The design and assessment of concrete elements are normally performed at the sectional (1D-element) or point (2D-element) level. This procedure is described in all standards for structural design, e.g., in (EN 1992-1-1), and it is used in everyday structural engineering practice. However, it is not always known or respected that the procedure is only acceptable in areas where the Bernoulli-Navier hypothesis of plane strain distribution applies (referred to as B-regions). The places where this hypothesis does not apply are called discontinuity or disturbed regions (D-Regions). Examples of B and D regions of 1D-elements are given in (Fig. 1). These are, e.g., bearing areas, parts where concentrated loads are applied, locations where an abrupt change in the cross-section occurs, openings, etc. When designing concrete structures, we meet a lot of other D-Regions such as walls, bridge diaphragms, corbels, etc.

\[ \textsf{\textit{\footnotesize{Fig. 1\qquad Discontinuity regions (Navrátil et al. 2017)}}}\]

In the past, semi-empirical design rules were used for dimensioning discontinuity regions. Fortunately, these rules have been largely superseded over the past decades by strut-and-tie models (Schlaich et al., 1987) and stress fields (Marti 1985), which are featured in current design codes and frequently used by designers today. These models are mechanically consistent and powerful tools. Note that stress fields can generally be continuous or discontinuous and that strut-and-tie models are a special case of discontinuous stress fields.

Despite the evolution of computational tools over the past decades, Strut-and-Tie models are essentially still used as hand calculations. Their application for real-world structures is tedious and time-consuming since iterations are required, and several load cases need to be considered. Furthermore, this method is not suitable for verifying serviceability criteria (deformations, crack widths, etc.).

The interest of structural engineers in a reliable and fast tool to design D-regions led to the decision to develop the new Compatible Stress Field Method, a method for computer-aided stress field design that allows the automatic design and assessment of structural concrete members subjected to in-plane loading.

The Compatible Stress Field Method is a continuous FE-based stress field analysis method in which classic stress field solutions are complemented with kinematic considerations, i.e., the state of strain is evaluated throughout the structure. Hence, the effective compressive strength of concrete can be automatically computed based on the state of transverse strain in a similar manner as in compression field analyses that account for compression softening (Vecchio and Collins 1986; Kaufmann and Marti 1998) and the EPSF method (Fernández Ruiz and Muttoni 2007). Moreover, the CSFM considers tension stiffening, providing realistic stiffnesses to the elements, and covers all design code prescriptions (including serviceability and deformation capacity aspects) not consistently addressed by previous approaches. The CSFM uses common uniaxial constitutive laws provided by design standards for concrete and reinforcement. These are known at the design stage, which allows the partial safety factor method to be used. Hence, designers do not have to provide additional, often arbitrary material properties as are typically required for non-linear FE-analyses, making the method perfectly suitable for engineering practice.

To foster the use of computer-aided stress fields by structural engineers, these methods should be implemented in user-friendly software environments. To this end, the CSFM has been implemented in IDEA StatiCa Detail; a new user-friendly commercial software developed jointly by ETH Zurich and the software company IDEA StatiCa in the framework of the DR-Design Eurostars-10571 project.

Main assumptions and limitations for CSFM

The CSFM assumes fictitious, rotating, stress-free cracks that open without slip (Fig. 2a), and considers the equilibrium at the cracks together with the average strains of the reinforcement. Hence, the model considers maximum concrete (σ_c₂_r) and reinforcement stresses (σ_sr) at the cracks while neglecting the concrete tensile strength (σ_c₁_r = 0), except for its stiffening effect on the reinforcement. The consideration of tension stiffening allows the average reinforcement strains (ε_m) to be simulated.

\( \textsf{\textit{\footnotesize{Fig. 2\qquad Basic assumptions of the CSFM: (a) principal stresses in concrete; (b) stresses in the reinforcement direction;}}}\) \( \textsf{\textit{\footnotesize{(c) stress-strain diagram of concrete in terms of maximum stresses with consideration of compression softening;}}}\) \( \textsf{\textit{\footnotesize{(d) stress-strain diagram of reinforcement in terms of stresses at cracks and average strains; (e) compression softening}}}\) \( \textsf{\textit{\footnotesize{law; (f) bond shear stress-slip relationship for anchorage length verifications.}}}\)

Despite their simplicity, similar assumptions have been demonstrated to yield accurate predictions for reinforced members subjected to in-plane loading (Kaufmann 1998; Kaufmann and Marti 1998) if the provided reinforcement avoids brittle failures at cracking. Furthermore, the non-consideration of any contribution of the tensile strength of concrete to the ultimate load is consistent with the principles of modern design codes, which are mostly based on plasticity theory.

However, the CSFM is not suited for slender elements without transverse reinforcement since relevant mechanisms for such elements as aggregate interlock, residual tensile stresses at the crack tip, and dowel action – all of them relying directly or indirectly on the tensile strength of the concrete – are disregarded. While some design standards allow the design of such elements based on semi-empirical provisions, the CSFM is not intended for this type of potentially brittle structure.

Concrete

The concrete model implemented in the CSFM is based on the uniaxial compression constitutive laws prescribed by design codes for the design of cross-sections, which only depend on compressive strength. The parabola-rectangle diagram specified in EN 1992-1-1 (Fig. 2c) is used by default in the CSFM, but designers can also choose a more simplified elastic ideal plastic relationship. When assessing according to the ACI code, it is possible to use only the parabola-rectangle stress-strain diagram. As previously mentioned, the tensile strength is neglected, as it is in classic reinforced concrete design.

The effective compressive strength is automatically evaluated for cracked concrete based on the principal tensile strain (ε₁) by means of the k_c₂ reduction factor, as shown in Fig. 2c and e. The implemented reduction relationship (Fig. 2e) is a generalization of the fib Model Code 2010 proposal for shear verifications, which contains a limiting value of 0.65 for the maximum ratio of effective concrete strength to concrete compressive strength, which is not applicable to other loading cases.

The 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, in addition to the factor of cracked concrete (k_c₂) defined in (Fig. 2e), 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:

\[f_{c,red} = k_c \cdot f_{c} = \eta _{fc} \cdot k_{c2} \cdot f_{c}\]

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

where:

k_cis the global reduction factor of the compressive strength

k_c₂ is the reduction factor due to the presence of transverse cracking

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

Reinforcement

The idealized bilinear stress-strain diagram for the bare reinforcing bars typically defined by design codes (Fig. 2d) 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). A user-defined stress-strain relationship can also be defined.

Tension stiffening is accounted for 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).

Bond model

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. 2f, with f_bd being the design 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 in further.

Tension stiffening

The implementation of tension stiffening distinguishes between cases of stabilized and non-stabilized crack patterns. In both cases, the concrete is considered fully cracked before loading by default.

\( \textsf{\textit{\footnotesize{Fig. 3\qquad Tension stiffening model: (a) tension chord element for stabilized cracking with distribution of bond shear,}}}\) \( \textsf{\textit{\footnotesize{steel and concrete stresses, and steel strains between cracks, considering average crack spacing); (b) pull-out assumption}}}\) \( \textsf{\textit{\footnotesize{for non-stabilized cracking with distribution of bond shear and steel stresses and strains around the crack; (c) resulting}}}\) \( \textsf{\textit{\footnotesize{tension chord behavior in terms of reinforcement stresses at the cracks and average strains for European B500B steel;}}}\) \( \textsf{\textit{\footnotesize{(d) detail of the initial branches of the tension chord response.}}}\)

Stabilized cracking

In fully developed crack patterns, tension stiffening is introduced using the Tension Chord Model (TCM) (Marti et al. 1998; Alvarez 1998) – Fig. 3a – which has been shown to yield excellent response predictions in spite of its simplicity (Burns 2012). The TCM assumes a stepped, rigid-perfectly plastic bond shear stress-slip relationship with τ_b= τ_b₀ =2 f_ctm for σ_s ≤ f_y and τ_b =τ_b₁ = f_ctm for σ_s> f_y. Treating every reinforcing bar as a tension chord – Fig. 3b and Fig. 3a – the distribution of bond shear, steel, and concrete stresses and hence the strain distribution between two cracks can be determined for any given value of the maximum steel stresses (or strains) at the cracks.

For s_r = s_r₀, a new crack may or may not form because at the center between two cracks σ_c₁ = f_ct. Consequently, the crack spacing may vary by a factor of two, i.e., s_r = λs_r₀, with l = 0.5…1.0. Assuming a certain value for λ, the average strain of the chord (ε_m) can be expressed as a function of the maximum reinforcement stresses (i.e., stresses at the cracks, σ_sr). For the idealized bilinear stress-strain diagram for the reinforcing bare bars considered by default in the CSFM, the following closed-form analytical expressions are obtained (Marti et al. 1998):

\[\varepsilon_m = \frac{\sigma_{sr}}{E_s} - \frac{\tau_{b0}s_r}{E_s Ø}\]

\[\textrm{for}\qquad\qquad\sigma_{sr} \le f_y\]

\[{\varepsilon_m} = \frac{{{{\left( {{\sigma_{sr}} - {f_y}} \right)}^2}Ø}}{{4{E_{sh}}{\tau _{b1}}{s_r}}}\left( {1 - \frac{{{E_{sh}}{\tau_{b0}}}}{{{E_s}{\tau_{b1}}}}} \right) + \frac{{\left( {{\sigma_{sr}} - {f_y}} \right)}}{{{E_s}}}\frac{{{\tau_{b0}}}}{{{\tau_{b1}}}} + \left( {{\varepsilon_y} - \frac{{{\tau_{b0}}{s_r}}}{{{E_s}Ø}}} \right)\]

\[\textrm{for}\qquad\qquad{f_y} \le {\sigma _{sr}} \le \left( {{f_y} + \frac{{2{\tau _{b1}}{s_r}}}{Ø}} \right)\]

\[ \varepsilon_m = \frac{f_s}{E_s} + \frac{\sigma_{sr}-f_y}{E_{sh}} - \frac{\tau_{b1} s_r}{E_{sh} Ø}\]

\[\textrm{for}\qquad\qquad\left(f_y + \frac{2\tau_{b1}s_r}{Ø}\right) \le \sigma_{sr} \le f_t\]

where:
E_sh the steel hardening modulus E_sh = (f_t – f_y)/(ε_u – f_y /E_s) ,

E_s modulus of elasticity of reinforcement,

Ø reinforcing bar diameter,

s_rcrack spacing,

σ_sr reinforcement stresses at the cracks,

σ_s actual reinforcement stresses,

f_yyield strength of reinforcement.

The Idea StatiCa Detail implementation of the CSFM considers average crack spacing by default when performing computer-aided stress field analysis. The average crack spacing is considered to be 2/3 of the maximum crack spacing (λ = 0.67), which follows recommendations made on the basis of bending and tension tests (Broms 1965; Beeby 1979; Meier 1983). It should be noted that calculations of crack widths consider a maximum crack spacing (λ = 1.0) in order to obtain conservative values.

The application of the TCM depends on the reinforcement ratio, and hence the assignment of an appropriate concrete area acting in tension between the cracks to each reinforcing bar is crucial. An automatic numerical procedure has been developed to define the corresponding effective reinforcement ratio (ρ_eff = A_s/A_c,eff) for any configuration, including skewed reinforcement (Fig. 4).

\( \textsf{\textit{\footnotesize{Fig. 4\qquad Effective area of concrete in tension for stabilized cracking: (a) maximum concrete area that can be activated;}}}\) \( \textsf{\textit{\footnotesize{(b) cover and global symmetry condition; (c) resultant effective area.}}}\)

Non-stabilized cracking

Cracks existing in regions with geometric reinforcement ratios lower than ρ_cr, i.e., the minimum reinforcement amount for which the reinforcement is able to carry the cracking load without yielding, are generated by either non-mechanical actions (e.g. shrinkage) or the progression of cracks controlled by other reinforcement. The value of this minimum reinforcement is obtained as follows:

\[{\rho _{cr}} = \frac{{{f_{ct}}}}{{{f_y} - \left( {n - 1} \right){f_{ct}}}}\]

where:

f_y reinforcement yield strength,

f_ct concrete tensile strength,

n modular ratio, n = E_s / E_c .

For conventional concrete and reinforcing steel, ρ_cr amounts to approximately 0.6%.

For stirrups with reinforcement ratios below ρ_cr, cracking is considered to be non-stabilized and tension stiffening is implemented by means of the Pull-Out Model (POM) described in Fig. 3b. This model analyzes the behavior of a single crack considering no mechanical interaction between separate cracks, neglecting the deformability of concrete in tension and assuming the same stepped, rigid-perfectly plastic bond shear stress-slip relationship used by the TCM. This allows the reinforcement strain distribution (ε_s) in the vicinity of the crack to be obtained for any maximum steel stress at the crack (σ_sr) directly from equilibrium. Given the fact that the crack spacing is unknown for a non-fully developed crack pattern, the average strain (ε_m) is computed for any load level over the distance between points with zero slip when the reinforcing bar reaches its tensile strength (f_t) at the crack (l_ε,_avg in Fig. 3b), leading to the following relationships:

The proposed models allow the computation of the behavior of bonded reinforcement, which is finally considered in the analysis. This behavior (including tension stiffening) for the most common European reinforcing steel (B500B, with f_t / f_y = 1.08 and ε_u = 5%) is illustrated in Fig. 3c-d.

Design tools for reinforcement

Workflow and goals

The goal of reinforcement design tools in the CSFM is to help designers determine the location and required amount of reinforcing bars efficiently. The following tools are available to help/ guide the user in this process: linear calculation and topology optimization.

Reinforcement design tools consider more simplified constitutive models than the models used for the final verification of the structure. Therefore, the definition of the reinforcement in this step should be considered a pre-design to be confirmed/refined during the final verification step. The use of the different reinforcement design tools will be depicted in the model shown in Fig. 5, which consists of one end of a simply supported beam with variable depth subjected to a uniformly distributed load.

\[ \textsf{\textit{\footnotesize{Fig. 5\qquad Model used to illustrate the use of the reinforcement design tools.}}}\]

Linear analysis

The linear analysis considers linear elastic material properties and neglects reinforcement in the concrete region. It is, therefore, a very fast calculation that provides a first insight into the locations of tension and compression areas. An example of such a calculation is shown in Fig. 6.

\[ \textsf{\textit{\footnotesize{Fig. 6\qquad Results from the linear analysis tool for defining reinforcement layout}}}\]

\[ \textsf{\textit{\footnotesize{(red: areas in compression, blue: areas in tension).}}}\]

Topology optimization

Topology optimization is a method that aims to find the optimal distribution of material in a given volume for a certain load configuration. The topology optimization implemented in Idea StatiCa Detail uses a linear finite element model. Each finite element may have a relative density from 0 to 100 %, representing the relative amount of material used. These element densities are the optimization parameters in the optimization problem. The resulting material distribution is considered optimal for the given set of loads if it minimizes the total strain energy of the system. By definition, the optimal distribution is also the geometry that has the largest possible stiffness for the given loads.

The iterative optimization process starts with a homogeneous density distribution. The calculation is performed for multiple total volume fractions (20%, 40%, 60%, and 80%), which allows the user to select the most practical result. The resulting shape consists of trusses with struts and ties and represents the optimum shape for the given load cases (Fig. 7).

\[ \textsf{\textit{\footnotesize{Fig. 7\qquad Results from the topology optimization design tool with 20\% and 40\% effective volume}}}\]

\[ \textsf{\textit{\footnotesize{(red: areas in compression, blue: areas in tension).}}}\]

Analysis model of IDEA StatiCa Detail

Introduction to finite element implementation

The CSFM considers continuous stress fields in the concrete (2D finite elements), complemented by discrete “rod” elements representing the reinforcement (1D finite elements). Therefore, the reinforcement is not diffusely embedded into the concrete 2D finite elements but explicitly modeled and connected to them. A plane stress state is considered in the calculation model.

Fig. 8 Visualization of the calculation model of a structural element (trimmed beam) in Idea StatiCa Detail.

\[ \textsf{\textit{\footnotesize{Fig. 6\qquad Visualization of the calculation model of a structural element (trimmed beam) in Idea StatiCa Detail.}}}\]

Both entire walls and beams, as well as details (parts) of beams (isolated discontinuity region, also called trimmed end), can be modeled. In the case of walls and entire beams, supports must be defined in such a way that an (externally) isostatic (statically determinate) or hyperstatic (statically indeterminate) structure results. The load transfer at the trimmed ends of beams is introduced by means of a special Saint-Venant transfer zone, which ensures a realistic stress distribution in the analyzed detail region.

Supports and load transmitting components

To model most of the situations during the construction process, many types of supports (Fig. 7) and components used for transferring load (Fig. 8) are available in the CSFM.

Supports

Point support can be modeled in several ways to ensure that stresses are not localized in one point but rather distributed over a larger area. The first option is a distributed point support (Fig. 7a), which uniformly distributes the load on the edge of the member over the specified width.

\[ \textsf{\textit{\footnotesize{Fig. 7\qquad Various types of supports:}}}\]

\[ \textsf{\textit{\footnotesize{(a) point distributed; (b) bearing plate; (c) line support; (d) patch support; (e) hanging.}}}\]

Patch support (Fig. 7d), on the other hand, can only be placed inside a volume of concrete with a defined effective radius. It is then connected by rigid elements to the nodes of the reinforcement mesh within this radius. Therefore, it is required to define a reinforcing cage around patch support.

For the more precise modeling of some real scenarios, there are two other options for point support. Firstly, there is point support with a bearing plate of defined width and thickness (Fig. 7b). The material of the bearing plate can be specified, and the whole bearing plate is meshed independently. Secondly, there is hanging support available (Fig. 7e), which can be used for modeling lifting anchors or lifting studs.

Line support (Fig. 7c) can be defined on an edge (by specifying its length) or inside an element (by a polyline). It is also possible to specify its stiffness and/or non-linear behavior (support in compression/tension or only in compression).

Read detailed descriptions in Types of supports in IDEA StatiCa Detail

Load transmitting components

The introduction of loads into the structure can also be modeled in several ways. For point loads, a bearing plate (Fig. 8a) can be used similarly as point support, distributing the concentrated load onto a larger area thanks to a steel plate with defined width and thickness.

Fig. 10 Various types of load transfer components: (a) bearing plate; (b) patch load; (c) hanging; (d) partially loaded area.

\[ \textsf{\textit{\footnotesize{Fig. 8\qquad Various types of load transfer components:}}}\]

\[ \textsf{\textit{\footnotesize{(a) bearing plate; (b) patch load; (c) hanging; (d) partially loaded area.}}}\]

The point load can be applied either directly to the surface of the structure with a defined radius of action (load is applied to the concrete elements) or via a special transmitting device called patch load (Fig. 8b and Fig. 9). Patch load allows transmitting the load directly to the defined reinforcement located within the area of the effective radius. To secure the correct functionality of the patch load, a group of rebars that will be interconnected with the load is necessary to define (in the reinforcement properties). When the interconnected reinforcement is not defined, the load transfer mechanism is the same as for the point load placed on a member surface, and the load is transferred by the constraints to the concrete elements, not directly to the reinforcement.

Fig. 11 Patch load: (a) load application; (b) load transferred through reinforcement; (c) load transferred through concrete.

\[ \textsf{\textit{\footnotesize{Fig. 9\qquad Patch load: (a) load application; (b) load transferred through rebars (a group of bars for the load transfer is defined);}}}\]

\[ \textsf{\textit{\footnotesize{(c) load transferred through concrete (a group of bars for the load transfer is not defined).}}}\]

Lifting anchors or lifting studs can be modeled by a hanging load (Fig. 8c). User can use a partially loaded area (Fig. 8d), which allows for increasing the load-bearing capacity of concrete in compression according to Eurocode (it is not possible to use this type of load transmitting component when ACI is set). The structure can also be loaded with line loads on the edges, by general polyline, or by surface loads. The Detail application is able to automatically consider a self-weight in the analysis.

Load transfer at trimmed ends of beams

In many cases, we need to model only some detail (part) of a structural member, such as beam support, opening in the middle of the beam, etc. This approach can lead to support configurations that are unstable but admissible in IDEA StatiCa Detail (including the case of no supports). However, in such cases, it is also necessary to model the section representing the connection to the adjoining B-region, including internal forces at this section that satisfy the equilibrium. In certain cases (e.g., when modeling beam support), these internal forces can be determined automatically by the program.

Between the B-region and the analyzed discontinuity region, a Saint-Venant transfer zone is automatically created to ensure a realistic stress distribution in the analyzed region. The width of the transfer zone is determined as half of the section’s depth. As the only purpose of the Saint-Venant zone is to achieve a proper stress distribution in the rest of the model, no results from this area are displayed in verification, and no stop criteria are considered here.

The edge of the Saint-Venant zone that represents the trimmed end of the beam is modeled as rigid, i.e., it may rotate but must rest plane. This is done by connecting all the FEM nodes of the edge to a separate node at the centre of inertia of the section using a rigid body element (RBE2). The internal forces of the element may then be applied at this node, as shown in Fig. 10.

Fig. 12 Transfer of internal forces at a trimmed end.

\[ \textsf{\textit{\footnotesize{Fig. 10\qquad Transfer of internal forces at a trimmed end.}}}\]

Geometric modification of cross-sections

Reduction of the cross-section is automatically performed for structures defined as a beam or frame joint (defined by the x-axis and a cross-section). This modification is automatically applied on cross-sections with very wide flanges (Fig. 11) and is based on the assumption that a compression stress field would expand from the wall at a 45° angle, so the aforementioned reduced width would be the maximum width capable of transferring loads

Note that the method of determining the effective width flange implemented in CSFM is different from the one stated in 5.3.2.1 EN 1992-1-1 (2015) or in 9.2.4.4 ACI 318-19. Besides geometry, Eurocode-based effective width flange is explicitly affected by the span lengths and boundary conditions of a structure.

Fig. 13 Width reduction of a cross-section: (a) user input; (b) FE model – automatically determined reduced width of a flange.

\[ \textsf{\textit{\footnotesize{Fig. 11\qquad Width reduction of a cross-section: (a) user input; (b) FE model – automatically determined reduced flange width.}}}\]

In the case of haunches lying in the horizontal plane (Fig. 12), each haunch is divided into five sections along its length. Each of these sections is then modeled as a wall with a constant thickness, which is equal to the real thickness in the middle of the respective section.

Fig. 14 Horizontal haunch: (a) user input; (b) FE model – a haunch automatically divided into five sections.

\[ \textsf{\textit{\footnotesize{Fig. 12\qquad Horizontal haunch: (a) user input; (b) FE model – a haunch automatically divided into five sections.}}}\]

Finite element types

The non-linear (inelastic) finite element analysis model is created by several types of finite elements used to model concrete, reinforcement, and the bond between them. Concrete and reinforcement elements are first meshed independently and then connected to each other using multi-point constraints (MPC elements). This allows the reinforcement to occupy an arbitrary, relative position in relation to the concrete. If anchorage length verification is to be calculated, bond and anchorage end spring elements are inserted between the reinforcement and the MPC elements.

\[ \textsf{\textit{\footnotesize{Fig. 13\qquad Finite element model: reinforcement elements mapped to concrete mesh using MPC elements and bond elements.}}}\]