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.

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. 

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.

$$\textsf{\textit{\footnotesize{Fig. 1.3.1\qquad Mohr-Coulomb Plasticity Model }}}$$

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. 

$$\textsf{\textit{\footnotesize{Fig. 1.3.2\qquad Meridian plane and tension cut-off}}}$$

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 1.3.3.

$$\textsf{\textit{\footnotesize{Fig. 1.3.3\qquad Mohr's circles for concrete}}}$$

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 7.

$$\textsf{\textit{\footnotesize{Fig. 1.3.4\qquad Mohr's circles for concrete implemented in IDEA StatiCa Detail}}}$$

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 1.3.3, where the real angle of internal friction is used, and Figure 1.3.4, 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

Equilibrium equations

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

$$\textsf{\textit{\footnotesize{Fig. 1.4.1\qquad Equilibrium equations and graphical representation on infinitesimal element}}}$$

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. 

$$\textsf{\textit{\footnotesize{Fig. 1.4.2\qquad Linearly elastic isotropic compliance matrix}}}$$

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. 

$$\textsf{\textit{\footnotesize{Fig. 2.1.1\qquad Rendering of the calculation model for concrete block and out-of-plane wall}}}$$

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.

$$\textsf{\textit{\footnotesize{Fig. 2.2.1\qquad Finite element model: reinforcement elements mapped to concrete mesh using MPC and bond elements}}}$$

Concrete

Concrete is analyzed using mixed tetrahedral elements with nodal rotations. The tetrahedral elements allow us to mesh regions of any topology while the implemented formulation guarantees accurate deformation results (without spurious shear stress known as the shear lock effect) even for the coarse mesh which would not be suitable for linear tetrahedral elements formulation. 

Full integration is utilized. It means that each element is equipped with four integration points situated within the volume. Such an integration yields a precise strain and stress field, allowing for sufficient evaluation and presentation of the results across the whole volume. Subsequently, the stop criteria are established based on the value in the integration point.

Reinforcement

Rebars are modeled by two-node 1D “rod” elements (CROD), which only have axial stiffness. These elements are connected to special “bond” elements that were developed in order to model the slip behavior between a reinforcing bar and the surrounding concrete. These bond elements are subsequently connected by MPC (multi-point constraint) elements to the mesh representing the concrete. This approach allows the independent meshing of reinforcement and concrete, while their interconnection is ensured later.

Bond elements

The anchorage length is verified by implementing the bond shear stresses between concrete elements (3D) and reinforcing bar elements (1D) in the finite element model. For this purpose, the “bond” finite element type was developed.

The bond element is defined as a shell finite element connected to elements representing reinforcement by the first layer and by the second layer to concrete mesh via multi-point constraints (MPC elements). It should be noted that the bond element is always displayed in this article with a non-zero height, which is, however, defined as infinitesimal in the model.

The behavior of this element is described by the bond stress, τ_b, as a bilinear function of the slip between the upper and lower nodes, δu, see (Fig. 2.2.2).

$$\textsf{\textit{\footnotesize{Fig. 2.2.2\qquad (a) Conceptual illustration of the deformation of a bond element; (b) shear-deformation function}}}$$

The elastic stiffness modulus of the bond-slip relationship, Gb, is defined as follows:

$$G_b = k_g \cdot \frac{E_c}{Ø}$$

k_g            coefficient depending on the reinforcing bar surface (by default kg = 0.2)

E_c            modulus of elasticity of concrete (taken as Ecm in case of EN)

Ø             the diameter of the reinforcing bar

The design values (factored values) of ultimate bond shear stress, f_bd, provided in the respective selected design codes EN 1992-1-1 or ACI 318-19 are used to verify the anchorage length. The hardening of the plastic branch is calculated by default as Gb/105.

Anchorage spring

The provision of anchorage ends to the reinforcing bars (i.e., bends, hooks, loops…), which fulfills the prescriptions of design codes, allows the reduction of the basic anchorage length of the bars (l_b,net) by a certain factor β (referred to as the ‘anchorage coefficient’ below). The design value of the anchorage length (lb) is then calculated as follows:

$$\textsf{\textit{\footnotesize{Fig. 2.2.3\qquad Model for the reduction of the anchorage length: a) Anchorage force along the anchorage length of }}}$$ $$\textsf{\textit{\footnotesize{the reinforcement bar, b) slip-anchorage force constitutive law}}}$$

The reduction of the anchorage length is included in the finite element model by means of a spring element at the end of the bar (Fig. 2.2.3a), which is defined by the constitutive model shown in (Fig. 2.2.3b). The maximum force transmitted by this spring (F_au) is:

$$F_{au} = \beta \cdot A_s \cdot f_{yd}$$

where :

β             the anchorage coefficient based on anchorage type

A_s            the cross-section of the reinforcing bar

f_yd           the design value (factored value) of the yield strength of the reinforcement

Base plate

The base plate is modeled as an elastic shell element. The steel material used for base plates is defined in the Materials tab. The only physical property is the modulus of elasticity E.

$$\textsf{\textit{\footnotesize{Fig. 2.3.1\qquad The base plate material definition}}}$$

The base plate can be loaded by the point load (Fx, Fy, Fz, Mx, My, Mz) and the group of forces (Fx, Fy, Fz), mainly used for loading models exported from the IDEA StatiCa Connection. Note that point loads and point moments directly load the corresponding node of the base plate. It means that there is no redistribution, only by the stiffness of the base plate. 

This implementation allows importing load effects from the IDEA StatiCa Connection that are applied to the base plate at the location of the individual weld finite elements with the value and direction determined from the general stress of that weld finite element. More can be read in the corresponding chapter of this document.

The second loading option is the Stub — representing a short portion of the column above the base plate. The stub is modeled as an elastic shell element structure and behaves as a physically accurate interface between the internal forces and the plate. The user selects a cross-section for the stub from a standard section database. The 6-component internal force set (forces and moments) is applied at a single point on the bottom face of the stub — i.e. the base of the column. Constraints transfer the forces to the top face of the stub, from where they are naturally redistributed through the stub into the base plate, anchors, and concrete.

$$\textsf{\textit{\footnotesize{Fig. 2.3.2\qquad The load transfer through the stub}}}$$

Shear transfer mechanism (from base plate to concrete block)

Frictional compression-only contact is defined between the baseplate and concrete. For the shear transfer user can choose from three options:

• By anchors
• By friction
• By shear lug

The software does not allow the combination of these shear transfer mechanisms. 

The friction coefficient should be input as a designed (factored) value. In case the resultant shear force F_xy exceeds the pressure force F_z times the frictional coefficient μ, the calculation will stop, and not all the loads will apply to the model. The condition is written as follows:

$$\frac {F_{xy}}{ \mu \cdot F_{z}}\le 1$$

This can be seen in the following example, where two load cases are considered. 

• LC1 - Permanent type - F_z = 100 kN
• LC2 - Variable type- F_x = 100 kN

$$\textsf{\textit{\footnotesize{Fig. 2.3.3\qquad Load input for example explaining shear transfer by friction}}}$$

In the first calculation step, all the permanent load is applied. Then the variable load is gradually applied until it reaches the value of the pressure load times the friction coefficient.

$$\textsf{\textit{\footnotesize{Fig. 2.3.4\qquad Results from example explaining shear transfer by friction}}}$$

The graph in Figure 2.3.5 defines the behavior of the frictional contact between the baseplate and concrete.

$$\textsf{\textit{\footnotesize{Fig. 2.3.5\qquad Force-displacement graph describing the behavior of frictional contact}}}$$

The value of F_zμ differs for each increment of the calculation, whereas the value of maximal shear deformation u_xy is constant. 

If the compressive normal force F_z and the shear force F_xy are input in one load case type (e.g. only permanent), and the condition of F_xy / (F_zμ) ≤ 1 is not fulfilled, no load will be applied to the model because the condition is not fulfilled in any increment of the calculation.

The shear lug is connected with the concrete mesh by constraints allowing only compression only normal stress transfer. 

$$\textsf{\textit{\footnotesize{Fig. 2.3.6\qquad Shear lug transfer of shear mechanism}}}$$

The shear lug is modeled from elastic shell elements, where the modulus of elasticity E defines the material.

The results are not evaluated and displayed for the base plate as well as for the shear lug.

 Base plate options  (stand-off, grout)

The following set of stand-off options, fully aligned with the Connection application, is available.

• Direct
• Mortar joint – nuts from the top
• Mortar joint – nuts from the top and bottom
• Gap

The mortar layer is modeled as a shell element, with its stiffness taken into account. Note that shell elements are incompressible in the direction of their thickness. This helps to redistribute local forces to the concrete and is valid for typical bedding thicknesses used in practice - 25-50 mm.

The distinction between nuts only from the top (pinned interconnection between anchor and base plate) vs. top and bottom (fixed interconnection between anchor and base plate) strongly influences the shear capacity from the point of view of concrete bearing.

Anchors

The finite elements representing anchors are modeled to be able to transfer normal and shear forces to the concrete, also taking into account the bending stiffness of the anchors. To model the slip between the anchor and the surrounding concrete, the same bond and MPC elements are used as for the reinforcement. With the difference that:

• For post-installed (adhesive) anchors, it is necessary to specify the design bond strength.
• For Washer plates and Headed studs, the bond is neglected along the shank of the anchor. All axial load is then transferred to the concrete through the washer plate or head of the anchor.

Bond within the concrete cover - For anchors connected to the concrete by bond along the shank, i.e. cast-in reinforcement anchors and post-installed (adhesive) anchors, the bond elements are not active within the concrete cover. The application determines the cover automatically as the distance from the concrete surface to the face of the surface reinforcement (Fig. 2.3.7a). Along this length, the bond strength between the anchor and the concrete is zero, and the anchor is connected to the concrete by bond elements only below the cover, over the remaining part of its embedment depth (Fig. 2.3.7b). All load transferred by bond is therefore introduced into the concrete enclosed by the surface reinforcement, and the bond stresses of the anchor within the cover are zero in the results.

The reason is the different failure behaviour of the concrete cover. 3D CSFM uses a plastic material model for concrete that allows stresses to redistribute (see Chapter 1.2). The concrete cover near a free surface, however, fails in a brittle manner governed by fracture mechanics, without any plastic reserve. If bond were active within the cover, the model would build up a compression field in a region that cannot reliably sustain it in reality, and it would overestimate the contribution of the cover to the anchor resistance. This error grows with the thickness of the cover. In addition, the cover is the part of the model exposed to the environment (carbonation, chlorides, aggressive agents), so its properties may degrade during the service life of the structure. Neglecting the bond within the cover is therefore a conservative assumption, consistent with the basic assumptions of 3D CSFM.

This assumption applies only to anchors that transfer the load to the concrete by bond along the shank. Headed studs and anchors with washer plates transfer the axial load through the head or plate, with zero bond along the shank, so they are not affected. Ordinary reinforcement is not affected either. The influence on the resistance is most pronounced for anchors whose resistance is governed by bond and which are installed in members with a large concrete cover, because a larger part of the embedment depth does not contribute.

$$\textsf{\textit{\footnotesize{Fig. 2.3.7\qquad Anchor with bond along the shank}}}$$

Anchors can be interconnected with base plates. For this interconnection, a fully nonlinear constraint is used to connect the anchor's end and a base plate node. This constraint allows us to control all degrees of freedom to ensure, for example, that the anchors transfer no compression force from the base plate, or that no shear is transferred by the anchor when modeling a shear lug, etc. 

Interconnection with base plate properties for anchors allows the user to control whether the anchor will be connected with the base plate by the previously mentioned constraint and how. 

$$\textsf{\textit{\footnotesize{Fig. 2.3.8\qquad Interconnection with base plate settings}}}$$

The Transfer of shear checkbox can be used to control whether the anchor and base plate will be connected or not in terms of shear. Note that it is not supported to combine shear transfer mechanisms, so for transfer by friction and shear lug, this checkbox is irrelevant. On the other hand, for shear transfer using anchors, this field gives the option to exclude some anchors from shear transfer.

The Transfer of axial forces checkbox can be used to control whether the anchor and base plate will be connected or not in terms of the axial direction. This is mainly used for the export from the Connection feature (see the corresponding chapter). For manual modeling, it makes sense to have this checkbox always checked.

When the checkbox is unchecked, the anchor is disconnected in both tension and compression (in the case of a model exported from the Connection application, the connection is replaced by a pair of forces). If the checkbox is checked, the anchor is always connected to the plate in tension, but the connection in compression is controlled by the anchor type and the type of stand-off. For more information, see Figure 2.3.8.

Cut threads

Controlled by a checkbox in anchor properties and has 2 purposes:

1. Defines how the anchor connects to the base plate:

• For headed studs and cast-in reinforcement connected to the Base plate (not for Cast-in plates), it distinguishes between a bolt connection (pinned) and a welded connection (fixed) — visible in the 3D scene.
• Note that the way of anchor-to-plate connection has a significant influence on the shear resistance from the point of view of bearing of the concrete.

$$\textsf{\textit{\footnotesize{Fig. 2.3.9\qquad Cut threads options}}}$$

2. For Eurocode, the resistance of the anchor with cut threads is reduced according to EN 1993-1-8 3.6.1 (3). It can be set in Project settings. For Threaded rods and Washer plates, it is recommended to keep this setting on at all times.

Axial and rotational interconnection between Anchor and Base plate

As already mentioned in this chapter, depending on the type of anchor, the stand-off setting, and whether or not cut threads are considered, anchors are connected to the base plate in different ways. In terms of rotational connection, this can be Hinged / Fixed. In terms of axial connection, this can be Tension / Tension + Compression. The rotational connection types strongly influence the shear capacity from the point of view of concrete bearing. In a 3D scene, it is easy to tell whether an anchor is connected as fixed or hinged based on the presence of nuts, see Figure 2.3.10.

$$\textsf{\textit{\footnotesize{Fig. 2.3.10\qquad Rotational constraints}}}$$

The following table shows all possible combinations of base plate connections with anchors and the corresponding rotational and axial connections.

$$\textsf{\textit{\footnotesize{Fig. 2.3.11\qquad Axial and rotational constraints between an anchor and a base plate}}}$$

Cast-in plates

Cast-in plate is a special case of a base plate. It is modeled analogously with the following differences:

Since the plate is embedded inside a concrete block, no type of stand-off can be specified. The depth of the slab embedding is neglected. The plate, modeled by shell elements, is placed directly on the concrete surface. Therefore, the side surfaces of the slab are not considered to be supported by the concrete.

It is only possible to use Reinforcement and Headed studs, which, like classic anchors, can be set to be connected to the slab in the axial and shear directions. Practical experience and some national documents indicate the need to design Headed studs only for shear and Reinforcement for axial load. From the perspective of axial and rotational constraints, anchors are always connected as Fixed and Tension + Compression.

$$\textsf{\textit{\footnotesize{Fig. 2.3.12\qquad Axial and rotational constraints between an anchor and a base plate}}}$$

The finite elements are implemented internally, and the analysis model is generated automatically without any need for proficient user interaction. An important part of this process is meshing.

Concrete

All concrete members are meshed together. A recommended element size is automatically computed by the application based on the size and shape of the structure and taking into account the diameter of the largest reinforcing bar. Moreover, the recommended element size guarantees that a minimum of four elements are generated in thin parts of the structure, such as slender columns or thin walls, to ensure reliable results in these areas. Designers can always select a user-defined concrete element size by modifying the multiplier of the default mesh size.

Reinforcement

The reinforcement is divided into elements with approximately the same length as the concrete element size. Once the reinforcement and concrete meshes are generated, they are interconnected with bond elements, as shown in Fig. 2.2.1.

Refinement

Concrete mesh is automatically refined around anchors, around shear lugs, and under the stub for loading. The size of the refined mesh is approximately twice smaller than the basic concrete mesh. The radius of the refined area is defined approximately as the element size multiplied by two.

A standard full Newton-Raphson (NR) algorithm is used to find the solution to a non-linear FEM problem. 

Generally, the NR algorithm does not often converge when the full load is applied in a single step. A usual approach, which is also used here, is to apply the load sequentially in multiple increments and use the result from the previous load increment to start the Newton solution of the subsequent one. For this purpose, a load control algorithm was implemented on top of the Newton-Raphson. In the case that the NR iterations do not converge, the current load increment is reduced to half its value, and the NR iterations are retried.

A second purpose of the load-control algorithm is to find the critical load, which corresponds to certain “stop criteria” – specifically the maximum strain in concrete, the maximum slip in bond elements, the maximum displacement in anchorage elements, and the maximum strain in reinforcing bars. The critical load is found using the bisection method. In the case where the stop criterion is exceeded anywhere in the model, the results of the last load increment are discarded and a new increment of half the size of the previous one is calculated. This process is repeated until the critical load is found with a certain error tolerance.

For concrete, the stop criterion was set to a 5% strain in compression (i.e., around an order of magnitude larger than the actual failure strain of concrete) and 7% in tension at the integration points of shell elements. In tension, the value was set to allow for the limit strain in reinforcement, which is usually around 5% without accounting for tension stiffening, to be reached first. In compression, the value was chosen from among several alternatives as one that is large enough for the effects of crushing to be visible in the results, but small enough so as not to cause too many problems with numerical stability.

$$\textsf{\textit{\footnotesize{Fig 2.5.1\qquad Constitutive law of bond and anchorage elements used for anchorage length verification: a) Bond shear stress}}}$$ $$\textsf{\textit{\footnotesize{slip response of bond element, b) force-displacement response of an anchorage element}}}$$

For reinforcement, the stop criterion is defined in terms of stresses. Since stresses at the crack are modeled, the criterion in tension corresponds to the reinforcement tensile strength accounting for the safety coefficient. The same value is used for the criterion in compression.

The stop criterion in bond elements and anchorage springs is α·δ_umax, where δ_umax is the maximum slip used in code checks and α = 10.

Other stop criteria for anchoring:

• Pull out of headed anchors (maximal contact compression stress at the top face of the head of the anchor). 
• Maximal shear force that can be transferred by the anchor from the point of view of the bearing of concrete.

These two criteria are dependent on the selected code. You can find more information about them in the sections explaining the code-dependent parts of structural analysis in the application.

The results are presented independently for concrete and for reinforcement elements. The stress and strain values in concrete are calculated at the integration points of volume elements. However, as it is not practical to present the data in such a manner, the results are presented by default in nodes, like the maximum value of compressive stress from adjacent Gauss integration points in connected elements. It should be noted that this representation might locally underestimate the results at compressed edges of members in a case where the finite-element size is similar to the depth of the compression zone.

The results for the reinforcement finite elements are either constant for each element (one value – e.g., for steel stresses) or linear (two values – for bond results). For auxiliary elements, such as elements of bearing plates, only deformations are presented.

The IDEA StatiCa Detail model does not always have to be modeled from scratch or from a template. There is also an option to import the model, including load effects, from IDEA StatiCa Connection. In Connection, the steel superstructure above the concrete block is analyzed using a nonlinear 3D model, while the concrete block itself is represented in a simplified way by a Winkler foundation. In Detail, on the other hand, the reinforced concrete block is modeled explicitly and checked in detail.

When transferring the model, only the base plate, anchors, and concrete block are imported into Detail – the steel member itself (and its global stiffness) is not. In the Connection model, this steel member is connected to the base plate by a weld. The stresses in the weld finite elements are integrated and converted into a set of equivalent forces that load the base plate in Detail. In this way, the effect of the missing steel member is represented by weld forces applied directly to the base plate.

$$\textsf{\textit{\footnotesize{Fig. 2.7.1\qquad Loads imported from IDEA StatiCa Connection}}}$$

Due to the different definition of stiffness between Connection and Detail (missing steel member, different material models, and concrete representation), a direct connection between the base plate and anchors in Detail would generally lead to a different redistribution of loads and, therefore, different tensile forces in the anchors. To avoid this, the anchors are imported axially disconnected from the base plate. Instead of transferring axial forces through the physical contact, the anchor tensions obtained from the Connection are applied directly to the anchors in Detail. At the same time, an equal and opposite force is applied to the base plate at each anchor location, so that the global equilibrium of the model is preserved. This pair of forces (one acting on the anchor, the other on the base plate) represents the interaction between the base plate and the anchor without allowing additional redistribution of axial forces in Detail. These two opposite forces are illustrated in Figure 2.7.1.

However, the shear forces are still transferred by the connection between the base plate and the anchors (or shear lug, or friction). This is possible because a constraint is used to connect the base plate and the anchors in shear, allowing us to control the relevant degrees of freedom of this interconnection. In Detail, the user can therefore modify the shear load path – for example, by releasing shear in two of four anchors and keeping only the edge anchors engaged in shear – while the axial forces remain as imported from Connection.

For cast-in plates, we adopted a different approach. Several European design recommendations require that only the reinforcement bars are considered to resist axial forces, while headed studs are assumed to transfer shear only. Since IDEA StatiCa Connection cannot internally separate axial forces in reinforcement anchors from those in headed studs during the export, the anchors of cast-in plates are imported into Detail fully connected, also in the axial direction. This allows the user to activate a design option in Detail where reinforcement anchors carry only axial tension and headed studs carry only shear. In this workflow, the axial force that was originally assigned to the headed studs has to be redistributed onto the reinforcement anchors within the Detail model. Such redistribution would not be possible if we used the pair-of-opposite-forces approach described above, which is why cast-in plates are handled differently.

Ultimate limit state

The different verifications required by specific design codes are assessed based on the direct results provided by the model. ULS verifications are carried out for concrete strength, reinforcement strength, and anchorage (bond shear stresses).

To ensure a structural element has an efficient design, it is highly recommended to run a preliminary analysis that takes into account the following steps:

• Choose a selection of the most critical load combinations.
• Calculate only Ultimate Limit State (ULS) load combinations.
• To expedite the calculation time and address any issues, consider using a coarse mesh by increasing the multiplier of the default mesh size in the Setup (Fig. 3.1.1). If the model performs well, revert the multiplier back to a factor of 1.

$$\textsf{\textit{\footnotesize{Fig 3.1.1\qquad Mesh multiplier}}}$$

Such a model will calculate very quickly, allowing designers to review the detailing of the structural element efficiently and re-run the analysis until all verification requirements are fulfilled for the most critical load combinations. Once all the verification requirements of this preliminary analysis are fulfilled, it is suggested that the complete ultimate load combinations be included and the use of fine mesh size (the mesh size recommended by the program). Users can change mesh size by the multiplier, which can reach values from 0.5 to 5 (Fig. 3.1.1).

The basic results and verifications (stress, strain, and utilization (i.e., the calculated value/limit value from the code)), as well as the direction of principal stresses in the case of concrete elements) are displayed by means of different plots where compression is generally presented in red and tension in blue. Global minimum and maximum values for the entire structure can be highlighted as well as minimum and maximum values for every user-defined part. In a separate tab of the program, advanced results such as tensor values, deformations of the structure, and reinforcement ratios (effective and geometric) used for computing the tension stiffening of reinforcing bars can be shown. Furthermore, loads and reactions for selected combinations or load cases can be presented.

Concrete - ULS

The concrete model implemented in 3D CSFM is based on the uniaxial compression constitutive laws prescribed by EN 1992-1-1 for the design of cross-sections, which only depend on compressive strength. The parabola-rectangle diagram specified in EN 1992-1-1 Cl. 3.1.7 (1) (Fig. 4.1.1a) is used by default in 3D CSFM, but designers can also choose a more simplified elastic ideal plastic relationship according to EN 1992-1-1 Cl. 3.1.7 (2) (Fig. 4.1.1b). The tensile strength is neglected, as it is in classic reinforced concrete design.

$$\textsf{\textit{\footnotesize{Fig 4.1.1\qquad The stress-strain diagrams of concrete for ULS: a) parabola-rectangle diagram; b) bilinear diagram}}}$$

The implementation of 3D CSFM in IDEA StatiCa Detail does not consider an explicit failure criterion in terms of strains for concrete in compression (i.e., after the peak stress is reached, it considers a plastic branch with ε_cu2 (ε_cu3) in a value of 5% while EN 1992-1-1 assumes ultimate strain less than 0.35%). This simplification does not allow the deformation capacity of structures failing in compression to be verified. However, their ultimate capacity f_cd according to EN 1992-1-1 3.1.3 is properly predicted when the increase in the brittleness of concrete as its strength rises is considered by means of the $\eta_{fc}$ reduction factor defined in fib Model Code 2010 as follows:

$$f_{cd}={\alpha_{cc}} \cdot \frac{f_{ck,red}}{γ_c} = {\alpha_{cc}} \cdot \frac{\eta _{fc} \cdot f_{ck}}{γ_c}$$

$${\eta _{fc}} = {\left( {\frac{{30}}{{{f_{ck}}}}} \right)^{\frac{1}{3}}} \le 1$$

where:

α_cc is the coefficient taking account of long-term effects on the compressive strength and of unfavorable effects resulting from the way the load is applied. It is according to EN 1992-1-1 Cl. 3.1.6 (1). The default value is 1.0.

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

Reinforcement

By default, the idealized bilinear stress-strain diagram for the bare reinforcing bars defined in EN 1992-1-1, section 3.2.7 (Fig. 4.1.2) is considered. The definition of this diagram only requires the basic properties of the reinforcement to be known during the design phase (strength and ductility class). Whenever known, the actual stress-strain relationship of the reinforcement (hot-rolled, cold-worked, quenched, and self-tempered, …) can be considered. The reinforcement stress-strain diagram can be defined by the user, but in this case, it is impossible to assume the tension stiffening effect (it is impossible to calculate crack width). Using the stress-strain diagram with a horizontal top branch does not allow for the verification of structural durability. Therefore, manual verification of standard ductility requirements is necessary.

$$\textsf{\textit{\footnotesize{Fig. 4.1.2 \qquad Stress-strain diagram of reinforcement: a) bilinear diagram with an inclined top branch; b) bilinear diagram}}}$$ $$\textsf{\textit{\footnotesize{with a horizontal top branch.}}}$$

Tension stiffening (Fig. 4.1.3) is accounted for automatically by modifying the input stress-strain relationship of the bare reinforcing bar in order to capture the average stiffness of the bars embedded in the concrete (ε_m).

$$\textsf{\textit{\footnotesize{Fig. 4.1.3\qquad Scheme of tension stiffening.}}}$$

The Compatible Stress Field Method is compliant with modern design codes. As the calculation models only use standard material properties, the partial safety factor format prescribed in the design codes can be applied without any adaptation. In this way, the input loads are factored, and the characteristic material properties are reduced using the respective safety coefficients prescribed in design codes, exactly as in conventional concrete analysis. Values of material safety factors prescribed in EN 1992-1-1 chap. 2.4.2.4 and factors for anchors prescribed in EN 1992-4, EN 1993-1-8, and EN 1994-1-1 are set by default, but the user can change safety factors in the Code and calculation settings (Fig. 4.2.1).

$$\textsf{\textit{\footnotesize{Fig. 4.2.1\qquad The setting of  material safety factors in Idea StatiCa Detail.}}}$$

Load safety factors have to be defined by the user in Combination rules for each non-linear combination of load cases (Fig. 4.2.2). For all templates implemented in Idea StatiCa Detail, partial safety factors are already predefined.

$$\textsf{\textit{\footnotesize{Fig. 4.2.2\qquad The setting of  load partial factors in Idea StatiCa Detail.}}}$$

By using appropriate user-defined combinations of partial safety factors, users can also compute with 3D CSFM using the global resistance factor method (Navrátil, et al. 2017), but this approach is hardly ever used in design practice. Some guidelines recommend using the global resistance factor method for non-linear analysis. However, in simplified non-linear analyses (such as 3D CSFM), which only require those material properties that are used in conventional hand calculations, it is still more desirable to use the partial safety format.

The different verifications required by EN 1992-1-1 are assessed based on the direct results provided by the model. ULS verifications are carried out for concrete strength, reinforcement strength, and anchorage (bond shear stresses).

Strength - Concrete

The concrete strength in compression is evaluated as the ratio between the maximum Equivalent principal stress σ_c,eq obtained from FE analysis and the limit value σ_c,lim = f_cd.

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 1992-1-1 Cl. 3.1.7 (1).

This expression is derived from the implementation of the Mohr-Coulomb plasticity theory, conservatively assuming the angle of internal friction φ = 0°.

Strength - Reinforcement

The strength of the reinforcement is evaluated in both tension and compression as the ratio between the stress in the reinforcement at the cracks σ_sr and the specified limit value σ_s,lim:

$σ_{s,lim} = \dfrac{k \cdot f_{yk}}{γ_s}\qquad\qquad\textsf{\small{for bilinear diagram with inclined top branch}}$

$σ_{s,lim} = \dfrac{f_{yk}}{γ_s}\qquad\qquad\,\,\,\,\textsf{\small{for bilinear diagram with horizontal top branch}}$

where:

f_yk        is the yield strength of the reinforcement according to EN 1992-1-1 Cl. 3.2.3,

k          is the ratio of tensile strength f_tk to the yield stress,
            $k = \dfrac{f_{tk}}{f_{yk}}$

γ_s             is the partial safety factor for reinforcement.

Strength - Anchors

Anchors are checked for normal stresses in a similar way to reinforcement, where the limit value σ_s,lim is determined.

In addition, the N_Ed and V_Ed values are specified for anchors, which are checked against N_Rd,s and V_Rd,s according to the selected code. The code is chosen depending on the type of anchor used in Project settings.

$$\textsf{\textit{\footnotesize{Fig. 4.3.1\qquad EN 1992-1-1 Figure 8.2 - Anchor check - Design code selection}}}$$

Since different approaches are chosen for checking anchors in different standards, the user can choose the following standards for individual anchor types:

• Anchors made of bolt material in tension and/or shear - EN 1992-4, EN 1993-1-8
• Headed studs in tension and/or shear - EN 1992-4, EN 1994-1-1
• Anchors made in tension and/or shear - EN 1992-4, EN 1992-1-1
• Anchors in compression and/or bending - EN 1993-1-1

Tension check according to EN 1992-4 - 7.2.1.3

$$N_{Rd,s} = \frac{c \cdot A_s \cdot f_{uk}}{\gamma_{Ms}}$$

where:

• c – reduction for cut threads  
• f_uk – minimum tensile strength of the bolt
• A_s – anchor bolt tensile stress area (reduced by the thread in the case of bolt material)
• $\gamma_{Ms} = 1.2 \cdot \dfrac{f_{uk}}{f_{yk}} \ge 1.4$ – partial safety factor for stee 
• f_yk – minimum yield strength of the bolt

Tension check according to EN 1993-1-8 - 3.6.1

$$N_{Rd,s} = F_{t.Rd} = \frac{c \cdot k_2 \cdot f_{ub} \cdot A_s}{\gamma_{M2}}$$ 

where:

• c – decrease in tensile resistance of bolts with cut thread according to EN 1993-1-8 – Cl. 3.6.1. (3) 
• k₂ = 0.9 – factor for non-countersunk anchors 
• f_ub – anchor bolt ultimate tensile strength 
• A_s – anchor bolt tensile stress area (reduced by the thread in the case of bolt material)
• γ_M2 =1.25 – partial safety factor for bolts (EN 1993-1-8, Table 2.1) 

Tension check according to EN 1992-1-1 - 3.2.7

$$N_{Rd,s} = \frac{kf_{yk}}{\gamma_{S}}$$

where:

• $k=(f_{t}/f_{y})$ is given in Annex C
• f_yk - characteristic yield strength
• γ_M2 =1.15 – partial safety factor for reinforcement

Compression check according to EN 1993-1-1 - 6.3

Used for all anchors subjected to a normal compression force, regardless of their material or stand-off type. 

$$F_{c,Rd}=\frac{\chi\,A_s f_y}{\gamma_{M2}}$$

Where:

• $\chi=\dfrac{1}{\Phi+\sqrt{\Phi^2-\bar{\lambda}^2}}\le 1$ – buckling reduction factor
• $\Phi=0.5\left[1+\alpha\left(\bar{\lambda}-0.2\right)+\bar{\lambda}^2\right]$ – value to determine buckling reduction factor χ
• $\alpha=0.49$ – imperfection factor for buckling curve c (belonging to the full circle)
• $\bar{\lambda}=\sqrt{\dfrac{A_s f_y}{N_{cr}}}$ – relative slenderness

  • A_s – the anchor area reduced by threads
• $N_{cr}=\dfrac{\pi^2 E I}{L_{cr}^2}$  – Euler's critical force
• $I=\dfrac{\pi d_s^4}{64}$ – moment of inertia of the bolt

  • d_s – anchor diameter reduced by threads
• $L_{cr}=2\,l$  – buckling length; it is assumed on the safe side that the bolt is fixed in the concrete and able to rotate at the base plate freely 
• $l=l_{a}$ – length of the bolt element equal to half the base plate thickness + gap + half the bolt diameter; it is assumed on the safe side that the washer and a nut are not clamped to the concrete surface (ETAG 001 – Annex C – Cl. 4.2.2.4), see Figure 4.3.2.

Shear check according to EN 1992-4 - 7.2.2.3

For stand-off = direct, the shear without lever arm is assumed (EN 1992-4 – Cl. 7.2.2.3.1):

$$V_{Rd,s} = \frac{k_6 \cdot A_s \cdot f_{uk}}{\gamma_{Ms}}$$

For stand-off = mortar joint, the shear with lever arm is assumed (EN 1992-4 – Cl. 7.2.2.3.2):

$$V_{Rd,s} = \frac{\alpha_M \cdot M_{Rk,s}}{\gamma_{Ms} \cdot l_a}$$

where:

• k₆ = 0.6 for anchors with fuk ≤ 500 MPa; k₆ = 0.5 otherwise
• A_s – shear area of anchor reduced by threads
• f_uk – anchor bolt ultimate strength
• α_M = 2 – full restraint is assumed (EN 1992-4 – Cl. 6.2.2.3)
• $M_{Rk,s} = M^{0}_{Rk,s} \left(1 - \dfrac{N_{Ed}}{N_{Rd,s}} \right)$ – characteristic bending resistance of the anchor decreased by the tensile force in the anchor
•  $M^{0}_{Rk,s} = 1.2 \cdot W_{el} \cdot f_{ub}$ – characteristic bending resistance of the anchor (ETAG 001, Annex C – Equation (5.5b))
• $W_{el} = \dfrac{\pi d^{3}}{32}$ – section modulus of the anchor
• d – anchor bolt diameter; if the shear plane in a thread is selected (which always is for threaded rod), the diameter reduced by threads is used; otherwise, nominal diameter, d_nom, is used
• N_Ed – tensile force in the anchor
• N_Rd,s – tensile resistance of the anchor
• $l_{a} = 0.5\, d_{\mathrm{nom}} + t_{\mathrm{mortar}} + 0.5\, t_{\mathrm{bp}}$ – lever arm
• t_mortar – thickness of mortar (grout)
• t_bp – thickness of the base plate
• $\gamma_{Ms} = 1.0 \cdot \dfrac{f_{uk}}{f_{yk}} \ge 1.25$ for $f_{uk} \le 800 \text{ MPa}$ and  $\dfrac{f_{yk}}{f_{uk}} \le 0.8$; γ_Ms = 1.5 otherwise – partial safety factor for steel failure (EN 1992-4 – Table 4.1)

Shear check according to EN 1993-1-8 - 6.2.2

Anchor shear steel resistance is determined according to EN 1993-1-8 – 6.2.2 (7) regardless of direct or mortar joint stand-off. The grout strength and thickness should be according to Cl. 6.2.5 (7).

$$V_{Rd,s} = F_{v,b,Rd} = \min \left\{ F_{1v,b,Rd} ,\, F_{2v,b,Rd} \right\}$$

where:

$$F_{1v,b,Rd} = \frac{\alpha_v \cdot f_{ub} \cdot A}{\gamma_{M2}}$$

• α_v = 0.6 for grades 4.6, 5.6, 8.8, and 0.5 for grades 4.8, 5.8, 6.8, 10.9
• f_ub – ultimate tensile strength of the bolt material
• A – tensile stress area of the bolt, A = A_s, where As is the tensile stress area of the bolt (reduced by the thread)
• γ_M2 – safety factor - EN 1993-1-8 – Table 2.1

$$F_{2v,b,Rd} = \frac{\alpha_b \cdot f_{ub} \cdot A_s}{\gamma_{M2}}$$

• ​ $\alpha_b = 0.44 - 0.0003\, f_{yb}$
• α_b is a coefficient depending on the yield strength of the anchor bolt
• f_yb – anchor yield strength; 235 MPa ≤ f_yb ​≤ 640 MPa
• f_ub – anchor tensile strength 
• A_s – tensile stress area (reduced by the thread)

Shear check according to EN 1993-1-1 - 6.2.6

These code checks are applied to anchors that are connected to the base plate with a gap or a directly loaded anchor with a projected length more than 0.5 times their diameter.

$$V_{pl,Rd}=\frac{A_v f_y/\sqrt{3}}{\gamma_{M2}}$$

where:

• A_V = 0.844 A_s – shear area
• A_s – bolt area reduced by threads
• f_y – bolt yield strength
• γ_M2 – partial safety factor (defined in the Project settings)

Shear check according to EN 1994-1-1 - 6.6.3.1

$$V_{Rd,s} = P_{Rd} = \frac{0.8 \, f_u \, \pi \, d^2}{4 \, \gamma_v}$$

where:

• γ_v is the partial factor for shear connection per EN 1994-1-1 chap. 2.4.1.2. The recommended value for γ_v is 1.25
• d is the diameter of the shank of the stud, 16 mm ≤ d ≤ 25 mm;
• f_u is the specified ultimate tensile strength of the material of the stud, but not greater than 500 MPa.

In EN 1994-1-1, clause 6.6.3.1 also provides Equation (6.19), which limits the shear resistance of a stud by the punching (bearing) capacity of the concrete. In IDEA StatiCa Detail, this failure mode is not checked by a separate code formula in the post-processing. Instead, it is built directly into the nonlinear finite element analysis as a stop criterion: the analysis is terminated before the shear force in an anchor reaches the corresponding P_Rd
from Equation (6.19). This approach is used because Equation (6.19) is valid only for headed studs welded to the steel plate and for stud diameters in the range 16 mm ≤ d ≤ 25 mm, as specified in 6.6.3.1.

To cover a wider range of practical cases, we created a series of 3D reference models in Abaqus with anchor diameters from 8 mm to 50 mm and concrete strengths from C16/20 to C50/60. The studs were modeled either welded rigidly to the base plate or connected by a pinned (hinged) joint. The material models and contact parameters in Detail were then calibrated against these Abaqus simulations, which were themselves verified against Equation (6.19) within its validity range. This stop criterion is valid for all anchor types and all EN codes.

Bending check according to EN 1993-1-1 - 6.2.5

$$M_{pl,Rd}=\frac{W_{pl} f_y}{\gamma_{M2}}$$

Where:

• $W_{pl}=\dfrac{d_s^{3}}{6}$ – section modulus of the bolt

  • d_s – anchor diameter reduced by threads
• f_y – material yield strength
• γ_M2 – partial safety factor (defined in the Project settings)

Acting design bending moment M_Ed - If the shear load acts with a lever arm, a bending moment acting on the fastener shall be accounted for. The design bending moment acting on the fastener is calculated according to EN 1993-1-1 Formula (6.1):

$$M_{Ed}=V_{Ed}\cdot\frac{l_a}{\alpha_M}$$

where:

• V_Ed - is the shear load acting on the fastener under consideration
• l_a = a₃ + e₁

  • a₃ = 0.5d_nom, where d_nom is the anchor diameter
  • e₁ - is the distance between the shear load and the concrete surface, neglecting the thickness of any levelling grout
• α_M = 2 – full restraint is assumed (EN 1992-4 – Cl. 6.2.2.3)

$$\textsf{\textit{\footnotesize{Fig. 4.3.2\qquad Buckling length}}}$$

Interaction of tension and shear in anchor steel

The interaction of tension and shear per EN 1993-1-8 is implicitly included in the anchor shear check.

The interaction of tension and shear per EN 1992-4 is determined separately for steel and concrete failure modes according to Table 7.3. The interaction in steel is checked for each anchor separately.

$$\left( \frac{N_{Ed}}{N_{Rd,s}} \right)^{2}+\left( \frac{V_{Ed}}{V_{Rd,s}} \right)^{2}\le 1$$

EN 1994-1-1 states in Article 6.6.3.2 that if the anchor tensile force is greater than 0.1P_Rd, the check is not covered by this standard. In such a case, the interaction is assessed in accordance with EN 1992-4 in the application. In such a case, the shear check should not be considered according to EN 1994-1-1.

Interaction of tension or compression and bending in anchor steel EN 1993-1-1 - 6.2.1

$$\frac{N_{Ed}}{N_{Rd}}+\frac{M_{Ed}}{M_{Rd}}\le 1$$

where:

• N_Ed – tensile (positive) or compressive (negative sign) design force
• N_Rd – tensile (positive, F_t,Rd) or compressive (negative sign, F_c,Rd) design resistance
• M_Ed – design bending moment
• M_Rd = M_pl,Rd – design bending resistance

Pull-out check for headed anchors (Washer plates and Headed studs)

For headed anchors, an additional stop criterion is implemented to check the concrete bearing (crushing) above the anchor head - pull-out. During the analysis, the compressive force transferred through the head-to-concrete contact is monitored and compared with the limit value given by EN 1992-4, Clause 7.2.1.5 (pull-out failure of headed fastenings).

$$N_{Rd,p} = k_2 \cdot A_h \cdot f_{ck} / \gamma_{Mp}$$

where:

• A_h is the load bearing area of the head of the fastener (without the shank area). 
• f_ck is the characteristic compressive strength of concrete - EN 1992-1-1 Cl. 3.1.2
• γ_Mp is taken in the application as γ_Mp = γ_c with the default value of 1.5
•  k₂​ is always taken as 7.5, i.e. the value for cracked concrete. This is consistent with the CSFM approach used in Detail, where the tensile strength of concrete is neglected and the concrete is assumed to be cracked in tension. 

Once the contact force reaches this code-based limit, the stop criterion is triggered and the analysis is terminated before the design pull-out resistance is exceeded.

Anchorage -  Bond stress

The bond shear stress is evaluated independently as the ratio between the bond stress τ_b calculated by FE analysis and the ultimate bond strength f_bd, according to EN 1992-1-1 chap. 8.4.2:

$$\frac{τ_{b}}{f_{bd}}\le 1$$

$$f_{bd} = 2.25 \cdot η_1\cdot η_2\cdot f_{ctd}$$

where:

• f_ctd      is the design value of concrete tensile strength according to EN 1992-1-1 Cl. 3.1.6 (2). Due to the increasing brittleness of higher-strength concrete, f_ctk,0.05 is limited to the value for C60/75 according to EN 1992-1-1 Cl. 8.4.2 (2)
• η₁       is a coefficient related to the quality of the bond condition and the position of the bar during concreting (Fig. 4.3.3).
• η₁ = 1.0 when ‘good’ conditions are obtained and
• η₁ = 0.7 for all other cases and for bars in structural elements built with slip-forms, unless it can be shown that ‘good’ bond conditions exist
• η₂        is related to the bar diameter:
  η₂ = 1.0 for Ø ≤ 32 mm
  η₂ = (132 - Ø)/100 for Ø > 32 mm

$$\textsf{\textit{\footnotesize{Fig. 4.3.3\qquad EN 1992-1-1 Figure 8.2 - Description of bond conditions.}}}$$

In IDEA StatiCa Detail, the bond conditions are taken into account according to Fig. 4.3.3 c) and d). The direction of concreting can be set in the application for each project item as follows:

$$\textsf{\textit{\footnotesize{Fig. 4.3.4\qquad Direction of concreting}}}$$

These verifications are carried out with respect to the appropriate limit values for the respective parts of the structure (i.e., in spite of having a single grade both for concrete and reinforcement material, the final stress-strain diagrams will differ in each part of the structure due to tension stiffening and compression softening effects).

Anchorage - Total force

Total force F_tot and Limit force F_lim

The total force F_tot is a result of the finite element analysis and can be defined in two ways.

$$F_{tot}=A_{s}\cdot \sigma_{s}$$

where A_s is the area of the reinforcement bar and σ_s is the stress in the bar.

Or as a sum of the anchorage force F_a and the bond force F_bond.

$$F_{tot}=F_{a}+F_{bond}$$

where F_a is the actual force in the anchorage spring and F_bond is the bond force that can be obtained by integrating the bond stress τ_b along the length of reinforcement bar l.

$$F_{bond}=C_{s} \cdot \int_{0}^{l}\tau_{b}\left( x \right)dx$$

C_s is the circumference of the reinforcement bar.

The limit force F_lim is the maximum force in the element of the rebar considering the ultimate strength of the rebar and also anchoring conditions (bond between concrete and reinforcement and anchorage hooks, loops, etc.).

$$F_{lim}=min\left( F_{lim,bond}+F_{au},F_{u} \right)$$

$$F_{u}=k\cdot f_{yd}\cdot A_{s}$$

$$F_{au}=\beta\cdot k\cdot f_{yd}\cdot A_{s}$$

$$F_{lim,bond}=C_{s}\cdot l \cdot f_{bd}$$

where C_s is the circumference of the reinforcement bar, and l is the length from the beginning of the rebar to the point of interest.

$$\textsf{\textit{\footnotesize{Fig. 4.3.5\qquad Definition of the limit force Flim}}}$$

$$F_{lim,2}=F_{lim,1}+F_{lim,add}$$

where F_lim,add is the additional force calculated from the magnitude of the angle between neighboring elements. F_lim,2 must always be lower than F_u.

Anchorage types at the end of Reinforcement (Anchors and Rebars)

The available anchorage types in 3D CSFM include a straight bar (i.e., no anchor end reduction), bend, hook, loop, welded transverse bar, perfect bond, and continuous bar. All these types, along with the respective anchorage coefficients β, are shown in Fig. 4.3.6 for longitudinal reinforcement and in Fig. 4.3.7 for stirrups. The values of the adopted anchorage coefficients are in accordance with EN 1992-1-1 section 8.4.4 Tab. 8.2. It should be noted that in spite of the different available options, 3D CSFM distinguishes three types of anchorage ends: (i) no reduction in the anchorage length, (ii) a reduction of 30% of the anchorage length in the case of a normalized anchorage, and (iii) perfect bond.

$$\textsf{\textit{\footnotesize{Fig. 4.3.6\qquad  Available anchorage types and respective anchorage coefficients for longitudinal reinforcing bars in the 3D CSFM:}}}$$

$$\textsf{\textit{\footnotesize{(a) straight bar; (b) bend; (c) hook; (d) loop; (e) welded transverse bar; (f) perfect bond; (g) continuous bar.}}}$$

$$\textsf{\textit{\footnotesize{Fig. 4.3.7\qquad  Available anchorage types and respective anchorage coefficients for stirrups.}}}$$

$$\textsf{\textit{\footnotesize{Closed stirrups: (a) hook; (b) bend; (c) overlap. Open stirrups: (d) hook; (e) continuous bar.}}}$$

In order to comply with EN 1992-1-1, the anchorage spring should be used in the calculation, the anchorage spring is modified by the β coefficient, so the user must use one of the available anchorage types when defining the reinforcement start and end conditions. 

Concrete - Strength

The concrete model implemented for strength calculations in the CSFM is based on the parabolic-plastic stress-strain curve for concrete based on the Portland CementAssociation’s parabolic stress-strain curve described in PCA’s Notes on ACI 318-99 Building Code Requirements for Structural Concrete, Figure 6-8. The tensile strength is neglected, as it is in classic reinforced concrete design.

$$\textsf{\textit{\footnotesize{Fig. 5.1.1\qquad The stress-strain diagram of concrete for Strength analysis}}}$$

The implementation of the CSFM in IDEA StatiCa Detail does not consider an explicit failure criterion in terms of strains for concrete in compression (i.e., after the peak stress is reached it considers a plastic branch with ε_c0 in maximum value 5% while ACI 318-19 Cl. 22.2.2.1 assumes ultimate strain less than 0.3%). This simplification does not allow the deformation capacity of structures failing in compression to be verified. However, the strength is properly predicted when the increase in the brittleness of concrete as its strength rises is considered by means of the $\eta_{fc}$ reduction factor defined in fib Model Code 2010 as follows:

$$f'_{c,lim}=\alpha_{1}\cdot\phi_{c}\cdot \eta _{fc}\cdot f'_{c}$$

$${\eta _{fc}} = {\left( {\frac{{30}}{{{f'_{c}}}}} \right)^{\frac{1}{3}}} \le 1$$

where:

α₁ is the Reduction factor of concrete compressive strength defined in ACI 318-19 Cl. 22.2.2.4.1. When using a parabola-rectangle stress-strain diagram, it is necessary to reduce the maximum compressive stress by this factor. This averages the stress distribution in the compression zone in such a way that the resulting compressive strength is less than or equal to the compressive strength calculated using a stress-strain diagram with a decreasing plastic branch. 

Φ_c is the strength reduction factor for concrete. The default value is set according to ACI 318-19 Table 24.2.1 (b)(f).

f'_c is the concrete cylinder strength (in MPa for the definition of $\eta_{fc}$).

Reinforcement

A perfectly elasto-plastic stress-strain diagram with a defined yield point for the non-prestresses reinforcement is considered. See ACI 319-19 Cl. 20.2.1. The definition of this diagram only requires the basic properties of the reinforcement to be known - strength and modulus of elasticity.

The reinforcement stress-strain diagram can be also defined by the user, but in this case, it is impossible to assume the tension stiffening effect. 

$$\textsf{\textit{\footnotesize{Fig. 5.1.2 \qquad Stress-strain diagram of reinforcement}}}$$

where:

Φ_s is the strength reduction factor for reinforcement. Where the default value is set according to ACI 318-19 Table 24.2.1.

f_y is the yield strength of reinforcement

E_s modulus of elasticity of reinforcement

10% is selected as the limit strain at which the calculation is stopped. This is considered safe based on ASTM A955/A955M-20c Article 7.

Tension stiffening (Fig. 5.1.3)  is accounted for automatically by modifying the input stress-strain relationship of the bare reinforcing bar in order to capture the average stiffness of the bars embedded in the concrete (ε_m).

$$\textsf{\textit{\footnotesize{Fig. 5.1.3\qquad Scheme of tension stiffening.}}}$$

The Compatible Stress Field Method is compliant with modern design codes. As the calculation models only use standard material properties, the partial safety factor format prescribed in the design codes can be applied without any adaptation. In this way, the input loads are factored, and the characteristic material properties are reduced using the respective strength reduction factors, exactly as in conventional concrete analysis.

Values of strength reduction factors are prescribed in ACI 318-19 chapter 21 and for anchors in ACI 318-19 chapter 17 and AISC 360-16 chapter D, E, F, G. 

$$\textsf{\textit{\footnotesize{Fig. 5.2.1\qquad The setting of strength reduction factors in IDEA StatiCa Detail.}}}$$

Load factors for Strength combinations shall be defined according to ACI 318-19 Table 5.3.1.

Except as stated in Chapter 34, service-level load combinations are not defined in ACI 318-19. It is recommended to use combination rules based on Appendix C of ASCE/SEI 7-16. For all templates, load factors are already predefined.

$$\textsf{\textit{\footnotesize{Fig. 5.2.3\qquad The setting of load factors in IDEA StatiCa Detail.}}}$$

The different verifications required by ACI 318-19 are assessed based on the direct results provided by the model. Verifications are carried out for concrete strength, reinforcement strength, and anchorage (bond shear stresses).

Strength - Concrete

The concrete strength in compression is evaluated as the ratio between the maximum Equivalent principal stress f_c,eq (also σ_c,eq in previous text) obtained from FE analysis and the limit value f'_c,lim.

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

$$f_{c,eq} = \sigma_{c3} - \sigma_{c1}$$

The f_c,eq value can, therefore, be directly compared with uniaxial strength limits. This expression is derived from the implementation of the Mohr-Coulomb plasticity theory, conservatively assuming the angle of internal friction φ = 0°.

Strength - Reinforcement

The strength of the reinforcement is evaluated in both tension and compression as the ratio between the stress in the reinforcement at the cracks f_s and the specified limit value f_y,lim.

$$f_{y,lim} = \phi_{s} \cdot f_{y}$$

Strength - Anchors

Anchors are checked for normal stresses in a similar way to reinforcement, where the limit value f_y,lim is determined. 

To make it easier to navigate the following text, we will first divide anchoring into three groups in terms of code-checking according to ACI or AISC.

Group 1

• Anchorage Types

  • Cast-in plate
  • Base plate - Stand-off = direct 
  • Base plate - Stand-off = Mortar joint - thickness of mortar less than 0.5 times the anchor diameter
  • Single anchor with projected length less than 0.5 times anchor diameter
• Anchor code-checks (ACI / AISC)

  • Tension/compression

    • All anchor types in tension – ACI 318-19 chap. 17.6.1.2  
    • All anchor types in compression – AISC 360-16 chap. E
  • Shear without lever arm

    • Bolt material – ACI 318-19 chap. 17.7.1.2 (b)
    • Headed studs – ACI 318-19 chap. 17.7.1.2 (a)
    • Reinforcement – ACI 318-19 chap. 17.7.1.2 (b)
  • Interaction of tension and shear - ACI 318-19 chap. 17.8

Group 2

• Anchorage Types

  • Base plate - Stand-off = Mortar joint - thickness of mortar more than 0.5 times the anchor diameter
• Anchor code-checks (ACI / AISC)

  • Tension/compression

    • All anchor types in tension – ACI 318-19 chap. 17.6.1.2  
    • All anchor types in compression – AISC 360-16 chap. E
  • Shear with lever arm

    • Bolt material – ACI 318-19 chap. 17.7.1.2 (b) + chap. 17.7.1.2.1.
    • Headed studs – ACI 318-19 chap. 17.7.1.2 (a) + chap. 17.7.1.2.1.
    • Reinforcement – ACI 318-19 chap. 17.7.1.2 (b) + chap. 17.7.1.2.1.
  • Interaction of tension and shear - ACI 318-19 chap. 17.8

Group 3

• Anchorage types

  • Base plate - Stand-off = gap
  • Single anchor with projected length more than 0.5 times the anchor diameter
• Anchor code-checks (ACI / AISC)

  • Tension/compression (with buckling)

    • All anchor types in tension – ACI 318-19 chap. 17.6.1.2
    • All anchor types in compression – AISC 360-16 chap. E3
  • Bending

    • For all anchor types – AISC 360-16 chap. F11
  • Shear

    • For all anchor types – AISC 360-16 chap. G
  • Interaction of axial force and bending

    • $\dfrac{N}{P_n}+\dfrac{M}{M_n}\le 1$ 

Tensile resistance of anchor according to ACI 318-19 chap. 17.6.1.2

$$\phi N_{sa}=\phi_{a,t}\,A_{se,N}\,f_{uta}$$

where:

• ϕ_a,t  – strength reduction factor for anchors in tension according to ACI 318-19 chap. 17.5.3 (a)
• A_se,N – tensile stress area (reduced by thread)
• f_uta – specified tensile strength of anchor steel and shall not be greater than 1.9 f_ya and 860 MPa

Shear resistance of anchor according to ACI 318-19 chap. 17.7.1.2 (a)

The steel strength in shear for headed studs is determined as:

$$\phi V_{sa}=\phi_{a,V}\,A_{se,V}\,f_{uta}$$

where:
ϕ_a,v – strength reduction factor for anchors in tension according to ACI 318-19 chap. 17.5.3 (a)
A_se,V – tensile stress area (reduced by thread)
f_uta – specified tensile strength of anchor steel and shall not be greater than 1.9 f_ya and 860 MPa

Shear resistance of anchor according to ACI 318-19 chap. 17.7.1.2 (b)

The steel strength in shear for anchors from bolt material and reinforcement is determined as:

$$\phi V_{sa}=\phi_{a,V}\,0.6\,A_{se,V}\,f_{uta}$$

where:

• ϕ_a,v  – strength reduction factor for anchors in tension according to ACI 318-19 chap. 17.5.3 (a)
• A_se,V – tensile stress area (reduced by thread)
• f_uta – specified tensile strength of anchor steel and shall not be greater than 1.9 f_ya and 860 MPa

Shear resistance of anchor connected to a base with mortar - ACI 318-19 chap. 17.7.1.2.1

If anchors are used with built-up grout pads (Group 2), the design strength calculated in accordance with 17.7.1.2 shall be multiplied by 0.80.

Interaction in tension and shear according to ACI 318-19 chap. 17.8

It shall be permitted to neglect the interaction between tension and shear if (a) or (b) is satisfied.
(a) N_ua/(ϕN_n) ≤ 0.2
(b) V_ua/(ϕV_n) ≤ 0.2 

If N_ua/(ϕN_n) > 0.2 for the governing strength in tension and V_ua/(ϕV_n) > 0.2 for the governing strength in shear, then Eq. (17.8.3) shall be satisfied.

$$\frac{N_{ua}}{\phi N_n}+\frac{V_{ua}}{\phi V_n}\le 1.2$$

Compressive resistance of anchor according to AISC 360-16 chap. E3

$$P_n =\phi_{a,c}\, F_{cr}\, A_{g}$$

where:

• ϕ_a,t  – the strength reduction factor for anchors in compression according to AISC 360-16 chap. E1
  
• (a) When: $\dfrac{L_c}{r} \le 4.71\sqrt{\dfrac{E}{F_y}}\quad$  or     $\dfrac{F_y}{F_e}\le 2.25$

  • $F_{cr}=\left(0.658^{\,F_y/F_e}\right)F_y$
    
• (b) When: $\dfrac{L_c}{r} > 4.71\sqrt{\dfrac{E}{F_y}}\quad$  or     $\dfrac{F_y}{F_e}> 2.25$

  • $F_{cr}=0.877F_e$
    
• A_g​ – gross cross-sectional area of member
• E – modulus of elasticity of steel
• $F_e=\dfrac{\pi^2 E}{\left(\dfrac{L_c}{r}\right)^2}$ - elastic buckling stress
• F_y – specified minimum yield stress of the type of steel being used
• $r=\sqrt{\dfrac{I}{A_s}}$ – radius of gyration
• $I=\dfrac{\pi d_s^4}{64}$ – moment of inertia of the bolt 

Bending resistance of anchor according to AISC 360-16 chap. F11

$$M_n=\phi_{a,b}\, Z\, F_y\, \le 1.6\,\phi_{a,b}\, S_x\, F_y$$

where:

• $Z=\dfrac{d_s^{3}}{6}$ – plastic section modulus of the bolt
• $S_x=\dfrac{2I}{d_s}$ – elastic section modulus of the bolt

Shear resistance of anchor according to AISC 360-16 chap. G

$$V_n=\phi_{a,v}\,0.6\,A_v\,F_y$$

where:

• A_V = 0.844As – the shear area
• A_s – the bolt area reduced by threads

Concrete crushing at the anchor–concrete interface

The anchor shear resistance is also limited from the point of view of concrete crushing at the anchor–concrete interface. The limit values and the method for determining them are described in detail in the article - Shear behaviour of anchors in reinforced concrete. Once the contact force reaches this limit, the stop criterion is triggered, and the analysis is terminated before the resistance is exceeded.​ 

Pull-out check for headed anchors (Washer plates and Headed studs)

For headed anchors, an additional stop criterion is implemented to check the concrete bearing (crushing) above the anchor head - pull-out. During the analysis, the compressive force transferred through the head-to-concrete contact is monitored and compared with the limit value given by ACI 318-19, Clause 17.6.3.2.2a (pull-out failure of headed fastenings).

$$N_{pn} = \Phi \cdot \Psi_{c,p} \cdot 8 \cdot A_{brg} \cdot f'_c$$
​

where:

• $\Phi$ is the strength reduction factor - Table 17.5.3(c)
• A_brg net bearing area of the head of stud, anchor bolt, or headed deformed bar (without the shank area). 
• f'_c is the specified compressive strength of concrete
• $\Psi_{c,p}$ is the pullout cracking factor according to 17.6.3.3, and is always taken as 1.0, i.e. the value for cracked concrete. This is consistent with the CSFM approach used in Detail, where the tensile strength of concrete is neglected and the concrete is assumed to be cracked in tension.

Once the contact force reaches this code-based limit, the stop criterion is triggered, and the analysis is terminated before the pull-out resistance is exceeded.​ 

Anchorage -  Bond stress

The bond shear stress is evaluated independently as the ratio between the bond stress τ_b calculated by FE analysis and the bond strength f_bu.

Although the bond strength is not explicitly defined in ACI 318-19, the calculation of the development length can be found in Section 25.4.2. However, since the bond strength is the basic input for determining the development length, see R25.4.1.1 and ACI Committee 408 1966, the bond strength can be calculated as follows:

Let us assume that if we anchor the reinforcement bar into a concrete block to the development length l_d or greater, pulling out the reinforcement will lead to rupture of the reinforcement and not to pulling out of the concrete. This can be written with the following formula.

$$\pi\cdot d_{b} \cdot l_{d} \cdot f_{bu}=f_{y}\cdot A_{s}$$

where:

d_b is the diameter of the reinforcement bar, l_d is the development length, f_bu is the bond strength, f_y is the yield strength of the reinforcement, and A_s is the area of the reinforcement rebar.

From the preceding, the formula for calculating bond strength can be easily derived:

$$f_{bu}=\frac{f_{y}\cdot A_{s}}{\pi\cdot d_{b} \cdot l_{d} }$$

The development length l_d is then determined according to ACI 318-19 Table 25.4.2.3 as follows:

$$l_{d}=\left( \frac{f_{y}\cdot\psi_{t}\cdot\psi_{e}\cdot\psi_{g}}{C\cdot\lambda\sqrt{f'_{c}}} \right)\cdot d_{b}$$

where:

C = 25 (2.1 for metric) for no. 6 and smaller bars and deformed wires, C = 20 (1.7 for metric) for no. 7 and larger bars, λ = 1.0 for normal weight concrete, ψ_t, ψ_e, ψ_g are determined according to ACI 318-19 Table 25.4.2.3. 

Only uncoated or zinc-coated (galvanized) reinforcement is supported, so ψ_e = 1.0. ψ_g is automatically determined from the reinforcement grade, and ψ_t is automatically derived from the position of the reinforcement in the model and from the direction of concreting that can be set in the application for each project item as follows.

$$\textsf{\textit{\footnotesize{Fig. 5.3.1\qquad Direction of concreting}}}$$

These verifications are carried out with respect to the appropriate limit values for the respective parts of the structure (i.e., in spite of having a single grade both for concrete and reinforcement material, the final stress-strain diagrams will differ in each part of the structure due to tension stiffening and compression softening effects).

Anchorage -  Total force

Total force F_tot and limit force F_lim

The total force F_tot is a result of the finite element analysis and can be defined in two ways.

$$F_{tot}=A_{s} \cdot f_{s}$$

where A_s is the area of the reinforcement bar and f_s is the stress in the bar.

Or as a sum of the anchorage force F_a and the bond force F_bond.

$$F_{tot}=F_{a}+F_{bond}$$

where F_a is the actual force in the anchorage spring and F_bond is the bond force that can be obtained by integrating the bond stress τ_b along the length of reinforcement bar l.

$$F_{bond}=C_{s} \cdot \int_{0}^{l}\tau_{b}\left( x \right)dx$$

C_s is the circumference of the reinforcement bar.

The limit force F_lim is the maximum force in the element of the rebar considering the strength of the rebar and also anchoring conditions (bond between concrete and reinforcement and anchorage hooks, loops, etc.).

$$F_{lim}=min\left( F_{lim,bond}+F_{au},F_{u} \right)$$

$$F_{u}=f_{y,lim}\cdot A_{s}$$

$$F_{au}=\beta\cdot f_{y,lim}\cdot A_{s}$$

$$F_{lim,bond}=C_{s}\cdot l \cdot f_{bu}$$

where C_s is the circumference of the reinforcement bar, and l is the length from the beginning of the rebar to the point of interest.

$$\textsf{\textit{\footnotesize{Fig. 5.3.2\qquad Definition of the limit force Flim}}}$$

$$F_{lim,2}=F_{lim,1}+F_{lim,add}$$

where F_lim,add is the additional force calculated from the magnitude of the angle between neighboring elements. F_lim,2 must be always lower than F_u.

The available anchorage types in CSFM include a straight bar (i.e., no anchor end reduction), 90-degree hook, 180-degree hook, perfect bond, and continuous bar. All these types, along with the respective anchorage coefficients β, are shown in Fig. 5.3.3 for longitudinal reinforcement. The values of the adopted anchorage coefficients are derived from the comparison of the equation from section ACI 318-19 25.4.3.1 and equations taken from section ACI 318-19 25.4.2.3. It should be noted that, in spite of the different available options, CSFM distinguishes three types of anchorage ends: (i) no reduction in the anchorage length, (ii) a reduction of 30% of the anchorage length in the case of a normalized anchorage, and (iii) perfect bond.

$$\textsf{\textit{\footnotesize{Fig. 5.3.3\qquad  Available anchorage types and respective anchorage coefficients for longitudinal reinforcing bars in CSFM:}}}$$

$$\textsf{\textit{\footnotesize{(a) straight bar; (b) 90-degree hook; (c) 180-degree hook; (d) perfect bond; (e) continuous bar}}}$$

The anchorage coefficient for stirrups is always - β = 1.0.

In order to comply with ACI, the anchorage spring should be used in the calculation, the anchorage spring is modified by the β coefficient so the user must use one of the available anchorage types when defining the reinforcement start and end conditions. 

Concrete - Strength

The concrete model implemented for strength calculations in 3D CSFM is based on the AASHTO LRFD strength-design assumptions of equilibrium and strain compatibility. In accordance with AASHTO LRFD (2024) Article 5.6.2.1, the tensile strength of concrete is neglected.

$$\textsf{\textit{\footnotesize{Fig. 6.1.1\qquad The stress-strain diagram of concrete for Strength analysis}}}$$

The implementation of the 3D CSFM in IDEA StatiCa Detail does not consider an explicit failure criterion in terms of strains for concrete in compression (i.e., after the peak stress is reached, it considers a plastic branch with ε_c0 in maximum value 5%, while AASHTO LRFD (2024) Article 5.6.2.1 assumes ultimate strain less than 0.3%). This simplification does not allow the deformation capacity of structures failing in compression to be verified. However, the strength is properly predicted when the increase in the brittleness of concrete as its strength rises is considered by means of the $\eta_{fc}$ reduction factor defined in fib Model Code 2010 as follows:

$$f'_{c,lim}=\alpha_{1}\cdot\phi_{c}\cdot \eta _{fc}\cdot f'_{c}$$

$${\eta _{fc}} = {\left( {\frac{{30}}{{{f'_{c}}}}} \right)^{\frac{1}{3}}} \le 1$$

where:

α₁ is the Reduction factor of concrete compressive strength defined in AASHTO LRFD (2024) Article 5.6.2.2. When using a parabola-rectangle stress-strain diagram, it is necessary to reduce the maximum compressive stress by this factor. This averages the stress distribution in the compression zone in such a way that the resulting compressive strength is less than or equal to the compressive strength calculated using a stress-strain diagram with a decreasing plastic branch.

Φ_c is the strength reduction factor for concrete. The default value is set according to AASHTO LRFD (2024) Article 5.5.4.2.

f'_c is the concrete cylinder strength (in MPa for the definition of $\eta_{fc}$).

Reinforcement

A perfectly elasto-plastic stress-strain diagram with a defined yield point for the non-prestresses reinforcement is considered. See AASHTO LRFD (2024) Article 5.4.3. The definition of this diagram only requires the basic properties of the reinforcement to be known - strength and modulus of elasticity.

The reinforcement stress-strain diagram can be also defined by the user, but in this case, it is impossible to assume the tension stiffening effect. 

$$\textsf{\textit{\footnotesize{Fig. 6.1.2 \qquad Stress-strain diagram of reinforcement}}}$$

where:

Φ_s is the strength reduction factor for reinforcement. Where the default value is set according to AASHTO LRFD (2024) Article 5.5.4.2.

f_y is the yield strength of reinforcement

E_s modulus of elasticity of reinforcement

10% is selected as the limit strain at which the calculation is stopped. This is considered safe based on ASTM A955/A955M-20c Article 7.

Tension stiffening (Fig. 6.1.3)  is accounted for automatically by modifying the input stress-strain relationship of the bare reinforcing bar in order to capture the average stiffness of the bars embedded in the concrete (ε_m).

$$\textsf{\textit{\footnotesize{Fig. 6.1.3\qquad Scheme of tension stiffening.}}}$$

The Compatible Stress Field Method is compliant with modern design codes. As the calculation models only use standard material properties, the partial safety factor format prescribed in the design codes can be applied without any adaptation. In this way, the input loads are factored, and the characteristic material properties are reduced using the respective strength reduction factors, exactly as in conventional concrete analysis.

Values of strength reduction factors are prescribed in AASHTO LRFD (2024) Article 5.5.4 and for anchors in ACI 318-19 Chapter 17 and AASHTO LRFD (2024) Article 6.5.4.2.

$$\textsf{\textit{\footnotesize{Fig. 6.2.1\qquad The setting of strength reduction factors in IDEA StatiCa Detail.}}}$$

Load factors and load combinations shall be defined according to AASHTO LRFD Bridge Design Specifications (2024), Article 3.4.1 and Tables 3.4.1-1 to 3.4.1-6. AASHTO LRFD explicitly specifies the Strength limit-state load combinations (Strength I to Strength V), including the corresponding load factors for each case.

$$\textsf{\textit{\footnotesize{Fig. 6.2.2\qquad The setting of load factors in Idea StatiCa Detail.}}}$$

The different verifications required by AASHTO are assessed based on the direct results provided by the model. Verifications are carried out for concrete strength, reinforcement strength, and anchorage (bond shear stresses).

Strength - Concrete

The concrete strength in compression is evaluated as the ratio between the maximum Equivalent principal stress f_c,eq (also σ_c,eq in previous text) obtained from FE analysis and the limit value f'_c,lim.

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

$$f_{c,eq} = \sigma_{c3} - \sigma_{c1}$$

The f_c,eq value can, therefore, be directly compared with uniaxial strength limits. This expression is derived from the implementation of the Mohr-Coulomb plasticity theory, conservatively assuming the angle of internal friction φ = 0°.

Strength - Reinforcement

The strength of the reinforcement is evaluated in both tension and compression as the ratio between the stress in the reinforcement at the cracks f_s and the specified limit value f_y,lim.

$$f_{y,lim} = \phi_{s} \cdot f_{y}$$

Strength - Anchors

Anchors are checked for normal stresses in a similar way to reinforcement, where the limit value f_y,lim is determined. 

To make it easier to navigate the following text, we will first divide anchoring into three groups in terms of code-checking according to AASHTO or ACI.

Group 1

• Anchorage Types

  • Cast-in plate
  • Base plate - Stand-off = direct 
  • Base plate - Stand-off = Mortar joint - thickness of mortar less than 0.5 times the anchor diameter
  • Single anchor with projected length less than 0.5 times anchor diameter
• Anchor code-checks (AASHTO / ACI)

  • Tension/compression

    • All anchor types in tension – ACI 318-19 chap. 17.6.1.2  
    • All anchor types in compression – AASHTO article 6.9.2
  • Shear without lever arm

    • Bolt material – ACI 318-19 chap. 17.7.1.2 (b)
    • Headed studs – ACI 318-19 chap. 17.7.1.2 (a)
    • Reinforcement – ACI 318-19 chap. 17.7.1.2 (b)
  • Interaction of tension and shear - ACI 318-19 chap. 17.8

Group 2

• Anchorage Types

  • Base plate - Stand-off = Mortar joint - thickness of mortar more than 0.5 times the anchor diameter
• Anchor code-checks (AASHTO / ACI)

  • Tension/compression

    • All anchor types in tension – ACI 318-19 chap. 17.6.1.2  
    • All anchor types in compression – AASHTO article 6.9.2
  • Shear with lever arm

    • Bolt material – ACI 318-19 chap. 17.7.1.2 (b) + chap. 17.7.1.2.1.
    • Headed studs – ACI 318-19 chap. 17.7.1.2 (a) + chap. 17.7.1.2.1.
    • Reinforcement – ACI 318-19 chap. 17.7.1.2 (b) + chap. 17.7.1.2.1.
  • Interaction of tension and shear - ACI 318-19 chap. 17.8

Group 3

• Anchorage types

  • Base plate - Stand-off = gap
  • Single anchor with projected length more than 0.5 times the anchor diameter
• Anchor code-checks (AASHTO / ACI)

  • Tension/compression (with buckling)

    • All anchor types in tension – ACI 318-19 chap. 17.6.1.2
    • All anchor types in compression – AASHTO LRFD article 6.9.2
  • Bending

    • For all anchor types – AAHTO LRFD article 6.12.2.2.7
  • Shear

    • For all anchor types – AASHTO LRFD article 6.10.9
  • Interaction of axial force and bending

    • $\dfrac{N}{P_n}+\dfrac{M}{M_n}\le 1$ 

Tensile resistance of anchor according to ACI 318-19 chap. 17.6.1.2

$$\phi N_{sa}=\phi_{a,t}\,A_{se,N}\,f_{uta}$$

where:

• ϕ_a,t  – strength reduction factor for anchors in tension according to ACI 318-19 chap. 17.5.3 (a)
• A_se,N – tensile stress area (reduced by thread)
• f_uta – specified tensile strength of anchor steel and shall not be greater than 1.9 f_ya and 860 MPa

Shear resistance of anchor according to ACI 318-19 chap. 17.7.1.2 (a)

The steel strength in shear for headed studs is determined as:

$$\phi V_{sa}=\phi_{a,V}\,A_{se,V}\,f_{uta}$$

where:
ϕ_a,v – strength reduction factor for anchors in tension according to ACI 318-19 chap. 17.5.3 (a)
A_se,V – tensile stress area (reduced by thread)
f_uta – specified tensile strength of anchor steel and shall not be greater than 1.9 f_ya and 860 MPa

Shear resistance of anchor according to ACI 318-19 chap. 17.7.1.2 (b)

The steel strength in shear for anchors from bolt material and reinforcement is determined as:

$$\phi V_{sa}=\phi_{a,V}\,0.6\,A_{se,V}\,f_{uta}$$

where:

• ϕ_a,v  – strength reduction factor for anchors in tension according to ACI 318-19 chap. 17.5.3 (a)
• A_se,V – tensile stress area (reduced by thread)
• f_uta – specified tensile strength of anchor steel and shall not be greater than 1.9 f_ya and 860 MPa

Shear resistance of anchor connected to a base with mortar - ACI 318-19 chap. 17.7.1.2.1

If anchors are used with built-up grout pads (Group 2), the design strength calculated in accordance with 17.7.1.2 shall be multiplied by 0.80.

Interaction in tension and shear according to ACI 318-19 chap. 17.8

It shall be permitted to neglect the interaction between tension and shear if (a) or (b) is satisfied.
(a) N_ua/(ϕN_n) ≤ 0.2
(b) V_ua/(ϕV_n) ≤ 0.2 

If N_ua/(ϕN_n) > 0.2 for the governing strength in tension and V_ua/(ϕV_n) > 0.2 for the governing strength in shear, then Eq. (17.8.3) shall be satisfied.

$$\frac{N_{ua}}{\phi N_n}+\frac{V_{ua}}{\phi V_n}\le 1.2$$

Compressive resistance of anchor according to AASHTO LRFD Article 6.9.2

$$P_r =\phi_{a,c}\, P_{n}$$

where:

• ϕ_a,c  – the strength reduction factor for anchors in compression according to AASHTO LRFD article 6.5.4.2

• If $\dfrac{P_o}{P_e} \le 2.25$,  then: $P_{n}=\left(0.658^{\,P_o/P_e}\right)P_o$,   Otherwise: $P_n=0.877\,P_e$

• A_g​ – gross cross-sectional area of member (in²)
• F_y – specified minimum yield stress of the type of steel being used (ksi)
• $P_e=\dfrac{\pi^2 E}{\left(\dfrac{K\,l}{r_s}\right)^2}A_g$ - elastic critical buckling resistance (kip)
• E – modulus of elasticity of steel (ksi)
• K = 2 – effective length factor according to Article 4.6.2.5 
• l – unbraced length in plane of bucling (in) 
• $r_s=\sqrt{\dfrac{I}{A_g}}$ – radius of gyration
• $I=\dfrac{\pi d_s^4}{64}$ – moment of inertia of the bolt 

Bending resistance of anchor according to AASHTO LRFD Article 6.12.2.2.7

$$M_n=\phi_{a,b}\, Z\, F_y\, \le 1.6\,\phi_{a,b}\, S_x\, F_y$$

where:

• $Z=\dfrac{d_s^{3}}{6}$ – plastic section modulus of the bolt
• $S_x=\dfrac{2I}{d_s}$ – elastic section modulus of the bolt