Concrete - Strength
The concrete model implemented for strength calculations in CSFM is based on the parabolic-plastic stress-strain curve for concrete based on the Portland Cement Association’s parabolic stress-strain curve described in PCA’s Notes on ACI 318-99 Building Code Requirements for Structural Concrete, Figure 6-8. The tensile strength is neglected, as it is in classic reinforced concrete design.
\[ \textsf{\textit{\footnotesize{Fig. 38\qquad The stress-strain diagram of concrete for Strength analysis}}}\]
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 εc0 in maximum value 5%, while ACI 318-19 Cl. 22.2.2.1 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 (kc2 defined in (Fig. 39)), 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 ACI 318-19 Cl. 22.2.2.4.1. 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 ACI 318-19 Table 24.2.1 (b)(f).
kc2 is the reduction factor due to the presence of transverse cracking.
f'c is the concrete cylinder strength (in MPa for the definition of \( \eta_{fc} \)).
\[ \textsf{\textit{\footnotesize{Fig. 39\qquad The compression softening law.}}}\]
kc2 is a reduction factor based on the same assumptions as the nodal zone coefficient βn given in ACI 318-19 Table 23.9.2, 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 which 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. 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 (as required by ACI). Therefore, the simplified models used for serviceability are only valid if all verification requirements are fulfilled.
\[ \textsf{\textit{\footnotesize{Fig. 40\qquad Concrete stress-strain diagrams implemented for serviceability analysis: short- and long-term verifications.}}}\]
Long-term effects
The long-term behavior of the structure, such as long-term deflections or calculation of crack widths caused by sustained loads, is influenced by concrete creep. The ACI 318-19 in paragraph 24.2.4.1.3 defines the time-dependent factor for sustained loads – ξ representing creep effect for specified sustained load duration.
In the Detail application, the modulus of elasticity Ec is adjusted to determine the long-term behavior of the structure through the factor ξ. The adjusted modulus of elasticity is referred to as Ec,eff – see Figure 40.
Assuming that the deformation of the element is expressed by strain, it can be written that:
\[\epsilon_{tot} = \epsilon_{0} + \epsilon_{creep} = \epsilon_{0} \cdot (1+\xi)\]
where:
ε0 is a short-term strain (without the influence of creep) and εcreep is a strain caused by creep.
Using Hooke's law, we can write:
\[E_{c,eff} = \frac{f_{c}}{\epsilon_{tot}}\]
Substituting for \(\epsilon_{tot} = \epsilon_{0} \cdot (1+\xi)\) and \(\epsilon_{0} = f_{c} / E_{c}\) we get:
\[E_{c,eff} = \frac{E_{c}}{1+\xi}\]
Sustained load duration for determination of the factor ξ can be set individually for each service long-term combination.
\[ \textsf{\textit{\footnotesize{Fig. 41\qquad Sustained load duration}}}\]
The time-dependent deflections, stresses, and crack widths are then calculated with a modified material model where the effect of compression refinement is taken into account automatically by the nature of the FE analysis. It is, therefore, not necessary to further multiply them by the factor defined in 24.2.4.1.1.
Short-term effects
To conduct short-term verifications, another calculation is performed in which all loads are calculated without the time-dependent factor for sustained loads. Both calculations for long and short-term verifications are depicted in Fig. 40.
Reinforcement
A perfectly elasto-plastic stress-strain diagram with a defined yield point for the non-prestresses reinforcement is considered, see ACI 319-19 CL. 20.2.1. 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 be also defined by the user, but in this case, it is impossible to assume the tension stiffening effect (it is impossible to calculate crack width).
\[ \textsf{\textit{\footnotesize{Fig. 42 \qquad Stress-strain diagram of reinforcement}}}\]
where:
Φs is the strength reduction factor for reinforcement. Where the default value is set according to ACI 318-19 Table 24.2.1.
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. 43) 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).
\[ \textsf{\textit{\footnotesize{Fig. 43\qquad Scheme of tension stiffening.}}}\]