C8. Composite Sections (Beams/Slabs)

Composite sections are made up of the assembly, rigid or elastic, of elements of different nature (wood, steel, concrete, ...) and/or at different dates.

We study here the most common cases encountered in modeling:

building floors (slabs + beams),
bridges beams (prestressed, precast beams),
girders of mixed steel/concrete bridges,
mixed building floors (steel beams + reinforced concrete slab),

These elements complicate the calculation with different approaches depending on the case studied.

C.8.1 Floor beams of buildings

This chapter concerns the floors of buildings calculated from a global model.

C.8.1.1 Calculation principle

The difficulty with this type of analysis is to reconcile the finite element calculations with the design regulations for reinforced concrete.

Indeed, the reinforced concrete regulations (BAEL and EC2-1-1 §5.3.2.1) are based on precise rules on the effective flange widths, on the offset of the bending moment curves (which correspond to the formation of the connecting struts), and on the deformation diagram (consistency between the deformations of the slab and the beam).

However, the finite element models are based on Strength of Materials and not on these regulations.

In any case, the calculation of the reinforcement must be performed:

by considering the direction of the slab's span (in particular the prefabricated elements),
by using the efforts from the model,
by correcting them to account for the effective flange widths (non-participating zones of the slab weigh but do not add to the strength),
by correcting for additional eccentricities not modeled (vertical or horizontal - effect P-Δ),
by redistributing the bending moments,
by performing a regulatory calculation with these post-processed efforts

C.8.1.2 How to model the beam/slab floor

The first aspect concerns the floor modeling method. Indeed, several options are available:

to model only the beams, the loads being directly applied to the beams,
to model beams and slabs on the same average fiber,
to model the beams with an eccentricity with respect to the slabs.

Illustration of the last two approaches

The comparison between these cases is made by using the following example:

A structure with 2 spans of 8m each, beams with a cross-section of 25cm x 50cm spaced 2m, and slab with a thickness of 15cm.

Plan view of the slab

Cross-section

We will study the central beam:

Load = dead weight + permanent load (cladding) of 3 kN/m² + accidental load of 5 kN/m².

We study the ULS case (1.35 PL + 1.5 AL).

There are 3 types of modeling:

Case 1: the slab is not modeled, which means that the beam is calculated according to the usual methods for reinforced concrete
Case 2: the slab is modeled on the same average fiber as the beam
Case 3: the slab is modeled with an eccentricity with respect to the beams

Case study 1: modeling the beams only - the slab is not modeled

Modeling scheme

Bending moment diagram (kN.m) in the beam

The bending moments are equal to -264kN.m on central support and 149kN.m in the spans; they are consistent with the classical calculation methods; this requires reinforcement of 19cm²on central support and 10cm²for the spans.

Case Study 2: the slab and the beam are modeled on the same medium fiber

Modeling scheme

Bending moment diagram (kN.m) in the beam

The bending moments in the beam are equal to -166kN.m on the central support and 94kN.m in the spans; these forces are much lower (-37%!!) than those calculated in Case 1: the calculated reinforcement is only 11cm² at the central support and 6cm² for the spans.

Longitudinal bending moments in the slab

The bending moments of the slab are equal to -21kN.m/m on the central supports and 12kN.m/m in the spans, which results in a reinforcement on the central support of As = 5cm²/m, Ai = 0 and As = 0, Ai = 3cm²/m in the spans.

Case study 3: the beam is off-center with respect to the slab

Modeling scheme

Bending moment diagram (kN.m) in the beam

Axial effort diagram (kN) in the beam

The bending moments in the beam are equal to -47kN.m on the central support and 21kN.m in the spans, but they are accompanied by axial forces (tension on supports and compression in the spans); the calculated reinforcement is then 10cm² in the upper layer and 2cm² in the lower layer on supports, while there are no steels in the spans!!!

Longitudinal bending moments in the slab

Longitudinal axial efforts

The bending moments in the slab are equal to -17kN.m/ml on the central supports and 9 kN.m/ml in the spans, they happen simultaneously with normal efforts with peaks on the central supports.

The analysis shows that case 3, with the offsets, is unusable and incompatible with the normative verifications because normal forces and peak forces appear in the slab. Indeed, how can the bending moments be redistributed considering normal forces?

The tables below summarize the main results.

Required reinforcement in the central support and the spans for the 3 calculation methods

Cases 1 and 2 result in similar reinforcement areas, which seems to validate the modeling of the beam and the slab on the same mean fiber, but the conclusions of this example should not be generalized. Indeed, as shown in the deformation diagrams of the sections below, there is an inconsistency in the spans with tensioned rebars in the slab, located at the level of the compressed zone of the beam.

This example shows that:

modeling only the beams alone gives good results, but this option is difficult to apply in a global model (how to transmit horizontal forces for example?),
modeling an offset between beams and slabs allows good modeling of the floor stiffness, but it is not compatible with the normative verifications (how to redistribute the bending moment diagram when part of the bending moments appear as normal forces in the beams?),
the reinforcement of beams and slabs should not be calculated directly from the results of the global model.

C.8.1.3 General method for the design of a beam/slab floor

The calculations of slabs and beams must, on one hand, consider all the efforts calculated in the global model and, on the other hand, respect the normative requirements.

Let us take the example of a building subjected to horizontal forces (wind, earthquake, thermal, etc...):

Step 1: Create a global model of the building. This global model allows the calculation of the forces in the diaphragms formed by the floors, which results in the appearance of membrane efforts (normal and shear efforts) in the horizontal elements. These are the forces that we will use for the rest of the calculation: Nxx, Nyy, Nxy in the slabs, and Nx in the beams.

Step 2: Create a local model of the slab. Indeed, except in very particular cases, it is not possible to use the global model of a building to justify the slabs for several reasons:

phasing is generally not modeled,
pre-slabs are generally not modeled,
the position of loads in a global model does not necessarily respect the zones of influence of beams and slabs at the local scale,
from a regulatory point of view, punching, stress redistribution, stress discontinuities, etc., are not accounted for.

In the local model of the slab, its geometry is extracted from the geometry of the global model. For the sake of simplification, beams are generally replaced by linear supports, while slabs are modeled by shell elements (bending) subjected to weighted loadings. It is this small and flat model that will be studied for the normative verifications, possibly considering the phasing, pre-slabs, etc...

The bending moments (Mxx, Myy, Mxy) in the slabs, resulting from this local model, must be cumulated with the normal forces (Nxx, Nyy, Nxy) of the global model to calculate the reinforcements and carry out the normative verifications (pay attention to the combinations).

Step 3: Create a local model for the calculation of the beams. Indeed, for the same reasons as for the slabs, it is not possible to use the global model to determine the totality of the stresses in the beams.

The geometry is identical to that of the local model of the slab, except that the beams are of course preserved.

In this model, the slabs do not have to take up bending forces, they play the role of load transmission to the beams, therefore, they are modeled by distribution surfaces (refer to the documentation of the software used).

The resulting efforts in the beams must be added to the normal efforts of the global model, which allows proceeding then to the normative verifications on the beams (either manually or using dedicated software).

C.8.2 Case of Bridge structures (ribbed slabs)

This approach applies to bridges such as PRAD, VIPP, ...

For the calculation of bridge structures, if the Guyon-Massonnet method is omitted, it consists of calculating the structures:

in girder grids, i.e. crossing longitudinal bars, representing the section of the ribs + the effective flange width, and of transverse bars, modeling the slabs: the advantage is that we directly have torsors that can be used in the calculation of reinforced or prestressed concrete, the disadvantage may be the placement of loadings, especially moving loads,

Beam grid model

in beam grids using ladder beams - can be advantageous for a phased calculation, especially if one wants to model in detail creep or shrinkage effects
as in the third approach above (C.8.1.2), by modeling the ribs with beam elements and eccentric slab in the form of a FE shell: the main advantage of this approach lies in the easy application of the loads, the disadvantage is that one does not directly obtain torsors that can be used in reinforced concrete calculations.

It should be noted that the beams modeling the slab must be perpendicular (or almost perpendicular) to each other for the model to be valid.

To illustrate this approach, in particular the reinforcement of the slab and the ribs, one must start from the example discussed in C.8.1.

View of the model - 25cm x 35cm drop-beams and 15cm thick slabs. Spans 2 x 8m - beam spacing 2m.

The calculation of reinforcement directly from a reinforcement module is not recommended if the assumptions used for the design of reinforced concrete are to be considered. A small post-processor (a spreadsheet) is enough to calculate the bending moment and the flange width assigned to the rib, as shown below.

Application to the central beam of the model - section on the support and section on the spans (note: both spans are fully loaded, without considering the influence line):

Bending moments (only the drop-beams) - kN.m

Normal force (only the drop-beam) - kN

The methodology consists of applying the plane section remain plane assumption and calculating the (elastic) equilibrium of the internal forces. 1. The stresses diagram is extended to obtain the stress on the upper fiber (slab top) 2. The normal stresses on the composite section are zero: the integration of the normal stresses must be equal to zero, the effective flange width of the slabs is deduced 3. All the geometrical and stress parameters are determined, then all that remains is to calculate the bending moment resulting from the stress diagram.

Application to the case of the supported section:

A reduced flange width (47cm) is observed, which is logical considering the shear drag effect. As=14.4cm² (ULS calculation).

Application for the span section case:

A larger effective flange width (139cm) is observed, which logically is larger than on the support. As=7.20cm² (ULS calculation).

If an automatic calculation is performed for the rib and the slab:

→ The software suggests reinforcement sections at locations that actually do not require them when performing a "manual" reinforced concrete calculation (it has been verified in parallel that no compressed steel section was necessary).

→ In the present case, the automatic approach leads to a slight reduction in reinforcement at the bottom layer in the span and an increase of reinforcement at the upper fiber.

Mapping of the reinforcement in the upper fiber of the slab

Reinforcement in the upper fiber of the slab at the central support, central beam (18.36cm² along 2m)

Reinforcement in the lower fiber of the slab

The efforts calculated using the beam grid model are as follows and lead to 7.2cm² in the spans and 15.4cm² on the central support:

Beam grid model (Mt = 140kN.m, Ma = -262kN.m) - without taking into account the effective flange widths

To conclude the above example, we realize that the automatic calculation of reinforcement is not satisfactory. Without considering minimum reinforcement, it leads to placing reinforcement in zones where the normative verifications would not require it and to under- and over-reinforcing some areas. Besides, let us recall once again that the automatic calculation does not consider the redistribution of the bending moments diagram, punching, or connecting struts... It is, once again, up to the engineer to analyze the results and to decide if they need to be considered or not. It should be noted that the problem of stress or reinforcement smoothing arises also when using slab or shell type finite elements.

For a beam grid calculation, one should refer to the SETRA Guide "Advice for the use of beam grid programs" - PRP 75 - a particular area of focus is how to consider the torsional inertias.

Other examples are given in Example C - Modeling of beam grids.

C.8.3 Mixed Steel-Concrete Beams and Slabs

Generally, the composite character of the sections is modeled. However, in some cases, the model may be limited to the main beam alone, without considering phasing, such as for pre-dimensioning. After calculations, the stresses of the steel beam are then used to dimension and verify the behavior of the composite beam according to the appropriate normative reference frame. The model does not detect that it is a composite beam and there is a small error on the stiffness, the acceptability of which must be evaluated.

When performing a normative verification of the beam, the mixed character and construction phasing should be considered.

Modeling Approaches:

For a more rigorous calculation, it is possible to model the composite beam:

either as a beam whose mechanical characteristics consider the connection of steel and concrete. This way, the difference in Young's moduli of the two materials is apprehended via an equivalence coefficient - in this case, it is said that the materials are homogenized, generally by reducing the concrete to a metal equivalence (a),
or as two superimposed beams, one, lower, metallic, the other, upper, made of concrete, at the altimetry of their respective centers of gravity. These beams are connected at their ends by rigid links. This can make it easier to consider differential shrinkage and creep. If the structure is modeled as a whole, the longitudinal concrete bars described above are, besides, connected by transverse bars in a way that forms a beam "grid" (b),
finally, one can also choose to replace the beam elements of the slabs by shell-type finite elements (c). In this case, the calculation of the torsors requires post-processing, ideally automatic, following the method described in C.8.2.

The approaches (b) and (c) should be applied for special cases. Indeed, although they may initially seem simpler, the pre and post-processing are always much longer than with a model of type (a), especially if a software dedicated to mixed calculations is used.

In order to account for phasing, one must, at each change of state, either modify the inertia of the homogenized beam (a) or activate the beams of the slab (b) or the shells (c), once the concrete has poured and the formwork is removed for example. Of course, the creep of the concrete must be considered, either by means of an equivalence coefficient or a creep law and cracked zones.

Example of a mixed structure modeled according to approach (a):

Example of a mixed structure whose slab is modeled with eccentric shell elements - approach (c). The steel beams are modeled in this case strictly according to the material distribution (metal only):

The use of software specifically developed for mixed calculations is always recommended whenever possible.

See in Part 3, example B - Mixed and Steel Beams.

C.8.4 Mixed floor (building)

A compound floor is composed of steel beams supporting reinforced concrete slabs (pre-slabs or not) or steel deck.

C.8.4.1 Weight and vertical loads: slab bearing direction

Stress calculations in a composite floor are performed by considering the resistance of steel beams alone. Concrete is then considered as a non-resistant dead load. More generally, these floors are made of collaborating steel deck that works in only one direction. These particularities require specific provisions in the models.

In the case of simple geometry, the concrete slab is not modeled and the loads are applied directly to the steel beams.

When the geometry is complex, the manual distribution of the loads on the profiles becomes too delicate, it is then necessary to distribute the loads using distribution surfaces. The most common software have this type of element which behaves like a very thin plate, without any bearing role but distributing the loads on the load-bearing beams. Alternatives allow to consider the directions of distribution of the steel deck, but the modeler must pay attention to respect the load-bearing directions, the verifications are essential. Let us take the example of the petals of the LUMA foundation in Arles made up of mixed floor with steel deck, they are represented in blue in the scheme below ...

The blue distribution surfaces are meshed like slabs but do not contribute to the strength of the structure.

Plan view of the floor with the load-bearing direction of the steel deck

Visualization of beam loads calculated directly by the software

The calculations are then carried out classically.

The calculation of the slabs is then done by a specific slab calculation (orthotropic) between beams.

C.8.4.2 Horizontal loads (wind, earthquake)

In general cases, the concrete slab is not connected to the structure, so it does not participate in the bracing of the floor. The floor is then braced by horizontal metal bracing.

However, in some complex cases, it may be necessary to brace the floors using the concrete slab. The modeling then becomes very complex:

the slab must be modeled with an eccentricity with respect to the average fiber of the profiles
it is necessary to model the connectors between the slab and the profiles
the slab/column connection node is different from the column/beam connection node

Mixed floor without slab modeling - Bracing provided by the profiles

Bracing provided by the slab