When a flexural load or shrinkage causes the applied tensile stresses to exceed the cracking stress, the stress is relieved at that location and a redistribution of stress occurs with a resulting increase in cross section curvature. As load increases, the number of cracks also increases. In the cross section calculations, at the crack locations the concrete is assumed to carry no tension. In the regions between the cracks the bonded tension reinforcement transfers tension back into the concrete. This phenomenon is normally referred to as tension stiffening. In a partially cracked concrete member, the mean curvature over a region lies between the uncracked curvature and the curvature at the crack locations.

A number of models exist for predicting the tension stiffening behavior. The tension stiffening model presented in the Eurocode 2-2004 Equation 7.19 is utilized in RAM Concept ’s load history calculations:

where

β	=	a coefficient taking account for the duration of loading = 1.0 for a short-term loading (characteristic or frequent service rule set) = 0.5 for sustained loads (quasi-permanent service rule set)
Mcr	=	the gross cross section cracking moment
Ma	=	the applied moment

This stress ratio is only the right hand side of equation 7.19 as we use this ratio to modify the uncracked results. In Eurocode 2 this stress ratio is subtracted from unity to be applied to the cracked results. As this formula does not consider axial forces which may be present (especially in post-tensioned structures), we have modified it to consider axial forces:

where

fcr	=	the concrete flexural tensile strength
fa	=	the cross-section tensile fiber stress (based on gross section properties)

If there is no axial force, then this formulation is identical to the eq. 7.19 formulation. If there are axial forces, this formulation is a reasonable (but not theoretically identical) extrapolation of the Eurocode formula. This value is always limited to be less than or equal to 1.0

Eurocode 2 states that β should be taken as 1 for short-term loading and 0.5 for long-term loading (see Clause 7.4.3). Some experts (see Scanlon and Bischoff and Gilbert references ¹) have concluded that β is intended to account for a reduced cracking moment due to additional stresses caused by internal reinforcement restraint to shrinkage. Since the internal reinforcement restraint to shrinkage is rigorously calculated in RAM Concept ’s load history calculations, the program uses β = 1 to avoid double counting that effect.

The modulus of rupture for the selected Design Code is used for the concrete flexural tension strength in the tension stiffening equation. RAM Concept calculates this rupture strength using the 28-day design concrete strength that is input in the Materials window (Criteria – Materials). Since the compressive strength increases over time, the program applies a correction factor to covert the modulus for rupture from 28-days to the actual time of loading in order to account for the reduced strength at early age loading (before 28 days). The adjustment factors used for each model are referenced in Table 104. When calculating the adjustment, RAM Concept uses the modulus of rupture calculated with f'ci in the Materials window as the lower bound for the early age modulus of rupture.

1. The modulus of rupture is calculated using the equation of defined in the selected design code. Any implemented design code can be used for each of the implemented creep/shrinkage models. The tabulated formula is a sample equation from only one of the implemented design codes.
2. The referenced clause is for detailed fatigue design. The $f_{\mathrm{cm}}(t)$ is used here as it is intuitive that concrete strength changes over time and the factor is exactly the same as the factor used in Eurocode 2-2004 Equation 3.2.
3. $f_{\mathrm{ctm}}$ is the 28-day modulus of rupture calculated using the equation defined in the selected design code.

In general, external restraint to shrinkage shortening can increase the cracking in the floor, thus increasing deflections. Failure to account for this effect can result in underestimation of deflection values. A crude means of accounting for this is through the “Shrinkage Restraint %” value in the Load History / ECR tab of the Calc Options dialog. This percentage is multiplied by the input free shrinkage strain value (as a function of time) to determine a hypothetical tension strain. This hypothetical tension strain is combined with the load induced strains which is then used to determine a hypothetical tension stress from the concrete stress strain curve. This hypothetical tension stress is used in the tension stiffening calculation. These stresses are not used in the cross section curvature calculations. As such, increasing this percentage will generally increase the amount of cracking predicted and used in the tension stiffening interpolation, but will not affect the calculated curvatures directly.

The Shrinkage Restraint % in the Load History / ECR tab of the Calc Options dialog may be selected based on one of the pre-set options mapped below or input by the user by entering a percentage directly in the field box.