Ultimate limit state analysis

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).

The concrete strength in compression is evaluated as the ratio between the maximum principal compressive stress σ_c3 obtained from FE analysis and the limit value σ_c3,lim. Then:

\[\frac{σ_{c3}}{σ_{c3,lim}}\]

\[σ_{c3,lim} = f_{cd} = α_{cc} \cdot \frac{f_{ck,red}}{γ_c} = α_{cc} \cdot \frac{k_c \cdot f_{ck}}{γ_c} = α_{cc} \cdot \frac{\eta _{fc} \cdot k_{c2} \cdot f_{ck}}{γ_c}\]

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

where:

f_ck        characteristic cylinder strength of concrete,

k_c2       compression softening factor (see 5.1.1),

γ_c             is the partial safety factor for concrete, γ_c = 1,5,

α_cc       is the coefficient taking account of long term effects on the compressive strength and of unfavourable effects resulting from the way the load is applied. The default value is 1,0.

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_sr,lim:

\[\frac{σ_{sr}}{σ_{sr,lim}}\]

\(σ_{c3,lim} = \frac{k \cdot f_{yk}}{γ_s}\qquad\qquad\textsf{\small{for bilinear diagram with inclined top branch}}\)

\(σ_{c3,lim} = \frac{f_{yk}}{γ_s}\qquad\qquad\,\,\,\,\textsf{\small{for bilinear diagram with horizontal top branch}}\)

where:

f_yk        yield strength of the reinforcement,

k          the ratio of tensile strength f_tk to the yield stress,
            \(k = \frac{f_{tk}}{f_{yk}}\)

γ_s             is the partial safety factor for reinforcement γ_s = 1,15.

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:

\[\frac{τ_{b}}{f_{bd}}\]

\[f_{bd} = 2.25 \cdot η_1\cdot η_2\cdot f_{ctd}\]

where:

 f_ctd      is the design value of concrete tensile strength. Due to the increasing brittleness of higher strength concrete, f_ctk,0,05 is limited to the value for C60/75,

 η₁       is a coefficient related to the quality of the bond condition and the position of the bar during concreting (Fig. 33):

             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

\[ \textsf{\textit{\footnotesize{Fig. 33\qquad Description of bond conditions.}}}\]

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).