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 EN 1992-1-1 in a similar manner to that specified for the ULS.

Stress limitation

The compressive stress in the concrete shall be limited in order to avoid longitudinal cracks. According to EN 1992-1-1 chap. 7.2 (2), longitudinal cracks may occur if the stress level under the characteristic combination of loads exceeds a value k₁f_ck. The concrete stress in compression is evaluated as the ratio between the maximum principal compressive stress σ_c3 obtained from FE analysis for serviceability limit states and the limit value σ_c3,lim. Then:

\[\frac{σ_{c3}}{σ_{c3,lim}}\]

\[σ_{c3,lim} = k_1\cdot f_{ck}\]

where:

f_ck        characteristic cylinder strength of concrete,

k₁         =0.6.

Unacceptable cracking or deformation may be assumed to be avoided if, under the characteristic combination of loads, the tensile stress in the reinforcement does not exceed k₃f_yk (EN 1992-1-1 chap. 7.2 (5)). The strength of the reinforcement is evaluated as the ratio between the stress in the reinforcement at the cracks σ_sr and the specified limit value σ_sr,lim:

\[\frac{σ_{sr}}{σ_{sr,lim}}\]

\[σ_{sr,lim} =  k_3\cdot f_{yk}\]

where:

f_yk        yield strength of the reinforcement,

k₃        =0.8.

Deflection

Deflections can only be assessed for walls or isostatic (statically determinate) or hyperstatic (statically indeterminate) beams. In these cases, the absolute value of deflections is considered (compared to the initial state before loading), and the maximum admissible value of deflections must be set by the user. Deflections at trimmed ends cannot be checked since these are essentially unstable structures where the equilibrium is satisfied by adding end forces, and hence deflections are unrealistic. Short-term u_z,st or long-term u_z,lt deflection can be calculated and checked against user-defined limit values:

\[\frac{u_ z}{u_{z,lim}}\]

where:

u_z         short- or long-term deflection calculated by FE analysis,

u_z,lim    limit value of the deflection defined by the user.

Crack width

Crack widths and crack orientations are calculated only for permanent loads, either short-term or long-term. The verifications with respect to limit values specified by the user according to the Eurocode are presented as follows:

\[\frac{w_ z}{w_{z,lim}}\]

where:

w         short- or long-term crack width calculated by FE analysis,

w_lim     limit value of the crack width defined by the user.

There are two ways of computing crack widths (stabilized and non-stabilized cracking). In the general case (stabilized cracking), the crack width is calculated by integrating the strains on 1D elements of reinforcing bars. The crack direction is then calculated from the three closest (from the center of the given 1D finite element of reinforcement) integration points of 2D concrete elements. 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.