# Limit states and crack width calculation

Assessment of the structure using the CSFM is performed by two different analyses: one for serviceability and one for ultimate limit state load combinations. The serviceability analysis assumes that the ultimate behavior of the element is satisfactory, and the yield conditions of the material will not be reached at serviceability load levels. This approach enables the use of simplified constitutive models (with a linear branch of concrete stress-strain diagram) for serviceability analysis to enhance numerical stability and calculation speed. Therefore, it is recommended the use the workflow presented below, in which the ultimate limit state analysis is carried out as the first step.

### Ultimate limit state analysis

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 which takes into account the following steps:

- Choose a selection of the most critical load combinations.
- Calculate only Ultimate Limit State (ULS) load combinations.
- Use a coarse mesh (by increasing the multiplier of the default mesh size in Setup (Fig. 19)).

*\[ \textsf{\textit{\footnotesize{Fig. 19\qquad Mesh multiplier.}}}\]*

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). User can change mesh size by the multiplier, which can reach values from 0.5 to 5 (Fig. 19).

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.

### Serviceability limit state analysis

SLS assessments are carried out for stress limitation, crack width, and deflection limits. Stresses are checked in concrete and reinforcement elements according to the applicable code in a similar manner to that specified for the ULS.

The serviceability analysis contains certain simplifications of the constitutive models which are used for ultimate limit state analysis. A perfect bond is assumed, i.e., the anchorage length is not verified at serviceability. Furthermore, the plastic branch of the stress-strain curve of concrete in compression is disregarded, while the elastic branch is linear and infinite. These simplifications enhance the numerical stability and calculation speed, and do not reduce the generality of the solution as long as the resultant material stress limits at serviceability are clearly below their yielding points (as required by standards). Therefore, the simplified models used for serviceability are only valid if all verification requirements are fulfilled.

## Crack width calculation and Tension stiffening

#### Crack width calculation

There are two ways of computing crack widths - stabilized and non-stabilized cracking. According to the geometrical reinforcement ratio in each part of the structure is decided, which type of crack calculation model will be used (TCM for stabilized cracking and POM for non-stabilized cracking model).

*\( \textsf{\textit{\footnotesize{Fig. 20 \qquad Crack width calculation: (a) considered crack kinematics; (b) projection of crack kinematics into the principal}}}\) \( \textsf{\textit{\footnotesize{directions of the reinforcing bar; (c) crack width in the direction of the reinforcing bar for stabilized cracking; (d) cases with}}}\) \( \textsf{\textit{\footnotesize{local non-stabilized cracking regardless of the reinforcement amount; (e) crack width in the direction of the reinforcing bar}}}\)\( \textsf{\textit{\footnotesize{for non-stabilized cracking.}}}\)*

While the CSFM yields a direct result for most verifications (e.g., member capacity, deflections…), crack width results are calculated from the reinforcement strain results directly provided by FE analysis following the methodology described in Fig. 20. A crack kinematic without slip (pure crack opening) is considered (Fig. 20a), which is consistent with the main assumptions of the model. The principal directions of stresses and strains define the inclination of the cracks (θ* _{r}* = θ

_{s}= θ

_{e}). According to (Fig. 20b), the crack width (

*w*) can be projected in the direction of the reinforcing bar (

*w*

*), leading to:*

_{b}\[w = \frac{w_b}{\cos\left(θ_r + θ_b - \frac{π}{2}\right)}\]

where θ* _{b}* is the bar inclination.

Please note, that the program displays values of θ* _{r}* and θ

*<*

_{b}*π/2*. It means that the previous equation works for cases, where the reinforcement and crack go through the different quadrants of the Cartesian coordinate system as shown in Fig. 20, where reinforcement goes through I. and III. quadrants and crack through II and IV. For cases where the reinforcement and crack go through the same quadrants, the equation has to be modified as follows:

\[w = \frac{w_b}{\cos\left(-θ_r + θ_b + \frac{π}{2}\right)}\]

The component *w** _{b}* is consistently calculated based on the tension stiffening models by integrating the reinforcement strains. For those regions with fully developed crack patterns, the calculated average strains (e

*) along the reinforcing bars are directly integrated along the crack spacing (*

_{m}*s*

*), as indicated in (Fig. 20c). While this approach to calculating the crack directions does not correspond to the real position of the cracks, it still provides representative values that lead to crack width results that can be compared to code-required crack width values at the position of the reinforcing bar.*

_{r}Special situations are observed at concave corners of the calculated structure. In this case, the corner predefines the position of a single crack that behaves in a non-stabilized fashion before additional adjacent cracks develop. These additional cracks generally develop after the serviceability range (Mata-Falcón 2015), which justifies calculating the crack widths in such a region as if they were non-stabilized (Fig. 21).

*\[ \textsf{\textit{\footnotesize{Fig. 21\qquad Definition of the region at concave corners in which the crack width is computed as if it were non-stabilized.}}}\]*

#### 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.

*\( \textsf{\textit{\footnotesize{Fig. 22\qquad Tension stiffening model: (a) tension chord element for stabilized cracking with distribution of bond shear,}}}\) *\( \textsf{\textit{\footnotesize{steel and concrete stresses, and steel strains between cracks, considering average crack spacing); (b) pull-out assumption}}}\) \( \textsf{\textit{\footnotesize{for non-stabilized cracking with distribution of bond shear and steel stresses and strains around the crack; (c) resulting}}}\) \( \textsf{\textit{\footnotesize{tension chord behavior in terms of reinforcement stresses at the cracks and average strains for European B500B steel;}}}\) \( \textsf{\textit{\footnotesize{(d) detail of the initial branches of the tension chord response.}}}\)

**Stabilized cracking**

In fully developed crack patterns, tension stiffening is introduced using the Tension Chord Model (TCM) (Marti et al. 1998; Alvarez 1998) – Fig. 22a – 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

_{b}_{0}=2

*f*

*for σ*

_{ctm}*≤*

_{s}*f*

*and τ*

_{y}*=τ*

_{b}

_{b}_{1}=

*f*

*for σ*

_{ctm}*>*

_{s }*f*

*. Treating every reinforcing bar as a tension chord – Fig. 22b and Fig. 22a – 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.*

_{y}For *s** _{r}* =

*s*

_{r}_{0}, a new crack may or may not form because at the center between two cracks σ

_{c}_{1}=

*f*

*. Consequently, the crack spacing may vary by a factor of two, i.e.,*

_{ct}*s*

*= λ*

_{r}*s*

_{r}_{0}, with l = 0.5…1.0. Assuming a certain value for λ, the average strain of the chord (ε

*) can be expressed as a function of the maximum reinforcement stresses (i.e., stresses at the cracks, σ*

_{m}*). 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):*

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

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

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

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

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

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

where:

*E** _{sh}* the steel hardening modulus

*E*

*= (*

_{sh}*f*

*–*

_{t}*f*

*)/(ε*

_{y}*–*

_{u}*f*

*/*

_{y}*E*

*) ,*

_{s}*E** _{s}* modulus of elasticity of reinforcement,

*Ø* reinforcing bar diameter,

s_{r}* ^{ }*crack spacing,

σ_{sr}* *reinforcement stresses at the cracks,

σ_{s}* *actual reinforcement stresses,

*f** _{y }*yield 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. 23).

*\( \textsf{\textit{\footnotesize{Fig. 23\qquad Effective area of concrete in tension for stabilized cracking: (a) maximum concrete area that can be activated;}}}\) \( \textsf{\textit{\footnotesize{(b) cover and global symmetry condition; (c) resultant effective area.}}}\)*

**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. 22b. 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

*) in the vicinity of the crack to be obtained for any maximum steel stress at the crack (σ*

_{s}*) directly from equilibrium. Given the fact that the crack spacing is unknown for a non-fully developed crack pattern, the average strain (ε*

_{sr}*) is computed for any load level over the distance between points with zero slip when the reinforcing bar reaches its tensile strength (*

_{m}*f*

*) at the crack (*

_{t}*l*

_{ε,}

*in Fig. 22b), leading to the following relationships:*

_{avg}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*

*= 1.08 and ε*

_{y}*= 5%) is illustrated in Fig. 22c-d.*

_{u}