E3. The specific case of reinforced concrete
E3. The specific case of reinforced concrete

Overview of the common methods for designing the reinforcement of plate elements
There are mainly 3 methodologies applicable to plates that can be consulted using the following references:

Johansen fracture line method:

Save M.A., Massonet C.E., De Saxcé G.  Plastic limit analysis of plates, shells and disks, Applied Mathematics and Mechanics Vol. 43.

This method applies to weakly reinforced plates (reinforcement ratio less than 0.7%), allowing the appearance of fracture lines to be evaluated and the use of the reinforcement opposing those fracture lines.

Sandwich method:

Marti, P., Design of concrete slabs for transverse shear, ACI Structural Journal, Vol. 87, pages 180 to 190.

This method intrinsically considers the behavior of plates, it can be found in annexes LL and MM of Eurocode 2 part 2,

Method of Capra and Maury:

Calcul automatique du ferraillage optimal des plaques et coques en béton armé, Annales de l'Institut Technique du Bâtiment et des Travaux Publics, n°367, December 1978,

This method is based on the equilibrium of the crosssection in combined bending and axial forces, it is detailed in the following section.

Example of the CapraMaury method
A set of facets is defined, centered at the point of computation of the finite element code. This can be a node, a Gauss point, or any point where forces are interpolated.
Its normal vector rotates from this point in the plane tangent to the mean sheet. The facet is marked by the angle θ between the OX axis and the normal vector of the element in its local coordinate system (see figure 2.1a). The angle θ is discretized regularly from 90º to +90º (here with a step of 5º). The Ox and Oy axes are the axes of the reinforcement layers.
Reference facet parallel to the beam’s crosssection in equilibrium subjected to combined bending and axial force
For each of these facets, the bending moment (M), the membrane force (N), and the shear force (V) are applied as a function of the effort tensors using the following equations:
By using combined bending calculations, it is possible to determine the reinforcement area in upper and lower layers A_{S}(θ) and A_{I}(θ) required in the θ direction to reach equilibrium in the chosen reinforcement concrete section.
The compressive strengths in the θ direction of the two reinforcement layers can be evaluated using the following expressions:
where f_{yd} is the maximum allowable stress of the steel (identical in both directions).
The strength is ensured if the resistive force is greater than the applied force, which is expressed as:
Thus, by considering an orthonormal coordinate system with AXS on the abscissa and AYS on the ordinate, one can finally solve for the upper and lower reinforcement:
for all angles θ
for all angles θ
and
and minimum.
The resistance inequalities define for each θ a halfspace limited by a straight line with negative slopes that reflects a validity domain, as shown in the following figure.
Resistance domain for a facet θ
By browsing through all the values of θ, one can obtain the validity domain shown in the following figure, delimited by the line ABCD.
Resistance domain for all facets
For each point P in the validity domain, the total area of reinforcement can be obtained by projecting the point P in Q onto the first bisector. The distance OQ then represents the value with A_{S} = A_{XS} + A_{YS}.
Therefore, it can be noticed that the optimum reinforcement corresponds to one of the 36 points (considering the facet rotation step chosen if one facet is taken every 10 degrees for example) of the boundary (illustrated by the 4 points of the line ABCD ...) whose projection on the first bisector is closest to the origin of the axes. Researching this point can be done using "dichotomy" type methods.

Connecting rod and tierod methods based on a finite element calculation result
In the presence of structural elements subjected to important point loads or presenting abrupt modifications of their section and geometry, the classical plane section methods of analysis are not satisfactory. These locations are generally reinforced using good practice rules based on experience or empirical guidelines. The strut and tie method (STM) is a rational design procedure for complex structural elements. The procedure is based on mechanics but is simple enough to be easily applied in design.
In general, the STM involves the idealization of a complex structural element into a simple structure capable of representing the path of stresses within the element.
The truss model consists of struts representing the compression fields in the concrete, ties representing the elastic steel reinforcement, and nodes representing the localized areas where the components interconnect or the areas where the elastic steel is anchored in the concrete. The struts and ties can carry only uniaxial forces. This mechanism must be stable and maintain the equilibrium with the applied loads.
The failure of the structure is dictated by the failure of one or more ties or by excessive compressive stresses within the struts or nodes. Ideally, only the first mode of failure should occur.
Example of D region and system modeling to obtain stress fields and then loads at the ST truss supports
This method is applied to the socalled D regions. To characterize these regions, the distribution of deformations over the depth of the element is considered to be nonlinear.
Consequently, the underlying hypotheses about the crosssection procedure are not valid. The principle of equivalence cannot be applied here.
According to Saint Venant's principle, an elastic stress analysis indicates that stresses due to axial forces and bending moments come near a linear distribution at a distance approximately equal to the depth of the element, h, away from the discontinuities.
In other words, a nonlinear stress distribution exists in the depth of a member from the location where the discontinuity is introduced. Then, it can be stated that the D regions are assumed to extend to a distance h from the applied load and support reactions. In general, a region of a structural member is assumed to be dominated by a nonlinear profile, or a D region, when the ratio extent/depth (a / h) is smaller than 2 or 2.5. The shear extent (a) is defined as the distance between the applied load and the nearest support in simple elements.
The approach consisting in defining a strutandtie truss can be summarized in Figure 10.
The general principle of design by STM
The verification methods are expressed in Eurocode 2 part 11.
The approach presented in Figure 10 is difficult to implement and depends strongly on the engineer who implements it.
More and more approaches are being developed to automate this procedure. In France, the algorithm with code_aster elements integrates a CALC_BT operator that renders the procedure semiautomatic based on:

an analysis of the local peak fields of the major and minor main stress fields,

a cutout of the D region modeled by Voronoi paving,

the projection of the mean directions of the main stresses in the Voronoi paving stones.

a set of optimization processes.
Reference example on the left  Solution obtained automatically by the CALC_BT operator of code_aster
This method requires a high level of experience and knowledge of the field.
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