrete and the ability to transfer stresses through interlocking between the lips of a crack when its opening is small.
Il est très vraisemblable que des redistributions de contraintes se fassent avant la rupture et que les pointes de contraintes seules ne suffisent pas à expliquer les faibles résistances mesurées
This hypothesis is supported by the presence of adhesion mechanisms resulting from the relative deformation of the reinforcement and concrete between cracks, known as tension stiffening, as well as the anchoring of bars within the concrete.
The transmission of forces from the reinforcement to the concrete occurs through compressed struts that rest on the ribs of the reinforcement, as outlined in Tepfers' model (Schenkel 1998, FIB 2000) These struts are counterbalanced by tensile rings that form around the reinforcement bars Experimental findings by Schenkel and Vogel indicate that these tensile stresses can lead to the spalling of the surrounding concrete if the cover thickness is insufficient (Schenkel, Vogel 1997, see Figure 2.36b).
Q 1 l b = 28 mm Ψ = coefficient tenant compte de l’effet du bétonnage sur la qualité du béton autour de l’armature, f cw = résistance à la compression sur cube b)
Figure 2.36: Essais Pull-out de Schenkel et Vogel (Schenkel, Vogel 1997): principe des essais et b) résultats
Figure 2.35b suggests a correlation between the longitudinal deformation of the reinforcement and the bursting strength of the surrounding material, indicating an interplay between these mechanisms and the mechanisms of force recovery from deviation.
Intichar et al (2004) acknowledge the impact of adhesion mechanisms on the burst resistance of coatings However, they only take this into account when the stress in the reinforcement, calculated during the cracked stage, varies due to fluctuating moments.
Intichar et al overlay tensile stresses resulting from adherence to stress points derived from elastic calculations However, the impact of adherence in a region of constant moment, which leads to crack formation, is not considered.
Existing models are theoretically unsatisfactory as they overlook traction constraints caused by adhesion However, they can still provide estimates of resistance in straightforward scenarios, such as constant moments and continuous reinforcement.
Il existe néanmoins certaines situations particulières, dans lesquelles l’adhérence joue un rôle prépondérant, qui ne peuvent pas être évaluées par ces modèles
Significant disruptions in the stress state of concrete can adversely affect the spalling resistance of the cover These disruptions may be caused by factors such as the plastic deformation of the reinforcement and the presence of a lap joint.
Le comportement de l’ộlộment courbe dans de telles situations doit ờtre maợtrisộ pour pouvoir répondre aux questions suivantes :
Est-il possible de développer la résistance totale et la capacité de rotation d’un élément courbe fléchi sans rupture fragile prématurée ?
Où doit-on disposer les joints de recouvrement pour ne pas influencer négativement le comportement de la structure ?
The initial question is crucial when considering the potential for significant plastic deformations in the structure The second question arises during the design of the reinforcement scheme.
Tous les essais réalisés à ce jour se concentrent sur le comportement des éléments courbes dans le domaine élastique de l’armature
An experimental campaign was conducted at the EPFL's Institute of Structures laboratory to enhance understanding of the phenomenon, particularly in the plastic domain, and to address the aforementioned questions.
Campagne d’essais
Des informations plus détaillées sur les essais effectués sont données en annexe Un résumé des résultats principaux nécessaires à l’argumentation est présenté dans les paragraphes suivants
La campagne d’essais comprend six poutres courbes dont la géométrie et le principe de chargement sont donnés à la figure 2.37
Figure 2.37: Géométrie et principe des éléments testés: a) élévation et charges b) section à mi- travée (ECP1-4 sans joint de recouvrement)
Le paramètre principal des essais est l’armature de flexion disposée en nappe inférieure
Deux sous-séries peuvent être distinguées :
4 poutres (ECP1-4) : armature principale continue avec des taux d’armature différents
2 poutres (ECP5-6) : armature principale avec un joint de recouvrement dans la partie centrale Deux détails ont été testés
La variation de ce paramètre est donnée au tableau 2.1
L’enrobage théorique est de c = 40 mm
Tableau 2.1: Armature principale pour les différents éléments
[mm] Joint de recouvrement ECP1 3 ỉ 26 1.53 222 -
1 b ef = s-nỉ, n = nombre de barres
Influence des déformations plastiques sur le comportement (ECP1-4)
La variation du taux d’armature a permis d’observer le comportement des éléments avant et après plastification des armatures La figure 2.38 présente de manière synthétique les résultats obtenus
The break occurred abruptly each time due to the bursting of the coating in the central part of the beam However, the level of stress, determined by the reinforcement, varied significantly from one case to another.
While the ECP1 element fractured during the elastic phase of the reinforcement, the ECP2-4 elements broke after reaching the reinforcement's plasticization Consequently, the phenomena associated with the plasticization of the reinforcement weaken the burst resistance of the surrounding material.
Ces essais confirment ainsi l’interaction des mécanismes d’adhérence (en stades élastique et plastique de l’armature) avec les mécanismes de reprise des poussées au vide
Effet de l'adhérence dû à la plastification a) b)
Figure 2.38 presents a synthesis of the ECP1-4 tests, illustrating the load curve Q versus the maximum elongation of the fiber located 0.05 m from the lower surface, measured by strain gauges (with a base measurement of l0 = 100 mm) attached to the side of the element Additionally, it shows the coefficient k in relation to the reinforcement ratio.
La figure 2.39 compare les essais réalisés avec les essais de Franz et Fein (Franz, Fein
1971), Neuner et Stửckl (Neuner, Stửckl 1981) et Intichar et al (Intichar et al 2004)
L’essai ECP1 (stade élastique) est très proche des essais réalisés par Franz et Fein et par
Intichar et al provide valuable insights into the phenomenon of coating rupture in the context of plastic deformations of reinforcement in their ECP2-4 trials The study illustrates a clear correlation between the deformation of the reinforcement and its rupture resistance, revealing a gradual loss of strength as deformation increases Notably, the resistance measured after significant plastic deformations is reduced to only 50% of the values recorded in the elastic regime.
Cet effet défavorable n’est pas mentionné dans la littérature scientifique Il n’est également pas pris en compte par les normes actuelles, dont la SIA 262 (SIA 2003b)
Franz et Fein 1971 Neuner et Stửckl 1981 Intichar et al 2002 Plumey et al 2006
Figure 2.39: Comparaison des essais ECP1-4 avec les essais existants
Influence du joint de recouvrement (ECP5-6)
The failure mode observed with a standard lap joint (ECP5, curved bars) is similar to that seen in other components However, the presence of a lap joint results in a decrease in burst resistance compared to elements without a joint.
The detail created with the ends of the straight bars (ECP6) results in a slightly different failure mode, but it does not show any improvement in strength In fact, a slight decrease in strength is observed.
In the presence of a lap joint, failure occurs within the elastic phase of the reinforcement Comparing these tests with the ECP1 element, which experiences failure without plastic deformation, provides a more accurate representation and insight into the impact of the lap joint.
Figure 2.40: Coefficient k pour les essais avec (ECP5-6) et sans joint de recouvrement (ECP1)
Figure 2.40 illustrates the comparison results, highlighting that the presence of a standard lap joint significantly reduces the average normalized rupture stress (coefficient k) The achieved value is only 74% of what is reached under similar conditions without a lap joint (ECP1) This reduction in strength is attributed to both the decrease in the net width that contributes to force redirection and the effects associated with the lap joint itself This adverse effect may be hypothetically linked to local stress state disturbances caused by the anchoring of reinforcements and the transfer of forces between bars Additionally, adhesion mechanisms are likely responsible for this loss of strength.
Développement d’un modèle théorique
Research and existing studies consistently demonstrate a strong interaction between the mechanisms of force recovery from deviations and the adhesion mechanisms.
A physical model that accurately replicates behavior in the plastic domain, or with an overlapping joint, must be developed by simultaneously considering these two previously separate phenomena.
L’application pratique d’un tel modèle s’oppose cependant à un obstacle de taille, celui de lier la déformation locale de l’armature à la déformation d’ensemble de la structure
Local phenomena play a crucial role in structural scaling, influenced by various parameters that are often not precisely known or are challenging to control during the design phases of a project, such as steel type and phenomenon dispersion Consequently, applying a theoretical model that is also valid in the plastic domain for sizing becomes a complex task.
Il existe néanmoins un intérêt scientifique à développer un tel modèle pour mieux comprendre les mécanismes dictant le comportement et établir des règles de dimensionnement cohérentes
This task is not included in this study as it seems too specific compared to the broader context of covered trench behavior The following paragraph presents simplified quantitative rules.
Proposition d’adaptation de la norme SIA 262
La SIA 262 préconise l’utilisation d’une résistance à la traction réduite qui correspond à
50% de la valeur caractéristique fractile 5% (k = 0.5, SIA 2003b, art 5.2.7.1) Les essais montrent cependant que cette valeur se situe du côté de l’insécurité
Cas sans joint de recouvrement
La valeur mesurée lorsque l’armature est en stade élastique est en effet inférieure à cette valeur et la situation s’aggrave encore lorsque des déformations plastiques se produisent
Sur la base des essais, il paraợt justifiộ d’adapter la norme en dộterminant la rộsistance de calcul à l’éclatement de l’enrobage par barre d’armature selon : ef ctd
U = ⋅ ⋅ (2.28) avec : k = 1 3 pour un calcul sans redistributions plastiques importantes ou k = 1 6 pour un calcul avec redistributions plastiques et c k , ct ctd f f γ 0 05
La figure 2.41 montre que ces propositions sont en accord avec les essais réalisés
Essais SIA 262 k, calcul élastique k, calcul plastique
Figure 2.41: Proposition d’adaptation de la norme SIA 262: comparaison avec les essais
La norme SIA 262 (art 5.2.7.1) requiert que la vérification soit faite à la plastification de l’armature Ceci est sujet à interprétation et doit donc également être précisé
The resistance to the bursting of the coating should not be exceeded, even with high yield strength values of the reinforcement Therefore, it is essential to use a design value for verification based on a higher characteristic value (95th fractile) In the absence of other information, this value can be approximately derived from the characteristic yield strength fractile.
5% en admettant que (écart-type de 25 MPa) : k s k
La sollicitation de calcul par barre d’armature peut alors s’écrire : f R
Le coefficient partiel appliqué à l’acier d’armature γ s = 1.15 multiplie la valeur caractéristique de sorte à obtenir une situation défavorable
La vérification est ensuite effectuée en comparant la résistance à la sollicitation selon : d
Si l’inéquation 2.31 n’est pas satisfaite, des mesures doivent être prises pour améliorer la situation Une possibilité est de disposer des étriers dimensionnés pour reprendre toute la sollicitation
La figure 2.42 présente la limite U Rd = U d pour un béton C30/37, un acier d’armature
B500B, une section d’épaisseur h = 0.4 m et pour des rayons de courbure des barres de
For a curvature radius of R = 5.15 m, which is a standard geometry for covered trenches in Switzerland, the maximum usable diameters for a typical spacing of s = 150 mm are ỉ = 22 mm and ỉ = 16 mm, respectively, for calculations without and with plastic redistributions The limitations become more stringent for smaller curvature radii.
Figure 2.42 illustrates a comparison between the proposed design and the SIA 162 and 262 standards, focusing on the maximum diameter of reinforcement bars based on spacing for a vault with a thickness of h = 0.4 m It highlights two scenarios with different bar radii: a) R = 5.15 m and b) R = 3.5 m, utilizing C30/37 concrete and B500B reinforcement in accordance with SIA 262.
La figure donne aussi les limites calculées à partir des normes SIA 162 (SIA 1993) et
According to an interpretation of these documents, the tensile strength of concrete (average rupture stress) and the yield limits of steel used for comparison are summarized in Table 2.2.
Tableau 2.2: Résistances à la traction du béton (contrainte moyenne à la rupture) et limite d’écoulement de l’acier pour la comparaison (béton C30/37)
SIA 162 SIA 262 Proposition f ct,d [MPa] 0.5/1.2 = 0.42 1 0.68 0.45 f sd [MPa] 460/1.2 = 383 1 435 690 f sd / f ct,d [-] 920 639 1533
Figure 2.42 indicates that the proposal appears significantly more conservative than SIA 162 and SIA 262 However, Table 2.2 reveals that this difference largely stems from the selection of the flow limit used for verification, which is much more safety-oriented in the proposal This value is open to interpretation in both SIA 162 and SIA 262.
A comparison of average stress constraints in concrete at failure indicates that the current proposal closely aligns with SIA 162 In contrast, SIA 262 appears overly lenient, demonstrating less caution than the previous SIA 162 standard.
Cas avec joint de recouvrement
Studies have demonstrated that the presence of a cover joint weakens the structure The SIA 262 standard acknowledges this effect but does not provide guidance on how to quantify it (SIA 2003b, art 5.2.6.4).
Les essais montrent que l’équation 2.28 reste applicable en réduisant cependant le facteur k d’un tiers par rapport au cas sans joint de recouvrement :
La largeur nette participant à la reprise des forces de déviation doit en plus être réduite pour tenir compte de la présence des barres supplémentaires
According to SIA 262 (SIA 2003b, art 5.2.6.2), lap joints should not be placed in highly stressed areas This regulation indicates that situations involving plastic deformations of the reinforcement and lap joints are not permissible.
Le recouvrement des barres d’armature et donc le schéma d’armature doivent être conỗus de sorte que cette vộrification ne devienne pas dộterminante.