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.
\[ \textsf{\textit{\footnotesize{Fig. 48\qquad The stress-strain diagram of concrete for Strength analysis}}}\]
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 εc0 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.
Φc 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.
\[ \textsf{\textit{\footnotesize{Fig. 49 \qquad Stress-strain diagram of reinforcement}}}\]
where:
Φs is the strength reduction factor for reinforcement. Where the default value is set according to AASHTO LRFD (2024) Article 5.5.4.2.
fy is the yield strength of reinforcement
Es 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. 50) 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 (εm).
\[ \textsf{\textit{\footnotesize{Fig. 50\qquad Scheme of tension stiffening.}}}\]