6.1 Material models (AASHTO)

Concrete - Strength

The concrete model implemented for strength calculations in 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.

\[ \textsf{\textit{\footnotesize{Fig. 57\qquad The stress-strain diagram of concrete for Strength analysis}}}\]

The implementation of 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 ε_c0 in maximum value 5%, while AASHTO LRFD (2024) Article 5.6.2.1 assumes ultimate strain of 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, in addition to the factor of cracked concrete (k_c2 defined in (Fig. 57)), 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:

\[f'_{c,lim}=\alpha_{1}\cdot\phi_{c}\cdot k_{c}\cdot f'_{c}\]

\[k_{c}=\eta_{fc}\cdot k_{c2}\]

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

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.

Φ_c is the resistance factor for concrete. The default value is set according to AASHTO LRFD (2024) Article 5.5.4.2.

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

f'_c is the concrete cylinder strength (in MPa for the definition of \( \eta_{fc} \)).

\[ \textsf{\textit{\footnotesize{Fig. 58\qquad The compression softening law.}}}\]

k_c2 is a reduction factor based on the same assumptions as the concrete efficiency factor ν given in AASHTO LRFD (2024) 5.8.2.5.3a and Table 5.8.2.5.3a-1, except that in CSFM, the presence of a principal tensional stress perpendicular to the principal compressional stress is checked for each finite element (not only for nodes of the Strut and Tie model).

Concrete – Serviceability

The serviceability analysis contains certain simplifications of the constitutive models that are used for strength analysis. The plastic branch of the stress-strain curve of concrete in compression is disregarded, while the elastic branch is linear and infinite. The compression softening law is not considered. 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. (consistent with the AASHTO LRFD service limit state approach). Therefore, the simplified models used for serviceability are only valid if all verification requirements are fulfilled.

\[ \textsf{\textit{\footnotesize{Fig. 59\qquad Concrete stress-strain diagrams implemented for serviceability analysis: short- and long-term verifications.}}}\]

Long-term effects

The long-term constitutive law (the red curve in Fig. 59) is used for crack width calculation, total deflection, and stress limitation of prestressed members when the long-term effect is selected in the top ribbon. In the IDEA StatiCa Detail application, the effective modulus of elasticity is used for long-term effects verification, as mentioned in AASHTO LRFD (2024) C5.12.5.3.6-1.

\[E_{eff} = \frac{E_{c}}{1+\psi}\]

where:
E_c is the modulus of elasticity defined in AASHTO LRFD (2024) article 5.4.2.4
ψ is the creep coefficient defined in AASHTO LRFD (2024) article 5.4.2.3.2

Creep factors are defined by the user in material properties.

Short-term effects

To conduct short-term verifications, another calculation is performed in which all loads are calculated without the creep factor. Both calculations for long and short-term verifications are depicted in Fig. 59.

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 – the strength and modulus of elasticity.

The reinforcement stress-strain diagram can also be defined by the user, but in this case, it is impossible to assume the tension stiffening effect (it is impossible to calculate crack width). 

\[ \textsf{\textit{\footnotesize{Fig. 60 \qquad Stress-strain diagram of reinforcement}}}\]

where:

Φ_s is the resistance factor for reinforcement. Where the default value is set according to AASHTO LRFD (2024) Article 5.5.4.2.

f_y is the yield strength of reinforcement

E_s 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. 61)  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 (ε_m).

\[ \textsf{\textit{\footnotesize{Fig. 61\qquad Scheme of tension stiffening.}}}\]