For each iterative step in the calculation, we compute a new effective stiffness. This new stiffness takes the form of the Aeff and Ieff given by the strain and curvatures, which in turn are given from calculations performed with current section forces. The values used are the strain and curvature remaining after we have deducted creep, shrinkage, temperature and so on. This means that as the calculation iterates, these values will dwindle until the cracking has stabilized.

We then have

Aeff = N / Ec / ε 

Ieff = M / Ec / κ 

where

  • Aeff = effective area
  • Ieff = effective moment of inertia
  • Ec = modulus of elasticity for concrete
  • ε = strain in the member's centroidal curve
  • κ = curvature of the member.

Should section calculations be non-linear due to the stress-strain curve of the materials (this mainly applies to Ultimate Limit State), the effective stiffness will be smaller than it would be under linear elastic conditions, since it is adapted to the increase in strain and curvature. 

Since the Aeff and Ieff above have physical significance as section properties, they must not be allocated negative values. This means that measures must be taken when either N or M equals zero, in order to avoid division by zero. The original values for A and I will be used in such a situation. 

During cracking, the centroid of the section is moved from its original location in the uncracked section. The strain must be calculated in the cracked state in order to get a strain value that corresponds to Aeff. This means that a cracked section gives 

ε = εcg,uncr + κ ⋅ zcg,cr 

Note that the curvature does not change.

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PRE-Stress Documentation