1.2 Main assumptions and limitations for CSFM in 3D

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_c1r, σ_c2r, σ_c3r ≤ 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. 1.2.1e), leading to a conservative design due to the plasticity surface resembling the Tresca model, which is independent of the first stress invariant.

$\textsf{\textit{\footnotesize{Fig. 1.2.1\qquad Basic assumptions of the 3D CSFM: (a) principal stresses in concrete; (b) stresses in the reinforcement direction;}}}$ $\textsf{\textit{\footnotesize{(c) stress-strain diagram of concrete in terms of maximum stresses; (d) stress-strain diagram of reinforcement}}}$ $\textsf{\textit{\footnotesize{in terms of stresses at cracks and average strains; (e) Mohr's circles for concrete model in 3D CSFM; (f) bond shear stress-slip}}}$ $\textsf{\textit{\footnotesize{relationship for anchorage length verifications.}}}$

Concrete 

The presented material model is a multisurface plasticity model given by the combination of the Mohr-Coulomb 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.

$$\textsf{\textit{\footnotesize{Fig. 1.2.2\qquad Mohr-Coulomb multi-surface plasticity model for friction angle 0 degree}}}$$

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. 1.2.1d), 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 1.2.1b).

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. 1.2.1f), 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, considering the bending stiffness. 

The following types of anchors are available:

• Cast-in-place anchors

  • Reinforcement
  • Washer plate
  • Headed stud
• Cast-in-place reinforcement

  • Reinforcement
  • Threaded rods

Cast-in-place - Reinforcement

Modeled as ribbed reinforcement embedded in concrete. Bond strength is calculated according to selected code rules in the same way as for standard reinforcement. At the anchor end, an Anchorage type can be defined, working identically to reinforcement - an anchorage spring is applied with the β-factor set according to the chosen code. Three geometric shapes are available: Straight, L-shape, U-shape.

$$\textsf{\textit{\footnotesize{Fig. 1.2.3\qquad Cast-in reinforcement anchor - shapes}}}$$

Cast-in-place - Washer plate and Headed stud

The washer plate and the head of the headed stud are modeled as a plate-shell element from the corresponding material attached directly to the anchor shank. It transfers load to the concrete through compression-only contact. Available shapes: circular and square (only circular for headed stud), with customizable dimensions. The washer plate and head model is elastic and is not checked for resistance. 

At the finite element model level, the pull-out of the anchor is directly checked. The compression contact has stop criteria set so that it is not able to transfer greater contact stress to the concrete than prescribed by the selected standard. In practical terms, this means that if the anchor were to be loaded with a force that does not comply with the pull-out assessment, the result would be premature termination of the calculation because this stop criterion would be exceeded during further loading.

The anchor shank has zero bond strength – all load is transferred to the concrete through the plate or head into the concrete.

Post-installed - Reinforcement and Threaded rod

Designed as bars installed into drilled holes and bonded with adhesive. The engineer specifies the design bond strength directly from the technical specification of the adhesive product.

Material model of anchors made of bolt material

For anchors made of bolt material (threaded rods and anchors with washer plates), two properties of the material model differ from the nominal values of the bolt material: the modulus of elasticity and the ductility. Both describe the behaviour of the bolt assembly and the geometry of the test specimens on which the nominal values are measured. They do not depend on the design code, so they are applied identically for all design codes. Both the nominal stress-strain diagram of the bolt material and the diagram modified by the coefficients C_E and C_A are displayed in the Materials tab (Fig. 1.2.4).

Elastic stiffness. The modulus of elasticity of the bolt material is reduced by the coefficient C_E:

$$E_{b} = C_{E} \cdot E , C_E=0.8$$

• E - modulus of elasticity of bolt
• E_b - modulus of elasticity of the anchor used in the analysis

This approach was adopted on the basis of the latest research (not yet published; the reference will be added after publication), in which the effective modulus of elasticity of the bolt assembly was calibrated by finite element analysis against an extensive database of bolt tests. This is confirmed by already published experimental results of Janković et al. (2025): in 66 tensile tests of 1000 mm long threaded rods (M20 to M30, grades 8.8 and 10.9), the measured initial stiffness corresponded to 0.8·E·A_s/L_b, with a mean ratio of predicted to measured stiffness of 0.98 and a coefficient of variation of 7.9 %. In rods of this length, the influence of the nuts and the thread engagement is negligible, so the reduction represents an effective modulus of elasticity of the threaded rod itself. Incidentally, the same factor is contained in the stiffness of a bolt in tension in EN 1993-1-8 (coefficient 1.6 for a bolt row of two bolts).

Ductility. The ductility of bolt material is characterised by the percentage elongation after fracture A, typically around 10 %. These values are measured on short test pieces, where local necking forms a large part of the measured elongation. The length of the neck does not depend on the length of the specimen, so the percentage elongation decreases as the specimen gets longer. For anchors embedded several hundred millimetres into the concrete, it is substantially lower than the specified value. The application therefore uses the reduced ductility:

$$A_{red} = A/C_A$$

$$C_{A} = L_s / (0.177 \cdot L_s + 2.19)$$

• A - percentage elongation after fracture of the bolt material
• A_red - reduced ductility, used as the strain stop criterion of the anchor
• C_A - ductility reduction coefficient
• L_s - embedded length of the anchor [mm]

When the strain in the anchor reaches A_red, the stop criterion is met and the analysis is terminated, in the same way as the other stop criteria of 3D CSFM.

The equation was adopted on the basis of the latest research (not yet published; the reference will be added after publication), which provides an approximation of the plastic deformation of a bolt at its ultimate resistance, calibrated on an extensive database of bolt tests: Δ_u,p = A · (0.177 · L + 2.19), where L is the deforming length of the bolt in mm. The term proportional to the length represents the uniform elongation of the rod, and the constant term represents the local contribution of necking. In the application, the embedded length of the anchor L_s is taken as this length, and dividing Δ_u,p by L_s gives the strain A/C_A. The coefficient C_A increases with length and approaches the limit 1/0.177 = 5.65. For embedded lengths of 200 mm and more, it exceeds 5.3, so the ductility of the anchor is reduced to roughly one fifth of the nominal value, e.g. from 12 % to about 2.2 % at L_s = 300 mm.

\[ \textsf{\textit{\footnotesize{Fig. 1.2.4\qquad Stress-strain diagram of the bolt material in the Materials tab: nominal diagram and the diagram modified by the coefficients C_E and C_A}}}\]

More information about connecting individual anchor types to the base plate or cast-in plate can be found in the chapter Finite elements types - Load transferring devices.