D2. Different categories of structural elements 

D2. Different categories of structural elements 

Here are presented the peculiarities of these elements regarding FE calculations. 

1. Reinforced concrete elements

Considering the delayed phenomena – Usually, the project engineer is interested in the redistribution of stresses and the delayed deformations that accompany the aging of the materials. 

To achieve this, the static calculation by time increments is particularly well adapted and largely sufficient. The shrinkage and the creep of concrete for instance depend only on the time that has elapsed since the pouring phase. Using the incremental calculation, they will be introduced as imposed deformations at each node of the mesh. It is possible to compute beforehand, for each time increment the shrinkage and creep related deformation map: the FE software incorporates the initial state seeking an equilibrium of the elastoplastic material. 

Nevertheless, attention should be paid to the interaction between the delayed effects and the construction phases, as it will be explained later.

Consideration of cracking - When concrete cracks in tension, cracks develop towards the nearest face, as well as along the steel-concrete interface ([Goto], figure below).

Internal cracking of the reinforced concrete

When considering a tensioned connecting rod of cracked reinforced concrete as a homogenized continuous medium, the behavior law N(ε) (with N the normal tensile stress) follows the trend of the figure below.

Model of the tension stiffening effect

The Model Code 1990 of the CEB-FIP expresses an analytical formulation. This relationship can be used by modeling the section of cracked concrete as a bar element whose stiffness would be updated in increments (depending on the value of normal stress or elongation, since the relationship is invertible). It can also be used as a law for the behavior of a fiber of a multi-fiber element.

Cast Concrete - For cast concrete areas (e.g. by passive reinforcement), it is possible to take into account the residual post-crush strength, as indicated in EN1992-1, using Sargin's law:

with εc1 the peak value such that

with

For

The EN1992-1 §3.1.9 also suggests an increment of strength and deformation of concrete when it is subjected to a confinement stress of σ2=σ3 such that: 

However, this law is only one-dimensional and is only valid under monotonic loading. Therefore, it can only be used with the finite element method when modeling concrete confined by a bar element (so the modulus would be incrementally modified), or as a constitutive law of a fiber of a multi-fiber beam element.

2. Pre-stressed Concrete Elements in Pre-tensioning

Prestressing by pre-tensioning, which is characteristic of industrially prefabricated concrete products (beams, pre-slabs, honeycomb slabs, girder, etc.), consists of tensioning cables (wires, strands or bars) on manufacturing benches, then pouring the concrete before slackening the cables when a minimum concrete strength is reached (called concrete sag resistance). The amount tension in the cable (which must not exceed the maximum prestressing force allowed by design codes), the number of cables, and the strength of the concrete are adjusted according to the loads that the prestressed floor or element must resist.

When the cables are loosened, instantaneous losses, of the order of 8% for a prestressed pre-slab for example, must be taken into account (losses due to the entry of anchors, to the relaxation of the prestressing reinforcement during the period between the tensioning of the cables and the transfer of the constraint, to the elastic shortening of the concrete under the compressive stress imposed by the constraint) when designing in the provisional phase of construction with a prestressing transmission length to be considered starting from the end of the precast concrete element.

In the longer term, delayed losses due to shrinkage of the concrete, relaxation of the steel, or creep of the concrete, which will finally reach a total loss of 20% for a prestressed pre-slab, must also be considered.

In many FE computational software, prestressing can be incorporated in beam-type finite elements representing the cables connected to volumetric finite elements representing the concrete. Depending on the study area, it may be necessary to consider the transmission length of the prestresses in the cables, for example in the end zones. The distribution of the actual prestressing force (considering the instantaneous or delayed losses as a function of the moment in the service life of the product where the calculation/verification is to be performed) is then variable along this transfer length. A linear distribution of this prestressing force is allowed in most computational software and remains safe for dimensioning purposes compared to the more realistic parabolic distribution.

For reasons of complexity and need (designs limited to elastic deformations), the modeling of these elements is generally carried out with linear assumptions (constitutive laws of concrete and linear elastic steel, perfect contact between concrete and steel...). However, for refined studies, non-linear hypotheses can be followed according to the need such as damage type constitutive laws for concrete, elasto-plastic type constitutive laws for steel, and the introduction of steel-concrete interface elements.

3. Pre-stressed Concrete Elements in Post-tensioning

Elastic losses - Post-tensioning of the prestressing cables is accompanied by instantaneous losses: friction, anchor recoil, and loss due to elastic elongation.

Delayed effects: creep, shrinkage, and relaxation - Delayed effects are considered using an incremental calculation. Creep and shrinkage can be introduced as incremental volume deformations given at each node of the mesh.

4. Steel Structures

Choosing the type of analysis

Due to their high slenderness, steel structures are very deformable. As a result, the traditional assumption of reaching stress equilibrium in the initial configuration is not always valid. It is then necessary to establish the internal stress distribution in the deformed configuration.

The sensitivity to these non-linear effects, sometimes called second-order effects, is expressed through the critical multiplier αcr: the load multiplier leading to the Eulerian instability of the structure. In the current version of Eurocode 3:

• if α_cr > 10, non-linear effects can be neglected. If the global structural analysis incorporates the plasticity of the elements the limit value of αcr is increased up to 15,

• if 4 < α_cr < 10, they must be taken into account. This can however be done with a classical linear elastic analysis by amplifying the transverse forces,

• if α_cr < 4, the non-linear analysis is mandatory.

In the last two cases, if the global geometrical irregularities (or imprecisions) and the element irregularities influence the global behavior, they must be considered.

Element irregularities include:

• geometrical irregularities: transverse and torsional,

• material irregularities: rolled elements or welded elements have self-balanced residual stress distributions created by their manufacturing process.

The latter can be represented by an equivalent geometric irregularity, whose value can be found in the current standards.

These irregularities must be included in any analysis of a structural member incorporating non-linear effects.

In principle, structural or Strength of materials models (bars, beams, plates, and shells) are ideally suited for structural steel calculations.

Wireframe structures

Elastic analysis

For modeling purposes, it is necessary to analyze precisely:

• the nature of the connections between the different structural elements,

• the load propagation from each part to the others.

Plastic analysis

When the elementary ductility and/or the assembly ductility is ensured, it is possible to carry out structural analyses incorporating plasticity. Different methods can be used (EC3, §5.4.3):

• elasto-plastic analysis, where the plasticized areas are modeled as plastic hinges,

• non-linear plastic analysis, taking into account the partial plasticization of the bars along the plastic zones,

• rigid-plastic analysis, in which the elastic behavior of the bars between hinges is neglected.

Torsion

Steel elements are rarely massive, they are composed of thin layers to form profiles, open or closed. Especially in the first case, the response to torsional stresses is both in uniform torsion, known as Saint Venant torsion, and in non-uniform torsion, leading to the buckling of the section. The latter phenomenon is usually not considered in commercial software, even though it can have a significant effect on the response of structures. In this case, two solutions are possible:

• surface modeling of the element: this solution is not applicable if the analysis is performed on a structure with more than a few elements,

• if the situation is similar to that of an I-beam, where the non-uniform torsion can be represented by the alternating bending of the flanges (sometimes called biflexion, Figure), one can proceed to bifilar modeling of the element where the two footings are represented by two distinct elements, connected by transverse elements representing the web. As a result, the specific bending of each flange, and thus the non-uniform torsion, is represented. For instance, this is the case for double girder composite bridges. An application is presented in chapter 3.

Decomposition of torsion into uniform and non-uniform parts: simplified biflexion hypothesis

2D or 3D elements 

The linear elastic analysis of two- or three-dimensional steel elements does not present any specific problems, so the general rules apply.

On the other hand, a non-linear analysis may be required to study instability phenomena. Indeed, structures composed of steel plates, whether flat or curved, are subject to buckling phenomena.

In the case of flat elements, called plates, buckling is a relatively stable phenomenon: the initiation of buckling of the plate does not lead to failure, the maximum load is reached after buckling. This is called post-critical behavior.

In the case of curved elements (shells), instability leads to the immediate and often brutal failure of the structure. From a numerical point of view, in the non-linear static analysis, this leads to a rapid decrease of the load beyond the maximum.

In both cases, the maximum load is strongly dependent on the initial deformation applied, in amplitude and shape. The amplitude is fixed by standards. The shape is usually chosen affine to the first mode of instability. However, this choice is not necessarily the most detrimental. It is advised to supplement it with local modes when the structure presents panels of strongly different dimensions.

For example, in the case of an orthotropic deck, the deformation affine to the global buckling mode must be supplemented by deformations affine to the buckling of the under panels.

In the case of shell-type structures, the problem is even more critical. It is advised, once a first calculation has been performed according to the above assumptions, to adopt in a second calculation a shape of initial deformation affine to the deformation obtained at failure.

5. Compound structures: steel-concrete

The alternative of a compound steel-concrete construction is sometimes preferred for certain types of industrial buildings and bridges with small to medium spans (midspan < 100 m). The combination of these two materials by making them "work" in their strength domains (concrete in compression and steel in tension) can result in a strong and lightweight design. To achieve this result, the connection between these two materials must be correctly designed. A distinction is made between:

• mixed slabs: solid slab + reinforcement,

• mixed beams: solid or mixed slab + steel profile + connectors,

• mixed columns: steel profile + (concrete filling or concrete coating).

Overall structural analyses are usually carried out in the elastic domain by homogenizing the section, or by representing the two materials separately. This second way of proceeding can lead to difficulties in the processing of the results. Indeed, it requires recalculating the stresses on the globalized section to apply the current standards. 

When more detailed analyses are needed, given the diversity of materials involved in this type of combination both geometrically and in terms of non-linear behaviors, 3D finite element models are necessary to conduct local analyses, which include the processing of the various interfaces (using contact finite elements for example). For studies at the scale of the structure or the elements, relatively high-performance 2D models have been developed over the last decade, such as those based on a fiber cross-section cut-out (fiber model) to enable the section stiffness to be estimated by numerical integration.

The assembly of composite bridge girders is another relatively complex detail. Regardless of the assembly solution chosen (butt-strap joint, joist trimmer, or diaphragm), 3D models using solid finite elements are preferred to simplified 2D models, but they are more computationally expensive.

Cracking - Composite girders are usually made of a steel section that is connected to a reinforced concrete floor or deck by means of connectors. As a result, in areas undergoing a positive moment, where the slab is compressed, strength and stiffness are particularly consequential, whereas in areas undergoing a negative moment, cracking results in significantly lower mechanical characteristics. This cannot be neglected in the overall structural analysis. Different levels of modeling are allowed in the codes:

• Flat-rate approaches: for example, Eurocode 4 recommends considering a crack length equal to 15% of the span, on either side of the supports. It also suggests a flat-rate stiffness for composite columns. Eurocode 8 recommends adopting an average stiffness over the whole length of the beam. These differences in approach are justified by the different shapes of the moment diagrams under classical solicitations, mainly gravitational, and seismic stresses,

• Approaches defining a cracking zone by analysis of the stress envelopes: Eurocode 4 recommends to consider as cracked any section where the stress exceeds twice the average tensile strength under the envelope of the characteristic stresses calculated, assuming that the structure is not cracked by adopting a long-term concrete modulus.

• It is also possible to use non-linear analyses.

Connection

Except in the case of non-linear analysis, it is not useful to model the connection. Current standards allow us to consider the effect of a partial connection on the strength of the elements.

For modeling purposes, beam-type finite elements (2D or 3D) are generally used to model the point connection. If a distributed connection hypothesis is considered, appropriate models are available in the literature.

Collaborative widths

Steel beams are connected to particularly wide members, and shear dragging can result in a non-uniform stress distribution across the width of the slab.

In wireframe models, this phenomenon is usually taken into account by modeling a slab of reduced width, at constant stress, since it is required for beam models.

Strictly speaking, since shear dragging is related to the transmission of shear forces through the connection to the slab, it is therefore dependent on the shape of the moment diagram. The collaborating width should vary from one combination to another. However, the standards allow a single width to be adopted for all calculations.

The variability of collaborative widths will be considered if the slab is modeled using surface type elements.

Delayed effects and shrinkage

Since the delayed effects of concrete influence the stiffnesses and stress distribution throughout the structure, they must be taken into account. It should be noted that the Eurocode adjusts the value of the steel-concrete equivalence coefficient according to the type of loading. The shrinkage solicitation induces a self-balanced stress distribution throughout the section; thus it should also be taken into account.

Non-linear analyses

Non-linear computational models of composite structures assume the material and geometric irregularities hypotheses used when modeling concrete and steel materials separately.

As mentioned earlier, the connection must be modeled with its own stiffness. However, it should be noted that its numerical processing remains difficult. Usually, it is assumed that the concrete and steel parts cannot be separated transversally and that only longitudinal sliding is possible. The finite element formulation of such a connection can lead to the "locking" phenomenon when the sliding is blocked by the looseness of the transverse stiffness projection preventing the uplift when the equilibrium is reached in the deformed configuration. Therefore, when this type of element is used, it is advised to verify the consistency of the reaction forces in the connection with the tensile/compressive forces of the steel elements.

6. Braces, tension cables, and suspension cables

Introduction to the calculations - Some cable elements can be modeled as bars of the equivalent section. This is the case for vertical tension cables, or prestressing cables, guided in their duct. The curvature of these cables at equilibrium is practically independent of their linear mass. In other cases, the cables tend to be bent by their own weight: the vertical direction is a specific direction that plays a role in the stiffness of these elements.

When, at the scale of the cable's span, its curvature becomes important, it strongly influences the stresses and strains it can transmit to the rest of the structure. As an example, for braces, the stiffness of the cable depends mainly on its deflection and elongation. Deflection affects the stiffness of the cable; thus, the tension/deflection relationship of the cable is inherently non-linear.

The cable elements present in most codes are based on the [Gimsing] model, which assumes that the cables deform as a parabolic curve, an assumption that is valid if the deflection/span ratio is less than 1/12. This model decouples the elastic elongation from the bending of the rope.

The catenary effect - When the deflection of a rope is greater than 1/12th of the span, it is no longer possible to decouple the elastic elongation from the bending because the tension lever arm becomes the dominant factor in the expression of the bending moment. Several computational models suggest catenary elements to account for this geometrical effect. These elements connect two nodes of the mesh so that a single element is sufficient to model the wire rope.

Nodal displacements of a catenary element

For this type of element, the nodal unknowns are the vertical displacement (in the direction of gravity) and the horizontal displacement (see figure 4). The associated reactions are the variation of the horizontal component of the cable tension, and the vertical reaction at the anchors. Since the relationship between nodal displacements and these reactions depend on the cable tension, finding the equilibrium becomes a non-linear problem, although the structure is globally elastic.