In practice, engineers may encounter different types of finite elements (from simple 1D bar elements to more complicated 3D brick elements) that are used in a variety of applications for the analysis and design of structural elements. A common feature of most of the computations in practice tends to be the linear behavior of the models, the advantages of which are undoubtedly speed, clarity, and simply the fact that for a large variety of problems, this solution is quite sufficient.

Especially in the world of concrete structures, it often happens that the linear approach is not sufficient simply because after the first cracks appear in the loaded element, the stresses are redistributed and the problem becomes significantly non-linear.

For these cases, it is necessary to choose one of the more sophisticated approaches. For 1D cases, analytical methods defined directly in codes can often be found. For example, popular Strut and Tie models can be built for 2D planar elements and discontinuity regions (D-regions), or the more sophisticated stress field method implemented in IDEA StatiCa Detail, CSFM, can be used.

However, if the engineer encounters a problem that cannot be simplified into planar behavior, the options are very limited. Of course, a 3D Strut and Tie model can be built or semi-scientific software can be used for accurate analysis. These procedures are often time-consuming, not code-compliant, and require an engineer knowledgeable in advanced modeling methods.

For this reason, IDEA StatiCa has developed and implemented the 3D CSFM (Compatible Stress Field Method) in the Detail application. 3D CSFM extends the established CSFM into a third dimension, offering a fast and code-compliant solution that is primarily applicable to the everyday engineer, giving them a unique new ability to safely tackle the complex details of concrete structures.

Main assumptions and limitations

3D CSFM defines the concrete behavior based on the Modified Mohr-Coulomb plasticity theory for monotonic loading. The method considers principal concrete stresses in compression and reinforcement stresses (σ_sr) at the cracks while neglecting the concrete tensile strength (tension cut-off), except for its stiffening effect on the reinforcement (Tension stiffening).

σ_c₁_r, σ_c₂_r, σ_c₃_r ≤ 0 MPa

The reinforcement bars are linked to concrete volume finite elements through bond elements, allowing for slip between the concrete and reinforcement. It should be noted that 3D CSFM is not suitable for simulating plain concrete due to the absence of tension, which may result in misleading deformation and model divergence. Generally, the Mohr-Coulomb theory includes two fundamental properties governing the evolution of the plasticity surface in compression and partially in tension: the internal friction angle φ and cohesion parameter c. 3D CSFM assumes a zero angle of internal friction (Fig. 1e), leading to a conservative design due to the plasticity surface resembling the Tresca model, which is independent of the first stress invariant.

\( \textsf{\textit{\footnotesize{Fig. 1\qquad Basic assumptions of the 3D CSFM: (a) principal stresses in concrete; (b) stresses in the reinforcement direction;}}}\) \( \textsf{\textit{\footnotesize{(c) stress-strain diagram of concrete in terms of maximum stresses; (d) stress-strain diagram of reinforcement}}}\) \( \textsf{\textit{\footnotesize{in terms of stresses at cracks and average strains; (e) Mohr's circles for concrete model in 3D CSFM; (f) bond shear stress-slip}}}\) \( \textsf{\textit{\footnotesize{relationship for anchorage length verifications.}}}\)

Concrete

The presented material model is a multisurface plasticity model given by the combination of the Mohr-Coloumb and Rankine models for monotonic loading. It’s important to note that this model does not address unloading, therefore, state variables are not stored, as they would be in classical plasticity models used for cyclic loading.

\[ \textsf{\textit{\footnotesize{Fig. 2\qquad Mohr-Coulomb multi-surface plasticity model for friction angle 0 degree}}}\]

As already mentioned, the material model is intended for use in applications that calculate the response of reinforced concrete (not suitable for plain concrete). This is due to the exclusion of concrete in tension. Therefore, the model is not even suitable for structural elements where the design rules for reinforced concrete such as minimum reinforcement ratio, maximum bar spacing, etc., are not fulfilled. It should also be added that, for numerical stability reasons, a very small tensile capacity is defined in the model. The tensile part is restricted by planes corresponding to the Rankine model.

3D CSFM in IDEA StatiCa Detail does not consider an explicit failure criterion in terms of strains for concrete in compression (i.e., it considers an infinitely plastic branch after the peak stress is reached). This simplification does not allow the deformation capacity of structures failing in compression to be verified. However, their ultimate capacity is properly predicted when the increase in the brittleness of concrete as its strength rises is considered by means of the 𝜂_𝑓𝑐 reduction factor defined in fib Model Code 2010 as follows:

\[f_{c,red} = \eta _{fc} \cdot f_{c}\]

\[{\eta _{fc}} = {\left( {\frac{{30}}{{{f_{c}}}}} \right)^{\frac{1}{3}}} \le 1\]

where:

f_c is the concrete cylinder characteristic strength (in MPa for the definition of \( \eta_{fc} \)).

The f_c,red is then compared with the Equivalent Principal Stress σ_c,eq in concrete, which will be defined further, of course, with consideration of all safety factors prescribed by code.

A detailed description of the concrete model can be found at the following link:

Concrete Material Model for 3D Detail

Reinforcement

The bilinear stress-strain diagram for reinforcement bars, as defined by design codes (Fig. 1d), represents an idealized model. This model necessitates knowledge of the basic properties of the reinforcement during the design phase, specifically the strength and ductility class. Alternatively, users have the option to define a customized stress-strain relationship.

Tension stiffening is considered by modifying the stress-strain relationship of the bare reinforcing bar to capture the average stiffness of the bars embedded in the concrete (ε_m) (Fig 1b).

Anchorage

Bond-slip between reinforcement and concrete is introduced in the finite element model by considering the simplified rigid-perfectly plastic constitutive relationship presented in (Fig. 1f), with f_bd being the design value (factored value) of the ultimate bond stress specified by the design code for the specific bond conditions.

This is a simplified model with the sole purpose of verifying bond prescriptions according to design codes (i.e., anchorage of reinforcement). The reduction of the anchorage length when using hooks, loops, and similar bar shapes can be considered by defining a certain capacity at the end of the reinforcement, as will be described further.

Anchors

The element of the anchor is defined as being able to transfer normal tensile or compression forces as well as shear forces also considering the bending stiffness. However, only normal stress in the anchors is evaluated.

Two types of anchors are available:

Adhesive anchor
Cast-in-place reinforcement

The Cast-in-place reinforcement behavior is the same as classical reinforcement (Anchorage type, bond, etc.) For Adhesive anchors, it is possible to define the design value Bond strength directly. This value should be read from the technical sheet of the fabricator.

Mohr-Coulomb plasticity theory implementation in 3D CSFM

In the following chapter, we will take a look at how the Mohr-Coulomb theory is implemented in 3D CSFM. We will explain how the confinement effect (triaxial stress) is considered and how the Equivalent Principal Stress σ_c,eq is calculated, which is used to determine the load-bearing capacity from the point of view of concrete.

Introduction to the theory

Mohr–Coulomb theory is a mathematical model describing the response of brittle materials, to shear and normal stress. Most of the classical engineering materials follow this rule in at least a part of their shear failure envelope. Generally, the theory applies to materials for which the compressive strength far exceeds the tensile strength.

\[ \textsf{\textit{\footnotesize{Fig. 3\qquad Mohr-Coulomb Plasticity Model }}}\]

In structural engineering, it is used to determine failure load as well as the angle of fracture for displacement of fracture surface in concrete and similar materials. Coulomb's friction hypothesis is used to determine the combination of shear and normal stress that will cause a fracture of the material. Mohr's circle is used to determine which principal stresses will produce this combination of shear and normal stress and the angle of the plane in which this will occur. According to the principle of normality, the stress introduced at failure will be perpendicular to the line describing the fracture condition.

\[ \textsf{\textit{\footnotesize{Fig. 4\qquad Meridian plane and tension cut-off}}}\]

It can be shown that a material failing according to Coulomb's friction hypothesis will show the displacement introduced at failure forming an angle to the line of fracture equal to the angle of friction. This makes the strength of the material determinable by comparing the external mechanical work introduced by the displacement and the external load with the internal mechanical work introduced by the strain and stress at the line of failure. By conservation of energy, the sum of these must be zero and this will make it possible to calculate the failure load of the construction.

Implementation in 3D CSFM

In general, for a given angle of internal friction of the concrete, which is around φ = 30-40° in Reference [1], [2], [3], [4], the tensile and compressive strengths of the concrete Mohr's circles can be constructed as in Figure 5.

\[ \textsf{\textit{\footnotesize{Fig. 5\qquad Mohr's circles for concrete}}}\]

Where f_c is concrete strength in compression, f_ct is concrete strength in tension, φ is the angle of internal friction, and σ_c₁, σ_c₃ are the principal stresses of concrete under triaxial compression.

It can be noticed that as the principal stress σ_c₃ increases, the maximal possible difference between the values of σ_c₃ and σ_c₁, which we define as maximal σ_c,eq (see below), also increases. This difference corresponds to twice the deviatoric stress defined in the literature as a radius of the mohr circles.

In 3D CSFM implemented in IDEA StatiCa Detail, the angle of internal friction is considered as φ = 0°, as shown in Figure 6.

\[ \textsf{\textit{\footnotesize{Fig. 6\qquad Mohr's circles for concrete implemented in IDEA StatiCa Detail}}}\]

The practical consequence of this implementation is that the maximum difference between σ_c₃ and σ_c₁ is constant as σ_c₃ increases.

Equivalent Principal Stress expresses the equivalent uni-axial stress for a general tri-axial stress state.

\[\sigma_{c,eq} = \sigma_{c3} - \sigma_{c1}\]

The σ_c,eq value can, therefore, be directly compared with uniaxial strength limits according to codes.

\[\frac{\sigma_{c,eq} }{ \sigma_{c,lim}} \le 1\]

Where σ_c_,lim is the design (factored) uniaxial strength of concrete f_c.

Comparing Figure 5, where the real angle of internal friction is used, and Figure 6, which shows the Mohr-Coulomb theory implementation with zero angle of internal friction, it can be seen that the approach chosen for the calculations in Detail is very conservative for the assessment of triaxial stress state.

For a better understanding of the areas affected by tri-axial compression stress, the expression of the increase of the effective material strength due to tri-axial compression has been added to the IDEA StatiCa Detail application as a ratio σ_c₃/σ_c,lim. You can find this ratio in the Strength code check.

In the Auxiliary results, the user can also find the κ factor, which explains the tri-axiality in a different way.

\[\kappa = \frac{ \sigma_{c3}}{ \sigma_{c,eq}}\]

The concrete strength check can be then rewritten as:

\[\frac{\sigma_{c,eq} }{ \sigma_{c,lim}} = \frac{\sigma_{c,3} }{ \kappa \cdot \sigma_{c,lim}} \le 1\]

It follows from the previous that if the element is under hydrostatic stress - σ_c₃=σ_c₂=σ_c₁, the Equivalent Principal Stress σ_c,eq will have the zero value, and the kappa factor will reach infinity.

More can be found here: Tri-axial stress – the active confinement effect

General mechanics assumptions for 3D CSFM

Equilibrium equations

The theory of small deformations enables the assembly of the equilibrium equation based on the undeformed volume using a first-order approach.

\[ \textsf{\textit{\footnotesize{Fig. 7\qquad Equilibrium equations and graphical representation on infinitesimal element}}}\]

Compatibility equations

A solid body comprises infinitesimal volumes or material points, each of which is interconnected without gaps or overlaps. Mathematical conditions must be adhered to in order to prevent the occurrence of gaps or overlaps when a continuum body undergoes deformation.

Constitutive equations

The constitutive equations governing the behavior of 3D elements play a pivotal role in the analysis of material behavior in structural mechanics. These equations are formulated to accommodate the non-linear isotropic behavior, which is valid for solid block members in IDEA StatiCa Detail.

When dealing with a 3D wall, it is essential to account for the orthotropic behavior across its thickness, paying particular attention to the tension in the concrete due to the absence of transverse reinforcement. The orthotropy is caused by allowing the tension in concrete in an out-of-plain direction. The material properties such as module elasticity and Poisson ratio remain the same.

\[ \textsf{\textit{\footnotesize{Fig. 8\qquad Linearly elastic isotropic compliance matrix}}}\]

Analysis model of IDEA StatiCa 3D Detail

An introduction to finite element implementation

3D CSFM considers continuous stress fields in the concrete (3D finite elements), complemented by discrete “rod” elements representing the reinforcement (1D finite elements). Therefore, the reinforcement is not diffusely embedded into the concrete 3D finite elements but explicitly modeled and connected to them.

\[ \textsf{\textit{\footnotesize{Fig. 9\qquad Rendering of the calculation model for concrete block and out-of-plane wall}}}\]

Finite element types

The non-linear (inelastic) finite element analysis model is created by several types of finite elements used to model concrete, reinforcement, and the bond between them. Concrete and reinforcement elements are first meshed independently and then interconnected using multi-point constraints (MPC elements). This allows the reinforcement to occupy any position not limited to nodes of tetrahedral mesh. To verify anchorage length, bond, and anchorage end spring elements are inserted between the reinforcement and the MPC elements.

\[ \textsf{\textit{\footnotesize{Fig. 10\qquad Finite element model: reinforcement elements mapped to concrete mesh using MPC and bond elements}}}\]